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# Thread: WLC's Argument Against an Actual Infinity

1. ## Re: WLC's Argument Against an Actual Infinity

Originally Posted by CLIVE
You know, I don't think that someone who characterizes philosophy as "mental masturbation" is really all that familiar with the field.
Well, your description robbed the field of any teeth or meaning.
Thus, I was characterizing your description of the field.
I don't believe your description is accurate.

Originally Posted by CLIVE
I'm saying that they're mostly interested in the philosophical statements made in the articles; that's what they're certainly going to peer-review. The accuracy of the scientific statements--say, that such-and-such an experiment had an error tolerance of such-and-such many standard deviations, or the various mathematical technicalities--aren't as important as the philosophical analysis.
And that level should effect the point that was made in the thread?
Is that what the reference was being noted for?

Point is, I don't believe your point is sufficient to reject the source for how it was used.

Originally Posted by CLIVE
Ideally, every error will be caught and corrections issued. But journals have a finite amount of resources, and a limited time in which to review submissions before they go to print. If a statement is made in passing about the price of tea in China, the journal could go through the effort of contacting an expert in Chinese commodities pricing to review the claim, but so long as the falsity of the statement doesn't have substantial impact on the article's overall thesis, there's not really a point in going through the effort of reviewing the claim.
Sure, sure.. and how does that relate here? was the source sited for something on that level?
Is that the basis for dismissing them as a valid source for the kind of claims being made in the thread?

To me, your chewing on the edges and not really discrediting them in the sense they were used.

Originally Posted by CLIVE
Nothing you've said has supported E. You need to show that philosophy of science journals actually do the kind of peer-review that you're talking about--i.e., articles that include physics claims are peer-reviewed by physicists, articles that include biology claims are peer-reviewed by biologists, etc.

You're just arguing that they should do this. You haven't shown that they actually do.
Yes, I'm saying it is logically necessary for their job description.
What I'm doing is creating the boundaries for which the objection must fall into.
The objection is however not mine. Unless you are willing to discredit the entire field, or find some specific flaw in my logic so as to challenge the boundaries I've argued for.
Then the dismissal which simply ASSUMES that, is not valid and as I have argued more likely false.

Originally Posted by clive
So first, philosophy of science isn't science. It's philosophy. Philosophy of science asks and attempts to answer questions like [paraphrasing]:

What counts as science?
And you think they can answer that question without needing to know what the subject matter is?
Such that they can determine if the theory of relativity is science or not.. while having no qualifications or competence in knowing WHAT the theory is?

See that is where the "truth" lies.

Originally Posted by CLIVE
What scientific theories are reliable?
Again, scientific theories which they are being accused or assumed to be unqualified to understand or comment on?

Originally Posted by CLIVE
What is the purpose of science?
True, but not relevant here.

Originally Posted by CLIVE
Philosophy of science is not attempting to produce a "theory of reality", i.e. a set of rules or laws that describe how objects interact in the actual universe.
That is not the sense in which I used that phrase.
I clearly pointed to it REFLECTING reality.

Originally Posted by CLIVE
This is not "fantasy" in any kind of negative sense; these are questions about the intellectual foundation of science, the interpretation of science, the role and scope of science, etc. Philosophers of science do not posit physical models without caring whether those models describe reality--except perhaps as a thought experiment,

Originally Posted by CLIVE
I agree that in order to properly do philosophical analysis on a theory, you must be familiar with the content of the theory--or at least, with the content that is relevant to the analysis you're doing.

That is not to say that the theory in question must be true.
Good then there is no reason to reject the source on the merit that it was in the wrong journal for the purpose it was used for.

Originally Posted by CLIVE
Now, with regard to E (what I described earlier as your "empirical" claim), philosophers of science do reference physical models; however, in practice philosophers of science are usually only familiar with general descriptions / features of physical models, not with the technical details (and with physical models, the details are just about the only things that matter).
So your assuming their are ignorant.
care to support that? I mean if your objection rests on that, then you must have reason to think it is true in this specific case.
Which if you would support it, it is a far cry improved than simply rejecting a source.

Originally Posted by CLIVE
Describing General Relativity as a theory that posits moral relativism is an error that would probably get caught. Technical errors, however, while likely to be found during peer-review among physicists, are known to get through peer review in philosophy of science journals.
Wouldn't you call that part of the peer review process? If so, haven't you supported that the source is subjected to that process?
Or is the case that no "real" science journal has editor notes added retracting papers?

I think rather you have proved my point.
Certainly it may be a "lesser" support, but it isn't a support that can be rejected as unqualified in totality.

-------------

Originally Posted by GP
You don't seem to understand where the burden of proof lies or what amount of strength of a claim I need to make is; I only need to point out it is not necessarily true that philosophy of science journals act as a scientific peer-review process. In order for your argument to go through, you need to demonstrate that it is the case that for this specific journal that claim is false.
First, You don't seem to understand my objection or how I have specifically challenged the strength and validity of your claim, but I do appreciate that you managed to dismiss me without offering offensive strawmen arguments.

Second, see clives evidence that the journal is subjected to peer review, as evidenced by the corrections made due to that process.

Third, considering I have specifically addressed what you are saying here, I suppose I should just read your response as "Na-huh".

Originally Posted by GP
However, I see little reasoning in continuing to debate this with you. I'm satisfied that Squatch has rescinded Shanahan as (one of) his primary sources. I don't see the need to beat a dead horse.
My point was in relation to Clives recent claim
Originally Posted by CS POST 353
Sure, but if a physics claim is part of that philosophical debate, then physics sources should be used to support it.
I have argued for the limitations on that, and in the process pointed to some flaws in the reasons offered for rejecting a source based on it's appearance in a "science philosophy" journal. My point stands on it's own.

2. ## Re: WLC's Argument Against an Actual Infinity

Originally Posted by MT
First to the line of burden of proof. I am challenging your line of reasoning for REJECTING a source. That is a positive claim and as shown above you have not actually made a relevant or challenging statement. IE your challenge is too general a statement to cast actual shadows on the quality of the source. You can show it as being less qualified, but not UN-qualfied.
Again, insert my argument of why the field requires some competence in the facts to be even a valid field.
Please note that I am perfectly fine and univested in that field, so please feel free to discredit it.
Originally Posted by MindTrap028
First, You don't seem to understand my objection or how I have specifically challenged the strength and validity of your claim, but I do appreciate that you managed to dismiss me without offering offensive strawmen arguments.

Second, see clives evidence that the journal is subjected to peer review, as evidenced by the corrections made due to that process.

Third, considering I have specifically addressed what you are saying here, I suppose I should just read your response as "Na-huh".

My point was in relation to Clives recent claim

I have argued for the limitations on that, and in the process pointed to some flaws in the reasons offered for rejecting a source based on it's appearance in a "science philosophy" journal. My point stands on it's own.

1.) The standard name is "the philosophy of science." It emphasizes the correct part: "the philosophy of X", and not "the study of X." You should avoid getting them confused, and what is involved in the study of X, rather than the philosophy of X.

2.) I'll take a leaf out of Squatch's book: I've already addressed this and I see no reason to continue to discussion with you. You've asked questions and they were answered, largely in post #356 and in several of Clive's posts (in particular the one you're responding to). Your questions have actually been explained and answered in several different methods, so I have little reason to think that adding another version of the answer is going to change anything.

Again, more to the point, Squatch has already abandoned this as a primary source for his assertions about the validity of LR, so, again, I see no reason to beat a dead horse.

3. ## Re: WLC's Argument Against an Actual Infinity

Originally Posted by MindTrap028
Well, your description robbed the field of any teeth or meaning.
Thus, I was characterizing your description of the field.
I don't believe your description is accurate.
Philosophy isn't science. Philosophers of science do philosophy.

And that level should effect the point that was made in the thread?
Is that what the reference was being noted for?

Point is, I don't believe your point is sufficient to reject the source for how it was used.
The claim was that the journal--Foundations of Physics--does physics peer-review. Merely because it is a philosophy of science journal doesn't mean it does physics peer review.

Sure, sure.. and how does that relate here? was the source sited for something on that level?
Is that the basis for dismissing them as a valid source for the kind of claims being made in the thread?

To me, your chewing on the edges and not really discrediting them in the sense they were used.
The basis for questioning Foundations of Physics was that it is a philosophy journal (specifically, a philosophy of physics), not a physics journal, so it was unclear what if any physics peer review FoP does.

Yes, I'm saying it is logically necessary for their job description.
What I'm doing is creating the boundaries for which the objection must fall into.
The objection is however not mine. Unless you are willing to discredit the entire field, or find some specific flaw in my logic so as to challenge the boundaries I've argued for.
Then the dismissal which simply ASSUMES that, is not valid and as I have argued more likely false.
First, you are saying it's logically necessary. That don't make it so.

Second, you have no training or experience in philosophy or science, and yet you're certain you know both how it is done and how it should be done. Since your opinion carries no weight in light of your complete lack of expertise in the fields of science and philosophy, what argument do you have to offer that philosophers of science do and should subject all their claims to peer-review?

And you think they can answer that question without needing to know what the subject matter is?
Such that they can determine if the theory of relativity is science or not.. while having no qualifications or competence in knowing WHAT the theory is?
Again, broad strokes vs. technical details. How many philosophers of science know differential geometry? You literally can't do General Relativity without it, and yet philosophers (like William Lane Craig, listed by Wikipedia as a philosopher of science) routinely philosophize about the impact of these theories without being experts in the theory themselves. David Hume was a philosopher of science because of his insight about the scientific method, not because he knew how to do integrals, or what the proper units are for quantities like power, energy, mass, etc.

Again, scientific theories which they are being accused or assumed to be unqualified to understand or comment on?
You don't have to be an expert in General Relativity to do philosophy about science, including the metaphysical assumptions or implications of general relativity. You don't need to be able to do tensor analysis to talk about whether GR is falsifiable, or to re-evaluate Kant's philosophy based on the non-Euclidean space due to General Relativity. But you sure as hell need to be able to do tensor analysis if you're peer-reviewing the physics presented.

True, but not relevant here.
MT, that was a list of questions that illustrate the kinds of inquiry that philosophers of science engage in. It makes no sense to say that a question is "true", so I think you've very badly misunderstood something.

That is not the sense in which I used that phrase.
I clearly pointed to it REFLECTING reality.
The point is that philosophy of science isn't concerned about coming up with "theories of reality". Philosophy of science is concerned about the philosophical aspects of science and the scientific method.

Good then there is no reason to reject the source on the merit that it was in the wrong journal for the purpose it was used for.
(1) You're assuming that because proper philosophy of science would entail getting all the relevant statements about physics and physical models correct, that Foundations of Physics always does proper philosophy of science. That is, because a job should be done a certain way, then it is done that way. This assumption is plainly, obviously false, and I'm a bit shocked you'd so blatantly engage in fallacious reasoning.

(2) You're also assuming that if a physical model is accurately referenced in a given philosophy of science article, then that physical model must be viable. This is also plainly false; even if Foundations of Physics ensures that only true references were made as to the content of the proposed LR model, it doesn't follow that the LR model is a decent physical model, or even that physicists consider LR a decent physical model.

So your assuming their are ignorant.
care to support that? I mean if your objection rests on that, then you must have reason to think it is true in this specific case.
Which if you would support it, it is a far cry improved than simply rejecting a source.
Browsing the wikipedia category of philosophers of cosmology, 5 out of the 10 members have phd's in philosophy, but no phd in any science field:

http://en.wikipedia.org/wiki/Michael..._(philosopher)
http://en.wikipedia.org/wiki/Tim_Maudlin
http://en.wikipedia.org/wiki/Roberto_Torretti

Of the 42 philosophers of science from 1980 to today listed here, 34 have no phd in physics, biology, mathematics, chemistry, or any other science (Chomsky has a phd in linguistics; Stegmuller has a phd in economics):

http://en.wikipedia.org/wiki/Richard_Boyd
http://en.wikipedia.org/wiki/Nancy_C..._(philosopher)
http://en.wikipedia.org/wiki/Daniel_Dennett
http://en.wikipedia.org/wiki/John_Dupr%C3%A9
http://en.wikipedia.org/wiki/John_Earman
http://en.wikipedia.org/wiki/Noam_Chomsky
http://en.wikipedia.org/wiki/William_Lane_Craig
http://en.wikipedia.org/wiki/Bas_van_Fraassen
http://en.wikipedia.org/wiki/Ronald_Giere
http://en.wikipedia.org/wiki/Peter_Godfrey-Smith
http://en.wikipedia.org/wiki/Ian_Hacking
http://en.wikipedia.org/wiki/Sandra_Harding
http://en.wikipedia.org/wiki/Philip_Kitcher
http://en.wikipedia.org/wiki/Larry_Laudan
http://en.wikipedia.org/wiki/Isaac_Levi
http://en.wikipedia.org/wiki/Peter_Lipton
http://en.wikipedia.org/wiki/Helen_Longino
http://en.wikipedia.org/wiki/Ernan_McMullin
http://en.wikipedia.org/wiki/Stephen_C._Meyer
http://en.wikipedia.org/wiki/John_E._Murdoch
http://en.wikipedia.org/wiki/Nancey_Murphy
http://en.wikipedia.org/wiki/Alex_Rosenberg
http://en.wikipedia.org/wiki/Wesley_C._Salmon
http://en.wikipedia.org/wiki/Brian_Skyrms
http://en.wikipedia.org/wiki/Patrick_Suppes
http://en.wikipedia.org/wiki/David_Stove
http://en.wikipedia.org/wiki/Wolfgang_Stegm%C3%BCller
http://en.wikipedia.org/wiki/Elliott_Sober
http://en.wikipedia.org/wiki/Kim_Sterelny
http://en.wikipedia.org/wiki/Richard_Swinburne
http://en.wikipedia.org/wiki/Sandra_Mitchell
http://en.wikipedia.org/wiki/Lawrence_Sklar
http://en.wikipedia.org/wiki/Gerard_Verschuuren

This is something you could have learned on your own, if you bothered to learn about a subject before forming an opinion.

Wouldn't you call that part of the peer review process? If so, haven't you supported that the source is subjected to that process?
Or is the case that no "real" science journal has editor notes added retracting papers?
Sure, it's part of the process. But it shows that Foundations of Physics lets critical technical errors get through its editing phase, contrary to your sure-and-certain declaration that philosophy of science journals subject their articles to rigorous peer review among physicists.

I think rather you have proved my point.
Certainly it may be a "lesser" support, but it isn't a support that can be rejected as unqualified in totality.
The whole issue has been whether any of the sources were published in a journal that does physics peer-review, not whether the sources should be "rejected as unqualified in totality". You're moving the goalposts.

4. ## Re: WLC's Argument Against an Actual Infinity

Originally Posted by CLIVE
Philosophy isn't science. Philosophers of science do philosophy.
A subject they are not qualified to grasp or express properly?

Originally Posted by CLIVE
The claim was that the journal--Foundations of Physics--does physics peer-review. Merely because it is a philosophy of science journal doesn't mean it does physics peer review.
My challenge is the amount of separation you can place between peer reviewing science, and peer reviewing the philosophy of it.

Your acting as though they are unrelated, and I have shown how they are specifically related.

Originally Posted by CLIVE
The basis for questioning Foundations of Physics was that it is a philosophy journal (specifically, a philosophy of physics), not a physics journal, so it was unclear what if any physics peer review FoP does.
Per your above, the reason for out right rejecting the source was (as you just said) they are basically unrelated fields.
I have argued that to be false.

Originally Posted by CLIVE
First, you are saying it's logically necessary. That don't make it so.
I have an argument, you can address it at your leisure, but you can't dismiss it in this manner.

Originally Posted by CLIVE
Second, you have no training or experience in philosophy or science,
You really are stuck on the intellectual snobbery thing.
I haven't appealed to my credentials so this rebuttal is irrelevant.
But by all means, thump your diploma if it makes you feel better, or if you think that makes my argument false. It would not follow logically, but as long as your ego is stroked.

Originally Posted by CLIVE
Since your opinion carries no weight in light of your complete lack of expertise in the fields of science and philosophy, what argument do you have to offer that philosophers of science do and should subject all their claims to peer-review?
Selective memory must be fun.

Originally Posted by CLIVE
Again, broad strokes vs. technical details. How many philosophers of science know differential geometry? You literally can't do General Relativity without it, and yet philosophers (like William Lane Craig, listed by Wikipedia as a philosopher of science) routinely philosophize about the impact of these theories without being experts in the theory themselves. David Hume was a philosopher of science because of his insight about the scientific method, not because he knew how to do integrals, or what the proper units are for quantities like power, energy, mass, etc.
Were they on the board of reviewers for the journal in question?
If not, then it isn't really relevant.
You still have not challenged my main point, which is they must be competent to an extent, especially to the extent that their philosophical claims rely on.
Which at the very least is a firm grasp of the topic. Which would make them a valid source for the topic.

Originally Posted by CLIVE
You don't have to be an expert in General Relativity to do philosophy about science, including the metaphysical assumptions or implications of general relativity. You don't need to be able to do tensor analysis to talk about whether GR is falsifiable, or to re-evaluate Kant's philosophy based on the non-Euclidean space due to General Relativity. But you sure as hell need to be able to do tensor analysis if you're peer-reviewing the physics presented.
And no one on the board was qualified to do that? or is the field itself inherently unqualified to do that?

Originally Posted by CLIVE
MT, that was a list of questions that illustrate the kinds of inquiry that philosophers of science engage in. It makes no sense to say that a question is "true", so I think you've very badly misunderstood something.
I think it absolutely wonderful how you ignored my answer to the other 2 questions, as they illustrated my objection.
I suppose we are at the point of the debate where you just shut out whatever you don't like.
I mean, you just finished demanding that I offer an argument.. that I have been offering.

Originally Posted by CLIVE
The point is that philosophy of science isn't concerned about coming up with "theories of reality". Philosophy of science is concerned about the philosophical aspects of science and the scientific method.
There is a distinction between what I have said regarding a "reflecting reality" and what you are saying "Theories OF reality".
They have to reflect the reality of the scientific theory they are specifically discussing (for one).

Originally Posted by CLIVE
(1) You're assuming that because proper philosophy of science would entail getting all the relevant statements about physics and physical models correct, that Foundations of Physics always does proper philosophy of science. That is, because a job should be done a certain way, then it is done that way. This assumption is plainly, obviously false, and I'm a bit shocked you'd so blatantly engage in fallacious reasoning.
Nice straw-man.. set-em up and knock them down.
My argument is about a qualified source.
I have not argued that because it is in the journal it must be true. Or that because it is in the journal it must not have any mistakes.
I'm arguing that if the journal is in good standing in it's field, and that field requires the evaluation and proper understanding of specific sciences(which I have shown it does), then one must do more than simply dismiss them as being in the wrong journal.

I am fine with you discrediting the entire field, you should just realize that is what your doing.

Originally Posted by CLIVE
(2) You're also assuming that if a physical model is accurately referenced in a given philosophy of science article, then that physical model must be viable. This is also plainly false; even if Foundations of Physics ensures that only true references were made as to the content of the proposed LR model, it doesn't follow that the LR model is a decent physical model, or even that physicists consider LR a decent physical model.
Your statement that it is false, is incorrect.

Suppose I accurately stated the theory of LR and I showed PHILOSOPHICALLY that it was equivalent to (some other), then It doesn't really matter what physicist think, my argument would be correct, and the implication is that LR is as viable as the other theory I showed it was equivalent to.

Originally Posted by CLIVE
Browsing the wikipedia category of philosophers of cosmology, 5 out of the 10 members have phd's in philosophy, but no phd in any science field:
How many of those were on the board of the journal in question?
And exactly how good a source is Wiki on this topic? Is wiki the arbiter of the state of the field?

Originally Posted by CLIVE
Of the 42 philosophers of science from 1980 to today listed here, 34 have no phd in physics, biology, mathematics, chemistry, or any other science (Chomsky has a phd in linguistics; Stegmuller has a phd in economics):
And how many of them were on the board of the journal in question?

Originally Posted by CLIVE
This is something you could have learned on your own, if you bothered to learn about a subject before forming an opinion.
I don't see the point of learning about the qualifications of people who are not on the board of reviewers in the journal being dismissed.
Or is that you dismiss the journal because Craig (not on the board) is not qualified?

Foundation of physics board of editors
http://www.springer.com/physics/hist...editorialBoard
Chief Editor:
Gerard ’t Hooft, Utrecht University, The Netherlands

Managing Editor:
Fedde Benedictus, Utrecht University, The Netherlands

Associate Editors:
Paul Busch, University of York, UK;
Dennis Dieks, Utrecht University, The Netherlands;
Erik Verlinde, University of Amsterdam, The Netherlands;
Brigitte Falkenburg, Technische Universität Dortmund, Germany

Editorial Board:
Jeffrey Bub, University of Maryland, MD, U.S.A.;
Xavier Calmet, University of Sussex, UK;
Arthur Fine, University of Washington, Seattle, U.S.A;
Robert Geroch, University of Chicago, IL, U.S.A.;
GianCarlo Ghirardi, University of Trieste, Italy;
Sheldon Goldstein, Rutgers University, NJ, U.S.A.;
Daniel Greenberger, The City College of CUNY, NY, U.S.A.;
Alan Kostelecky, Indiana University, IN, U.S.A.;
Tim Maudlin, Rutgers University, Bloomington, U.S.A.;
D. Carlo Rovelli, Centre de Physique Theorique de Luminy, Marseilles, France;
Abner Shimony, Boston University, MA, U.S.A.;
C. Anton Zeilinger, University of Vienna, Austria;
Wojciech H. Zurek, Los Alamos National Laboratory, NM, U.S.A.

Founding Editors: Wolfgang Yourgrau; Henry Margenau

Past Board Members:
Asim O. Barut; Peter G. Bergmann; David Bohm; Nikolai N. Bogolubov; Louis de Broglie; Robert H. Dicke; Vladimir Fock; Murray Gell-Mann; Lajos Jánossy; André Mercier; Alwyn van der Merwe; Christian Möller; Louis Néel; Lars Onsager; Rudolf Peierls; Asher Peres; Ilya Prigogine; Nathan Rosen; Arthur Ruark; Abdus Salam; John l. Synge; Albert Szent-Gyorgy; Jean-Pierre Vigier; Mikhail Vol'kenshtein

First name on the list
Jeff Bub PHD Mathematicla physics http://carnap.umd.edu/philphysics/bub.html

Second name
Xavier Calmet http://www.sussex.ac.uk/profiles/242816
List of his citations(a measure forwarded by one of you earlier) http://scholar.google.com/citations?user=Y4Ca4hgAAAAJ

So let me stop right there, because I don't see those guys on the list you provided.
Why are our lists different? Why would you try to discredit a journal, with people who are not on it's board?
It is now out right humorous that you would lecture me on research and qualifications, and then botch it so badly.

The second guy "Xavier", he may not have a PHD in physics(, but his citation list seems to suggest that others don't dismiss him as quickly as you would. ) Either way my argument is supported.

Originally Posted by CLIVE
Sure, it's part of the process. But it shows that Foundations of Physics lets critical technical errors get through its editing phase, contrary to your sure-and-certain declaration that philosophy of science journals subject their articles to rigorous peer review among physicists.
My" sure as certain" statement was proven, because that paper had it's support pulled.
Why?
Because the validity of the philosophical claims is directly related to the content of the theories being considered.
Which was my point.

Originally Posted by CLIVE
The whole issue has been whether any of the sources were published in a journal that does physics peer-review, not whether the sources should be "rejected as unqualified in totality". You're moving the goalposts
No, moving the goal post fallacy is when I set a goal for you to reach, and then you reach it and I move the goal.
I am here challenging the placement of your goal posts.
Get it right.

5. ## Re: WLC's Argument Against an Actual Infinity

Originally Posted by GP
Again, it's simply wrong to say otherwise; in post #250 I actually took nearly every single (if not, then every single) author you mentioned, and ran them through HEP Inspires to see their peer-reviewed publications;
And you say you didn't find a single paper published in a peer-reviewed journal in that search? And none of my authors were physicists?

Originally Posted by GP
If you had such a source and you felt that I neglected it, then why didn't you direct me to these "better sources" when I asked you for them multiple times?
Let me ask this GP. Let's say you wrote a post in a thread on say economics and someone responded with "TLDR" and "this guy screams crackpot" would you be overly invested in attempting to point out his errors?

Originally Posted by CliveStaples
No, no, no. You think that the definition for causet has to specify what the partial order is--i.e., what elements of C x C are in the partial order?

Rideout-Sorkin call "locally finite posets" causets. As long as you know that (S,<) is a locally finite poset, then Rideout-Sorkin would call (S,<) a causet.
I believe you are incorrect here. There seems no coherent reason to accept that a partial order defined by non-causal relations would conform to the physical interpretations they are attempting to model. Unless you are going to argue that this theory is simply an abstract mathematical concept rather than an attempt to model physical reality, the fact that causal sets require causation is required.

This is further supported by the expanded definitions offered in the papers referenced here.

The elements of a causal set are taken to represent spacetime events, while its binary relation is taken to encode causal relations between pairs of events.
Axioms Paper

If a; b are elements of a causet x, we interpret the order a < b as meaning that b is in the causal future of a.
Labeled Causet paper

For a fuller introduction to causal sets, see [3, 4, 5, 6, 7]. (For recent examples of other discrete models incorporating a causal ordering see [8, 9, 10].)
R/S Paper

By the ‘causal structure’ on a manifold, one often means abstractly a set of binary relations between points, with characteristics that intuitively capture
the notion of ‘is lightlike related’ or ‘is timelike related’ or some other suitable sense (see Kronheimer and Penrose [4] for a precise, technical foray). In this
way, one can define the causal structure without the explicit presence of a metric. However, when it comes time to express the physical meaning, or even just the geometric meaning in the attempt to associate a metric to the structure, one must conclude as the references cited here do, that ‘p relates to q’ means that p connects to q via a suitable causal curve, as defined by the Lorentzian geometry.
The Class of Causally-Concerned Objects Can Confuse paper (pretty sure we linked this earlier, if not let me know).

This last condition guarantees that the “parameter time” of our stochastic process is compatible with physical temporality, as recorded in the order relation ≺ that gives the causal set its structure. In a broader sense, general covariance itself is also an aspect of internal temporality, since it guarantees that the parameter time adds nothing to the relation ≺... For example, the events of Minkowski space (in any dimension) form a poset whose order relation is the usual causal order
R/S Paper 2, Continuum Limits.

Clive, let me ask you a simple question. What do the lines in the Haas Diagrams mean? What does < mean when relating two elements in these papers? What do these relationships mean in a physical interpretation in our universe?

Originally Posted by CS
Which of the lines in the third image denote spatial relations, which denote temporal relations, and which denote causal relations?
As I noted in my last response, the lines (or the < if displayed in another format) represent causal relationships. If the point differs along the y axis here (time) then it has a temporal component to it. If it is along the X axis it has a spatial component to it. Two elements can be causally related if:

They are related only across the temporal axis.

or

They are related across both temporal and physical axes.

Two elements cannot be related if they are only related across the spatial axis.

Originally Posted by CS
Gregarious and timid children are defined in terms of the partial order of the causet. There's really no need to talk about these "spatial" and "temporal" relations you're bringing up, because all of the relevant relations are captured by the partial order.
After re-reading this statement a few times either I'm dramatically misreading what you are saying here or you seem to have missed the entire point of these theories. If you mean what the plain text reading of your statement would imply, then it seems odd that R/S would discuss this issue for 10% of their paper, specifically saying "The child formed by adjoining an element which is to the future of every element of the parent will be called the timid child. The child formed by adjoining an element which is spacelike to every other element will be called the gregarious child." If there is really no need to talk about spatial and temporal relations here, why do they go out of their way to do so?

These papers are discussing relationships between points in our physical universe, these points exist in physical and temporal dimensions so the discussion of those dimensions would seem to be cogent to the question of whether or not the universe has a finite age.

Originally Posted by CS
Are you saying that you can pick a set of points S such that there is a point p satisfying:

(1) p is to the future of S [by the above definition]
(2) p is spacelike to all elements of S (i.e., for all y in S, neither y < p nor p < y)

That seems true.
Ok, so given that, can we agree that a point could be both a) to the future of all other points in S (ie a timid child) and b) not causally related to any point in S (ie not connected via a line or a <)?

Originally Posted by CS
Again, in R/S the children are causets, not an element of a causet.
Let me ask this. If I were to say that we had two elements, a and b in a cuaset. And I were to say that a < b, meaning that b is to the causal future of a (lets assume no intermediate elements) would it be correct or incorrect of me to say that b is a child of a the parent?

Originally Posted by CS
How would it imply that there is reverse causation?
It would either imply reverse causation or create a paradox. You are using information about x1000 to determine the value of x999 when that information cannot exist unless the value of x999 is known. So you are using y to figure out x so that you can use x to figure out y.

Originally Posted by CS
This seems like an incomplete thought.
You're correct, it seems to have deleted my additional text for some reason (actually it seems to have hidden a portion of it in your post and repeated it about a dozen times, click reply to your post and you'll see some odd text repeated in there). Odd (after further review, it looks like vbulletin has an issue with using the < sign with text around it without spaces, ie a < b if you removed the spaces). Essentially I was just giving an example to a point you seem to agree with.

If I were to label the elements with colors {blue, red, green, orange} Inferences about the elements based on the wavelength ordering of those colors wouldn't be valid since that isn't an order I used to relate the colors to the order in the causet. This might be ridiculously obvious to be pointing out, but I think it is important to prevent confusion concerning physical interpretations.

Originally Posted by CS
(2) holds via the proof by contradiction I gave.
If I recall correctly, that was a proof on integers, not on natural numbers.

Originally Posted by CS
Honestly, Squatch, I don't even think you have an argument at this point--at least, one that isn't circular.
Just as I think you are ignoring an obvious conclusion. But setting that aside, I still haven't seen an objection against my argument in post 333

1) There is a specific theory that only allows for one method of set creation (iterative addition of finite values).
2) That method of set formation is insufficient to explain the existence of an infinite set.
3) Therefore there are no infinite sets within the confines of that theory.

Originally Posted by CS
I hope you didn't miss the point of mine.
I saw it, the problem is that it is not a material objection to anything I've said. Your final statement:

Originally Posted by CS
The only way you get a gregarious child, if you're constructing children using the above processes, is to choose the empty partial stem in (C,<).
Would seem to be a "I agree with what you said" statement. If you select an empty partial stem (IE you don't include the element's causal past) then you have a gregarious child. If you were to select a non-empty partial stem (ie causally prior elements) then it would not be gregarious.

So all this page after page wrangling could be simplified to "gregarious children" do not reflect a continuation of a causal chain of elements and therefore would be inappropriate to include in any discussion of a measurement of something's age via causal chains.

Originally Posted by CS
I don't know what "exist along some other relation" means.
Again, I think we need to remember that these causets and relationships have a physical meaning. Each element in the causet relates to a point in the universe, so while not every point within a causet must be related via causation, it is related in the sense that it exists (or existed) within a physical universe.

Originally Posted by CS
I don't know. Perhaps it's browner than the other point? Perhaps it's raining at that point?
So if I were to say, a,b,c,d are all points in a spatial temporal universe and that point a is not earlier or later than b, you would not conclude that point b must be a point at the same time as a, but at different spatial coordinates?

And if not, what, specifically, do you think R/S mean when they say a point that is "spacelike" to every other point?

Originally Posted by CS
Okay, but the dimension of what? A spacetime regime? You're talking about a mathematical dimension, right?
I think the definition offered in the Axiom paper is the best one to ground this discussion. "The elements of a causal set are taken to represent spacetime events, while its binary relation is taken to encode causal relations between pairs of events." So given that these elements are taken as space time events, they would need to be represented in spatial and temporal dimensions.

Originally Posted by CS
Well, we're looking at causal relationships; causally maximal elements shouldn't be assumed to be contemporaneous with each other.
It doesn't really matter either way. They can be contemporaneous to each other in relation to some absolute reference frame or they can be relative to each other, neither interpretation really affects the question. Even if each maximal element only relates to a local "now" and could be further ahead or behind (in a purely temporal not a causal sense) any other maximal element.

Lets assume a very simple universe. One physical dimension made up of one single point and one temporal dimension. This will eliminate any need to worry about relative or absolute "nows."

That universe is at x,tn

xn is defined as "now."

(x,tn+1) being a maximal element not yet born, we can conclude there is a longest chain in that universe correct?

Originally Posted by CliveStaples
The claim was that the journal--Foundations of Physics--does physics peer-review. Merely because it is a philosophy of science journal doesn't mean it does physics peer review.
Then why is it's review board made up entirely of physicists?

Originally Posted by CS
This is something you could have learned on your own, if you bothered to learn about a subject before forming an opinion.
And I think you would have been better served to pick a better source than Wiki. What you got was the pundit list made up of some good names and some names that really have nothing to do with the subject outside of popular press (Chomsky for example is in no way a philosopher of science).

What would have served you better would be to review where actual peer-review is going on.

If you want to claim that Foundations of Physics is a philosophy of science journal than your entire argument is rebutted by the make up of the review board.

If however you would like to argue a different philosophy of science journal, lets look at the Philosophy of Science journal, whose board can be found here: http://www.press.uchicago.edu/journa...l?journal=phos

I searched 15 random people on the board. 12 had degrees in fields other than philosophy (in conjunction with a philosophy degree). 1 (no other degree that I could find) also taught applied mathematics, 3 others with different degrees also taught course work in this field. An extremely cursory look showed 7 of them having published in non-philosophy journals.

In fact, a deeper look at the listed articles of both the editors and the journal itself would seem to indicate a deeper working relationship between the two groups. There are papers in this journal concerning the validity of the statistical measures used to the inferences offered in two different other academic works. Reviewing some of the board member's papers I see papers published in field journals directly as well as in philosophy journals concerning interpretation issues primarily (most of the titles appear to revolve around technical issues concerning whether the inferences of papers were justified, whether experiments were underpowered, etc.). There would seem to be a lot more dialogue between these fields than you are prepared to grant, or perhaps than you realize.

6. ## Re: WLC's Argument Against an Actual Infinity

Originally Posted by Squatch347
And you say you didn't find a single paper published in a peer-reviewed journal in that search? And none of my authors were physicists?
That's a great question, Squatch. Did I miss one of your sources? This is a good kind of question you can take me to task on when you address this:

"How about this, Squatch: How about you give me one source (the source that you think is your best source) that you think supports your claim about Lorentzian Relativity being a valid model, and I'll go through the entire source and explain the problems and strengths of the paper? Would that be fair?"

If it wasn't clear, I'm removing your ability to claim that I'm being a big, bad, unfair, elitist, arrogant academic by letting you pick the paper which you feel best supports your case. I'm not even demanding that you give me a peer-review source. You want to give me a completely BS paper? Go for it. I will be ruthless. Want to give me a credible paper that supports your position in an intelligable and cogent manner? Good, show me up. But either way, feel free to do something other than grandstand.

Originally Posted by Squatch
Let me ask this GP. Let's say you wrote a post in a thread on say economics and someone responded with "TLDR" and "this guy screams crackpot" would you be overly invested in attempting to point out his errors?
I did a lot more than just say "this guy is a crackpot" in post #250. In case you've forgotten that, on top of the other issues that I've raised and you've been blowing off for four pages now, you've already conceded that I showed you that one of your sources overtly went against your position in post #250. So to characterize #250 as just "calling your sources crackpots" is just a blatant falsehood.

But again, this is a great issue to show me up on when you respond to my request.

Well, MT, I should congratulate you on finally defending your claim; I don't know if you finally accepted that the onus was on you to provide evidence for your claim, but the results are the same. Now then, since you've finally defended your claim, I'll respond to you:

Originally Posted by MindTrap028
Foundation of physics board of editors
http://www.springer.com/physics/hist...editorialBoard

First name on the list
Jeff Bub PHD Mathematicla physics http://carnap.umd.edu/philphysics/bub.html

Second name
Xavier Calmet http://www.sussex.ac.uk/profiles/242816
List of his citations(a measure forwarded by one of you earlier) http://scholar.google.com/citations?user=Y4Ca4hgAAAAJ

So let me stop right there, because I don't see those guys on the list you provided.
Why are our lists different? Why would you try to discredit a journal, with people who are not on it's board?
It is now out right humorous that you would lecture me on research and qualifications, and then botch it so badly.

The second guy "Xavier", he may not have a PHD in physics(, but his citation list seems to suggest that others don't dismiss him as quickly as you would. ) Either way my argument is supported.
1.) Well, let's be clear here about the first point, which is that Clive was listing top cited philosophers of science. The people on the editorial board are not well-known philosophers of science, although there was some cross-over (such as Tim Maudlin, who by the way was on of the people Clive included as having scientific training). In fact, most of them are scientists or mathematicians (10 out of 14 and 2 out of 14, respectively), which would explain their absence from his list.

2.) The first error of yours is that the editorial board often do not, if ever, review individual papers that are published in the journal. In general, they will never see any of them unless they choose to review after publication. Their purpose is largely to tell the associate editors what kind of papers they should accept for publication to the journal, what kinds of topics the journal should focus on, and so on.

3.) Who actually handles the peer-review of individual papers? This is the Associate Editors, they are the ones who handle the peer-review process of papers:

"[T]he [associate editor]’s main task is to make editorial recommendations to the [Chief Editor] about what decision should be made on submitted papers.

To accomplish this, the [associate editor] has a seemingly simple set of responsibilities: to obtain referee reports for each paper they are assigned, and use these to make their recommendation for the paper, in a timely fashion."

In other words, you ought to be looking at the list of Associate Editors for Foundations of Physics. They're the ones whose job is actually to review papers for their content, to find referees, and to submit their recommendation to the Chief Editor.

4.) The Associate Editors are:

a.) Paul Busch (Mathematical Physicist, PhD in Physics; predominant work is on mathematical quantum information theory)
b.) Dennis Dieks (Philosopher of Science, PhD in Philosophy; predominant work is on interpretations of quantum mechanics)
c.) Erik Verlinde (Physicist, PhD in Physics; predominant work on string theory)
d.) Brigitte Falkenburg (Philosopher of Science, PhD in Philosophy; predominant work has been in "Kantian cosmology")

In other words, only 50% of the people who review physics papers are physics experts. More importantly, only one of them is qualified to understand the error made by Shanahan, which is Verlinde. That means 75% of the authors have no training in the physics that Shanahan was getting involved in. (However, it's unclear if the division of labor is even 25% each associate editor. It's possible that Falkenberg and Dieks are the primary associate editors and the other two only occasionally are editors for papers. It would explain a lot.)

5.) You are right, on paper they should be doing a lot better with the two scientific editors that they have. However, as to how effectual they are at catching these kinds of errors, I need only remind you of the so-called "Einstein-Cartan-Evans theory", which was proven to be crackpot. If you notice, it's not like one of his papers made it through the peer-review process. He was published for years in Foundations of Physics, with over fifteen papers. You think one of their associate editors might have caught it after fifteen papers over the course of nearly four years, wouldn't you? No, I think it is unreasonable to think that they actually engage in scientific peer-review. They are doing exactly what I said they were; in fact, we'll discover soon what the Chief Editors opinions were on this subject.

So, it appears that their track record with Quantum Field Theory is quite poor. And I'm not surprised; to quote Edward Witten, the greatest living field theorist (seconded only possibly by Stephen Weinberg):

"People tend to think that the most difficult thing that physicists grapple with is Einstein's work, but actually that's not true. Since quantum theory --especially since Dirac on relativistic quantum theory-- the most difficult theory in modern physics, by far, is Quantum Field Theory. It's an extremely difficult subject. Well, it was the work of most of the twentieth century to understand it better; we're still [trying to understand] it better. But you could say that the modern understanding took half a century, from Dirac to what we call asymptotic freedom."

The fact that philosophers don't understand these issues isn't surprising. The fact that Shanahan's work was already addressed (as I talked about in post #356) by early advents in QFT isn't entirely shocking, because you don't learn this stuff until you get a PhD and if you choose to focus your PhD in QFT. It's an extremely complicated subject and physicists are far from fully understanding, so it's not surprising that philosophers don't understand it, especially given that you literally need to get a PhD exclusively in QFT before you can make reliable statements about it. It's just one of those things where you can only BS your way through it for a tiny stretch of time before you need to know the math and do serious math.

6.) The impact factor of Foundations of Physics is 1.170. Well, I'll quote Chad on this:

"The best indication you can go off of is a journals "impact factor" which is a ratio of the number of citations to number of publications. The higher the impact factor, the more citations a journal receives per paper...which typically indicates the quality and importance of the research. At the very top are journals like Nature, Science, and Cell. These have impact factors in the 20s-30s. Meaning that about every paper published gets on average that many citations. Really good journals specific to fields (in Biology at least) are typically in the range of 7-10. Good solid journals are often in the range 3-7. Lower end journals, which still typically have good peer review and are reputable are in the 1-3 range. Once you get below 1 or even the lower ranges of 1, that means there are almost no citations for the papers published and I would very much question the journal itself."

That speaks for itself.

7.) Does the Chief Editor of Foundations of Physics view the journal as a peer-review science journal? Well, let's see:

"Between 2003 and 2005, the former Journal “Foundations of Physics Letters” (now subsumed into Foundations of Physics) has accepted and published a series of 15 papers by M.W. Evans. A partial list of these papers is given below [1–13]. Together they would form a book that was intended to unleash a revolutionary paradigm switch in theoretical physics, rendering well-established results of quantum ﬁeld theory and general relativity, including the Standard Model, superstring theory, and much of cosmology, obsolete. The magic word is ECE (Einstein-Cartan-Evans) theory, and the theory is claimed to have ignited frantic activities on the Internet.

In fact however, these activities have remained limited to personal web pages and are absent from the standard electronic archives, while no reference to ECE theory can be spotted in any of the peer reviewed scientiﬁc journals. This issue of Foundations of Physics now publishes three papers (G.W. Bruhn, F.W. Hehl, and F.W. Hehl and Y.N. Obukhov) that critically analyse the ECE theory and its claims. M.W. Evans has declined the invitation to respond, referring to his web pages, http://atomicprecision.com. Taking into account the ﬁndings of Bruhn, Hehl and Obukhhov, the discussion on ECE theory in the journal Foundations of Physics will be concluded herewith unless very good arguments are presented to resume the matter.
"

-Gerardus 't Hooft (PM me for a PDF if you like, but this was the full main body of the letter)

Notice how he says that ECE theory has been published in Foundations of Physics and then follows it with the statement that ECE has never been published in a peer-review scientific journal? No, 't Hooft (i.e. the Chief Editor of Foundations of Physics) states that it does not conduct scientific peer-review. I don't know who better to say that then he.

8.) Does the fact that they have 50% science trained, associate editors mean that they are a "scientific peer-review journal", as MT indicates and 't Hooft rebukes above? Clearly not. Why then don't the physicists read these papers and check for technical errors at the level of a professional physicist? This is likely because Foundations of Physics is a philosophy journal, and as their mission statement says, they do not require authors to have conducted a sound scientific analysis. To quote them, the Foundations of Physics has the following the mission statement:

"Foundations of Physics is an international journal devoted to the conceptual bases and fundamental theories of modern physics and cosmology, emphasizing the logical, methodological, and philosophical premises of modern physical theories and procedures.

We welcome papers on the interpretation of quantum mechanics, quantum field theory, thermodynamics and statistical mechanics, special and general relativity as well as cosmology. Also, we think it is time for the experts on quantum gravity, quantum information, string theory, M-theory, and brane cosmology to ponder the foundations of these approaches.

New insights are gained only by intense interactions with professionals all over the globe, and by solidly familiarizing oneself with their findings. Fortunately there are many authors with a deep understanding of the topics they are discussing who are willing to take the opportunity to present their ideas in our journal, and their clever inventiveness continues to surprise us. Acceptance of a paper may not necessarily mean that all referees agree with everything, but rather that the issues put forward by the author were considered to be of sufficient interest to our readership, and the exposition was clear enough that our readers, whom we assume to be competent enough, can judge for themselves."

I think that speaks for itself, as well.

Well, I think that I've said enough about this journal. If Squatch still wants my appraisal of Shanahan's work and what the editors should have caught, then I'm willing to give it. However, I doubt that it's going to be Squatch's "best source."

7. ## Re: WLC's Argument Against an Actual Infinity

Originally Posted by MindTrap028
A subject they are not qualified to grasp or express properly?
The fact is that many philosophers of science don't have phds in a scientific field, and yet they still do work in philosophy of science. Big names in philosophy of science too, like David Hume and Daniel Dennett,

My challenge is the amount of separation you can place between peer reviewing science, and peer reviewing the philosophy of it.
Someone without a phd in physics should probably not be doing peer-review on physics claims.

Your acting as though they are unrelated, and I have shown how they are specifically related.
I never acted or said that philosophy of science and science were unrelated. That's just a strawman.

Per your above, the reason for out right rejecting the source was (as you just said) they are basically unrelated fields.
I have argued that to be false.
Well, you've said it's false. And you've said that you know about philosophy of science. But since you have no training or experience in philosophy, science, or philosophy of science, your opinion doesn't carry much weight.

And since you're clearly unfamiliar with the CVs of a pretty big swath of philosophers of science (those who lack phds in science fields), I'd say that your statements about the discipline are based on ignorance, not knowledge.

You really are stuck on the intellectual snobbery thing.
I haven't appealed to my credentials so this rebuttal is irrelevant.
But by all means, thump your diploma if it makes you feel better, or if you think that makes my argument false. It would not follow logically, but as long as your ego is stroked.
Ah, I see the mistake you're making.

You can make a claim like "This is how you do philosophy of science." If you want people to believe it on your say-so, then your opinion has to mean something. You want people to believe your statements on your say-so; you aren't citing any textbooks on philosophy of science, you aren't citing the statements of any philosophers of science discussing how to do philosophy of science.

You're just some random guy giving his opinion about a field in which you have no training or experience.

Were they on the board of reviewers for the journal in question?
If not, then it isn't really relevant.
You still have not challenged my main point, which is they must be competent to an extent, especially to the extent that their philosophical claims rely on.
Which at the very least is a firm grasp of the topic. Which would make them a valid source for the topic.
You're shifting the goalposts again.

There are a few issues here:

(1) How philosophy of science "should" be done
(2) How philosophy of science is actually done
(3) How Foundations of Physics does philosophy of science

Foundations of Physics doesn't have to do philosophy of science the way philosophy of science is generally done (although then their peer-review for philosophy of science might come into question), nor does Foundations of Physics have to do philosophy of science how it "should" be done.

You've accurately ascertained that I was not making claims about (3). I was talking about (1) and (2); you can in principle do philosophy about General Relativity without being an expert in tensor analysis; this relates to (1). As an example, I cited William Lane Craig, who is not an expert in physics nor an expert in General Relativity, and yet does philosophy of science and philosophizes about General Relativity (and Special Relativity, which he is also not an expert in); this relates to (2), although to be rigorous I should have given an argument to support the tacit assumption that WLC is not "atypical" or shunned by his fellow philosophers of science for philosophizing about physics without having a physics phd.

Philosophers of science are probably better sources of knowledge for science than, say, someone untrained in science and untrained in philosophy of science. The kind of understanding of science you need to do philosophy of science is less than the kind of understanding of science you need to do science. This is why a phd in philosophy of science doesn't include the coursework necessary to get a phd in, say, physics or biology. It's why there's philosophy of science degrees, and not philosophy of physics, philosophy of biology, philosophy of psychology, philosophy of neuroscience, philosophy of chemistry, etc.

And no one on the board was qualified to do that? or is the field itself inherently unqualified to do that?
The field isn't inherently unqualified. That's just a strawman you've concocted.

Rather, the field merely isn't inherently qualified to peer-review physics. Some philosophers of science absolutely are qualified to peer-review physics, and perhaps the referees at Foundation of Physics are all so qualified.

But I haven't seen any evidence of that (I'd love to be wrong, because that would make the discussion more interesting); if there is evidence, I've either missed it being presented here or it hasn't been presented here.

And without that evidence--again, to avoid confusion, I mean evidence that their referees are both qualified to do physics peer-review and do physics peer-review in their capacity as referees for Foundations of Physics--I don't have a good reason to trust that Foundations of Physics does physics peer-review.

I think it absolutely wonderful how you ignored my answer to the other 2 questions, as they illustrated my objection.
I suppose we are at the point of the debate where you just shut out whatever you don't like.
I mean, you just finished demanding that I offer an argument.. that I have been offering.
Those weren't questions meant for you to answer, MT. Those were questions that illustrate what kinds of concerns philosophers of science are interested in.

Your answers to them are irrelevant to the point at hand (although if you think you've "answered" the central questions of philosophy of science in the sense of making any other answer untenable, then you're probably beyond hope).

There is a distinction between what I have said regarding a "reflecting reality" and what you are saying "Theories OF reality".
They have to reflect the reality of the scientific theory they are specifically discussing (for one).
Sure, but they only have to know things like "In general relativity, the speed of light is constant", or "In special relativity, there are frames of reference, and no absolute frame". Philosophers of science routinely work off of statements like these without understanding their technical, mathematical expressions or basis (see e.g. William Lane Craig).

Nice straw-man.. set-em up and knock them down.
My argument is about a qualified source.
I have not argued that because it is in the journal it must be true. Or that because it is in the journal it must not have any mistakes.
I'm arguing that if the journal is in good standing in it's field, and that field requires the evaluation and proper understanding of specific sciences(which I have shown it does), then one must do more than simply dismiss them as being in the wrong journal.
It depends on what you mean by "evaluation and proper understanding". You can discuss the philosophical aspects of Special Relativity--e.g., how the Lorentz invariance in Special Relativity or Minkowski spacetime is compatible or incompatible with the A-theory or B-theory of time, which are philosophical constructs--without understanding all the technical details.

Have you read any philosophy of science journals? Have you read any philosophers of science?

What is your basis for your opinion about what kind of understanding of physics is necessary to do philosophy about things like General Relativity?

What makes you think that philosophers of science generally have that necessary understanding? After all, just because MindTrap thinks that understanding X is necessary to do philosophy of science doesn't mean that all or most philosophers of science understand X.

I am fine with you discrediting the entire field, you should just realize that is what your doing.
It discredits the field of philosophy of science to cite the work, CV, and methodology of giants in the field of philosophy of science? Your problem seems to be with philosophy of science, not with me.

I'll again note the irony in talking about the requirements and foundations of a field that you have no training or experience in. When Hume talked about the philosophical foundations of science, Hume was an expert in philosophy. You're not an expert in philosophy, science, or philosophy of science, and yet you think that if philosophers of science don't do it the way you want them to, their entire field is discredited.

Suppose I accurately stated the theory of LR and I showed PHILOSOPHICALLY that it was equivalent to (some other), then It doesn't really matter what physicist think, my argument would be correct, and the implication is that LR is as viable as the other theory I showed it was equivalent to.
Let me make sure I'm understanding your argument correctly.

When you talk about the "viability" of a theory, do you mean physical viability--i.e., the theory's predictions are consistent with all experimental data up to date, etc.? If so, then the kind of equivalence that you're talking about--model equivalence--is a mathematical concept. And mathematical proofs should be peer-reviewed by mathematicians with the appropriate specialization--e.g., model theory. Having a degree in philosophy of science does not endow you with a sufficient understanding of model theory to be able to peer-review statements in model theory.

How many of those were on the board of the journal in question?
Irrelevant and goalpost-shifting, my point is about the qualifications of philosophers of science. Do you agree that a person doing peer-review for physics should have an understanding of physics at least at the phd level?

And exactly how good a source is Wiki on this topic? Is wiki the arbiter of the state of the field?
No, but wiki's a good place to start to learn about what philosophers of science generally have degrees / expertise in, what kind of work they do, etc.

And how many of them were on the board of the journal in question?
Goalpost shifting; part of this discussion is about the qualifications of philosophers of science to do peer review in science fields. Most of the philosophers of science I referenced had no advanced degree in a mathematical subject, and the ones who did had masters degrees (i.e., not expertise at the phd level).

I don't see the point of learning about the qualifications of people who are not on the board of reviewers in the journal being dismissed.
Or is that you dismiss the journal because Craig (not on the board) is not qualified?
No, I'm not talking about anything specific to this journal. I'm talking about how philosophy of science is done, what kind of science expertise philosophers of science can have and still do work in philosophy of science, etc. If I were critiquing the qualifications of the referees at Foundations of Physics, I'd probably be citing the CVs of people that I show to be referees at Foundations of Physics. Since I'm not doing that, it's probably safe to assume I'm not critiquing the qualifications of the referees at Foundations of Physics.

Foundation of physics board of editors
http://www.springer.com/physics/hist...editorialBoard

First name on the list
Jeff Bub PHD Mathematicla physics http://carnap.umd.edu/philphysics/bub.html

Second name
Xavier Calmet http://www.sussex.ac.uk/profiles/242816
List of his citations(a measure forwarded by one of you earlier) http://scholar.google.com/citations?user=Y4Ca4hgAAAAJ

So let me stop right there, because I don't see those guys on the list you provided.
Why are our lists different? Why would you try to discredit a journal, with people who are not on it's board?
It is now out right humorous that you would lecture me on research and qualifications, and then botch it so badly.
Our lists are different because my list was Wikipedia's list of philosophers of science from 1980 to today, and your list is editors at Foundations of Physics.

I was talking about philosophers of science and what kind of expertise in science philosophers of science have. You're looking at two people who are editors of Foundations of Physics.

Editors don't do peer-review. Editors send out the articles to reviewers to peer-review.

The second guy "Xavier", he may not have a PHD in physics(, but his citation list seems to suggest that others don't dismiss him as quickly as you would. ) Either way my argument is supported.
It seems you don't know what a reader is.

Are you familiar with the review/referee process at academic journals?

My" sure as certain" statement was proven, because that paper had it's support pulled.
Why?
Because the validity of the philosophical claims is directly related to the content of the theories being considered.
Which was my point.
And yet, the peer-review process at Foundations of Physics was insufficient to catch egregious mathematical flaws in a series of submissions over two years. And this is evidence we should trust their peer-review process to catch technical scientific errors?

No, moving the goal post fallacy is when I set a goal for you to reach, and then you reach it and I move the goal.
I am here challenging the placement of your goal posts.
Get it right.
You went from talking about the foundations of philosophy of science (how it should be done) to qualifications of philosophers of science (how it is done) to the editorial staff of Foundations of Physics. You're now only accepting arguments about Foundations of Physics. Sorry, MT, you don't get to pretend you never made your earlier statements. You don't get to ignore the evidence contradicting your claims about the expertise of philosophers of physics.

Well, I mean, you can--obviously, since you are--but I don't have to play along.

8. ## Re: WLC's Argument Against an Actual Infinity

Originally Posted by Squatch347
I believe you are incorrect here. There seems no coherent reason to accept that a partial order defined by non-causal relations would conform to the physical interpretations they are attempting to model. Unless you are going to argue that this theory is simply an abstract mathematical concept rather than an attempt to model physical reality, the fact that causal sets require causation is required.
Their causets are defined in terms of sets and partial orders (with additional restrictions, e.g. local finiteness, finiteness, etc., depending on the author). The dynamics are defined in terms of causets, with certain restrictions (the dynamics must model what we understand to be physical processes).

Again, to cite the language in the R/S paper:

A causal set (or “causet”) is a locally ﬁnite, partially ordered set (or “poset”). We represent
the order-relation by ‘≺’ and use the irreﬂexive convention that an element does not precede
itself. [p.3]

To my knowledge, R/S do not refine or change the definition of causet later in the paper. They do restrict their attention to certain kinds of causets, e.g. finite causets.

It seems that you want R/S causet to mean something other than a locally finite, partially ordered set. But none of that affects the validity or soundness of my proof, or my conclusion--that if you want R/S causets (construed generally, i.e. not necessarily my definition of R/S causet) to admit what I call R/S labelings, then you must use a definition of R/S causet other than a locally finite, partially ordered set.

This is further supported by the expanded definitions offered in the papers referenced here.
The elements of a causal set are taken to represent spacetime events, while its binary relation is taken to encode causal relations between pairs of events.
Axioms Paper
If a; b are elements of a causet x, we interpret the order a < b as meaning that b is in the causal future of a.
Labeled Causet paper
For a fuller introduction to causal sets, see [3, 4, 5, 6, 7]. (For recent examples of other discrete models incorporating a causal ordering see [8, 9, 10].)
R/S Paper
By the ‘causal structure’ on a manifold, one often means abstractly a set of binary relations between points, with characteristics that intuitively capture
the notion of ‘is lightlike related’ or ‘is timelike related’ or some other suitable sense (see Kronheimer and Penrose [4] for a precise, technical foray). In this
way, one can define the causal structure without the explicit presence of a metric. However, when it comes time to express the physical meaning, or even just the geometric meaning in the attempt to associate a metric to the structure, one must conclude as the references cited here do, that ‘p relates to q’ means that p connects to q via a suitable causal curve, as defined by the Lorentzian geometry.
The Class of Causally-Concerned Objects Can Confuse paper (pretty sure we linked this earlier, if not let me know).
This last condition guarantees that the “parameter time” of our stochastic process is compatible with physical temporality, as recorded in the order relation ≺ that gives the causal set its structure. In a broader sense, general covariance itself is also an aspect of internal temporality, since it guarantees that the parameter time adds nothing to the relation ≺... For example, the events of Minkowski space (in any dimension) form a poset whose order relation is the usual causal order
R/S Paper 2, Continuum Limits.
None of these quotes substantiate your claim. They aren't putting physical requirements on the partial order; rather, so far as I can tell, the physical requirements are put on the dynamics (e.g., "3. Physical requirements on the dynamics", p.8 R/S).

Clive, let me ask you a simple question. What do the lines in the Haas Diagrams mean? What does < mean when relating two elements in these papers? What do these relationships mean in a physical interpretation in our universe?
Those are three questions, and the last is not at all simple, but I'll take a crack

1. What do the lines in the Haas Diagrams mean?

I'll assume you mean Hasse diagram.

A Hasse diagram is a way to graphically represent a finite partially-ordered set. ["Graphically represent" here can be construed either generally, meaning "represent with a drawing or illustration", or technically, meaning "represent as a graph", here a directed graph.]

Let (S,<) be a partially-ordered set, and let D be a Hasse diagram of (S,<).

The vertices (or "points") in D are the elements in S.

If u,v are vertices in D (and therefore elements in S), then a line upward from u to v in D means:

(i) u < v
(ii) There is no z in S such that u < z < v

2. What does < mean when relating two elements in these papers?

Typically, < is a symbol used to refer to the standard order of a common number set, like , , , or . In the R/S paper, for example, < is used only to refer to the standard order on or :

(i) p.3, ℕ, in the statement "label(x) < label(y)", since label(.) is defined to be a non-negative integer
(ii) p.7, , in the statement "i < j", where i and j are labels and thus non-negative integers
(iii) p.7, , in the statement "p < pcrit", where p and pcrit are both elements of the closed unit interval [0,1] and thus real numbers

In the Axioms paper, < is used only to refer to the standard order on or ℝ:
(i) p. 52, , in the statements " 0 < ρ < 1 " and " Stρ(x) ∶= {t ∈ MR ∣ d(t, x) < ρ} "
(ii) p.59, , in the statement "(, <)"
(iii) p. 71, , in the statement " 0 j < n "
(iv) p. 115, , in the statement " t' < t''"

3. What do these relationships mean in a physical interpretation in our universe?

If you're talking about the relations on , , , and ℝ, the common interpretation of " x < y " is "x is smaller than y" (so long as x,y > 0), or "x is to the left of y on the standard number line". So I guess I'd say that < is interpreted as "is smaller than" or "is to the left of (on a standard number line)".

As I noted in my last response, the lines (or the < if displayed in another format) represent causal relationships. If the point differs along the y axis here (time) then it has a temporal component to it. If it is along the X axis it has a spatial component to it.
Normally, Hasse diagrams don't care about left/right positioning; If D is a Hasse diagram of (S,<), you can move the vertices of D around, so long as all the "upward" lines still obey rules (i) and (ii) from part 1 above. This is similar to the concept of a graph isomorphism, with the added "upwardness" restriction.

So it seems you're adding something to the Hasse diagram, since you care about left/right positioning. As before, I imagine you're sort of "overlaying" or embedding the Hasse diagram into a graph of space vs. time.

I'm not sure what you mean by "if the point differs along the y axis here (time) then it has a temporal component to it". If the point differs from what? The origin? What does it mean to "differ along" an axis?

[These are serious questions, I'm not asking them in a mocking fashion. I just want to know what your definitions are.]

Taken literally (with the common definition of "along", as in the question, "A particle moves along the x-axis"), your next statement ("If it is along the X axis it has a spatial component to it") says only that points lying directly on the X-axis have a spatial component.

Can you give me a further explanation of these definitions?

Two elements can be causally related if:

They are related only across the temporal axis.

or

They are related across both temporal and physical axes.

Two elements cannot be related if they are only related across the spatial axis.
I'm not sure what you mean by "related only across the temporal axis", because I'm not sure what you mean by "related across the temporal axis". Your definitions above talk about a point having a temporal component or a spatial component, but nothing about what "relation across the temporal axis" means.

"Across the temporal axis" might mean "on opposite sites of the temporal axis". If we think of the standard Cartesian coordinate system ( x ℝ) with the x-axis as space and the y-axis as time, two points are "across the y-axis" from each other if one is on the left side of the y-axis (i.e., its y-coordinate is negative) and the other is on the right side of the y-axis (i.e., its y-coordinate is positive).

Can you give me a further explanation of these definitions?

After re-reading this statement a few times either I'm dramatically misreading what you are saying here or you seem to have missed the entire point of these theories. If you mean what the plain text reading of your statement would imply, then it seems odd that R/S would discuss this issue for 10% of their paper, specifically saying "The child formed by adjoining an element which is to the future of every element of the parent will be called the timid child. The child formed by adjoining an element which is spacelike to every other element will be called the gregarious child." If there is really no need to talk about spatial and temporal relations here, why do they go out of their way to do so?

These papers are discussing relationships between points in our physical universe, these points exist in physical and temporal dimensions so the discussion of those dimensions would seem to be cogent to the question of whether or not the universe has a finite age.
I wasn't quite sure at first what R/S meant by "to the future of" and "is spacelike to", but now I'm fairly confident that those relations are defined solely in terms of the partial order ≺ on C.

Specifically, let x,y ∈ C:

(DEF1)
y is to the future of x x ∈ past(y) [where past(y) = {c ∈ C: c ≺ y}] x ≺ y
(DEF2) y is spacelike to x (x ≺ y or y ≺ x) (x ≺ y) and (y ≺ x)

[The second ⇔ statements follow in an obvious way from the statement before them, so their proofs are omitted.]

Note that DEF1 and DEF2 are defined with respect to the given partial order, ≺.

Given a partial order ≺ on C, you could define a temporal relation # and a spatial relation $on C as follows: (TEMP) x # y y is to the future of x (WRT ≺) (SPAT) x$ y y is spacelike to x (WRT ≺)

# turns out to be a partial order; $is a little more complicated. Here are my issues: 1. # is equivalent to ≺ The statement that # is equivalent to ≺ means that: x # y x ≺ y The proof breaks down into two parts: (x # y x ≺ y) Suppose x # y. Then y is to the future of x WRT ≺. By DEF1, x ≺ y. QED. (x ≺ y x # y) Suppose x ≺ y. Then y is to the future of x. By TEMP, x # y. QED. Taking p = x # y and q = x ≺ y, we have proved p⇒q and q ⇒p. Therefore, we have p⇔q and the result follows. 2. # is non-transitive This statement means that: (TRANS) (x$ y) and (y $z) (x$ z)

does not always hold.

Take for example C = {a,b,c}, ≺ = {(a,b)}.

Since (b ≺ c) and (c ≺ b), we have b $c. Since (a ≺ c) and (c ≺ a), we have c$ a.
If $were transitive, we must have b$ a.
However, since a ≺ b, we have (b $a). Therefore,$ is not transitive.

3. Different spacetime relations can have the same Hasse diagram

Take C = {a,b,c} and ≺ = {(a,b)}. This means that c is causally "remote" from a and b; c doesn't cause a or b, and neither a nor b cause c.

However, c can differ in spacetime coordinates from a and b by as much as you like without changing the Hasse diagram of the causal relations.

From issue (1), there's no point to talk about #, since it's just the same as ≺. # "adds" nothing.
Issue (2) isn't really saying much, to be honest. It seems a bit counter-intuitive to me, though, that x could be spacelike to y, and y spacelike to z, but x not spacelike to z. The conclusion I'd draw is that I'd be careful applying intuition to spacelike relations (at least spacelike relations that are similar to the one defined by SPAT).
From issue (3), you can't infer spacetime relations ("near", "far", "before", "after", "to the left", "to the right", etc.) from the Hasse diagram.

Ok, so given that, can we agree that a point could be both a) to the future of all other points in S (ie a timid child) and b) not causally related to any point in S (ie not connected via a line or a <)?
I think there's a disagreement on definitions here. I've defined "to the future of" in DEF1 above, and I believe this is the definition in the R/S paper. To wit:

Note that the timid child Cb is the only child whose additional, maximal element x is such that c ≺ x for all c from the parent.

You could have "redrawn" the gregarious child Cc by putting the "new" element (that is not connected to any element from the parent) above all the other elements; as long as this is the only change you make (e.g., you move no other points and add no other arrows), then you haven't changed the Hasse diagram of Cc.

Look at the following two children from the image:

The child on the left is identified as the timid child on page 5 of the R/S paper. The child on the right is not timid, even though its maximal element is "above" the elements from the parent.

If you are interpreting "above" as "to the future of", then the child on the right should also be a timid child. However, it is not.

Furthermore, on page 5 of the R/S paper, we have the following statement:

The past of the new element (a subset of C) will be referred to as the precursor set of the transition (or sometimes just the “precursor of the transition”).

On page 14 of the same paper, we have the following statement:

a transition whose precursor set is the entire parent (‘timid’ transition)

A timid transition is one whose precursor set (i.e., the past of the new, maximal element x) is the entire parent, i.e. past(x) = C, so c ≺ x for all c from the parent.

Let me ask this. If I were to say that we had two elements, a and b in a cuaset. And I were to say that a < b, meaning that b is to the causal future of a (lets assume no intermediate elements) would it be correct or incorrect of me to say that b is a child of a the parent?
I'd say it's an abuse of notation / terminology. Technically, A and B (I'm changing to capital letters for readability) could also be causets such that B is the child of A in the R/S sense of "child", i.e. B can be formed by adding a maximal element to A.

But you could use "B is a child of A" to mean "A ≺ B". This definition of "child" is different than R/S's definition of child (i.e., B can be formed by adding a maximal element to A), so that's why I'd call it an abuse of notation/terminology. But abuse of notation/terminology can be useful (e.g. treating differentials as actual fractions).

I don't see the usefulness of calling B a "child" of A (or A a "parent" of B) when A ≺ B. We already have concise notation for that concept: A ≺ B. Why bother with an abuse of terminology when it's longer, adds nothing, and is potentially confusing?

It would either imply reverse causation or create a paradox. You are using information about x1000 to determine the value of x999 when that information cannot exist unless the value of x999 is known. So you are using y to figure out x so that you can use x to figure out y.
I'll give you an example. Suppose I told you that there's a sequence
S = x0, x1, x2, ... = {xn}n∈

such that xk+1 = xk + 2. So given a "point" in the sequence, xk, you know how to find the next "point", xk+1.

If I told you that x0 = 0, then you could immediately deduce that

S = 0, 2, 4, ... = {2*n}n∈

The deduction might go like this:

Let f(x) = x + 2.

(LEM1) For all k ∈ ,xk = fk(x0) = f(f(...f(x0)...)), with k applications of f [by convention, f0(x) = x].

(Base Case)
k=0
Then x0 = f0(x0), so the statement holds.

(Inductive Case)
Suppose xk = fk(x0) for some k ≥ 0
Then xk+1 = xk + 2 = f(xk) = f(fk(x0)) = fk+1(x0)
So the statement holds for k+1.
QED.

(LEM2) For all k ∈ , fk(x0) = x0 + 2*k

(Base Case)
k=0
Then f0(x0) = x0 = x0 + 2*0, so the statement holds.

(Inductive case)
Suppose fk(x0) = x0 + 2*k for some k ≥ 0.
Then fk+1(x0) = f(fk(x0)) = f(x0+2*k) = x0 + 2*k + 2 = x0 + 2*(k+1).
So the statement holds for k+1.
QED.

(THM) For all k ∈ , xk = x0 + 2*k = 0 + 2*k = 2*k

This follows from LEM1 and LEM2, along with the initial value specification x0 = 0.

However, if I told you that x1 = 2, you could also deduce that

S = 0, 2, 4, ... = {2*n}n∈

The deduction might go like this:

By THM, for all k in N:
xk = x0 + 2*k
x0 = xk - 2*k
x0 = x1 - 2*1 = 2 - 2*1 = 2 - 2 = 0

And in fact, given the value of any xk you can deduce the value of every "point" in S due to the structure of the sequence (i.e., that xk+1 = xk + 2).

You're correct, it seems to have deleted my additional text for some reason (actually it seems to have hidden a portion of it in your post and repeated it about a dozen times, click reply to your post and you'll see some odd text repeated in there). Odd (after further review, it looks like vbulletin has an issue with using the < sign with text around it without spaces, ie a < b if you removed the spaces). Essentially I was just giving an example to a point you seem to agree with.

If I were to label the elements with colors {blue, red, green, orange} Inferences about the elements based on the wavelength ordering of those colors wouldn't be valid since that isn't an order I used to relate the colors to the order in the causet. This might be ridiculously obvious to be pointing out, but I think it is important to prevent confusion concerning physical interpretations.
If you defined a partial order on {blue, red, green, orange}, it might work.

If I recall correctly, that was a proof on integers, not on natural numbers.
Hmm, I'm not sure what you mean here.

My proof was that (,<) (the set of integers with their standard order) admits no R/S labeling, i.e. that no map L:(,<) → (ℕ,<) satisfies

z1 < z2 L(z1) < L(z2)

for all z1, z2.

Just as I think you are ignoring an obvious conclusion. But setting that aside, I still haven't seen an objection against my argument in post 333
1) There is a specific theory that only allows for one method of set creation (iterative addition of finite values).
2) That method of set formation is insufficient to explain the existence of an infinite set.
3) Therefore there are no infinite sets within the confines of that theory.
First, what are the arguments for (2)?

Second, (3) is a bit ambiguous. What does "within the confines of that theory" mean?

Let me illustrate my point with an example.

Here's some pretty intuitive definitions that might come in handy: Suppose you have a set of axioms A = {A1, A2, A3}. Any system where every axiom in A holds is a model of A. If you can deduce a contradiction from the axioms in A, then A is inconsistent. Otherwise, A is consistent.

Suppose you have another set of axioms A' = {A1, A2, A3, A4}, and that A' is consistent. Clearly every deduction using axioms in A can be deduced in A', since A' includes all those axioms. If every deduction using axioms in A' can also be deduced in A, then A' and A are equivalent.

Every model of A' is a model of A. If A' is equivalent to A, then every model of A is a model of A'.

Now, to the example:

Suppose your axioms are something like S = "You can build sets using [these rules]". If you add a rule to make a new axiom set S', then as long as you don't contradict the deductions from S's axioms (i.e., the new rule doesn't forbid building any of the sets from S), S' is consistent. [I should prove this, to be rigorous, but

So: Are models of S' "within the confines" of S? Models of S' are all models of S, and I tend to think that "within the confines of S" means "is a model of S", so I'd say "yes, models of S' are within the confines of S".

But what I'm interested in is what you mean by "within the confines". Can you address this issue by considering this example?

I saw it, the problem is that it is not a material objection to anything I've said. Your final statement:

Would seem to be a "I agree with what you said" statement. If you select an empty partial stem (IE you don't include the element's causal past) then you have a gregarious child. If you were to select a non-empty partial stem (ie causally prior elements) then it would not be gregarious.
(1) It's more accurate to say the empty partial stem, since there's only one.
(2) The partial stem you choose defines which elements are causally prior to the new maximal element, which I'll call x. The elements causally prior to x are precisely the elements in the partial stem you choose.
(3) Are you not accepting my criticism regarding your "a different partial stem (and therefore anti-chain)" statement? You didn't retract or correct your statement, so I suspect you're just ignoring my criticism.

So all this page after page wrangling could be simplified to "gregarious children" do not reflect a continuation of a causal chain of elements and therefore would be inappropriate to include in any discussion of a measurement of something's age via causal chains.
Again, gregarious children are causets. I think here you're referring to the new maximal element in the gregarious child. And I don't know what "inappropriate to include in any discussion of a measurement of something's age via causal chains" means.

Suppose we're trying to determine the age of a causet (C,<), and we decide to consider the length of chains in C. Let Cg be the gregarious child of C.

Possible Interpretation #1: You're saying that the gregarious child of C is irrelevant to C's age in terms of lengths of chains.

This seems wrong.
Whenever the length of the longest chain in C exists, it is equal to the length of the longest chain in Cg. So it seems that Cg isn't irrelevant to C.

Possible Interpretation #2: Whenever past(x) is empty for x ∈ C, then x is irrelevant to C's age.

This seems true, except for the case where C's age is 0 (i.e., past(x) is empty for all x ∈ C) [although whether you call "x" a chain of length 0 or length 1 is a matter of definition; we haven't actually defined what the length of a chain is yet].

Again, I think we need to remember that these causets and relationships have a physical meaning. Each element in the causet relates to a point in the universe, so while not every point within a causet must be related via causation, it is related in the sense that it exists (or existed) within a physical universe.
This seems like a radical statement. Every causet represents a universe that physically exists?

So if I were to say, a,b,c,d are all points in a spatial temporal universe and that point a is not earlier or later than b, you would not conclude that point b must be a point at the same time as a, but at different spatial coordinates?

And if not, what, specifically, do you think R/S mean when they say a point that is "spacelike" to every other point?

I'd assume a,b,c,d are all points that have a time coordinate. I'll write the time coordinate of x as xt.

I'd probably assume that the time coordinate is totally ordered, i.e. at < bt or at > bt or at = bt. Given that the first two are false, I'd deduce that the last must be true.

I think you're assuming that these spacetime relations are uniquely defined by the causal order. This is not true; as I said earlier, two different spacetime arrangements S1 and S2 of a set of events E can have the same Hasse diagram.

I think that R/S mean DEF2 as defined above in this post.

I think the definition offered in the Axiom paper is the best one to ground this discussion. "The elements of a causal set are taken to represent spacetime events, while its binary relation is taken to encode causal relations between pairs of events." So given that these elements are taken as space time events, they would need to be represented in spatial and temporal dimensions.
But those relations aren't uniquely defined by the causal order. The particular arrangement will depend upon a choice of basis / frame, cf. Twin paradox and relativity of simultaneity.

It doesn't really matter either way. They can be contemporaneous to each other in relation to some absolute reference frame or they can be relative to each other, neither interpretation really affects the question. Even if each maximal element only relates to a local "now" and could be further ahead or behind (in a purely temporal not a causal sense) any other maximal element.

Lets assume a very simple universe. One physical dimension made up of one single point and one temporal dimension. This will eliminate any need to worry about relative or absolute "nows."

That universe is at x,tn

xn is defined as "now."

(x,tn+1) being a maximal element not yet born, we can conclude there is a longest chain in that universe correct?
I don't think I understand your construction; I'll have to ask some questions to clarify my understanding.

You're saying that the universe consists of a one-dimensional "spacetime" (really, just "time") as well as a single "point".

(1) Is the point a point in spacetime (i.e., time), or is the point a physical object in the universe?? If the universe has only one point in time, then its time dimension is 0.

(2) What is "x,tn"? Is x the "point"? Is tn a single value from the universe's time dimension?
(3) What is "xn"? Is it equal to "x,tn"?
(4a) What is "(x,tn+1)"? Do the parentheses indicate this is a different kind of object than "x,tn"?
(4b) In what way is "(x,tn+1)" maximal? Is it maximal WRT to some partial order? If so, what partial order?

Then why is it's review board made up entirely of physicists?
Not quite true--I counted two non-physicists on its board--but to my understanding, editors send out articles for review. Do you know the extent to which the board of editors actively reviews articles for physics errors?

And I think you would have been better served to pick a better source than Wiki. What you got was the pundit list made up of some good names and some names that really have nothing to do with the subject outside of popular press (Chomsky for example is in no way a philosopher of science).
I cited Wikipedia because it seemed like a good place to start; literally nobody else had provided any sources to support their statements about philosophy of science or philosophers of science.

The Stanford Encyclopedia of Philosophy has quite a few references to Chomsky (78 on Philosophy of Linguistics), although I'm not sure to what extent you consider philosophy of linguistics a philosophy of science. The Stanford Encyclopedia of Philosophy takes this position [top of the Philosophy of Linguistics page]:

Philosophy of linguistics is the philosophy of science as applied to linguistics.

Seems to me like philosophy of linguistics is a philosophy of science, and that Chomsky is a philosopher of linguistics--and therefore a philosopher of science.

What would have served you better would be to review where actual peer-review is going on.

If you want to claim that Foundations of Physics is a philosophy of science journal than your entire argument is rebutted by the make up of the review board.
I'm not claiming that Foundations of Physics is a philosophy journal. I'm not even sure GP has made that claim (although perhaps he has).

Rather, the question is whether Foundations of Physics does physics peer-review. If it does, then you've clearly met GP's standard. If it doesn't, then hopefully one of your other sources was published in a journal that is peer-reviewed for physics.

It's not incumbent on GP or anyone else to show that Foundations of Physics doesn't do physics peer-review; if you're going to claim that it does [which perhaps you haven't--I'm never quite clear on which statements you admit you make and which you deny], then you must support your claim.

If however you would like to argue a different philosophy of science journal, lets look at the Philosophy of Science journal, whose board can be found here: http://www.press.uchicago.edu/journa...l?journal=phos

I searched 15 random people on the board. 12 had degrees in fields other than philosophy (in conjunction with a philosophy degree). 1 (no other degree that I could find) also taught applied mathematics, 3 others with different degrees also taught course work in this field. An extremely cursory look showed 7 of them having published in non-philosophy journals.

In fact, a deeper look at the listed articles of both the editors and the journal itself would seem to indicate a deeper working relationship between the two groups. There are papers in this journal concerning the validity of the statistical measures used to the inferences offered in two different other academic works. Reviewing some of the board member's papers I see papers published in field journals directly as well as in philosophy journals concerning interpretation issues primarily (most of the titles appear to revolve around technical issues concerning whether the inferences of papers were justified, whether experiments were underpowered, etc.). There would seem to be a lot more dialogue between these fields than you are prepared to grant, or perhaps than you realize.
Hmm, perhaps I'm mis-recalling this debate, but I'm at a loss as to which claims of mine you're addressing here.

I don't recall realizing that there was very little dialogue (or at any rate, less dialogue than you've claimed here) between science and philosophy of science, or being unprepared to grant that there is more such dialogue.
I don't recall claiming that the journal Philosophy of Science, or philosophy of science journals in general, tend to have very few board members possessing degrees in science fields.

My claims were things like:

Philosophy of science doesn't necessarily require a technical expertise in the theories being analyzed, or at any rate the kind of technical expertise needed to qualify as a science expert.
Philosophers of science don't require advanced degrees in scientific fields to do work.
Philosophy of science journals don't necessarily do physics peer-review.

9. ## Re: WLC's Argument Against an Actual Infinity

So I'm going to weigh into a couple of important points surrounding Squatch and Clive's causet discussion.

Originally Posted by Squatch347
It would either imply reverse causation or create a paradox. You are using information about x1000 to determine the value of x999 when that information cannot exist unless the value of x999 is known. So you are using y to figure out x so that you can use x to figure out y.
That's not what causality says, at least not precisely. As a good example, I can construct the trajectory of a particle, as a sequence, and determine it to be of the form (x_n, t_n). As long as you know the form of the sequence, e.g. x_n = f(t_n), then you only need to know where it was at at one point in time in order to reconstruct the entire trajectory. You don't need to know x_n+1 to know x_n. You just need to know any of them, for example x_n-1, x_n-2, x_n-3, etc, would also have been fine choices and there's obviously no causality violation there. (The key point here is that you know the general form of the trajectory it gives you a lot of predictive power with minimal input).

Causality issues arise when x_n+1 is actually required to know x_n. An example of this arises in study of the Abraham-Lorentz force law for accelerating, charged test particles, which is called pre-acceleration. Other true examples of causality violation are closed time-like curves, such as those in the hypothetical Gödel spacetime.

Originally Posted by Squatch347
I grabbed these quotes together because they seem to represent a relatively fundamental difference in understanding between us that would probably benefit us by resolving.

In the context of a R/S Causet, what is the physical meaning of the order relation that makes this a partially ordered set? The reason these are called "causal sets" is because the order is one of causal influence.

From our Axioms paper:
The elements of a causal set are taken to represent spacetime events, while its binary relation is taken to encode causal relations between pairs of events....
The binary relation < defines an interval finite partial order on C, called the causal order, with the physical interpretation that x < y in C if and only if the event represented by x exerts causal influence on the event represented by y.
In this way, one can define the causal structure without the explicit presence of a metric. However, when it comes time to express the physical meaning, or even just the geometric meaning in the attempt to associate a metric to the structure, one must conclude as the references cited here do, that ‘p relates
to q’ means that p connects to q via a suitable causal curve, as defined by the Lorentzian geometry.
http://www.physics.umd.edu/grt/jacob...ts/brendan.pdf (I cite him not as a definitive definition, but because he is quoting a classic paper I do not have full access too).
It is conjectured that, when suitable causal sets on a large number of elements are considered, there exist unique lorentzian manifolds (up to small changes in the metric), in which the causal sets appear as uniformly distributed points, with metric-induced causal relations which agree with the partial order relation, and which are approximately flat on the length scales determined by the density of embedded points. These manifolds are free of causality violations and time-orientable, and provide a causal macroscopic interpretation of the partial order relation.
http://www.researchgate.net/publicat...s_a_causal_set
The causal set program is one of a number of discrete spacetime approaches to the problem of quantum gravity. It proposes that the microstructure of spacetime is that of a partially ordered set, a causal set, in which the partial order encodes information about the causal structure of spacetime

http://scitation.aip.org/content/aip...1063/1.2905136

Hold on there, cowboy.

You are aware that spacetimes and, in particular, Minkowski spacetime (and any spacetime manifold) is 100% incompatible with your assertions that you believe in LR and your verison of LR is not an interpretation of SR? SR admits an interpretation in terms of Minkowski spacetime; your variant of LR does not admit an interpretation in terms of Minkowski spacetime by definition. Remember, you're the one who's been throwing a stink about any "non-mechanical" origin of relativistic effects. You do realize, don't you, that these geometric pictures provide a non-mechanical explanation of relativistic effects, and literally imply the direct existence of Lorentz invariance, right? They make the Minkowski interpretation of Special Relativity (You know, the thing you've spent 200 posts denying that it has any kind of validity), and are manifestly (by your own admission 200 posts ago) inconsistent with the A-theory of time.

It'd be great if you could clarify which theory you believe in. Oh, and by the way, your lame excuse of "it predicts almost the same thing" line will not work. Minkowski spacetime has exact Lorentz invariance (as do the other spacetimes). It is exactly Special Relativity. It has no absolute reference frame.

Originally Posted by Squatch
Why not do your own research? Why must I be the one to justify myself to you? That is the merit of my objection. Both you and GP have shown that you seem to view yourselves as the academic panel to which I must defend myself and which has the final say on a source's "worthiness." That is not the case here. I made a claim which I defended, if either of you wished to make an objection to that claim then make an argument countering the claim or showing how the source is incorrect.
So here's what you and MT don't understand very well. Both Clive and I have professional training in mathematics and mathematics & physics, respectively. This is a fact. But you guys don't seem to understand what that means, so let me be clear here:

We're not saying it makes us right about everything. This training does not give us the ability to know the absolute truth and make us beyond reproach. What it does do is give us the ability to know when something is wrong. Most of our undergraduates and my PhD have been spent learning how to identify problems and come up with solutions. But I will emphasize that learning how to identify problems section of that. This training tells us what arguments should look like, it tells us what kinds of sources are reasonable, it gives us the background information so we can evaluate the validity of claims, and it gives us the cultural knowledge of what an acceptably rigorous argument in the field is.

So when Clive and I see you and MT making statements that fly in the face of over a century (and in some cases, centuries) of mathematical theorems or experimentally verified physical theories, we don't need to be Richard Feynman to know that you're screwing up your physics or to be John van Neumann to know that you're screwing up your mathematics. Our objection to your statements aren't because you're both laypeople. Our problem lies with your continual desire to casually imply that we're wrong about what are often very basic facts, and when you casually assert falsehoods as facts. Or when you casually understand that you know better about X academic field when really you have no idea what goes on in that field.

When we ask you to do your own research, frankly, it's because you're trying to just assert things without bothering to check if they're actually reasonable or proofed for their veracity, or the arguments that you are presenting are openly sub par to the level of rigor considered acceptable in the field. The fact of the matter is, we don't care that you haven't been trained in these subjects; but we would like you to be a little bit more humble about the fact that guys haven't been trained. And by humble, I mean that you should be a little bit more willing to double and triple check that you're correct before you assert that someone --who is trained in the topic at hand-- is wrong, that you should make an serious effort to emulate the level of rigor acceptable in the fields that you are discussing (rather than the "well, my intuition tells me" bare assertions), and that you should try to understand why these trained people are nonplussed by your sources and your assertions. It should raise a red flag, and the flag it shouldn't raise is "Wow, I guess they were too misinformed/ignorant/biased/arrogant/elitist to recognize the obvious brilliance of my assertion/source."

I'm personally happy that you're interested enough to debate Clive and I on the topics of physics and mathematics. I'm much less happy that you're not showing any skepticism towards your information sources, and that you are basically defaulting to the position that Clive and I don't know what we're talking about when we disagree with you, when we correct you, or when we ask for clarification, and then the following all-too-frequent hissy fit about how we're just being arrogant, mean, overly-critical, a know-it-all, elitist, self-absorbed, or just plain full-of-it. We're are holding you to a much lower standard than we hold ourselves, in case you're wondering.

10. ## Re: WLC's Argument Against an Actual Infinity

Originally Posted by GoldPhoenix
If it wasn't clear, I'm removing your ability to claim that I'm being a big, bad, unfair, elitist, arrogant academic by letting you pick the paper which you feel best supports your case.
GP, I've already offered the argument and the sources. If you wish to debate them, go ahead and do so, but it serves no purpose for you to continue to ask me to refine an argument you couldn't be bothered to rebut.

Originally Posted by GP
In case you've forgotten that, on top of the other issues that I've raised and you've been blowing off for four pages now, you've already conceded that I showed you that one of your sources overtly went against your position in post #250.
I conceded that it showed evidence against LR, not against my claim. My claim was that this subject was being discussed. If anything it supports the point that LR is at least legitimate enough to warrant experimentation and research writing.

Originally Posted by CliveStaples
Again, to cite the language in the R/S paper:

A causal set (or “causet”) is a locally ﬁnite, partially ordered set (or “poset”). We represent
the order-relation by ‘≺’ and use the irreﬂexive convention that an element does not precede
itself. [p.3]
Please define what is meant by "precede" here. Precede in what sense? What is the partial order that R/S are imposing on this set?

Originally Posted by CS
None of these quotes substantiate your claim. They aren't putting physical requirements on the partial order;
I'm not sure how that conclusion can be reached given the quotes above. So when the Axioms paper says:

This phrase represents a specific version of what I refer to as the causal metric hypothesis, which is the idea that the properties of the physical universe, and in particular, the metric properties of classical spacetime, arise from causal structure at the fundamental scale.

It isn't talking about a causal relationship in the physical world?

Or when it says:

Causal set theory may be expressed in terms of six axioms: the binary axiom, the measure axiom, countability, transitivity, interval finiteness, and irreflexivity. The first three axioms, which fix the physical interpretation of a causal set,

It isn't talking about the binary axiom (which discusses the relationship between elements) describing a physical interpretation of the causal set?

Or even when it says:

The binary relation < defines an interval finite partial order on C, called the causal order, with the physical interpretation that x < y in C if and only if the event represented by x exerts causal influence on the event represented by y.

It isn't saying that we should interpret < in the physical world as a causal influence?

Or when the Foster paper says:

However, when it comes time to express the physical meaning, or even just the geometric meaning in the attempt to associate a metric to the structure, one must conclude as the references cited here do, that ‘p relates to q’ means that p connects to q via a suitable causal curve, as defined by the Lorentzian geometry.

We shouldn't intrpret the relationships described in the causet as being connected via a causal curve?

And when R/S say:

One can
then use this rule – technically a probability measure – to ask physically meaningful questions of the theory.

We shouldn't interpret this as them relating these causets to physical reality?

Or when they say:

One obvious reason is that the classical case, being much simpler, can help us to get used to a relatively unfamiliar type of dynamical formulation, bringing out the pertinent physical issues and guiding us toward physically suitable conditions to place on the theory.

That doesn't mean they are constraining the theory to conform to physical reality?

Really the bottom line question is Clive. If the authors presented here don't mean causal relationship when they discuss Causal Sets, what are elements related by when they express this z < x?

Originally Posted by CS
So I guess I'd say that < is interpreted as "is smaller than" or "is to the left of (on a standard number line)".
Hmm, you seem to be confusing the labeling criteria for the physical interpretation. I'm not talking about the number or sign or picture placed on an element, I'm talking about the physical relationship that is being described between two elements. A causet is a set of elements (spacetime events) that all exist within a spacetime context. These elements can be related by causal curves, (the labeling equivalent of which is < ) or they can be non-causally related (ie they exist in the same universe but have no causal effect on each other).

Do you disagree that the causets in these papers are meant to model the causal relationships between spacetime events?

Originally Posted by CS
Normally, Hasse diagrams don't care about left/right positioning...I imagine you're sort of "overlaying" or embedding the Hasse diagram into a graph of space vs. time...if the point differs along the y axis here (time) then it has a temporal component to it". If the point differs from what?...Can you give me a further explanation of these definitions?
So first, lets operate off of this image from the axioms paper for a common picture.

From this we can see that the Hasse diagram is, in fact, a defacto overly on a space time graph. R/S also refer generally to this process in that they describe the upward direction of a Hasse diagram as being related to the stochiastic growth process which is described as being the process of temporal passage.

Now, consider points x and A. They exist at different points along the Y axis right? Given that we see this axis is labelled as time we can interpret this difference as being that these two events exist at different points in time. Just as we could interpret point C existing at the same point in time, but at a different physical location than x.

So in the final panel we can see the same elements, but related via causal relationships. The elements that x has a causal influence on all exist at a subsequent time from x and in this case (though not necessarily) at a different physical location from x.

Originally Posted by CS
I wasn't quite sure at first what R/S meant by "to the future of" and "is spacelike to", but now I'm fairly confident that those relations are defined solely in terms of the partial order ≺ on C.
And given the language used above, does that partial order < have any physical significance?

Originally Posted by CS
A timid transition is one whose precursor set (i.e., the past of the new, maximal element x) is the entire parent, i.e. past(x) = C, so c ≺ x for all c from the parent.
I think that is a fair point and that you are right there. A timid child is only that definition if the new element is the causal future of all the elements of it's parent. I think we still have a disagreement on what < means, but sufficed to say the subsequent point (whether a timid child has a < relationship to all the maximal elements of its parent) I'll certainly concede.

Originally Posted by CS
I'd say it's an abuse of notation / terminology.
Hmm, so Prof. Gudder would be abusing the notation on page 3 of the Labeled Causets paper when he said:

If a, b are elements of a causet x, we interpret the order a < b as meaning that b is in the causal future of a.

Originally Posted by CS
And in fact, given the value of any xk you can deduce the value of every "point" in S due to the structure of the sequence (i.e., that xk+1 = xk + 2).
And that fact relies on the imposition of an external structure (the natural numbers) which allows for you to make that deduction. IE you know the relationship of the natural numbers to each other which allows you to map the entire sequence. It isn't clear that any such structure exists within causets. While they can be labelled with natural numbers, those numbers don't dictate the structure of the causal relationship.

To approach this from another angle. Let us assume for a moment that I have shown sufficiently that the < relationship describes a causal relationship and that a < b your argument would seem to be implying that the effect (b) can causally precede the cause (a). IE I have the effect b, and then I can deduce the cause. That is about as close as I can image to reverse causation right?

Originally Posted by CS
Hmm, I'm not sure what you mean here.

My proof was that (,<) (the set of integers with their standard order) admits no R/S labeling, i.e. that no map L:(,<) → (ℕ,<) satisfies

z1 < z2 L(z1) < L(z2)

for all z1, z2.
Perhaps I still misunderstand you. This seems to be that conclusion that we cannot use ℤ as an R/S labeling. IE, you cannot map ℤ onto a naturally labeled causet and satisfy the given requirements.

If that is what you mean, I totally agree. But then why the earlier argument that there are causets that admit not natural labeling?

Originally Posted by CS
First, what are the arguments for (2)?
I think I've made quite a few to date. In PM and in post 322 I offered up the discussion from set theory where the author makes the argument that a process that is similar to what we are talking about here is insufficient to create infinite sets. Your objection was that he is talking about finite unions which I think misses the point. I wasn't attempting to limit any axioms that would allow for an infinite past, rather I was showing that if we start from the ground up and require justification for each axiom added rather than simply assuming it until proven incoherent that the conclusion warrants a finite universe until/unless we assume an infinite one (which seems a bit circular).

The difference between our two approaches seems to be that you are willing to accept all axioms until proven inconsistent and as such you see my non-inclusion of a specific axiom as an attempt to form a circular argument. Rather I am approaching it from the view that no axioms are permitted until justified and we have yet to justify that axiom. Without it's justification we cannot reasonably conclude that the past formed by the process is actually infinite.

Originally Posted by CS
Here's some pretty intuitive definitions that might come in handy: Suppose you have a set of axioms A = {A1, A2, A3}. Any system where every axiom in A holds is a model of A. If you can deduce a contradiction from the axioms in A, then A is inconsistent. Otherwise, A is consistent.

Suppose you have another set of axioms A' = {A1, A2, A3, A4}, and that A' is consistent. Clearly every deduction using axioms in A can be deduced in A', since A' includes all those axioms. If every deduction using axioms in A' can also be deduced in A, then A' and A are equivalent.

Every model of A' is a model of A. If A' is equivalent to A, then every model of A is a model of A'.
I mean confines in the sense that a conclusion of infinite sets (given A) would be unwarranted. Confines is used in the sense of boundaries. Given the axioms offered in the earlier premises one cannot conclude from that alone that the universe is infinitely old. IE in order to form the conclusion you would need to break the scope of the premises offered by offering either another axiom, in this case the assumption that an actual infinite exists, which would require the formulation of another method of set creation sufficient to create an infinite set if the new axiom set (A') is to be consistent with A.

Originally Posted by CS
(3) Are you not accepting my criticism regarding your "a different partial stem (and therefore anti-chain)" statement? You didn't retract or correct your statement, so I suspect you're just ignoring my criticism.
To be frank it is because your criticism don't seem material to my point. If we were to form the same new element x and select a non-empty partial stem, it would no longer be gregarious right?

Of course all of this is a rabbit hole discussion to a point I think we agree on. That if we were to use a concept from causets to discuss the "age" of the universe, we would use a chain, which by definition contains no gregarious children.

Originally Posted by CS
Suppose we're trying to determine the age of a causet (C,<), and we decide to consider the length of chains in C. Let Cg be the gregarious child of C.

Possible Interpretation #1: You're saying that the gregarious child of C is irrelevant to C's age in terms of lengths of chains.

This seems wrong.
How can a gregarious child be considered in a chain given that it is by definition not related by a causal < ?

Originally Posted by CS
This seems like a radical statement. Every causet represents a universe that physically exists?
That is a pretty far stretch of what I said. No, depending on the theory causets either represent a portion of different related elements within a universe (or the entire universe which has sub-causets) or the different possible outcomes in the case of the "sum of many worlds" theory.

Originally Posted by CS
You're saying that the universe consists of a one-dimensional "spacetime" (really, just "time") as well as a single "point".
No, I said that it consists of: "One physical dimension made up of one single point and one temporal dimension." IE we have two dimensions, one physical, one temporal, (x,t).

Any event in this universe can be described by a coordinate, (x,t). Part of the confusion I imagine was a typo. I meant to define "now" as "tn."

Given that the event (x,tn+1) is a maximal element and not yet born, this universe has a longest chain at "now" (tn), correct?

I define (x,tn+1) with respect to the causal order defined in R/S (which is a disagreement we have). I argue that each event in this universe must have a causal predecessor if it is to obey causality. We could form a causal set in this universe in the following manner (assuming (tn=4):

{(1,0) < (1,1) < (1,2) < (1,3)}

I hope this clarifies my example.

Originally Posted by CS
Not quite true--I counted two non-physicists on its board--but to my understanding, editors send out articles for review. Do you know the extent to which the board of editors actively reviews articles for physics errors?
I hope you aren't referring to Dennis Dieks and Brigitte Falkenburg, while GP perhaps did not notice, both of these editors have physics degrees as well. Regardless, we can agree that at very least, a majority of the board is made up of physicists correct?

It wouldn't seem odd to you that a panel of mostly physicists would read papers for comment and inclusion and not for technical review? If technical review was not part of this process, why have physicists and not professional editors?

I would also point to this document: http://www.lnf.infn.it/acceleratori/...-FOOP353R1.fdf which is a re-submission to the journal of a paper where technical and style corrections were made. The response to reviewer 1 is clearly of a technical nature where the reviewer found the evidence not fully supportive (or at least not clear) for the conclusion made.

Originally Posted by CS
Hmm, perhaps I'm mis-recalling this debate, but I'm at a loss as to which claims of mine you're addressing here.
The implicit assumption being made both to me and specifically to MT is that a philosophy of science publication can be discounted out of hand simply because it is from a philosophy of science journal. Aside from the salaciousness of that argument (as I pointed out before, it could be from a homeless dude, I don't care as long as the claim is about the truth value of the material rather than the authoritativeness of the author), that conclusion would seem to rely on the assumption that philosophers of science are writing papers ill informed about the subject they are writing about, which would seem to make the entire field a fraud and sham, which is perhaps the position being taken here, at lest implicitly.

My point by showing an actual board of an actual philosophy of science journal was to show that it is intentionally made up of experts in the fields in question to ensure that papers meet a standard set out by the journal.

For the journal offered above:

What we publish

Philosophy of Science aims to publish the best original work in philosophy of science, broadly construed. This work will advance the discussion in some area of philosophy of science or the philosophy of a particular science in a way that is of direct interest to experts in the field, and in a way that explicitly acknowledges and responds to existing work in the area. All submissions are peer reviewed.

Professional Standards

Submissions that do not take into account the existing literature on the topic of the paper and related topics are not publishable in Philosophy of Science.
http://journal.philsci.org/editorial-policies

So given that they are peer-reviewing the journal and that at least a significant portion of those reviewers have field specific degrees, what warrant is there for the assumption that these works are not consistent with existing work in the area?

11. ## Re: WLC's Argument Against an Actual Infinity

Originally Posted by Squatch347
GP, I've already offered the argument and the sources. If you wish to debate them, go ahead and do so, but it serves no purpose for you to continue to ask me to refine an argument you couldn't be bothered to rebut.
Your unwillingness to even give me a paper (I'm not asking you to even make an argument; I'm just asking you to give a paper), which you found the most convincing, speaks volumes about how much you don't have a valid argument or source.

I'm not sure who you think that you're convincing here, Squatch, because these tactics are fooling no one.

Originally Posted by Squatch
I conceded that it showed evidence against LR, not against my claim. My claim was that this subject was being discussed. If anything it supports the point that LR is at least legitimate enough to warrant experimentation and research writing.

Really? Really?

"[I am using] the Lorentzian interpretation of Special Relativity. This view on SR has become more prominent in recent years because it allows for absolute simultaneity."(Post #161)

Now, I really cannot understate this:

You basically just got done conceding that I've been right all along.

What claims did I object to you making? I objected to you making the claims that:

1.) "This is a serious debate between physicists."
2.) "LR is physics and not metaphysics."
3.) "LR is becoming a predominant view amongst physicists."

Now that these claims are manifestly false even to you, you're trying to back pedal and mealy-mouth your way out of them with statements like:

1.) "Well at least physicists are discussing Lorentzian ether models."
2.) "Look, I didn't say the debate was widespread; there are some physicists who're talking about it."
3.) "Well look, there's a false model of LR. That makes it scientific, right?"

Great job, Squatch.

Let's alert Webster's Dictionary that the definition of "a serious debate in the academic community" now means that a single paper on a theory was published and then it was summarily disproved.

Also, let's
alert them that the definition of "becoming a predominant view" now means that there's a couple of people actively engaged in disproving it.

Why can't you just concede outright that you were wrong about Lorentzian ether theory's role in modern physics? How hard is this? This is a highly technical issue. No one should or does expect you to understand theoretical physics.

Originally Posted by Squatch
I hope you aren't referring to Dennis Dieks and Brigitte Falkenburg, while GP perhaps did not notice, both of these editors have physics degrees as well. Regardless, we can agree that at very least, a majority of the board is made up of physicists correct?
What, you mean undergraduate degrees in physics? Yeah, you're right, that obviously means that they can give meaningful review on topical material not introduced until late in a graduate program.

(No, Squatch, that didn't escape my attention. Undergraduate physics degrees do not qualify you to give peer-review on every topic of physics. True story.)

Originally Posted by Squatch
My point by showing an actual board of an actual philosophy of science journal was to show that it is intentionally made up of experts in the fields in question to ensure that papers meet a standard set out by the journal.

For the journal offered above:
What we publish

Philosophy of Science aims to publish the best original work in philosophy of science, broadly construed. This work will advance the discussion in some area of philosophy of science or the philosophy of a particular science in a way that is of direct interest to experts in the field, and in a way that explicitly acknowledges and responds to existing work in the area. All submissions are peer reviewed.

Professional Standards

Submissions that do not take into account the existing literature on the topic of the paper and related topics are not publishable in Philosophy of Science.
http://journal.philsci.org/editorial-policies

So given that they are peer-reviewing the journal and that at least a significant portion of those reviewers have field specific degrees, what warrant is there for the assumption that these works are not consistent with existing work in the area?
I've addressed this thoroughly in post #386, including a statement from the Chief Editor that the journal does not engage in scientific peer-review. What more do you want, Squatch? A personalized e-mail with Gerardus 't Hooft's signature and a signed statement from every associate editor?

12. ## Re: WLC's Argument Against an Actual Infinity

Originally Posted by Squatch347
GP, I've already offered the argument and the sources. If you wish to debate them, go ahead and do so, but it serves no purpose for you to continue to ask me to refine an argument you couldn't be bothered to rebut.
GP's request is pretty reasonable, imo. He's not asking you to refine an argument. He's asking for (what you consider to be) your best source explicitly so that he can address/rebut it.

Why not just provide it? How does stonewalling advance this discussion?

Honestly, Squatch, if you wanted me to repeat a citation or an argument so that you could respond to it, I'd be happy to. Because I want you to respond to my arguments and citations. I want feedback, criticism, and analysis for my claims, citations, arguments, etc.

Don't you? Don't you want to see feedback, criticism, and analysis done on your best source? Name (and a link would be helpful) the source, and you can get just that.

Please define what is meant by "precede" here. Precede in what sense? What is the partial order that R/S are imposing on this set?
Here, they're referring to the reflexive property of partial orders.

The following definitions will be helpful:
(REL) A relation R on S is a subset of S x S = {(a,b): a,b S}. If (x,y) R, write xRy.
(REFL) A relation R on S is reflexive \forall s S: sRs
(ANTISYM) A relation R on S is antisymmetric \forall x,y S: xRy and yRx \implies x=y
(TRANS) A relation R on S is transitive \forall x,y,z S: xRy and yRz \implies xRz
(PO) A relation on S is a partial order on S is reflexive, antisymmetric, and transitive.

The archetypal partial order is the "less than or equal to" relation, (e.g. 2 3).

The "irreflexive convention" referred to in R/S says that although ≺ is a partial order and is thus reflexive, by convention you consider x ≺ x false, i.e. ≺ is irreflexive.

Technically what they're doing is constructing what's called a strict partial order from . A strict partial order is a relation that is irreflexive, antisymmetric, and transitive. The archetypal strict partial order is the "strictly less than" relation, <.

Every strict partial order corresponds uniquely to what is defined above in PO as a partial order (and which is sometimes referred to as a weak partial order), i.e. each partial order defines a strict partial order (by "forgetting" that is reflexive), and each strict partial order ≺ defines a partial order (by adding that x x).

The derivation of
≺ from and < from is instructive (the processes are essentially identical).

So they're basically just saying, "Instead of , we're working with <".

I'm not sure how that conclusion can be reached given the quotes above. So when the Axioms paper says:

This phrase represents a specific version of what I refer to as the causal metric hypothesis, which is the idea that the properties of the physical universe, and in particular, the metric properties of classical spacetime, arise from causal structure at the fundamental scale.

It isn't talking about a causal relationship in the physical world?
Sure, it is. But causets are defined in terms of sets and partial orders.

Or when it says:

Causal set theory may be expressed in terms of six axioms: the binary axiom, the measure axiom, countability, transitivity, interval finiteness, and irreflexivity. The first three axioms, which fix the physical interpretation of a causal set,

It isn't talking about the binary axiom (which discusses the relationship between elements) describing a physical interpretation of the causal set?
The Binary Axiom (p.22 of the Axioms paper) says:
Classical spacetime may be modeled as a set C, whose elements represent spacetime events, together with a binary relation ≺ on C, whose elements represent causal relations between pairs of spacetime events.

The Binary Axiom requires only that be a binary relation on C, i.e. that ≺ be a subset of C x C (an n-ary relation on C is a subset of Cn, and binary means 2-ary). This is weaker than requiring that ≺ be a partial order, since every partial order is a binary relation, but not every binary relation is a partial order.

Or even when it says:

The binary relation < defines an interval finite partial order on C, called the causal order, with the physical interpretation that x < y in C if and only if the event represented by x exerts causal influence on the event represented by y.

It isn't saying that we should interpret < in the physical world as a causal influence?
Oh, causets and their orderings are undoubtedly intended to be interpreted as modeling causal relationships between physical events.

But the interpretation of a causet is different than the definition of a causet. You can model the movement of a particle x in spacetime with a "position" function x(t) that changes over time t; the derivative of x(t) is interpreted as the velocity of x at t. But the definition of the derivative of x(t) is mathematical: the limit of a particular difference quotient ("limit" in its mathematical sense).

Or when the Foster paper says:

However, when it comes time to express the physical meaning, or even just the geometric meaning in the attempt to associate a metric to the structure, one must conclude as the references cited here do, that ‘p relates to q’ means that p connects to q via a suitable causal curve, as defined by the Lorentzian geometry.

We shouldn't intrpret the relationships described in the causet as being connected via a causal curve?
Again, this is an issue of interpretation vs definition. You have a model M, which is defined mathematically, and some system S; an interpretation of M is essentially a function that transforms statements about M to statements about S.

When it comes to Lorentzian geometry, I take Foster's word for it that such an interpretation is possible; whether such an interpretation is necessary (i.e., we should always interpret causets in terms of causal curves in Lorentzian geometry), I don't know. I presume that our current models, even those using Lorentzian geometry, are all inadequate vis-a-vis quantum gravity. Perhaps an adequate model will not admit of a "causal curve" interpretation of causets; perhaps it will. I don't have the expertise to say. You might ask GP if you're interested in that question.

And when R/S say:

One can
then use this rule – technically a probability measure – to ask physically meaningful questions of the theory.

We shouldn't interpret this as them relating these causets to physical reality?
Sure, this is an issue of interpreting causets, not defining them. I.e., transforming statements about causets to statements about, say, the physical universe.

Or when they say:

One obvious reason is that the classical case, being much simpler, can help us to get used to a relatively unfamiliar type of dynamical formulation, bringing out the pertinent physical issues and guiding us toward physically suitable conditions to place on the theory.

That doesn't mean they are constraining the theory to conform to physical reality?
This is actually a research question, i.e., "What kind of models should we be investigating?" The models, again, are mathematical. The models are interpreted once there's a way to transform questions (or statements) about reality into questions (or statements) about the model.

Really the bottom line question is Clive. If the authors presented here don't mean causal relationship when they discuss Causal Sets, what are elements related by when they express this z < x?
Their mathematical model exists independent of interpretation. [That's actually one of the biggest reasons that mathematics is so useful--it can admit many interpretations and apply to many different systems / circumstances.]

So when they're giving the mathematical definition of their model, you don't have to think about the interpretation. You just do the math.

Once you have your mathematical model set up, then you can start doing interpretation.

Hmm, you seem to be confusing the labeling criteria for the physical interpretation. I'm not talking about the number or sign or picture placed on an element, I'm talking about the physical relationship that is being described between two elements. A causet is a set of elements (spacetime events) that all exist within a spacetime context. These elements can be related by causal curves, (the labeling equivalent of which is < ) or they can be non-causally related (ie they exist in the same universe but have no causal effect on each other).
Again, if you're going to give an interpretation of a model, you need to specify that interpretation (i.e., how statements about the model correspond to statements about reality). It seems like you want to give a Minkowski interpretation of causets, since you're talking about spacetime.

I've been talking about the model, not any specific interpretation of it just yet.

Do you disagree that the causets in these papers are meant to model the causal relationships between spacetime events?
That they're meant to model causal relationships between spacetime events? Sure, that seems pretty obviously true.

So first, lets operate off of this image from the axioms paper for a common picture.

From this we can see that the Hasse diagram is, in fact, a defacto overly on a space time graph. R/S also refer generally to this process in that they describe the upward direction of a Hasse diagram as being related to the stochiastic growth process which is described as being the process of temporal passage.
Again, once you're in "Hasse diagram" land, you don't care about the particular arrangement of the vertices, you only care about the "upward" lines between them.

Now, consider points x and A. They exist at different points along the Y axis right? Given that we see this axis is labelled as time we can interpret this difference as being that these two events exist at different points in time. Just as we could interpret point C existing at the same point in time, but at a different physical location than x.

So in the final panel we can see the same elements, but related via causal relationships. The elements that x has a causal influence on all exist at a subsequent time from x and in this case (though not necessarily) at a different physical location from x.
Sure, but the causet that results from that last picture is based on the Hasse diagram, not the particular arrangement of the vertices (again, so long as any "upward" line from p to q still exists and is still "upward" from p to q).

So the causet doesn't define a unique spacetime arrangement of the events (although certain spacetime relations hold, as in SR; you can't have retrocausality, where x ≺ y (in the causet) but yt < xt (in the spacetime embedding/interpretation of C).

And given the language used above, does that partial order < have any physical significance?
Any given interpretation will endow the partial order with physical significance.

I think that is a fair point and that you are right there. A timid child is only that definition if the new element is the causal future of all the elements of it's parent. I think we still have a disagreement on what < means, but sufficed to say the subsequent point (whether a timid child has a < relationship to all the maximal elements of its parent) I'll certainly concede.
Noted.

Hmm, so Prof. Gudder would be abusing the notation on page 3 of the Labeled Causets paper when he said:

If a, b are elements of a causet x, we interpret the order a < b as meaning that b is in the causal future of a.
Not sure I understand. Gudder didn't use "parent", "child", or "future" differently that R/S did, or different from the definitions I gave in my previous post.

Gudder doesn't call b the "child" of a, or a the "parent" of b.

And that fact relies on the imposition of an external structure (the natural numbers) which allows for you to make that deduction. IE you know the relationship of the natural numbers to each other which allows you to map the entire sequence. It isn't clear that any such structure exists within causets. While they can be labelled with natural numbers, those numbers don't dictate the structure of the causal relationship.

To approach this from another angle. Let us assume for a moment that I have shown sufficiently that the < relationship describes a causal relationship and that a < b your argument would seem to be implying that the effect (b) can causally precede the cause (a). IE I have the effect b, and then I can deduce the cause. That is about as close as I can image to reverse causation right?
Reverse causation would be deducing that b < a.

What's going on here is that if you know the causal relationship between a and b, and you know b, sometimes you can deduce a.

Roughly speaking, if I know that you take one step forward every ten seconds, and that you've just taken your 8th step, I can deduce that your 7th step was ten seconds ago.

Perhaps I still misunderstand you. This seems to be that conclusion that we cannot use ℤ as an R/S labeling. IE, you cannot map ℤ onto a naturally labeled causet and satisfy the given requirements.

If that is what you mean, I totally agree. But then why the earlier argument that there are causets that admit not natural labeling?
You're misunderstanding.

An R/S labeling on a causet (C,) is a map L:(C,) (ℕ,<) that satisfies:

For all c1,c2 C: c1 c2 L(c1) < L(c2)

With this definition, it doesn't make sense to ask "Can ℤ be used as an R/S labeling?", except perhaps in the trivial case where you map to an isomorphic copy of (ℕ,<) in (,<).

In any case, even if you did ask that question, what you'd need for (,<) to be a labeling on a causet (C,≺) is a map M: (C,) (ℤ,<) that meets some criteria.

Rather, what I did was prove that there is no map L:(,<) → (ℕ,<) that satisfies:

For all z1, z2ℤ: z1 < z2 L(z1) < L(z2)

This means that there's no R/S labeling on (,<); or, if you prefer different language, that (ℕ,<) can't be used to label (,<) [in the given sense; obviously there is a bijection between them, just not an order-preserving one from (,<) to (ℕ,<)].

The issue here, other than establishing an order-theoretic difference between (,<) and (ℕ,<), is that (,<) meets the causet criteria established in the R/S paper; (,<) is a locally finite poset.

The conclusion I drew--although there might be more significant ones to be drawn--is that if you want your definition of causet to be such that every causet admits an R/S labeling [i.e., can be labeled by (ℕ,<)], then you'll need to use a different definition than R/S, since they permit (,<), which admits no such labeling.

I think I've made quite a few to date. In PM and in post 322 I offered up the discussion from set theory where the author makes the argument that a process that is similar to what we are talking about here is insufficient to create infinite sets. Your objection was that he is talking about finite unions which I think misses the point. I wasn't attempting to limit any axioms that would allow for an infinite past, rather I was showing that if we start from the ground up and require justification for each axiom added rather than simply assuming it until proven incoherent that the conclusion warrants a finite universe until/unless we assume an infinite one (which seems a bit circular).
Well, it depends on what you mean by requiring "justifications" for the axioms; axioms in some sense can't be justified within a given axiom system; they can be justified in a meta- sense, e.g. "We assume the axiom of choice because it lets us to useful things".

Here's my general, abstract issue with your argument:

Given a model M, if you want to prove that M doesn't contain infinite sets, you can't just say "M models A, and A doesn't include a way to get infinite sets, so M can't have infinite sets."

If you want to appeal to axiom systems, then you'd need to show that M models some axiom system that explicitly disallows infinite sets.

Otherwise, you'd need to go through each set in M and show that it is finite. (This can be done schematically, e.g. if you know a general form / representation of sets in M, and can show that each represented set has to be finite.)

Here's my specific concern with your argument:

What you're trying to do is show that given the way time accretes / propagates, our universe must have a finite age.
Here's what your argument actually is:

(a) Time accretes according to some process P.
(b) Time accretes in finite amounts at each step in this process.
(c) P can't produce an infinite total amount of accreted time.
(d) Therefore, the total time accreted must be finite

Your argument for (c), however, is along the lines:

(i) P only accretes an infinite amount of time if you assume infinite amounts, or infinite processes, are possible
(ii) Therefore, without that assumption, P cannot accrete an infinite amount of time.
(iii) Assuming that there are infinite amounts or infinite processes would be circular.
(iv) Therefore, P can't accrete an infinite amount of time.

Now, if your argument is rigorous, then the process P mentioned in (a) should be defined. If P isn't known to be finite, then it is illicit to assume it is finite; if P isn't known to be infinite, then it is illicit to assume it is infinite.

In general, if P isn't entirely well-defined (i.e., all of its features--finiteness, etc.), then your argument isn't even well-formed. If P isn't well-defined, then what (a) is really saying is:

(a1) Let S be the set of processes each of which possess properties (X,Y,Z); time accretes according to one of the processes in S.

If there are infinite processes that have properties (X,Y,Z) (including, say, the property that each additional amount of time that accretes is finite, to satisfy (b)), on what basis are you excluding them?

I don't think you can just say, "infinite processes don't exist unless we axiomatically assume we can construct infinite processes, which is an axiom I'm not willing to grant". The statement in (a1) is that time accretes according to some process with properties (X,Y,Z); either "process" is defined so that it's impossible to define an infinite process (say, if a "process" is defined to be finite), or "process" is defined so that it's possible to define an infinite process.

If "process" is defined in such a way that it's possible to construct infinite processes, then perhaps there is one such infinite process included in the S in (a1).

As a concrete example:
(P) A "process" is a collection of objects, along with a rule R that "moves you" from object to object.

[This definition is just an example for analytical purposes; I'm not trying to capture my own notion of "process"]

Does (P) allow for infinite processes? It depends on what counts as a "collection", right? If you allow finite collections, then there are finite processes; if you allow infinite collections, then there are infinite processes.

I think that (P) does allow for infinite processes; that is, you couldn't determine from (P) alone that all processes are finite; you'd need an additional axiom--say, "(C) All collections are finite"--to reach that conclusion.

Moving forward:

If you accept the framing device I've given above (i.e., the way I've characterized and analyzed your argument and the nature of your claims), then I think the best way to continue this discussion is to nail down just what is meant by "process". If, like I argued (P) does, your definition allows for infinite processes, then I think you will need to restructure your argument, and perhaps add additional justification for excluding infinite processes.

If you don't accept the framing device I've given above, can you explain what I've gotten wrong, and then (hopefully) propose an alternative framing device?

The difference between our two approaches seems to be that you are willing to accept all axioms until proven inconsistent and as such you see my non-inclusion of a specific axiom as an attempt to form a circular argument. Rather I am approaching it from the view that no axioms are permitted until justified and we have yet to justify that axiom. Without it's justification we cannot reasonably conclude that the past formed by the process is actually infinite.
This gets to the "framing device" issue I mentioned above. As a summary, if it can make sense to talk about infinite processes, and you don't know that the process P by which time accretes is finite (e.g., finitely many "steps", etc.), then to conclude that P is finite requires additional premises.

I mean confines in the sense that a conclusion of infinite sets (given A) would be unwarranted. Confines is used in the sense of boundaries. Given the axioms offered in the earlier premises one cannot conclude from that alone that the universe is infinitely old. IE in order to form the conclusion you would need to break the scope of the premises offered by offering either another axiom, in this case the assumption that an actual infinite exists, which would require the formulation of another method of set creation sufficient to create an infinite set if the new axiom set (A') is to be consistent with A.
Let's say that there are certain axioms that define what a process is; call this set of axioms P.

(PROC) Every model of P is a process

Proof: By definition of P, anything that meets all the axioms in P is a process.
If there were additional rules to qualify as a process, those would have to be included in P.
QED.

Let S be the set of Squatch-approved set construction axioms.

Define the following propositions:

(PFIN) Every model of P is finite
(PINF) There exists an infinite model of P

Given PROC, we can rephase the first premise:

(a2) Time accretes according to a model of P.

Now, it seems like you want a more restrictive first premise:
(a3) Time accretes according to a model of P that is also a model of S, and not a model of any other axiom system

(a3) seems much less intuitively-appealing to me than (a2).

To be frank it is because your criticism don't seem material to my point. If we were to form the same new element x and select a non-empty partial stem, it would no longer be gregarious right?
It is material to your understanding of causets, partial stems, and anti-chains. Are you not concerned with understanding causets, partial stems, or anti-chains?

The element x, however, isn't gregarious no matter what partial stem you choose, since x is an element of a causet, not a causet.

Of course all of this is a rabbit hole discussion to a point I think we agree on. That if we were to use a concept from causets to discuss the "age" of the universe, we would use a chain, which by definition contains no gregarious children.
Sure, but that's because gregarious children aren't elements in chains. The elements in a chain are elements of causets, while children (including gregarious children) are causets.

How can a gregarious child be considered in a chain given that it is by definition not related by a causal < ?
(1) A gregarious child is a causet, not an element of a causet. So a gregarious child (Cg,g) of (C,) can't be in a chain in C, since (Cg,g) isn't an element of C; nor is (Cg,g) an element in Cg, so it can't be in any chain in the gregarious child, either.

(2) If by "gregarious child" you mean "a minimal element of a causet", i.e., an element x such that past(x) is empty, then like I said: it depends on how you define a chain.

I. Definitions

Let (C,≺) be a partially ordered set.
(CHN) A subset S C is a chain in C S is totally ordered as a subspace of C [What I'm calling the subspace order is the restriction of ≺ to S, i.e. for all x,y in S: x ≺ y (in S) x ≺ y (in C)]
(TOT) C is totally ordered For all x,y C: x y or y x

So a chain in C is just a subset S of C where each of the elements in S are related to each other in C.

II. Does {x} count as a chain?

The answer might appear to be no, given the irreflexive convention, by the following reasoning:

(i) If {x} is a chain, then {x} is totally ordered.
(ii) If {x} is totally ordered, then for all p,q \in {x}: p
≺ q or q ≺ p.
(iii) Taking p = q = x, we have x ≺ x or x ≺ x.
(iv) Therefore, x ≺ x.

However, if this line of reasoning is valid, then there are no non-empty chains in C, by the following argument:

(1) Suppose S is a non-empty chain in C
(2) Since S is non-empty, there is some s \in S.
(3) Since S is a chain,
then for all p,q \in {x}: p ≺ q or q ≺ p.
(4) Taking p = q = s, we have s ≺ s or s ≺ s.
(5) Therefore, s ≺ s.

There are two obvious solutions:

Solution 1: Ignore the irreflexive convention; chains in (C,
≺) are determined by the weak partial order, not the strict partial order.
Solution 2: Change the definition of chain to exclude self-comparisons.

Both solutions entail that {x} is, indeed, a chain.

That is a pretty far stretch of what I said. No, depending on the theory causets either represent a portion of different related elements within a universe (or the entire universe which has sub-causets) or the different possible outcomes in the case of the "sum of many worlds" theory.
Okay, so not necessarily universes that actually exist.

No, I said that it consists of: "One physical dimension made up of one single point and one temporal dimension." IE we have two dimensions, one physical, one temporal, (x,t).
I'm not sure if a single point technically counts as a dimension, but perhaps you're speaking informally.

Any event in this universe can be described by a coordinate, (x,t). Part of the confusion I imagine was a typo. I meant to define "now" as "tn."
Okay, so tn is just some point in the time dimension that you're labeling "now".

Given that the event (x,tn+1) is a maximal element and not yet born, this universe has a longest chain at "now" (tn), correct?
I define (x,tn+1) with respect to the causal order defined in R/S (which is a disagreement we have). I argue that each event in this universe must have a causal predecessor if it is to obey causality. We could form a causal set in this universe in the following manner (assuming (tn=4):

{(1,0) < (1,1) < (1,2) < (1,3)}

I hope this clarifies my example.[/quote]

(1) I have no idea what you think our disagreement about "the causal order defined in R/S" is. Do you think I don't understand R/S's definition of causet, or do you think you don't?

(2) My understanding is that your universe defines a causet; subregions of the universe may correspond to other causets, but the universe as a whole corresponds to one and only one causet (up to isomorphism).

(3) If the causet you're looking at is {(1,t) : t = 0,1,2,3} with the relation you defined above, then clearly this causet has a longest chain. Every finite causet has a longest chain.

(4) If you're asking whether the causet that corresponds to the entire universe has a longest chain, then I'll need more information about the causal relationships between points. Heck, I'm not even sure what points exist--does (x,t) exist, where t is a real number less than n?

I hope you aren't referring to Dennis Dieks and Brigitte Falkenburg, while GP perhaps did not notice, both of these editors have physics degrees as well. Regardless, we can agree that at very least, a majority of the board is made up of physicists correct?
No, I was referring to Arthur Fine and someone else I can't recall. In any case, yes, the majority of the editorial board have advanced degrees in physics.

It wouldn't seem odd to you that a panel of mostly physicists would read papers for comment and inclusion and not for technical review? If technical review was not part of this process, why have physicists and not professional editors?

I would also point to this document: http://www.lnf.infn.it/acceleratori/...-FOOP353R1.fdf which is a re-submission to the journal of a paper where technical and style corrections were made. The response to reviewer 1 is clearly of a technical nature where the reviewer found the evidence not fully supportive (or at least not clear) for the conclusion made.
First, the editorial board aren't the ones doing the bulk of the review.

Second, the editorial board reviews papers based on the mission statement of the journal, not necessarily based on their own personal qualifications.

Neither of these seem "odd" to me, no.

The implicit assumption being made both to me and specifically to MT is that a philosophy of science publication can be discounted out of hand simply because it is from a philosophy of science journal. Aside from the salaciousness of that argument (as I pointed out before, it could be from a homeless dude, I don't care as long as the claim is about the truth value of the material rather than the authoritativeness of the author), that conclusion would seem to rely on the assumption that philosophers of science are writing papers ill informed about the subject they are writing about, which would seem to make the entire field a fraud and sham, which is perhaps the position being taken here, at lest implicitly.

My point by showing an actual board of an actual philosophy of science journal was to show that it is intentionally made up of experts in the fields in question to ensure that papers meet a standard set out by the journal.

For the journal offered above:
What we publish

Philosophy of Science aims to publish the best original work in philosophy of science, broadly construed. This work will advance the discussion in some area of philosophy of science or the philosophy of a particular science in a way that is of direct interest to experts in the field, and in a way that explicitly acknowledges and responds to existing work in the area. All submissions are peer reviewed.

Professional Standards

Submissions that do not take into account the existing literature on the topic of the paper and related topics are not publishable in Philosophy of Science.
http://journal.philsci.org/editorial-policies

So given that they are peer-reviewing the journal and that at least a significant portion of those reviewers have field specific degrees, what warrant is there for the assumption that these works are not consistent with existing work in the area?
(a) I'm not sure why you're quoting a journal that none of your sources are from.
(b) None of this is relevant to my arguments. I can't help what hidden agendas you decide to see in my statements, and I'm certainly not going to respond to them.

13. ## Re: WLC's Argument Against an Actual Infinity

Originally Posted by GoldPhoenix
Your unwillingness to even give me a paper
I've given you papers GP. If you wish to rebut them then do so.

Originally Posted by GP
Really?
Yes, really. You seem to have a very, very different view of what the word prominent means than the accepted definition. The statement you quote basically means that it has become more widely known or argued. You seem to infer that prominent means "replacing" SR, in which case I would encourage you to re-read my statements with the more accurate definition in mind.

Originally Posted by GP
No, Squatch, that didn't escape my attention. Undergraduate physics degrees do not qualify you to give peer-review on every topic of physics. True story.
So again, a person with a degree in physics and a technically focused graduate program doesn't "count" because you say so. And that is a logically valid rebutal because?

Originally Posted by GP
I've addressed this thoroughly in post #386, including a statement from the Chief Editor that the journal does not engage in scientific peer-review.
Yeah, you didn't really do that though GP. You showed that Prof. t' Hooft distanced the journal itself from ECE. What you seem to have missed is how the Foundations of Physics Letters was subsummed into the Journal. It was added as their equivalent of "letters to the editor" which you can find in many journals. Because it was published in that section of the journal does not mean that the entire journal is not peer-reviewed.

Let me ask you the same question as I asked Clive. If the review process is solely for format, why are the editors physicists rather than writers?

Originally Posted by CliveStaples
So they're basically just saying, "Instead of , we're working with <".
Agreed, but I'm not sure it fully answers the question, we seem to have just pushed it back a step. What is the partial order these are defined on? Are the orders just arbitrary? Do they result from a physical relationship?

Originally Posted by CS
Sure, it is.
Excellent, so we can agree that the relationship modeled through < is a causal relationship between elements in the Causal Metric Hypothesis?

This disagreement arose because you implied back in post 351 that a Causet can be a Causet even if the relationship between elements is defined on something other than a causal relationship. This is a bit more than just an interpretation issue, we are talking about distinguishes this hypothesis from a simple discussion of a partially ordered set that happens to be locally finite.

Originally Posted by CS
Once you have your mathematical model set up, then you can start doing interpretation.
That can only function in the event that the model has all the physical limitations included in it. Hence why two elements that are only related via a spatial relationship cannot have a causal < link.

I originally pointed this out because the authors of all of these papers are not doing what you seem to be describing here. Their papers frequently discuss the physical relationships being modeled, the physical requirements needed to constrain the model and the whether or not the physical interpretation causes the model to be nonsensical (the reason for reflexivity as a constraint is because it led to absurd physical outcomes).

Originally Posted by CS
Again, once you're in "Hasse diagram" land, you don't care about the particular arrangement of the vertices, you only care about the "upward" lines between them.
And do the elements diagrammed in the Hasse diagram lose that fundamental information then? So element x and element C no longer have spatial and temporal coordinates? Because a Hasse diagram does not include spatial and temporal axes does not mean that all other relationships between two elements ceases to exist once you diagram it in this form.

From the graphic I offered in my last post we can see two obvious points.

Element x and Element C exist at approximately the same temporal point, but at two different spatial points.

Element x and Element C are not related by a causal relationship.

In fact to take it a step further, you'll notice the light cone being drawn on the first two panels. And you'll notice that the causal relationships radiating from x do not exceed that light cone. I should point out that I didn't imply before or earlier that the causal relationships arise because of the causet (though at least one paper makes that argument), but rather the causal relationships within the causet are limited by certain physical principles.

Originally Posted by CS
Gudder doesn't call b the "child" of a, or a the "parent" of b.
My apologies, I only pasted half of the quote in my last response. Here is the full text, from page 3 of Labeled Causets:

If a, b are elements of a causet x, we interpret the order a < b as meaning that b is in the causal future of a. If a < b and there is no c with a < c < b, then a is a parent of b and b is a child of a.

Originally Posted by CS
What's going on here is that if you know the causal relationship between a and b, and you know b, sometimes you can deduce a.
Agreed, but if the value of b is reliant on the knowledge of a, then knowing b without a is not possible, correct? In both your examples you relied on an outside metric to deduce b (numbers and in this case time), but that metric doesn't exist here. There isn't some independent temporal measure we can use to deduce the causet at any given point.

Originally Posted by CS
You're misunderstanding....The conclusion I drew--although there might be more significant ones to be drawn--is that if you want your definition of causet to be such that every causet admits an R/S labeling
Clearly I misunderstood your initial point which I took to mean that there were causet that do not admit a labeling via natural numbers. This confusion seems to have arisen from our fundamental difference concerning what is a cuaset. I would maintain that ℤ is not a cuaset because it lacks the fundamental causal relationship required of the Cuasal Metric Hypothesis. It is certainly a locally finite, partially ordered set, but that doesn't make it a causet (as I maintain your definition from R/S is only a partial definition of what they meant). A causet is a "(locally) finite partially ordered set, in which the order is causally interpreted."

Originally Posted by CS
(a) Time accretes according to some process P.
(b) Time accretes in finite amounts at each step in this process.
(c) P can't produce an infinite total amount of accreted time.
(d) Therefore, the total time accreted must be finite

Your argument for (c), however, is along the lines:

(i) P only accretes an infinite amount of time if you assume infinite amounts, or infinite processes, are possible
The argument was more or less acceptable until this sub-support. Rather, this point was argued as:

(i) P accreted discrete, finite amounts of time per process step.
(ii) P results in an actual infinite iff the input to P is assumed to be actually infinite.
(iii) Assuming the input is infinite to argue for it being infinite without an additional mechanism that would generate that infinite is fallacious.

Proposed example

Imagine we had a machine which operated via three rules.

1) If the item placed in the machine is red, make it blue.
2) If the item placed in the machine is blue, make it red.
3) If no item is placed in the machine, out put a red item.
4) If the item placed in the machine is green, leave it green.

This machine, left to its own devices will create red and blue items. It will not produce a green item unless a green item, produced from some other machine is put into it.

If we were to further stipulate that there are no other machines out there besides this one, we could well deduce that the output of the machine is not green.

If that argument makes sense, we can go ahead and discuss the reasons given for the argument that "there are no other machines" and "rules 1&2 are true."

Originally Posted by CS
Thank you.

Originally Posted by CS
Sure, but that's because gregarious children aren't elements in chains.
I disagreed with the second sentence above, but it is irrelevant. As long as we can say that the longest chain is a proxy for the age of the universe, that is all the agreement necessary for this discussion.

Originally Posted by CS
So a gregarious child (Cg,g) of (C,) can't be in a chain in C, since (Cg,g) isn't an element of C; nor is (Cg,g) an element in Cg, so it can't be in any chain in the gregarious child, either.
Likewise, we can simply use the definition of chain from R/S in which every two elements of the chain are related via <. Since a gregarious child (under both of our definitions) contains elements that are not related via < a chain cannot contain a gregarious child.

IE we both agree that chains do not contain gregarious children.

Originally Posted by CS
Okay, so not necessarily universes that actually exist.
That depends on which model you are adopting. For simplicity's sake we can say that a causet can represent an entire universe or it can represent portion of that universe.

If you are getting at what I think you are getting at, I would also agree that you could make a causet model that does not actually represent this universe or a universe that exists. But that would make that causet model non-realistic (we can insert whatever term you would like, sufficed to say, that would make it a non-viable model of our universe). R/S discuss this in sectino 4.4 of their paper at length discussing how stochiastic growth must occur for it to result in a universe like the one we observe today.

Originally Posted by CS
I'm not sure if a single point technically counts as a dimension, but perhaps you're speaking informally.
I'm not sure why it wouldn't here. I cannot accurately describe that event without detailing that physical dimensional point right?

Originally Posted by CS
Okay, so tn is just some point in the time dimension that you're labeling "now"....

Yes, tn is defined as now in that it is the maximal element within the universe at the step considered.

So my example was meant to ask whether or not a causet that terminates at a specific point has a longest chain in your opinion.

We could adopt some relatively common sense deductions from that example. No event within the universal causet (the causet that contains all elements from that example universe) can be completely devoid of any causal relationships (ie it must obey causality). Events within the causet have occurred (have been born) and are therefore included within the causet. No event that has not been "born" can, by definition, be said to be accreted to the causet.

So, given that there is a maximal element within this universe (x,tn) beyond which elements have not been born and the comments made above, can we conclude that this example universe has a longest chain? Does this conclusion apply more broadly given the limitations laid out here?

Originally Posted by CS
No, I was referring to Arthur Fine and someone else I can't recall. In any case, yes, the majority of the editorial board have advanced degrees in physics.
Arthur Fine is also a referee on several Physics based journals as well, "European Physics Journal, International Journal of Theoretical Physics, Physical Review, Physics Letters A." That would seem to indicate that philosophers of science are a bit more important to physics peer-review than the comments made above would seem to indicate.

Originally Posted by CS
First, the editorial board aren't the ones doing the bulk of the review.

Second, the editorial board reviews papers based on the mission statement of the journal, not necessarily based on their own personal qualifications.
It is irrelevant if they are doing the bulk of the review or not. If they are doing review and they are physicists then for this journal to have the unreliability imputed to it by GP (primarily) they would need to be actively ignoring mistakes. Again, we aren't talking about them taking a different opinion, we are talking about what GP described as massive and obvious technical errors. Are we seriously to accept that editors who are trained physicists decided to ignore such egregious errors?

Do you have any reason to believe the referee board is vastly different in makeup from the editorial board? In a quick search, the only two referees I found were physicists by training and practice.

I also agree with you that they are responsible for including papers given the mission statement of the journal (though why they would include physicists if the mission was not about including papers relevant to actual physicists is still something unexplained), which in this case distinguishes papers of a non-technical nature (to be included in the letters section) from those technical ones that require peer-review.

You also didn't directly address the response letter I offered as evidence which clearly detailed review of a technical nature concerning the accuracy and warrant for results.

Originally Posted by CS
(a) I'm not sure why you're quoting a journal that none of your sources are from.
As I stated last time, I was showing that the boards of philosophy of science journals are not made up simply of people untrained in the academic fields being discussed, but are specifically picked to be competent to review the technical nature of the papers they accept.

14. ## Re: WLC's Argument Against an Actual Infinity

Squatch, you seem to have missed something from my previous post, so I'll just repeat it here:

GP's request is pretty reasonable, imo. He's not asking you to refine an argument. He's asking for (what you consider to be) your best source explicitly so that he can address/rebut it.

Why not just provide it? How does stonewalling advance this discussion?

Honestly, Squatch, if you wanted me to repeat a citation or an argument so that you could respond to it, I'd be happy to. Because I want you to respond to my arguments and citations. I want feedback, criticism, and analysis for my claims, citations, arguments, etc.

Don't you? Don't you want to see feedback, criticism, and analysis done on your best source? Name (and a link would be helpful) the source, and you can get just that.

Well? Don't you?

Originally Posted by Squatch347
I've given you papers GP. If you wish to rebut them then do so.
He's asking you to give what you consider to be your best source. What's the holdup? Do you not want GP to review your best source? Do you not know what your best source is?

Or do you just not feel like participating in this discussion anymore?

Agreed, but I'm not sure it fully answers the question, we seem to have just pushed it back a step. What is the partial order these are defined on? Are the orders just arbitrary? Do they result from a physical relationship?

1. What is the partial order these are defined on?

I'm not sure what the "these" are; I suspect you're meaning to ask one (or possibly both) of the following questions:

(a) What set are these partial orders defined on?
(b) What are the elements of the partial order?

To (a): It depends entirely on the causet you're talking about; given a causet (C,), the set that ≺ is defined on is C.

To (b): It depends entirely on the causet you're talking about; if a general (or arbitrary) causet (C,≺) is considered, then ≺ isn't specified [this is what makes proofs on general causets so strong; they hold for all causets that meet the hypotheses, regardless of the particular contents of ]. If a particular causet (C,≺) is specified, then you know both the content of C and the content of ≺.

2. Are the orders just arbitrary?

It depends on what causet you're talking about.

If you're talking about causets generally, then you don't really care about the contents of the partial orders, because you're considering causets generally.

Other times, you might be interested in talking about a particular class or subset of causets--perhaps those causets whose partial order has particular properties (such as being totally ordered, or being a gregarious child of another causet), or perhaps those causets whose set has particular properties (such as having finite cardinality).

3. Do they result from a physical relationship?

No.

In general, a causet doesn't necessarily encode actual causal relationships, e.g. causal relationships among events that actually occur in our universe--in the same way that a number doesn't necessarily encode an actual value that occurs in our universe (e.g., the number of atoms, etc.).

The research goal with causet theory is, of course, to identify those causets that do encode actual causal relationships, in order to model those relationships.

But causets are a mathematical construct, and as such do not result from any physical relationships whatsoever (depending on your philosophy of mathematics; if mathematics is just a particular language that humans speak, then of course every mathematical construct results from a physical relationship).

Excellent, so we can agree that the relationship modeled through < is a causal relationship between elements in the Causal Metric Hypothesis?
I'm not sure what "elements in the Causal Metric Hypothesis" means.

The Causal Metric Hypothesis, as quoted by you in post 390, states:
...[T]he properties of the physical universe, and in particular, the metric properties of classical spacetime, arise from causal structure at the fundamental scale.

So suppose we are interpreting causets as a model of causal structure at the fundamental scale.

This doesn't mean that every causet models a causal relationship in the physical universe. That would mean that for every causet, there is a corresponding set of physical events whose causal relationship is modeled by that causet.

Rather, it means that the causal structure at the fundamental scale explain properties of the physical universe. That is, if (C,) is a causet that models the causal structure of physical events at the fundamental scale, then (C,) explains the physical properties of the physical universe, in particular, the metric properties of classical spacetime.

That is, we can choose a causet that does model the causal structure of physical events in our universe.

As an analogy, we can use functions to model growth; that doesn't mean that every function models the growth of some actual process/object/quantity in our universe.

This disagreement arose because you implied back in post 351 that a Causet can be a Causet even if the relationship between elements is defined on something other than a causal relationship. This is a bit more than just an interpretation issue, we are talking about distinguishes this hypothesis from a simple discussion of a partially ordered set that happens to be locally finite.
If we're going with the R/S definition, a causet is just a locally finite poset. R/S doesn't require that a causet model causal relationships among actual physical events; R/S requires that a causet be a locally finite poset.

It is just an interpretation issue. There is the mathematical construct (locally finite posets) and the interpretation (causal relations of physical events).

That can only function in the event that the model has all the physical limitations included in it.
This seems like a very strong requirement. You can model an aspect of a physical system without needing to model every physical property of that system.

For example, at speeds very far away from the speed of light, the Newtonian model of physics is a pretty good model. It doesn't include all the physical limitations--e.g., relativity, quantum mechanics, etc.--but it can still be interpreted (here meaning "interpreted with little error", since the interpreted statements will not literally be true due to slight relativistic effects) as a model of physics when spacetime is "roughly" classical.

Hence why two elements that are only related via a spatial relationship cannot have a causal < link.
This is true by definition for causets; a spatial (which I take to mean "spacelike") relation between x and y is precisely the statement that x and y have no causal link in either direction.

I originally pointed this out because the authors of all of these papers are not doing what you seem to be describing here. Their papers frequently discuss the physical relationships being modeled, the physical requirements needed to constrain the model and the whether or not the physical interpretation causes the model to be nonsensical (the reason for reflexivity as a constraint is because it led to absurd physical outcomes).
Sure, because they're not just building the mathematical construct, they're concerned with whether the interpretation makes sense.

Again, when they're talking about the mathematical construct, they're doing math. When they're talking about the interpretation, they're doing interpretation.

For example, if you say, "I want to model the amount of money in my bank account with a function", you're going to choose a function that "makes sense". But while your decision to include that function in your model is based on the intended interpretation of the function--you want the function to be interpreted as the amount of money in your bank account--the definition of the function is entirely mathematical.

And do the elements diagrammed in the Hasse diagram lose that fundamental information then? So element x and element C no longer have spatial and temporal coordinates? Because a Hasse diagram does not include spatial and temporal axes does not mean that all other relationships between two elements ceases to exist once you diagram it in this form.
You're misunderstanding.

Yes, if you stipulate certain spacetime relationships among the elements (e.g., by embedding the points in spacetime), then there are certain spacetime relationships among the elements.

But causets don't come with a spacetime embedding; they only come with the Hasse diagram (i.e., the partial order). And for every Hasse diagram of a set of events E, there are many different spacetime embeddings of E.

From the graphic I offered in my last post we can see two obvious points.

Element x and Element C exist at approximately the same temporal point, but at two different spatial points.

Element x and Element C are not related by a causal relationship.

In fact to take it a step further, you'll notice the light cone being drawn on the first two panels. And you'll notice that the causal relationships radiating from x do not exceed that light cone. I should point out that I didn't imply before or earlier that the causal relationships arise because of the causet (though at least one paper makes that argument), but rather the causal relationships within the causet are limited by certain physical principles.
If you're talking about physical interpretation, then you know that because of relativity of simultaneity, if two events are spacelike to each other, whether they are "at the same temporal point" will depend on your reference frame. Different reference frames can change the spacetime arrangement of events; they can't change the causal relationships.

The causal relationships within the causet depend only on the Hasse diagram, i.e. the partial order that the causet comes with.

My apologies, I only pasted half of the quote in my last response. Here is the full text, from page 3 of Labeled Causets:
If a, b are elements of a causet x, we interpret the order a < b as meaning that b is in the causal future of a. If a < b and there is no c with a < c < b, then a is a parent of b and b is a child of a.
The definitions of "child" and "parent" in Labeled Causets is entirely different than their definitions in the R/S sequential growth dynamics paper. If we're going to use these terms ("child" and "parent"), then we should either choose which of the two paper's definitions wins out, or when necessary indicate which definition of "parent" and "child" are being used.

As a side note, if this kind of issue comes up again, it's pretty standard to say something like, "I'm not using the definitions 'parent' and 'child' from the R/S paper here, I'm using this other definition." It doesn't matter whose definition gets used, what matters is that everyone understands the definitions being used.

Agreed, but if the value of b is reliant on the knowledge of a, then knowing b without a is not possible, correct? In both your examples you relied on an outside metric to deduce b (numbers and in this case time), but that metric doesn't exist here. There isn't some independent temporal measure we can use to deduce the causet at any given point.

Clearly I misunderstood your initial point which I took to mean that there were causet that do not admit a labeling via natural numbers.
This, indeed, was my point.

As a side note, it seems that you're contradicting yourself. You said:
This seems to be that conclusion that we cannot use ℤ as an R/S labeling.

If this was your understanding of my point, then clearly you did not take my meaning to be that "there are causets that do not admit a labeling via natural numbers".

In any case, I'm glad that you seem to understand my argument now, that (,<) is a locally-finite poset that admits no R/S labeling.

This confusion seems to have arisen from our fundamental difference concerning what is a cuaset. I would maintain that ℤ is not a cuaset because it lacks the fundamental causal relationship required of the Cuasal Metric Hypothesis. It is certainly a locally finite, partially ordered set, but that doesn't make it a causet (as I maintain your definition from R/S is only a partial definition of what they meant). A causet is a "(locally) finite partially ordered set, in which the order is causally interpreted."
No, when R/S say:
A causal set (or “causet”) is a locally ﬁnite, partially ordered set (or “poset”).

And when Gudder says:
A finite partially ordered set is called a causet.

They are giving definitions. Nowhere later in their paper are these definitions refined; no additional properties are appended to these definitions. These definitions hold throughout the paper (which is standard practice; the point of Definition/Notation sections is to establish the meaning of terms and symbols that are used in the paper.

The argument was more or less acceptable until this sub-support. Rather, this point was argued as:

(i) P accreted discrete, finite amounts of time per process step.
(ii) P results in an actual infinite iff the input to P is assumed to be actually infinite.
(iii) Assuming the input is infinite to argue for it being infinite without an additional mechanism that would generate that infinite is fallacious.

Proposed example

Imagine we had a machine which operated via three rules.

1) If the item placed in the machine is red, make it blue.
2) If the item placed in the machine is blue, make it red.
3) If no item is placed in the machine, out put a red item.
4) If the item placed in the machine is green, leave it green.

This machine, left to its own devices will create red and blue items. It will not produce a green item unless a green item, produced from some other machine is put into it.

If we were to further stipulate that there are no other machines out there besides this one, we could well deduce that the output of the machine is not green.

If that argument makes sense, we can go ahead and discuss the reasons given for the argument that "there are no other machines" and "rules 1&2 are true."
Okay, so suppose you know that you got a green item from a machine M such that rules (1)-(4) hold for M.

Q: What was the color of your object before it went through the machine?

Perhaps, unbeknowst to you, M has an additional rule:

5) If the item placed in the machine is white, make it green

I disagreed with the second sentence above, but it is irrelevant. As long as we can say that the longest chain is a proxy for the age of the universe, that is all the agreement necessary for this discussion.
If you define the longest chain to be a proxy for the age of the universe, then I can agree that you've defined it as such.

I would probably not agree that the longest chain is a proxy (or a "good" proxy) for the age of the universe without seeing some reasoning why. You haven't done much analysis on it: You haven't really explained why it should be used, what the possible complications are, whether or not it matches our intuitions, etc.

But again, if that's your definition, then I can agree that it's your definition and construe your arguments accordingly.

Likewise, we can simply use the definition of chain from R/S in which every two elements of the chain are related via <.
This is equivalent to my solution 1, and still entails that {x} is a chain.

Since a gregarious child (under both of our definitions) contains elements that are not related via < a chain cannot contain a gregarious child.
This is very confused.

(1) A gregarious child does contain elements (at least one) that are not related.
(2) Chains can't contain gregarious children, because chains contain elements of causets, not causets.

The definition of chain from R/S (p.3), which you allude to, is:
(CHN) A chain is a linearly ordered subset of C (a subset, every two elements of which are related by ≺)

The question is, does this require self-comparisons? I.e., if S is a chain in C, and s S, must s ≺ s hold?

I'll use the following definitions:

(MM) An element x C is minimax x is minimal and maximal

Let S be a subset of C.
(OC) For all x,y S with x y: x ≺ y or y ≺ x
(SC) For all x S, x ≺ x

Note that an element x is minimax for all y C, ¬(y ≺ x) (otherwise x would fail to be minimal) and ¬(x ≺ y) (otherwise x would fail to be maximal). A gregarious child's new, maximal element must also be minimal, and is thus minimax.

It is unclear whether the "two elements" in CHN must be distinct from one another, leaving two and only two possibilities:

(A) CHN OC and SC
(B) CHN OC

(1) If CHN requires only OC, then for all c C, {c} is a chain

Proof:
If S = {c}, then trivially OC holds.

(2) If CHN requires OC and SC, then for all c C, {c} is a chain

This discussion is somewhat complicated by the irreflexive convention.

Going by the irreflexive convention, we assume ¬(x ≺ x) for all x C.
This in turn implies that if CHN requires SC, there are no non-empty chains.

Proof:
Suppose that
(i) CHN requires SC
(ii) ¬(x ≺ x) for all x C
(iii) S satisfies CHN
Since S satisfies CHN, and CHN requires SC, S satisfies SC.
Since S is non-empty, it contains some element s S.
Since S satisfies SC, we have s ≺ s.
This contradicts the assumption that ¬(x ≺ x) for all x C.
QED.

It seems reasonable to assume that we want there to be some non-empty chains; this means we must either reject either (i) or (ii) [as defined in the above proof].

If we reject (i)
, then CHN requires only OC; from (1), {c} satisfies OC, so {c} is a chain.

If we reject (ii), then since c ≺ c (since ≺ is a partial order and is thus reflexive), {c} satisfies SC; from (1), {c} satisfies OC. Since {c} satisfies OC and SC, {c} is a chain.

So whatever your choice about chains (whether a chain must satisfy OC and SC, or merely OC), if c C then {c} is a chain.
In particular, if x C is minimax, then {x} is chain.

IE we both agree that chains do not contain gregarious children.
Chains do not contain gregarious children because gregarious children are causets, not elements of causets.
Chains can contain minimax elements [although if a chain contains a minimax element, it cannot contain any other element].

That depends on which model you are adopting. For simplicity's sake we can say that a causet can represent an entire universe or it can represent portion of that universe.
Okay.

If you are getting at what I think you are getting at, I would also agree that you could make a causet model that does not actually represent this universe or a universe that exists. But that would make that causet model non-realistic (we can insert whatever term you would like, sufficed to say, that would make it a non-viable model of our universe). R/S discuss this in sectino 4.4 of their paper at length discussing how stochiastic growth must occur for it to result in a universe like the one we observe today.
"Non-viable model" is an apt description, since it's still a model.

I'm not sure why it wouldn't here. I cannot accurately describe that event without detailing that physical dimensional point right?
If your points all look like
{(x,t) : t T} (so the x is constant)

then every value of t uniquely specifies an (x,t). So you only need one parameter to specify a point in your universe. So it's one-dimensional.

This is similar to the dimensionality of, say, the x-axis. The x-axis is the set {(0,r) : r }, and is one-dimensional.

Yes, tn is defined as now in that it is the maximal element within the universe at the step considered.
Well, there's an issue here.

You have a causet (C,≺) that represents the universe at the "current" step; the next step will produce a new causet, (C',≺'), that is the child of (C,≺).

Now, in general your causet might have a bunch of maximal points, all causally remote from one another. You can't define "now" just in terms of maximality.

What I suspect you're using (x,tn) to refer to is the particular maximal element that was adjoined to the parent of the "current" causet.

So my example was meant to ask whether or not a causet that terminates at a specific point has a longest chain in your opinion.
Hmm. I'm not sure quite what this means.

Obviously, in the case of finite causets there are longest chains. So this question is really only applicable to infinite causets.

Specifically, what does it mean to "terminate at a specific point"? Does that mean that there's just one maximal element? Or does a causet terminate at all of its maximal elements?

If a causet has one maximal element, then the existence of a "longest" chain depends on your definition of "longer".

You might take the "length" of a chain to be its cardinality, i.e. how many points it contains, so:

(1) A chain S1 is 'longer' than a chain S2 if |S1| > |S2|.

This has certain counterintuitive results for infinite chains. If you take (,<) as your causet, where < is the standard ordering on , then both and 2 (the set of even numbers) are chains; however, it seems a bit counterintuitive to say that 2 is "just as long" as .

So you might expand your definition of 'longer' and say:
(1') A chain S1 is 'longer' than a chain S2 if |S1| > |S2| or if S1 S2

[i.e., S1 contains all the elements of S2, as well as elements not contained in S2]

At this point, it's unclear what "longest" means.

(1) and (1') each define a strict partial order (say, 1 and 2) on the set of chains.

So you could define "longest" in terms of i; certain chains will be maximal WRT i, and in certain cases there might be a maximum chain WRT i [i.e., a chain S such that if T != S is a chain, then T i S].

We could adopt some relatively common sense deductions from that example. No event within the universal causet (the causet that contains all elements from that example universe) can be completely devoid of any causal relationships (ie it must obey causality).
Hmm, I'm not sure what this actually means. "Can't be devoid of causal relationships" isn't the same as "doesn't violate causality", is it?

Let (C,≺) be a causet.

Do you mean either of the following:
(1) x C, y C : x ≺ y or y ≺ x
(2) x in C, y C : y ≺ x

Events within the causet have occurred (have been born) and are therefore included within the causet. No event that has not been "born" can, by definition, be said to be accreted to the causet.
Okay, I think I have a general idea of what you mean.

So, given that there is a maximal element within this universe (x,tn) beyond which elements have not been born and the comments made above, can we conclude that this example universe has a longest chain? Does this conclusion apply more broadly given the limitations laid out here?
Not enough information.

To my understanding, the "current" causet is:

C := {(x,t) : t T, t tn}

≺ := (x,a), (x,b) C: (x,a) ≺ (x,b) a < b

Observe that < is an ordering on T.
Unfortunately, without information about (T,<), it's impossible to say what the chains are in (C,). This is because the chains in C are just the chains in T. (C,≺) doesn't "add" any information that doesn't already exist in T. Picking elements to exist in (C,) is the same as picking subsets of T, so the whole (C,) formalism is extraneous.

For example, if T is finite, then there are finitely many chains in C, and thus there is a longest chain.

Arthur Fine is also a referee on several Physics based journals as well, "European Physics Journal, International Journal of Theoretical Physics, Physical Review, Physics Letters A." That would seem to indicate that philosophers of science are a bit more important to physics peer-review than the comments made above would seem to indicate.

It is irrelevant if they are doing the bulk of the review or not. If they are doing review and they are physicists then for this journal to have the unreliability imputed to it by GP (primarily) they would need to be actively ignoring mistakes. Again, we aren't talking about them taking a different opinion, we are talking about what GP described as massive and obvious technical errors. Are we seriously to accept that editors who are trained physicists decided to ignore such egregious errors?

Do you have any reason to believe the referee board is vastly different in makeup from the editorial board? In a quick search, the only two referees I found were physicists by training and practice.

I also agree with you that they are responsible for including papers given the mission statement of the journal (though why they would include physicists if the mission was not about including papers relevant to actual physicists is still something unexplained), which in this case distinguishes papers of a non-technical nature (to be included in the letters section) from those technical ones that require peer-review.

You also didn't directly address the response letter I offered as evidence which clearly detailed review of a technical nature concerning the accuracy and warrant for results.

As I stated last time, I was showing that the boards of philosophy of science journals are not made up simply of people untrained in the academic fields being discussed, but are specifically picked to be competent to review the technical nature of the papers they accept.
This is all irrelevant to my arguments. My point was only that if you want to say the peer-review was done by people qualified to do physics peer-review, you should probably talk about the people who actually do the reviewing.

Ignore GP's professional evaluation if you like. I don't care. Take it up with GP.

I don't care about your opinions on philosophy of science, philosophy of science journals, physics journals, or physics.

I don't have a position on whether philosophy of science journals often do physics peer-review, whether Foundations of Physics does physics peer-review, whether Arthur Fine does physics peer-review, or whether scientists really like philosophers of science and want them to participate in the science peer-review process.

15. ## Re: WLC's Argument Against an Actual Infinity

Originally Posted by CliveStaples
He's asking you to give what you consider to be your best source.
I offered my sources, I will respond to a rebuttal of those sources when offered.

Originally Posted by CS
To (a): It depends entirely on the causet you're talking about; given a causet (C,), the set that ≺ is defined on is C.

...
If you're talking about causets generally, then you don't really care about the contents of the partial orders, because you're considering causets generally.
....
So suppose we are interpreting causets as a model of causal structure at the fundamental scale.
Let me offer a few premises and you can tell me which one you disagree with.

1) The Causal Set Theory is an attempt to model fundamental spacetime structure.

2) Causal Sets are a feature of the Causal Set Theory

3) Elements of a causal sets represent spacetime events.

4) Binary relationships between elements are taken to encode causal relations between those elements.

Do you disagree with any of these premises?

I think you are assuming that because a set of elements and their relationship doesn't represent a specific, realized event and relationship that therefore the interpretation of the relationship is undefined except as an operator. That is inconsistent with the works published here. Just because a model doesn't represent an actualized state of affairs does not mean that the physical meaning of parts of that model are therefore undefined.

Let's use your example, let's say we have a function that models bacteriological growth. Let's assume that we have a simple model, y=2*x, where x is defined as the number of reproductive cycles and y as the current population.

Now it could well be (and probably is) that no actual bacteriological colony matches this model precisely. Does that then mean that the relationships within the model lose their physical interpretation? That the relationship of population growing over cycles is just a construct and that we should just interpret this as a two variables? No, of course not, x still has a meaning in the context of this theory, y still has a meaning in the context of this theory, a relationship (even if it is incorrect) is still being defined in the context of the theory.

Because a model is not an accurate state of actualized affairs does not mean that it loses physical meaning. Just that the physical meaning does not match empirical observation. Even an abstract model carries physical interpretation.

Likewise, Causal Sets have a physical interpretation in which elements are spacetime events and < represents a causal relationship between those events, even in a hypothetical or simplified model.

Originally Posted by CS
R/S doesn't require that a causet model causal relationships among actual physical events
R/S are writing a paper concerning a specific theory, Causal Set Theory. Causal Set Theory defines the relationship between elements as representing a causal relationship.

It is irrelevant whether or not the elements or the relations are specific, actual events, only that they are operating to mimic that kind of relationship.

Originally Posted by CS
You can model an aspect of a physical system without needing to model every physical property of that system.
Yep you absolutely can, but for every property you exclude, you increase the likelihood that the model varies from reality in different circumstances. IE the physical meaning of your model must be increasingly hedged. "This model assumes X, Y and Z"

Originally Posted by CS
This is true by definition for causets; a spatial (which I take to mean "spacelike") relation between x and y is precisely the statement that x and y have no causal link in either direction.
And why does that definition exist? What is the origin of that conclusion?

Originally Posted by CS
Yes, if you stipulate certain spacetime relationships among the elements (e.g., by embedding the points in spacetime), then there are certain spacetime relationships among the elements.

But causets don't come with a spacetime embedding; they only come with the Hasse diagram (i.e., the partial order). And for every Hasse diagram of a set of events E, there are many different spacetime embeddings of E.
So the points in panel three are different points than in panel one? And you are saying that there are elements in this Hasse diagram that have a different definition that what elements are defined as on page 1 of the axioms paper as (even if hypothetical) spacetime events? Or are you arguing that spacetime events do not have spatial and temporal coordinates inherent to them?

Originally Posted by CS
The causal relationships within the causet depend only on the Hasse diagram, i.e. the partial order that the causet comes with.
And from whence does that partial order come? Is is arbitrarily applied to the elements?

Originally Posted by CS
The definitions of "child" and "parent" in Labeled Causets is entirely different than their definitions in the R/S sequential growth dynamics paper. If we're going to use these terms ("child" and "parent"), then we should either choose which of the two paper's definitions wins out, or when necessary indicate which definition of "parent" and "child" are being used.
This is a false dichotomy. We only need to pick which one wins out if we assume the two are different. Given my understanding offered above there is no discrepancy at all. This also seems like a more plausible answer than that two sets of respected professors disagree on what is a relatively basic definition.

Once you realize that all of this is meant to be a mathematical representation of a real (even if abstract) event then the distinction you are trying to impart falls away. An element is a specific space time event. Spacetime events like all physical objects encode information, in this case about their causal relationship to other elements, but that information doesn't mean they aren't single events. A single event is timid in relation to a set of events or gregarious to a different set of events, but that doesn't mean that the event actually is the set of events whose relationship defines it as timid or gregarious.

You quoted, but didn't respond to a point I had made, not sure if you missed it or the point was just unintentionally quoted:

Agreed, but if the value of b is reliant on the knowledge of a, then knowing b without a is not possible, correct? In both your examples you relied on an outside metric to deduce b (numbers and in this case time), but that metric doesn't exist here. There isn't some independent temporal measure we can use to deduce the causet at any given point.

Originally Posted by CS
As a side note, it seems that you're contradicting yourself. You said:This seems to be that conclusion that we cannot use ℤ as an R/S labeling.

If this was your understanding of my point, then clearly you did not take my meaning to be that "there are causets that do not admit a labeling via natural numbers".
I fail to see how this is a contradiction. Why would the fact that the set of integers cannot be used as an R/S labeling necessarily imply that the subset of integers, the natural numbers cannot?

I also still hold the objection that ℤ is a causet, R/S labeling occurs within the Causal Set Theory, which relates elements via a binary relationship defined as causation. The individual elements of ℤ are not necessarily defined as being related by a binary relationship defined as causation.

Originally Posted by CS
Okay, so suppose you know that you got a green item from a machine M such that rules (1)-(4) hold for M.
In which case I would argue that the item was green before it went in.

If you wanted to argue that there was an additional rule, 5 as you define it you would need to provide evidence and rationale for its existence right?

Originally Posted by CS
I would probably not agree that the longest chain is a proxy (or a "good" proxy) for the age of the universe without seeing some reasoning why.
Good enough to measure whether the universe is infinitely long or not?

As for the reasons, I'll reiterate my thoughts on this issue.

Given that no spacetime event arises without a causal predecessor (causality) we should be able to take a single event that we define as "now" (or close enough to now to be irrelevant to our conclusion) and trace back a causal history, considering only timid relationships, such that the finiteness of the cardinality of the chain formed by that process matches the finiteness of the universe.

Originally Posted by CS
If your points all look like
{(x,t) : t T} (so the x is constant)

then every value of t uniquely specifies an (x,t). So you only need one parameter to specify a point in your universe. So it's one-dimensional.
Given that (x,t) is an event, that doesn't necessarily imply that x can only have one value, only that in our simplistic universe the events only happen on one specific value of x. IE because it is unchanging that does not mean that the x dimension is only 1 value in size.

Originally Posted by CS
Now, in general your causet might have a bunch of maximal points, all causally remote from one another. You can't define "now" just in terms of maximality.
What would the value of a maximal point that is causally remote from say (x,4) be?

Of course I could define now as: "the point with the largest value of t in the model universe that has been "born."

Originally Posted by CS
Specifically, what does it mean to "terminate at a specific point"?

If a causet has one maximal element, then the existence of a "longest" chain depends on your definition of "longer".
I would tentatively propose that to terminate at a specific point be defined as: "there exists an anti-chain composed of maximal elements of all existing chains."

As for longest, I would follow the definition of Bombelli on page 7 even if he was referring to the length of two points, which I've always taken to mean cardinality. I'm don't see a reason for concern given your infinite example because it isn't problematic if there is more than one "longest" chain. In both those scenarios (setting aside my objection to ℤ as a causet), the chain's cardinality is infinite and such the universe would be infinite and so you could pick either arbitrarily.

Originally Posted by CS
(1) x C, y C : x ≺ y or y ≺ x
1 would be more akin to what I am expressing.

Originally Posted by CS
Given that these causets are part of Causal Set Theory, I think you can safely assume that the ordering is of a causal nature. For example, the element (x,1) causes (x,2).

Originally Posted by CS
This is all irrelevant to my arguments.
Not at all, your point was that philosophers of science (as defined by the wiki article you picked) were incapable of doing peer-review. Since there was no evidence provided that those philosophers of science were engaged in peer-review it makes that list irrelevant. Rather, I took the review board of an actual journal and showed that it did contain people who: a) were philosophers of science, b) did peer review and c) were capable of doing technical peer review. Thus invalidating your comment: "Describing General Relativity as a theory that posits moral relativism is an error that would probably get caught. Technical errors, however, while likely to be found during peer-review among physicists, are known to get through peer review in philosophy of science journals.

For example, the journal that Squatch's article is actually from, Foundations of Physics, published articles on a new physical theory (ECE theory) that were found to contain central, fatal mathematical flaws. "

As a side note, your second statement is incorrect, as I pointed out to GP. Foundations of Physics didn't publish those theories, Foundations of Physic letters, a non-peer reviewed document meant as the equivalent to an editorial section (just as Physics Letters B sacrifices technical review for publication speed). That publication has been subsumed by Foundations of Physics as an independent section within the Journal. It's publication there does not invalidate the process used for a different section.

And further, if we are to cast out every journal that contains technical errors, peer-review becomes a dead process.

Originally Posted by CS
My point was only that if you want to say the peer-review was done by people qualified to do physics peer-review, you should probably talk about the people who actually do the reviewing.
Which I did, right?

16. ## Re: WLC's Argument Against an Actual Infinity

Originally Posted by Squatch347
I offered my sources, I will respond to a rebuttal of those sources when offered.
That is completely non-responsive to the point you're responding to.

Everyone knows you've given sources.

The question being put to you is which you consider to be your best source, so that one of your debate opponents can choose to do a detailed analysis of that source.

Do you see how rude, unresponsive, and unbecoming of an ODN staff member that is?

Also, you seem to have missed a few direct, non-rhetorical questions I asked you. I'll bold them and put them in red to make them more apparent:

GP's request is pretty reasonable, imo. He's not asking you to refine an argument. He's asking for (what you consider to be) your best source explicitly so that he can address/rebut it.

Why not just provide it? How does stonewalling advance this discussion?

Honestly, Squatch, if you wanted me to repeat a citation or an argument so that you could respond to it, I'd be happy to. Because I want you to respond to my arguments and citations. I want feedback, criticism, and analysis for my claims, citations, arguments, etc.

Don't you? Don't you want to see feedback, criticism, and analysis done on your best source? Name (and a link would be helpful) the source, and you can get just that.

Well? Don't you?

I look forward to your response on these matters; I anticipate it will exhibit the quality I've come to expect from ODN staff.

Let me offer a few premises and you can tell me which one you disagree with.

1) The Causal Set Theory is an attempt to model fundamental spacetime structure.

2) Causal Sets are a feature of the Causal Set Theory

3) Elements of a causal sets represent spacetime events.

4) Binary relationships between elements are taken to encode causal relations between those elements.

Do you disagree with any of these premises?
I agree with (1), (2), and (4).

Whether (3) actually holds depends on whether causet models admit such an interpretation; to my knowledge, there are promising indications that causet models do admit such an interpretation, but the model hasn't been fully developed yet. I'm not an expert in causet theory (or the "causet approach" or however you want to name this particular subfield of research), so my caution in declaring an interpretation valid doesn't indicate what a causet theory expert would say on the matter.

I think you are assuming that because a set of elements and their relationship doesn't represent a specific, realized event and relationship that therefore the interpretation of the relationship is undefined except as an operator. That is inconsistent with the works published here. Just because a model doesn't represent an actualized state of affairs does not mean that the physical meaning of parts of that model are therefore undefined.
A causet is a mathematical object, Squatch. If you want to talk about interpretation (i.e., how statements about causet models can be translated into statements about other systems), fine. But in every text I've seen so far (R/S, Axioms, Gudder), causets are described as mathematical objects:

A finite partially ordered set is called a causet. (Gudder 3)

A causal set (or “causet”) is a locally ﬁnite, partially ordered set (or “poset”). (R/S 3)
A causal set is a countable set C, equipped with a transitive, interval ﬁnite, irreﬂexive binary relation ≺, physically interpreted according to the binary axiom and the measure axiom. (Axioms 23)

The Axioms definition does include information about a "physical interpretation", i.e. a way to pass statements about causal sets through to statements about spacetime.

Let's use your example, let's say we have a function that models bacteriological growth. Let's assume that we have a simple model, y=2*x, where x is defined as the number of reproductive cycles and y as the current population.

Now it could well be (and probably is) that no actual bacteriological colony matches this model precisely. Does that then mean that the relationships within the model lose their physical interpretation? That the relationship of population growing over cycles is just a construct and that we should just interpret this as a two variables? No, of course not, x still has a meaning in the context of this theory, y still has a meaning in the context of this theory, a relationship (even if it is incorrect) is still being defined in the context of the theory.

Because a model is not an accurate state of actualized affairs does not mean that it loses physical meaning. Just that the physical meaning does not match empirical observation. Even an abstract model carries physical interpretation.

Likewise, Causal Sets have a physical interpretation in which elements are spacetime events and < represents a causal relationship between those events, even in a hypothetical or simplified model.
That's just a matter of choosing the system your interpretation you're mapping you. Instead of interpreting y = 2*x as a model of the growth pattern of some actually extant bacterial colony, you can interpret it as a model of the growth pattern of a hypothetical bacterial colony. I.e., instead of mapping to the actual world, you map to a set of hypothetical worlds that all have the same rules vis-a-vis bacterial growth.

R/S are writing a paper concerning a specific theory, Causal Set Theory. Causal Set Theory defines the relationship between elements as representing a causal relationship.

It is irrelevant whether or not the elements or the relations are specific, actual events, only that they are operating to mimic that kind of relationship.
Yes, causal set theory attempts to model causal relationships with what they define as causal sets. But causal sets are sets. Their properties are mathematically defined and interpreted physically.

Yep you absolutely can, but for every property you exclude, you increase the likelihood that the model varies from reality in different circumstances. IE the physical meaning of your model must be increasingly hedged. "This model assumes X, Y and Z"
Sure, but every model ever developed has been hedged; we don't yet have a "theory of everything".

And why does that definition exist? What is the origin of that conclusion?
Those seem like two different questions; are "definitions" really "conclusions"?

In any case, to address your first question, my understanding is that the authors decided which relationships to carry over from current physical models; in this case, they decided to carry over the "spacelike" relation from the relativistic spacetime+metric physical model of the universe.

So the points in panel three are different points than in panel one? And you are saying that there are elements in this Hasse diagram that have a different definition that what elements are defined as on page 1 of the axioms paper as (even if hypothetical) spacetime events? Or are you arguing that spacetime events do not have spatial and temporal coordinates inherent to them?
No, no, no. Let me try to clear this up:

The points specified in panel 1 and panel 2 are not just given as a set of events; they are given a spacetime embedding.

Let E be the set of "points", i.e. the black dots, in panels 1-3. Panel 1 and 2 don't just tell you that x E. They also give you an embedding f: E S x T, where if e ∈ E, then f(e) gives the "spacetime location" of e.

To see why this is the case, observe that each point in panels 1 and 2 occupy a certain specific location in spacetime.

Since panel 3 preserves the location of the points from panel 2, I think the inference the author wishes to be drawn is that these points are the same points from panels 1 and 2; that is, the set of points in panel 3 is equal to E.

The embedding f is not inherent to E, however. There are many different maps E S x T.

What you're talking about at isn't just E, it's the pair (E,f); that is, E equipped with a particular spacetime embedding. In this context, it makes sense to say that f is the embedding of (E,f).

I'm saying that there are many different embeddings of E that will produce a Hasse diagram isomorphic to panel 3--i.e., producing the exact same set of events with the exact same causal relations, but a different spacetime embedding.

And from whence does that partial order come? Is is arbitrarily applied to the elements?
These seem like two separate questions.

1. Where do partial orders come from?

They come from the definition of Cartesian product. The partial orders on a set S are the set of subsets of S x S that are reflexive, antisymmetric, and transitive.

2. Are partial orders applied arbitrarily to the elements of a set?

Here I'm not sure what it means to "apply" a partial order.

If you have a single element s of a poset (S,<), you can't really "apply" < to s in the same way that you might apply a function to s; the closest thing I can think of to function application for partial orders is to say that < "applied to" s is the set {t S: s < t or t < s}, i.e. the set of all elements in S related to s by <.

If you mean something like that, then I suppose you could say that partial orders are applied "arbitrarily" to the elements of the set, since you can choose when and where to apply < to some element of S.

This is a false dichotomy. We only need to pick which one wins out if we assume the two are different. Given my understanding offered above there is no discrepancy at all. This also seems like a more plausible answer than that two sets of respected professors disagree on what is a relatively basic definition.
You're saying that the following definitions aren't different:

(DEF3) Let (C,<) and (C',<') be causets satisfying:

(a) C' = C U {x} for some x C
(b) < <'
(c) x is maximal WRT <'

Then:

(i) (C',<') is a child of (C,<)
(ii) (C,<) is the parent of (C',<').

(DEF4) If x, y are elements of a causet (C,<) with x < y, then

(i) x is a parent of y
(ii) y is the child of x
As a caveat, I'm not entirely sure what you think it means for a pair of definitions to be different (i.e., how you're defining the difference relation for definitions).

But DEF3 is talking about a relation between different causets, and DEF4 is talking about a relation between different elements of a causet. They're comparing different objects.

If I said "R is a child of S", it matters whether DEF3 or DEF4 is being referenced.

If DEF3 is being referenced: We immediately deduce that R and S are causets, so R = (C,<), S = (D,$) where C is a set partially ordered by < and D is a set partially ordered by$ (and perhaps satisfying additional conditions, depending on what definition of causet is at work). Additionally, you know:

(1) C = D U {x} for some x C
(2) $⊆ < (3) x is maximal WRT < If DEF4 is being referenced: We immediately deduce that R and S are elements of some causet (A,%). Additionally, you know that (1') S % R These sets of information aren't equal to each other. In the case of DEF3, you know that R and S are causets; in the case of DEF4, you don't know that R and S are causets (maybe they are, maybe they aren't). Once you realize that all of this is meant to be a mathematical representation of a real (even if abstract) event then the distinction you are trying to impart falls away. An element is a specific space time event. Spacetime events like all physical objects encode information, in this case about their causal relationship to other elements, but that information doesn't mean they aren't single events. A single event is timid in relation to a set of events or gregarious to a different set of events, but that doesn't mean that the event actually is the set of events whose relationship defines it as timid or gregarious. I'm not sure what your point is here. Yes, elements are meant to represent spacetime events. A causet is meant to encode the causal relationships among the events it contains (although here I'd say that you're technically wrong; the elements of the causet don't encode any causal information; the partial order on the causet does). I don't know what it means for a "single event" (i.e., an element of C) to be "timid" or "gregarious" in relation to a different set of events. I do know what it means for a causet to be a timid or gregarious child. What does it mean for an element of C to be "timid" or "gregarious" in relation to a different set of events? [It's standard practice for the person making a claim to put in the extra time to explain what definitions are being used in that claim. If you're going to use a new term, or use an old term in a new way, it's courtesy to provide a definition for that use of the term.] You quoted, but didn't respond to a point I had made, not sure if you missed it or the point was just unintentionally quoted: Agreed, but if the value of b is reliant on the knowledge of a, then knowing b without a is not possible, correct? In both your examples you relied on an outside metric to deduce b (numbers and in this case time), but that metric doesn't exist here. There isn't some independent temporal measure we can use to deduce the causet at any given point. Let's be precise, here. My analogy was to the following situation: (1) There is some specific sequence of numbers, S = {xn} = {x0, x1, x2, ...} (2) You are given information about this sequence, in one of the following forms: (i) A function which when given as input an element xi S produces the next element xi+1 S (ii) The value of xj for some j My guess is that by "metric", you're referring to the function in (i). It turns out that there actually is an analog to (i) for causets. The functions for causets are the transition maps that take you from a parent causet to a child causet. I fail to see how this is a contradiction. Why would the fact that the set of integers cannot be used as an R/S labeling necessarily imply that the subset of integers, the natural numbers cannot? I'm totally at a loss here. (1) I never proved that the integers can't be used as an R/S labeling (whatever that means). (2) I never said that the natural numbers cannot be used as an R/S labeling (whatever that means). (3) My proof was that there's no R/S labeling on the integers. As I've repeated numerous times, my definition of an R/S labeling on a causet (C,) is as follows: (RSL) An R/S labeling on (C,≺) is a map L:(C,≺) (N,<) such that For all c1, c2 C: c1 ≺ c2 L(c1) < L(c2) My confusion in (1) and (2) stems from not understanding what is meant by the phrase "cannot be used as an R/S labeling". What does it mean for a set to be "used as an R/S labeling"? I also still hold the objection that ℤ is a causet, R/S labeling occurs within the Causal Set Theory, which relates elements via a binary relationship defined as causation. The individual elements of ℤ are not necessarily defined as being related by a binary relationship defined as causation. What is your definition of a causet, here? What is your source for this definition, or is it one of your own making? Here's the list of causet definitions that I'm at least somewhat familiar with: A finite partially ordered set is called a causet. (Gudder 3) A causal set (or “causet”) is a locally ﬁnite, partially ordered set (or “poset”). (R/S 3) A causal set is a countable set C, equipped with a transitive, interval ﬁnite, irreﬂexive binary relation ≺, physically interpreted according to the binary axiom and the measure axiom. (Axioms 23) With these definitions, it doesn't really make sense to ask if ℤ is a causet, because ℤ by itself doesn't have a partial order; it's just a set. For these definitions, you need a set along with a partial order on that set. Now, there's a fairly standard partial order on ℤ, usually referred to as , that makes (ℤ,) a possible candidate for being a causet. I'll go through the causet definitions one-by-one, and see whether (ℤ,) qualifies as a causet under each definition. (1) A finite partially ordered set is called a causet. (Gudder 3) (ℤ,) is a partially ordered set, but it is not finite. So (ℤ,) is not a Gudder causet. (2) A causal set (or “causet”) is a locally ﬁnite, partially ordered set (or “poset”). (R/S 3) (ℤ,) is a partially ordered set, and it is indeed locally finite. So (ℤ,) is an R/S causet. (3) A causal set is a countable set C, equipped with a transitive, interval ﬁnite, irreﬂexive binary relation ≺, physically interpreted according to the binary axiom and the measure axiom. (Axioms 23) ℤ is countable; (ℤ,<) equips ℤ with a transitive, interval finite binary relation (where < is the "strictly less than" relation). The only question is whether (ℤ,<) is "physically interpreted according to the binary axiom and the measure axiom". Therefore, so as long as you physically interpret (ℤ,<) using the binary axiom and the measure axiom, then (ℤ,<) is an Axiom causet. Since I don't know what you mean by "causet"; I'll call any object meeting your (as-of-yet undeclared) "causet" criteria an S-causet. I can't argue whether (ℤ,) is a S-causet without knowing the definition of an S-causet. But for the definitions found in Gudder, R/S, and Dribus (author of the Axiom paper), I think it's pretty clear in each case whether (ℤ,<) [or (ℤ,)] meets that definition. In which case I would argue that the item was green before it went in. If you wanted to argue that there was an additional rule, 5 as you define it you would need to provide evidence and rationale for its existence right? If we have no information about whether (5) is a rule, how is assuming that it isn't any more justified than assuming that it is? In either case, you're making an additional assumption. If you wanted to argue that the machine was guided only by rules (1)-(4) and none others, you would need to provide evidence and rationale for that conclusion, right? As an analogy to your example, imagine you have a revolver and you know that chambers 1 through 5 are empty. Do you have enough evidence to conclude that chamber 6 is empty? Do you have enough evidence to conclude that chamber 6 is more likely empty than loaded? Good enough to measure whether the universe is infinitely long or not? Just a notational quibble; infinitely "long" suggests to me a spatial dimension; perhaps "infinitely old" would be better, although it still makes sense to talk about temporal "length". Doesn't really matter what word you use, so long as the definition of it is clear. As for the reasons, I'll reiterate my thoughts on this issue. Given that no spacetime event arises without a causal predecessor (causality) we should be able to take a single event that we define as "now" (or close enough to now to be irrelevant to our conclusion) and trace back a causal history, considering only timid relationships, such that the finiteness of the cardinality of the chain formed by that process matches the finiteness of the universe. (1) Are you sure that you've accurately stated the requirement of causality? You're basically saying that there are no gregarious children (in the R/S sense). Is that borne out in the causal set literature? (2) I tend to agree (although I'm not conclusively certain) that if the longest chain in a causet has finitely many elements, then the universe represented by that causet has finite age. If that's correct, then a finite longest chain implies a universe with finite age. However, I'm not sure that the converse is true; I wonder if there are causets with infinite causal chains whose universes are only finitely old. Given that (x,t) is an event, that doesn't necessarily imply that x can only have one value, only that in our simplistic universe the events only happen on one specific value of x. IE because it is unchanging that does not mean that the x dimension is only 1 value in size. Then I'm not clear on what your universe is. You said: Lets assume a very simple universe. One physical dimension made up of one single point and one temporal dimension. [...] Any event in this universe can be described by a coordinate, (x,t). Let me see if I've got this right, then: You have a set of spatial coordinates X and a set of temporal coordinates T. Every event in the universe corresponds to some (x,t), where x X and t T. Is that correct? What would the value of a maximal point that is causally remote from say (x,4) be? I don't know. You haven't specified the causal relations yet; you've just specified the set of events, not the causal relations among them. Of course I could define now as: "the point with the largest value of t in the model universe that has been "born." What if there is no largest value of t? What if there are points in T that aren't comparable? I would tentatively propose that to terminate at a specific point be defined as: "there exists an anti-chain composed of maximal elements of all existing chains." The quantification in your statement is a little ambiguous. Did you mean one of the following: (DEF5) A causet (C,≺) is said to terminate at a specific point if there exists a subset S of C satisfying: (i) S is an antichain; (ii) For all s S, there exists a chain Ts in C such that s Ts and s is maximal in T (DEF6) A causet (C,≺) is said to terminate at a specific point if there exists a subset S of C satisfying: (i) S is an antichain; (ii) For all s S, and for all chains T in C, s T and s is maximal WRT T DEF5 says that each element of S is a maximal element in some chain in C. DEF6 says that each element of S is a maximal element in every chain in C. Potential problems with this definition: (1) You don't exclude the empty antichain The empty set is an antichain of every set; if you don't exclude the empty antichain, then every set has an antichain satisfying DEF5 and DEF6, so every causet terminates at a specific point. (2) DEF5 implies that the following causet terminates at a specific point: 0 1 < 2 3 < 4 < 5 6 < 7 < 8 < 9 10 < 11 < 12 < 13 < 14 ... [To be rigorous, I should specify the set and the partial order that constitute this causet; here, the set is , and the partial order is given by the pattern above; the partial order can be written explicitly, but it would be a bit messy, so if you don't object I'll omit its explicit presentation.] According to DEF5, this causet terminates at a specific point if there is an antichain each of whose elements are maximal elements in some chain. The set {2} satisfies these criteria, as do the sets {5}, {2,5}, {9}, {5,9}, {2,9}, {2,5,9}, {14}, etc. (3) DEF6 implies that if C terminates at a specific point, then C = {s}, i.e. the only causets that terminate at a specific point are causets containing only one element. Suppose (C,≺) terimates at a specific point according to DEF6. Then there is a non-empty antichain S C. Since S is non-empty, there is some s S. Let T be a chain in C. By DEF6, s T. Let c be any element in C. Then {c} is a chain. Hence s {x}, and thus s = x. Thus every element in C is equal to s, so C = {s}. As for longest, I would follow the definition of Bombelli on page 7 even if he was referring to the length of two points, which I've always taken to mean cardinality. I'm don't see a reason for concern given your infinite example because it isn't problematic if there is more than one "longest" chain. Bombelli is talking about two things: (1) The timelike distance between two points on the spacetime manifold, where the two points are related by (2) The length of the longest chain in the poset From what I gather about Bombelli's work here, you pick a discrete set of points from the manifold, and the causal relations among them define the poset (causet). Now, I'm not super familiar with Bombelli's work, but I take (2) to be referring to the longest chain beginning at one point and ending at the other. Suppose you pick a discrete set of points S such that: (a) x,y S such that x ≺ y (b) The timelike distance between x and y is very small (c) xi ∈ S, where i = 1,2,..., 1000 (d) For all i, xi ≺ xi+1≺ x (e) For all i, the timelike distance between xi and x is very large (f) There is no z S such that x ≺ z ≺ y (f) implies that {x,y} is the only chain beginning at x and ending at y, so the length of the longest chain beginning at x and ending at y is 2. The longest chain in S, however, is at least length 1002, since {x1, x2, ..., x1000, x, y} is a chain in S. I suspect it might introduce a large approximation error to estimate the timelike distance between x and y with the length of {x1, x2, ..., x1000, x, y} rather than the length of {x,y}. However, I'm not sure what the definitions in play are, so I can't say for certain. If the above is true, then what Bombelli is doing is not estimating the length of a causet, but rather estimating the length of the timelike distance between two points. One way you could define the length of the causet might be the supremum of the timelike distance between each pair of points in the causet (this is similar to the definition of the diameter of a set). This however relies on a spacetime embedding of those points, and without a canonical embedding this cannot lead to a coordinate-free / natural / intrinsic definition of causet length/age. You might try something like: The distance between any two related points x ≺ y in a causet is the length of the longest chain starting at x and ending at y. The length of a set S of points in a causet is the supremum of the distances between pairs of points in S. In both those scenarios (setting aside my objection to ℤ as a causet), the chain's cardinality is infinite and such the universe would be infinite and so you could pick either arbitrarily. Okay. Although here is where my previous speculation becomes important: I wonder if there are causets with infinite causal chains whose universes are only finitely old. If there are such causets, and if all universes with only finite causal chains have finite age, then infinite causal chains are a necessary but not sufficient condition for a universe to be infinitely old. If there are not such causets, and all universes with only finite causal chains have finite age, then infinite causal chains are a necessary and sufficient condition for a universe to be infinitely old. 1 would be more akin to what I am expressing. I'll just restate (1) here for convenience: (1) x C, y C : x ≺ y or y ≺ x I assume that you will additionally stipulate that the y in (1) must be different than x. This has the following implications: 1. There are no gregarious children (in the R/S sense). Proof: If (Cg,g) is a gregarious child of (C,), then Cg contains a maximal element x not related to any element in C. By (1), there must exist a y Cg - {x} such that y ≺ x or x ≺ y. Suppose such a y exists; since y Cg - {x}, y C. But then x is related to an element in C, contradicting the gregariousness of Cg. Thus (1) requires that C have no gregarious children. This implies that qn, the probability of a gregarious child developing (as defined in the R/S paper), is equal to 0. This, in turn, implies: (i) Each transition probability is equal to 0 or undefined. Since each transition probability involves factors of the form qi and 1/qj, if each qn = 0 then these expressions are undefined. This means there is no growth. (ii) Bell Causality fails. Since the relevant probabilities are undefined, they cannot be equal. 2. There are no causets consisting of only a single event. Proof: Suppose (C,) contained only a single element, say C = {c}. Then there are no elements y (≠ x) C with y c or c y, contradicting (1). This means that there is no causet representing the "beginning" of the universe (i.e., the universe consisting of a single, first event). Since every universe can be represented by a causet, this means the universe never consisted of a single, first event. Given that these causets are part of Causal Set Theory, I think you can safely assume that the ordering is of a causal nature. For example, the element (x,1) causes (x,2). Yes, that's one interpretation of causets (clearly, the intended one). But mathematical objects/constructs/models often admit many interpretations: 1 and 0 are interpreted as "true" and "false" in computer logic, functions are interpreted as values in bank accounts, levels of risk, levels of confidence, etc. But to your proposed universe, you haven't told me what the causal relationships are. For what values of x1,x2,t1,t2 does (x1,t1) cause (x2,t2) [i.e., (x1,t1) (x2,t2)]? You also haven't said anything about the set X and T (from which the elements xi and tj are taken). Are these ordered sets, i.e. can we compare elements in T? Elements in X? If so, is the ordering a partial order? A weak partial order? A strict partial order? A total order? Are X or T finite? Infinite? Not at all, your point was that philosophers of science (as defined by the wiki article you picked) were incapable of doing peer-review. Nope, wrong. I never argued that philosophers of science were incapable of doing peer-review. The only argument I offered was that philosophers of science aren't necessarily capable (not that they're necessarily incapable) of doing physics peer-review. The logical distinction between my argument and the argument you've attacked seems obvious. As a side note, your second statement is incorrect, as I pointed out to GP. Foundations of Physics didn't publish those theories, Foundations of Physic letters, a non-peer reviewed document meant as the equivalent to an editorial section (just as Physics Letters B sacrifices technical review for publication speed). What's your evidence for this? N.B. Asking for evidence does not amount to claiming that you are wrong. Asking for evidence does not amount to making a claim on this topic. Asking for evidence doesn't mean that I'm soliciting further argument on this topic. Asking for evidence is asking for evidence. Which I did, right? Eh, I don't really care. It's not even an argument you're having with me, it's an argument between you and GP. And frankly, with the way you've stonewalled and refused basic debate etiquette over the last 5 pages, I don't see much value in talking with you about your arguments unless they're directly related to my claims. 17. ## Re: WLC's Argument Against an Actual Infinity Originally Posted by CliveStaples Everyone knows you've given sources. Great, then perhaps they could argue the sources rather than continue to list a series of extraneous conditions? Originally Posted by CS Do you see how rude, unresponsive, and unbecoming of an ODN staff member that is? How is my refusal to comply with GP's unwarranted request to limit the discussion to a single source rather than as a body of sources reflective of my role as a staff member? You've brought this up a couple of times and so I think it needs to be addressed. How does my role as a staff member require me to jump through arbitrary hoops of other members when interacting with them as a member? Originally Posted by CS I agree with (1), (2), and (4). Whether (3) actually holds depends on whether causet models admit such an interpretation I don't think that that is accurate. I think you are confusing the claim "it represents an actualized event" with "it represents the concept of an event." Just as if you were to look at a very old map and see Atlantis. Because Atlantis is not an actual place (presumably) does not mean that the representation on the map is not meant to symbolize it. Likewise in the equation F=MA, M stands for mass, even if it is a hypothetical or non-actualized mass, such as all the physics 101 test questions about acceleration. Those questions have a hypothetical ramp, hypothetical ball, etc, but that doesn't mean that M doesn't represent mass. As such elements in causets represent space time events, even if the specific event represented is hypothetical. I need to add a further consequence of premise 2. Since Causal Sets are part of Causal Set Theory, we must accept the limitations placed on sub-disciplines within that field, one of which, as detailed in painstaking levels above, is that the order placed on elements within the theory represents a causal structure. Causets are, as you point out, partially ordered sets with local finiteness, but they are partially ordered sets in which the partial order is represents a causal structure. Originally Posted by CS Sure, but every model ever developed has been hedged; we don't yet have a "theory of everything". Agreed, what I was getting at is that if you wish to do that here, we need to be explicit about those caveats to prevent an erroneous conclusion. Originally Posted by CS my understanding is that the authors decided which relationships to carry over from current physical models; in this case, they decided to carry over the "spacelike" relation from the relativistic spacetime+metric physical model of the universe. So the makeup of the causet is dictated by physical law right? So we agree that the mathematical concept is constrained by physical law, in which case we can't simply assign relationships without a discussion of whether or not those relationships accord to physical law. So it would seem that causets are more than just mathematical objects. They are mathematical objects governed by physical laws. Originally Posted by CS The points specified in panel 1 and panel 2 are not just given as a set of events; they are given a spacetime embedding. What is the definition of "event" in your mind? Events, according to the quotes above are inherently space time in nature. The axes there are to clarify the nature of the relationships within spacetime, not to establish them. Point x has a temporal and spatial coordinate set regardless of the diagram it is overlayed onto. The fact that a Hasse Diagram does not contain inherent axes of that nature is no barrier to synthesizing the data from the panels. Originally Posted by CS If you mean something like that, then I suppose you could say that partial orders are applied "arbitrarily" to the elements of the set, since you can choose when and where to apply < to some element of S. I didn't really mean either. Let me try again with this phrasing; You made this statement (emphasis mine): "The causal relationships within the causet depend only on the Hasse diagram, i.e. the partial order that the causet comes with." I'm asking what dictates the partial order the causet comes with? Originally Posted by CS You're saying that the following definitions aren't different As I said in my last post. I think you are missing the meaning of the definitions because you are attempting to divorce the notation from the theory. Personally, I would think it should be a big warning sign if your understanding leads to the conclusion that colleagues disagree on basic definition. Take a step back and think about people as an example. My son is a person. An individual with a specific identity. Part of that identity is the fact that he is a child. But that fact doesn't mean that we have to be considered as a group in order for him to be a child. He is a child regardless of my nature or status. What kind of child (mine or someone else's) depends on us as a group, but not his status as a child. Likewise an event is a child if it causally depends on another event (as they all do) and a parent if another event causally depends on it. Expressing that causal relationship requires both elements to be expressed via a causet. The reason R/S use the terminology they do is because they are seeking to distinguish the various types of children that can be formed from a parent. That distinction only makes sense if you consider it in relation to the parent. Likewise, if I needed to discuss the relation ship between Steve and Amy in the context of siblings I would need to explicitly define each of them in relation to parental sets. IE are they full relatives, half-relatives, step relatives, etc. So when R/S talk about children as part of a causet they are doing so because they are distinguishing between the different relationships they can have with their parents, not arguing that they are inherently a causet. Originally Posted by CS My analogy was to the following situation: (1) There is some specific sequence of numbers, S = {xn} = {x0, x1, x2, ...} (2) You are given information about this sequence, in one of the following forms: (i) A function which when given as input an element xi S produces the next element xi+1 S (ii) The value of xj for some j My guess is that by "metric", you're referring to the function in (i). It turns out that there actually is an analog to (i) for causets. The functions for causets are the transition maps that take you from a parent causet to a child causet. No, I was referring to (ii). You are assuming it is possible to get The value of xj for some j But in every example given so far, that value was determined by appealing to some external order that you could use to deduce xj. My point was that no such external order exists here. How exactly would you determine a causet at any given point without using the causally prior causets? Originally Posted by CS What is your definition of a causet, here? What is your source for this definition, or is it one of your own making? Here's the list of causet definitions that I'm at least somewhat familiar with: A finite partially ordered set is called a causet. (Gudder 3) A causal set (or “causet”) is a locally ﬁnite, partially ordered set (or “poset”). (R/S 3) A causal set is a countable set C, equipped with a transitive, interval ﬁnite, irreﬂexive binary relation ≺, physically interpreted according to the binary axiom and the measure axiom. (Axioms 23) My problem with your analysis is that you assume both R/S and Axioms do not have a specific partial order in mind in their definitions. However, as I pointed out in this post earlier and in multiple quotes in earlier posts, these causets are part of the Causal Set Hypothesis and therefore the nature of the partial order is restricted to causal partial orders. A set of space-time elements whose relation < was meant to imply an order consisting of my preference would not be a causet in any of these papers. As for where I got this idea, you agree with me that Causets are a segment of the Causal Set Hypothesis. As such I would point out the definition of the latter in the Axioms paper (which would disagree with your conclusion concerning (ℤ,<) as you defined it): "Causal set theory is a promising attempt to model fundamental spacetime structure in a discrete order-theoretic context via sets equipped with special binary relations, called causal sets. The elements of a causal set are taken to represent spacetime events, while its binary relation is taken to encode causal relations between pairs of events." Your definition of the partial order on ℤ doesn't match the partial order defined here as causal relations between pairs of events. If you wish to suggest that the Causal Set Hypothesis includes partial orders that are not causal in nature, I think we would need to see a reason why. Originally Posted by CS If we have no information about whether (5) is a rule, how is assuming that it isn't any more justified than assuming that it is? In either case, you're making an additional assumption. Not at all. I'm highlighting two pieces of known data. 1) There is a green item. 2) There is a known mechanism for creating a green item. Your objection is "there could be another mechanism for creating a green item." Which is fine, but as a stand alone appeal it is a kind of appeal to ignorance fallacy (there is an additional mechanism because we don't know there isn't). If you wish to propose another mechanism then a reason needs to be offered to justify it. I think a better analogy would be finding a dinosaur bone (1) and knowing about fossilization (2) and then saying "well we don't know that it wasn't created via another process like aliens." Maybe it was put there by aliens, but then we need to justify why that is the mechanism rather than fossilization. We can't just discount fossilization on the possibility. Originally Posted by CS Just a notational quibble; infinitely "long" suggests to me a spatial dimension; perhaps "infinitely old" would be better, although it still makes sense to talk about temporal "length". Doesn't really matter what word you use, so long as the definition of it is clear. Fair point, agreed. Originally Posted by CS (1) Are you sure that you've accurately stated the requirement of causality? You're basically saying that there are no gregarious children (in the R/S sense). Is that borne out in the causal set literature? Not exactly, I'm saying that the requirements placed on the causets by physical law prevent there being a gregarious child (if the parent is considered to be the causet representing all elements in the universe). This is what I believe is the concern being discussed on page 24 of the axioms paper as the possible emergence of infinitely large universes over finite time periods, or more precisely an imbalance between a large spatial volume to a small temporal one. On a more abstract note, I would be hesitant to suggest that any of the authors here are suggesting at a gregarious child grouping is implying than an element arose causelessly. So why discuss gregarious children at all? My guess is that (as described in section 4.5) the various different possible causets and the status of elements in relation to those causets is what gives rise to the emergent order that makes this theory explanatory. IE the different possible groupings allow for a more dynamic outcome than would be otherwise possible. Hence the statement; "order plus number equals geometry." Originally Posted by CS However, I'm not sure that the converse is true; I wonder if there are causets with infinite causal chains whose universes are only finitely old. Hmm, while it would be tempting to agree with that possibility as a debate tactic (since it shifts the argument towards finite universes somewhat), I think that the converse is true. I'm not sure how a causet with an infinite number of causal steps could be finitely old. I'm happy to discuss it if you wish for the fun of discussing it. Could you elaborate on your misgivings of the converse? Originally Posted by CS You have a set of spatial coordinates X and a set of temporal coordinates T. Yes, but we are artificially limiting the range of possible spatial coordinates in consideration to a single one. Originally Posted by CS I don't know. You haven't specified the causal relations yet; you've just specified the set of events, not the causal relations among them. I did in post 390, but to restate: {... (x,1) < (x,2) < (x,3) < (x,4)} So hypothetically, how could you generate a "bunch of maximal points, all causally remote from one another?" Originally Posted by CS What if there is no largest value of t? What if there are points in T that aren't comparable? In the example 4 was the largest value of t having been born. You'll have to elaborate on what you mean by points in T that aren't comparable. Originally Posted by CS (DEF5) A causet (C,≺) is said to terminate at a specific point if there exists a subset S of C satisfying: (i) S is an antichain; (ii) For all s S, there exists a chain Ts in C such that s Ts and s is maximal in T 5 is closer, however I would rephrase ii to say: (ii) For all chains Ts in C, there exists a maximal element in Ts, s, such that s ∈ S IE it isn't that there exists a chain with a maximal element corresponding to our anti-chain, it is that the anti chain contains the maximal elements from all of our chains. Originally Posted by CS If (Cg,g) is a gregarious child of (C,), then Cg contains a maximal element x not related to any element in C. By (1), there must exist a y Cg - {x} such that y ≺ x or x ≺ y. Suppose such a y exists; since y Cg - {x}, y C. But then x is related to an element in C, contradicting the gregariousness of Cg. Thus (1) requires that C have no gregarious children. [/INDENT] This is only true if you assume C contains all elements within the poset. If C is a subset of all points in the universe this does not become a problem because the maximal element can be related to an element not in C. Originally Posted by CS This means that there is no causet representing the "beginning" of the universe (i.e., the universe consisting of a single, first event). Since every universe can be represented by a causet, this means the universe never consisted of a single, first event. Not necessarily. Remember the conditions this definition was introduced to. We are at some definite point, tn. Given that, for all values x that exist there is a y that fits that definition. Imagine our sample universe is points 1,2,3,4, with 4 being "now." All of those points fit the definition offered, and point 1 is the "beginning." Originally Posted by CS What's your evidence for this? It is discussed on the Foundations of Physics page: "The former letters publication Foundations of Physics Letters has been merged with Foundations of Physics. Short papers that demand a rapid publication process will be included in the special section 'Letters to the Editor'. Authors who wish to raise discussion points or other articles of a different format are welcome to submit these as well, if they care to explain their wishes with the submission." http://link.springer.com/journal/10701 18. ## Re: WLC's Argument Against an Actual Infinity There seems to have been some confusion! You've overlooked some direct, non-rhetorical questions again, Squatch! Hopefully you'll see them this time: GP's request is pretty reasonable, imo. He's not asking you to refine an argument. He's asking for (what you consider to be) your best source explicitly so that he can address/rebut it. Why not just provide it? How does stonewalling advance this discussion? Honestly, Squatch, if you wanted me to repeat a citation or an argument so that you could respond to it, I'd be happy to. Because I want you to respond to my arguments and citations. I want feedback, criticism, and analysis for my claims, citations, arguments, etc. Don't you? Don't you want to see feedback, criticism, and analysis done on your best source? Name (and a link would be helpful) the source, and you can get just that. Well? Don't you? Or perhaps you prefer a list form: (1) Why haven't you provided GP with the name of the source that you consider to be the strongest support for your position? (2) How has your refusal to name this source advanced the discussion in this thread? (3) Do you want feedback, criticism, and analysis done on your best source? Ironically, I've already put in more work trying to get you to answer questions about why you won't answer simple questions than the work it would have taken for you to answer the simple question in the first place. Who knew that answering simple questions was so far above the paygrade of ODN staff? Originally Posted by Squatch347 Great, then perhaps they could argue the sources rather than continue to list a series of extraneous conditions? Asking which of those sources you think is the best one is "a series of extraneous conditions"? It seems like a pretty natural question to me. You give a bunch of sources, there's some arguments that eventually dead-end, so to advance the discussion you start over. GP just wants to start with whichever source you think is the best one you've got. Apparently, this is an egregious demand on your time and energy. How is my refusal to comply with GP's unwarranted request to limit the discussion to a single source rather than as a body of sources reflective of my role as a staff member? He isn't doing that. He's not saying, "The only source that can be discussed by anyone is this single source." He's saying, "Hey, I want to give a really thorough critique of your very best source. Which one is it, by your reckoning?" You've brought this up a couple of times and so I think it needs to be addressed. How does my role as a staff member require me to jump through arbitrary hoops of other members when interacting with them as a member? Not every hoop. But this is a very reasonable hoop. It's like if a new member didn't understand your argument and asked you to rephrase it, or explain it more in depth. Are you required to rephrase your argument? No. Are you required to explain it more in depth? No. But ODN is a place that is meant to lead by example and help people reason better. So I would expect an ODN staff member, who should exemplify the best ODN has to offer, to be more responsive to such requests than your standard ODN member. I'd place GP's request for you to name the source that you consider the strongest to be a very reasonable, easily satisfied request. He's not asking you to make an argument that the source is the strongest. He's just asking you to say which source it is that you think is the strongest. Imagine if in the middle of this discussion between you and I, you asked me, "Hey, which of these papers that we've been referencing do you think best supports my argument?" I would probably make a good-faith effort to answer that question for you. I'm familiar with the sources already, so the only work I have to do is think for a few minutes about what kind of support these sources provide for your argument. Here, again, I expect more from ODN staff than from average members when it comes to supplying analysis of the strength of a member's argument, especially when the cost to the staff member of supplying that analysis is quite low. I don't think that that is accurate. I think you are confusing the claim "it represents an actualized event" with "it represents the concept of an event." Just as if you were to look at a very old map and see Atlantis. Because Atlantis is not an actual place (presumably) does not mean that the representation on the map is not meant to symbolize it. Likewise in the equation F=MA, M stands for mass, even if it is a hypothetical or non-actualized mass, such as all the physics 101 test questions about acceleration. Those questions have a hypothetical ramp, hypothetical ball, etc, but that doesn't mean that M doesn't represent mass. As such elements in causets represent space time events, even if the specific event represented is hypothetical. Yes, this is essentially my point here: That's just a matter of choosing the system your interpretation you're mapping you. Instead of interpreting y = 2*x as a model of the growth pattern of some actually extant bacterial colony, you can interpret it as a model of the growth pattern of a hypothetical bacterial colony. I.e., instead of mapping to the actual world, you map to a set of hypothetical worlds that all have the same rules vis-a-vis bacterial growth. I need to add a further consequence of premise 2. Since Causal Sets are part of Causal Set Theory, we must accept the limitations placed on sub-disciplines within that field, one of which, as detailed in painstaking levels above, is that the order placed on elements within the theory represents a causal structure. Causets are, as you point out, partially ordered sets with local finiteness, but they are partially ordered sets in which the partial order is represents a causal structure. (1) What is your argument in support of your premise? Which paper do you think places this limitation on the definition of causets? The R/S paper certainly doesn't; it defines a causet as a locally finite poset, and never refines this definition. The Gudder paper certainly doesn't; it defines a causet as a finite poset, and never refines this definition. The Axioms paper certainly doesn't; it places a restriction not on the set of causets, but on the set of interpretretations of causets; the Axioms paper requires that any physical interpretation of causets must satisfy the binary and measure axioms. I don't see any place in any of these papers where the author says, "In addition to being a finite / locally finite poset, the partial order on a causet must satisfy these additional conditions". The only even remotely plausible exception is the Axioms paper's definition, but there the requirement is placed on the interpretation, not on the partial order. In lots of places the authors restrict themselves to considering certain kinds of causets, but there's no declaration that the causets outside the scope of their inquiry aren't "really" causets. (2) What partial orders don't represent causal structure? For any set S with any partial order <, isn't there a possible world where there are a set of events corresponding to S whose causal relationships correspond to Agreed, what I was getting at is that if you wish to do that here, we need to be explicit about those caveats to prevent an erroneous conclusion. I don't understand the reason for making this point, Squatch. I'm not proposing a physical model, nor do I wish to. So the makeup of the causet is dictated by physical law right? So we agree that the mathematical concept is constrained by physical law, in which case we can't simply assign relationships without a discussion of whether or not those relationships accord to physical law. So it would seem that causets are more than just mathematical objects. They are mathematical objects governed by physical laws. No, causets are just mathematical objects. What you're getting at is that they're mathematical objects used in models that are designed to have physical interpretations. This is why, say, you require that causets be partially ordered sets, rather than just considering all sets to be "causets". But the causets are mathematical objects. Any restrictions placed on them are mathematical restrictions (even if they result from physical considerations), e.g. the requirement that a causet be a set that is partially ordered is a mathematical restriction, but one that is motivated by physical considerations. Again, this is an issue of model vs. interpretation. The model is defined mathematically. Take for example the R/S paper. They mathematically define causets, then mathematically define transition maps / dynamical laws. They then go about proving that their dynamical laws obey physical requirements. What is a physical requirement? To my understanding, a physical requirement of a model is a property P such that if a model lacks P, then there is no physical interpretation of that model. Now, here is where you could probably lodge an objection to certain kinds of causets. You could argue something like: "Yes, every locally finite poset is a causet. But there is a physical requirement that causets have properties X,Y,Z, and not every locally finite poset has properties X,Y,Z." Of course, you would need to support the claim that such a physical requirement exists and entails that causets have properties X,Y,Z. If this claim has not been shown already in the literature, then if you can indeed provide such support, you've probably got the makings of a publishable paper on your hands. What is the definition of "event" in your mind? Events, according to the quotes above are inherently space time in nature. The axes there are to clarify the nature of the relationships within spacetime, not to establish them. Point x has a temporal and spatial coordinate set regardless of the diagram it is overlayed onto. The fact that a Hasse Diagram does not contain inherent axes of that nature is no barrier to synthesizing the data from the panels. (1) Yes, events are inherent to spacetime; no, they are not inherent to the choice of coordinates. Coordinates depend on your reference frame. Technically, a reference frame is literally just a choice of basis. (2) You are making statements without providing reasoning or proof. If you know of a method to recover the coordinates of a set of events in a Hasse diagram, present that method. Just flatly stating that you can do so doesn't advance the discussion. I didn't really mean either. Let me try again with this phrasing; You made this statement (emphasis mine): "The causal relationships within the causet depend only on the Hasse diagram, i.e. the partial order that the causet comes with." I'm asking what dictates the partial order the causet comes with? Hmm, I'm not sure quite what you mean. The person defining the causet dictates it, I suppose. Given C = {a,b,c}, what "dictates" a choice of < = {(a,b), (a,c), (b,c)} as opposed to < = {(a,b)}? You choose set and partial order based on what set and partial order you want to talk about. If you want to talk about a causet that looks like a < b < c < d, then you choose C = {a,b,c,d}, < = {(a,b),(a,c),(a,d), (b,c), (b,d), (c,d)}. As I said in my last post. I think you are missing the meaning of the definitions because you are attempting to divorce the notation from the theory. Personally, I would think it should be a big warning sign if your understanding leads to the conclusion that colleagues disagree on basic definition. First, mathematicians/physicists/computer scientists re-appropriate terms all the time. This is one of the reasons why you include a "definitions" section in your paper. There are many terms have multiple definitions in the literature, and there are many definitions that have multiple terms in the literature. For instance, there is not universal agreement on whether includes 0. Many but not all computer science sources exclude 0; many but not all mathematics sources include 0. Second, the definitions are actually pretty similar in structure--x is a child of y iff x < y--the only question is what < you're looking at. If the < is the partial order on a causet, you get Gudder's definition. If the < is the inclusion partial order on the set of subsets of a causet, you get R/S's definition. Third, the fact that Gudder never uses "parent"/"child" in the way they're used in R/S, and that R/S never use "parent"/"child" in the way Gudder does, shows you that these terms are used differently to refer to different things. Take a step back and think about people as an example. My son is a person. An individual with a specific identity. Part of that identity is the fact that he is a child. But that fact doesn't mean that we have to be considered as a group in order for him to be a child. He is a child regardless of my nature or status. What kind of child (mine or someone else's) depends on us as a group, but not his status as a child. Likewise an event is a child if it causally depends on another event (as they all do) and a parent if another event causally depends on it. Expressing that causal relationship requires both elements to be expressed via a causet. The reason R/S use the terminology they do is because they are seeking to distinguish the various types of children that can be formed from a parent. That distinction only makes sense if you consider it in relation to the parent. Likewise, if I needed to discuss the relation ship between Steve and Amy in the context of siblings I would need to explicitly define each of them in relation to parental sets. IE are they full relatives, half-relatives, step relatives, etc. So when R/S talk about children as part of a causet they are doing so because they are distinguishing between the different relationships they can have with their parents, not arguing that they are inherently a causet. You're absolutely right that R/S want to be able to talk about the kind of "descendants" you can get from a parent causet. But they're talking about causets. R/S don't talk about children as "part" of a causet. They talk about children as a set of causets, specifically those causets that can be formed from the parent in a particular way. To use your analogy, suppose I defined "child" both as your genetic offspring and as your legal children, i.e. including adopted children. If I said, "Billy is Squatch's child", would it be clear whether I meant that Billy is your genetic offspring, or whether he was "merely" one of your legal children? If we were talking about family law, and genetic offspring are treated differently than legal children, wouldn't it make sense for us to nail down a definition of "children" to be used for the purposes of our discussion? Suppose we're interested in figuring out what the situation is with your genetic offspring. We might define "child" (when not preceded by "legal") to mean genetic offspring. No, I was referring to (ii). You are assuming it is possible to get The value of xj for some j But in every example given so far, that value was determined by appealing to some external order that you could use to deduce xj. My point was that no such external order exists here. This makes almost no sense to me. I'll try to clear things up. (i) and (ii) refer to what information is supplied to the person doing the deducing (the "Deducer"). In my example, the Deducer was given (i) and (ii). Specifically, the Deducer was told that xn+1 = xn + 2, which is "A function which when given as input an element xi S produces the next element xi+1 S", i.e. (i). Additionally, the Deducer was told the value of x0 in the first scenario, and the value of xk in the second scenario. I'm not sure what the "external order" is that you're talking about. Is the function in (i) the external order? Or is the provision of information to the Deducer the external order? How exactly would you determine a causet at any given point without using the causally prior causets? (1) I'm not sure what it means to "determine at causet at [] a given point". Can you define that? (2) If I hazard a guess at what you mean, then it depends on your interpretation of the causet model. If causets are universes, then each "birth" of a new, child causet is the production of a new universe. If causets are 'stages' of a universe, then each "birth" of a new, child causet is the next (wherever the concept of 'next' makes sense) 'stage' of the universe. If we have a causet model, then we know the dynamics; we know how each child gets produced by a parent. So, how do we know anything about any of the causets? Presumably we rarely know all the answers; in general, then, there will be a range of models that are consistent with our data. For a given 'child' causet, one and only one of the following must be true: (1) There is exactly one choice of parent that results in a model consistent with our data (2) There is more than one choice of parent that results in a model consistent with our data For a simplified example, if we're using R/S dynamics, suppose our data leads us to think that our current universe is modeled by the following causet: a < b < c < d Based on how R/S dynamics require that children are born, we know that the only possible parent of this set is a < b < c My problem with your analysis is that you assume both R/S and Axioms do not have a specific partial order in mind in their definitions. However, as I pointed out in this post earlier and in multiple quotes in earlier posts, these causets are part of the Causal Set Hypothesis and therefore the nature of the partial order is restricted to causal partial orders. A set of space-time elements whose relation < was meant to imply an order consisting of my preference would not be a causet in any of these papers. Do you have an argument, here? You're saying that R/S has a specific partial order in mind when they say: A causal set (or “causet”) is a locally ﬁnite, partially ordered set (or “poset”). First, what specific partial order do they have in mind? Doesn't it depend on the set? There's one and only one partial order? If there are a range of specific partial orders that are allowable, which partial orders are allowable and which are not? Second, where is it indicated that R/S have additional constrains on what partial orders a causal set can have, beyond local finiteness? Third, how do you know that the poset corresponding to your preference can't be interpreted as a causal order? The posets ({ice cream, pizza, cheeseburgers}, ice cream pizza cheeseburgers) and ({x,y,z}, x < y < z) are isomorphic; the fact that ≺ was defined by my preference and < was defined by my intuition regarding the causal connection of events x,y,z is irrelevant. How you label the events is irrelevant--whether you choose "x" or "ice cream" doesn't matter. What symbol you use for the relation--≺ or <--is irrelevant. Fourth, what are causal partial orders? Is every partial order a causal partial order? If not, then there's at least one partial order that is not a causal partial order; can you give an example along with a proof that it is not a causal partial order? As for where I got this idea, you agree with me that Causets are a segment of the Causal Set Hypothesis. (1) Do you mean the causal metric hypothesis? (2) I don't know how I could agree that causets are a segment of the Causal Set (or metric) Hypothesis, since I don't know what a "segment of the Causal Set (or metric) Hypothesis" is. What is a segment of a hypothesis? Do you mean that the string "causets" is a substring of the string referred to as the Causal Set (or metric) Hypothesis? As such I would point out the definition of the latter in the Axioms paper (which would disagree with your conclusion concerning (ℤ,<) as you defined it): "Causal set theory is a promising attempt to model fundamental spacetime structure in a discrete order-theoretic context via sets equipped with special binary relations, called causal sets. The elements of a causal set are taken to represent spacetime events, while its binary relation is taken to encode causal relations between pairs of events." Your definition of the partial order on ℤ doesn't match the partial order defined here as causal relations between pairs of events. You have to give arguments, Squatch. What's the proof that "my definition of the partial order on ℤ" (which I take to mean "the standard order relation on ℤ") fails to "encode causal relations between pairs of events"? If you wish to suggest that the Causal Set Hypothesis includes partial orders that are not causal in nature, I think we would need to see a reason why. What makes a partial order "causal in nature"? As far as I can tell, CMH says that the binary relation is taken to encode causal relations, i.e. when you interpret a model of these axioms, the binary relations are interpreted as causal relations between pairs of events. How does this exclude any partial orders? And if it does, which partial orders does it exclude? Not at all. I'm highlighting two pieces of known data. 1) There is a green item. 2) There is a known mechanism for creating a green item. Your objection is "there could be another mechanism for creating a green item." Which is fine, but as a stand alone appeal it is a kind of appeal to ignorance fallacy (there is an additional mechanism because we don't know there isn't). If you wish to propose another mechanism then a reason needs to be offered to justify it. It's not a fallacy to say "We don't know whether there is or isn't another mechanism for creating a green item". That's part of the specification of the problem. Calling this an "appeal to ignorance" fallacy is ludicrous; I'm not saying there is such a mechanism because we don't know if there is one. I'm saying we don't know if there is one. Denoting the machine as M, your argument goes like this: (1) It is unreasonable to assume the existence of a rule for M without evidence of that rule existing. (2) Every item that is produced by M is produced according to a rule for M. (3) M does not change the color of green objects. (4) Suppose G is a green object produced by M. (5) If it were supposed that G was a non-green object before being output by M, one would assume the existence of a rule for M without evidence of that rule existing. (6) Therefore, it is unreasonable to suppose that G was a non-green object before being output by M. (7) Therefore, it is reasonable to suppose that G was a green object before being output by M. Can you explain the reasoning on (5) and the deduction from (6) to (7)? The deduction (D) If x is a green output of M, then x was a green input for M is logically equivalent to (D') If x is not a green input for M, then x is not a green output of M by modus tollens. Since D is logically equivalent to D', inferring D is inferring D'. I present the following argument: (1) It is unreasonable to assume the existence of a rule for M without evidence of that rule existing. (2) Every item that is produced by M is produced according to a rule for M. (3) M does not change the color of green objects. (4) Suppose G is a green object produced by M. (5') If it were supposed that G was a green object before being output by M, one would assume the existence of a rule for M without evidence of that rule existing. (6) Therefore, it is unreasonable to suppose that G was a green object before being output by M. My argument for (5') is as follows: (i) (5) is true because the supposition that G was non-green as input assumes the existence of a rule of the form "M changes the color of some non-green objects to green". [premise] (ii) In general, if one supposes that an X-object was input and a Y-object was output, one assumes the existence of a rule of the form "M changes some X-objects to Y-objects". [premise] (iii) (5') is true because the supposition that G was green as input assumes the existence of a rule of the form "M changes all non-green objects to non-green objects." [from (ii) and the equivalence of D and D'] I think a better analogy would be finding a dinosaur bone (1) and knowing about fossilization (2) and then saying "well we don't know that it wasn't created via another process like aliens." Maybe it was put there by aliens, but then we need to justify why that is the mechanism rather than fossilization. We can't just discount fossilization on the possibility. Well, I think a better analogy would be finding a book and knowing about your own journaling and then saying, "I must have written this book myself as part of my journaling!" We can come up with restatements of our conclusions all day, in as friendly-to-our-position analogies as we like. Not exactly, I'm saying that the requirements placed on the causets by physical law prevent there being a gregarious child (if the parent is considered to be the causet representing all elements in the universe). This is what I believe is the concern being discussed on page 24 of the axioms paper as the possible emergence of infinitely large universes over finite time periods, or more precisely an imbalance between a large spatial volume to a small temporal one. Do you have any evidence for this? I found zero discussion of anything like the physical impossibility or implausibility of gregarious children on page 24 of the Axioms paper. The discussion at the bottom of page 24 relates to volumes, but the discussion is about what cardinalities of causal sets to include, not whether there can be gregarious children. On a more abstract note, I would be hesitant to suggest that any of the authors here are suggesting at a gregarious child grouping is implying than an element arose causelessly. This seems to violate the causal relation interpretation of causets. The new maximal element in a gregarious child, x, is such that there is no event y such that y ≺ x. Since all subsequent children of the gregarious child can only add maximal elements, there will never be an element z such that z ≺ x. This means that there is no event that caused x, was causally prior to x, etc. [My understanding is that those terms are all equivalent; if you think "x causes y", "x is causally prior to y", and the interpretation of "x ≺ y" have different meanings, can you clarify the distinctions?] So why discuss gregarious children at all? My guess is that (as described in section 4.5) the various different possible causets and the status of elements in relation to those causets is what gives rise to the emergent order that makes this theory explanatory. IE the different possible groupings allow for a more dynamic outcome than would be otherwise possible. Because they're assuming qn != 0, obviously. None of their analysis makes sense otherwise. Part of their analysis is uncovering a deep connection between non-timid children and the gregarious child: "Consider a parent and its children. Every such child, except for the timid child, participates in a Bell causality equation with the gregarious child." (R/S, p. 13) "Given the qn, the remaining transition probabilities (for the non-gregarious children) are determined by Bell causality and the sum rule, as we have seen." (R/S, p.14) All of the tn's in section 4.5 require that every qk != 0, i.e. the probability of a gregarious child is non-zero at every step. This follows from the definition of tn (10) on page 18 of R/S: Hmm, while it would be tempting to agree with that possibility as a debate tactic (since it shifts the argument towards finite universes somewhat), I think that the converse is true. I'm not sure how a causet with an infinite number of causal steps could be finitely old. I'm happy to discuss it if you wish for the fun of discussing it. Could you elaborate on your misgivings of the converse? It depends on how you're measuring age. If you're talking about the age of a universe, then you've got spacetime, and lines in spacetime connecting events--spacelike lengths, timelike lengths, etc., depending on the value the metric outputs for the given events. So you can define the age of the universe as, say, the least upper bound of the set of timelike distances between events in the universe. The question is whether you could embed an infinite causal chain in a finitely-old spacetime. Since I'm not quite up on my manifold theory, a super oversimplified version of this might go: Define our causet (C,<) as: ... < -2 < -1 < 0 I.e., the set of negative integers, ordered by the standard ordering on ℤ. You can embed this causet in the real interval [0,1] with the function f: C [0,1] defined by f(c) = 2^(c) = e^[c*ln(2)] This function preserves the ordering of (C,<): Let c1 < c2. c1ln(2) < c2ln(2) [since 2 > 1 and ln(x) is strictly increasing, ln(2) > ln(1), and ln(1) = 0, so ln(2) > 0] e^c1ln(2) < c2ln(2) [e^x is strictly increasing] 2^c1 < 2^c2 [a^b := e^[b*ln a] for a > 0] f(c1) < f(c2) [definition of f] QED. So if you think of [0,1] as the universe, and the distance metric as the standard Euclidean distance (here, d(x,y) = sqrt[(x-y)^2] = |x-y|), then [0,1] has a finite age, since sup{d(x,y) : x,y [0,1]} = 1. So we have an embedding of (C,<), which has an infinite chain, in a finite "universe" [0,1]. Yes, but we are artificially limiting the range of possible spatial coordinates in consideration to a single one. So if X x T = {(x,t) : x ∈ X, t ∈ T} is your set of spacetime coordinates, you're saying that it isn't that X = {x}, i.e. there's only one spatial coordinate, but rather that you're just talking about events that share the same spatial coordinate. Is that right? I did in post 390, but to restate: {... (x,1) < (x,2) < (x,3) < (x,4)} All you said in post 390 about the causal order was this: {(1,0) < (1,1) < (1,2) < (1,3)} That only tells me how these four elements relate to each other. If these are the only elements in (x,T) = {(x,t) : t T}, then the order properties are pretty obvious. If there are more elements than just (x,0), (x,1), (x,2), and (x,3) in (x,T), then the order properties might be different. This doesn't really tell me anything, other than 1,2,3,4 ∈ T. Is π T? If we're going to talk about events of the form (x,t), then I need to know what T is. So hypothetically, how could you generate a "bunch of maximal points, all causally remote from one another?" Well, every maximal point is causally remote from other maximal points. Suppose that x and y are maximal; if y ≺ x, then y isn't maximal, and if x ≺ y, then x isn't maximal. So y and x are causally remote. Since I don't know what the elements in T are, or how the elements in T are ordered, I can't really say anything about the order properties of (x,T). As I said before: To my understanding, the "current" causet is: C := {(x,t) : t T, t tn} ≺ := (x,a), (x,b) C: (x,a) ≺ (x,b) a < b If this is how you're defining ≺ on (x,T), then the order properties of (x,T) are just the order properties of T. As an example, if 3.5 T [you haven't told me that it is or isn't], then perhaps (x,3.5) is a maximal element [you haven't told me that (x,3.5) is in (x,T), let alone how it relates to other elements in (x,T)]. In the example 4 was the largest value of t having been born. You'll have to elaborate on what you mean by points in T that aren't comparable. Okay, so it seems like you've got some kind of order on T that you can use to compare elements of T to one another (e.g., "largest value of t"). So you have (T,$), where $is some, say, partial order. You haven't stipulated that T is totally ordered by$, i.e. for all ti, tj T, either ti $tj or tj$ ti.

If T isn't totally ordered, then there are incomparable elements, i.e. some ti, tj such that neither ti $tj nor tj$ ti.

If your causal ordering on (x,T) is defined by the ordering on T, i.e.

(x,ti) ≺ (x,tj) ti $tj [This may or may not be the causal ordering you're using on (x,T); you haven't specified one yet] Then: [When referring to a comparison in T,$ is the partial order being used; when referring to a comparison in (x,T), ≺ is the partial order being used.]

(1) ti, tj are incomparable in T (x,ti) and (x,tj) are incomparable (i.e., causally remote) in (x,T)

(1') ti, tj are comparable in T (x,ti) and (x,tj) are comparable (i.e., causally related) in (x,T)

(2) t is maximal in T ⇔ (x,t) is maximal in (x,T)

(3) S is a chain in T ⇔ (x,S) is a chain in (x,T)

5 is closer, however I would rephrase ii to say:

(ii) For all chains Ts in C, there exists a maximal element in Ts, s, such that s ∈ S

IE it isn't that there exists a chain with a maximal element corresponding to our anti-chain, it is that the anti chain contains the maximal elements from all of our chains.
A few potential issues with your definition:

(1) Your definition requires only that S not exclude elements that are maximal WRT some chain in C. Your definition does not require that S include only those elements. For example, if S is a set satisfying (ii), and x is an element that is not maximal WRT to any chain in C, then S U {x} also satisfies (ii).

(2) Your definition assumes that every chain in C has a maximal element. How do you know that?

This is only true if you assume C contains all elements within the poset. If C is a subset of all points in the universe this does not become a problem because the maximal element can be related to an element not in C.
I'm not clear on what you're saying here. The posets I'm talking about are (Cg,g) and (C,). C contains all the points in C, and Cg contains all the points in Cg (as well as all the points in C).

It seems like you're saying that there's some universal poset, say (U,≺), and that your requirement is only that no elements in U be minimax (i.e., both minimal and maximal).

If I've characterized your position accurately, then I have a few questions:

(1) How do you know that this requirement holds, i.e. that (U,≺) has no minimax elements?
(2) Does your concept of a universal poset involve causet dynamics? If (U,≺) is a child (in the R/S sense), then you're requiring that the probability that its parent have a gregarious child is 0, which forces the probability of a non-gregarious transition to also be 0; if (U,≺) is a parent, then I assume that its child must also contain no minimax elements, which again forces the probability of a gregarious child to be 0.

Not necessarily.

Remember the conditions this definition was introduced to. We are at some definite point, tn. Given that, for all values x that exist there is a y that fits that definition.
This seems like gibberish. Do you mean that "for all values x, there exists a y such that y = tn"?

Imagine our sample universe is points 1,2,3,4, with 4 being "now."

All of those points fit the definition offered, and point 1 is the "beginning."
I assume you intend your causal ordering to be 1 < 2 < 3 < 4.

Right, this is an example of a causet with no minimax elements. This causet does not represent a universe that consists of a single, first event; it represents a universe that consists of four events, with the given causal ordering.

The child of this causet will be of the form 1 < 2 < 3 < 4; x. The relations among {1,2,3,4} and {x} depend on (or determine, depending on how you look at it) which child is 'born'.

It is discussed on the Foundations of Physics page: "The former letters publication Foundations of Physics Letters has been merged with Foundations of Physics. Short papers that demand a rapid publication process will be included in the special section 'Letters to the Editor'. Authors who wish to raise discussion points or other articles of a different format are welcome to submit these as well, if they care to explain their wishes with the submission." http://link.springer.com/journal/10701
I'm sorry, I wasn't specific enough.

What is the evidence for this statement:

Foundations of Physics didn't publish those theories, Foundations of Physic letters, a non-peer reviewed document meant as the equivalent to an editorial section[, did].

The bracketed language is mine, and it captures the meaning that I assume you intended; your actual language was a sentence fragment. If this meaning differs from your intended meaning, please issue either a correction or a restatement.

19. ## Re: WLC's Argument Against an Actual Infinity

Originally Posted by GP
1.) Well, let's be clear here about the first point, which is that Clive was listing top cited philosophers of science. The people on the editorial board are not well-known philosophers of science, although there was some cross-over (such as Tim Maudlin, who by the way was on of the people Clive included as having scientific training). In fact, most of them are scientists or mathematicians (10 out of 14 and 2 out of 14, respectively), which would explain their absence from his list.
But the failure to check means the rejection was unjustified and done out of bias and not proper research.
Further, I don't see how it helps to point out that the ones reviewing the papers are trained in science, as evidence that science philosophers don't know what they are talking about.
Or are you now trying to make the point that because the reviewers are not trained in science philosophy then they are not qualified to judge the philosophy papers.

I think however what you are supporting is that there is more overlap then you give credit.

Originally Posted by GP
2.) The first error of yours is that the editorial board often do not, if ever, review individual papers that are published in the journal. In general, they will never see any of them unless they choose to review after publication. Their purpose is largely to tell the associate editors what kind of papers they should accept for publication to the journal, what kinds of topics the journal should focus on, and so on.
Originally Posted by NATURE
Editorial Board
The Editorial Board — composed of practising scientists in all relevant fields — manages the peer review process, and takes final decisions on whether papers should be accepted.
http://www.nature.com/srep/eap-ebm/index.html#eb

... should I go with Wiki? Or with the lauded "Nature"?

Originally Posted by GP
3.) Who actually handles the peer-review of individual papers? This is the Associate Editors, they are the ones who handle the peer-review process of papers:

"[T]he [associate editor]’s main task is to make editorial recommendations to the [Chief Editor] about what decision should be made on submitted papers.

To accomplish this, the [associate editor] has a seemingly simple set of responsibilities: to obtain referee reports for each paper they are assigned, and use these to make their recommendation for the paper, in a timely fashion."

In other words, you ought to be looking at the list of Associate Editors for Foundations of Physics. They're the ones whose job is actually to review papers for their content, to find referees, and to submit their recommendation to the Chief Editor.
But not if it were nature

Originally Posted by NATURE
The Editorial Advisory Panel of Scientific Reports works with the publishing team to recruit the Editorial Board. The Editorial Advisory Panel will provide editorial advice to the journal as and when needed, and comprises experts from all major fields within the clinical, biological, chemical, physical and earth sciences to ensure representation across all fields. The breadth and depth of their collective expertise will ensure Scientific Reports reacts to the varying needs of these research communities.
http://www.nature.com/srep/eap-ebm/index.html

So I would say that your accusation that I have made an error here needs better support than what you have offered.
Because I have demonstrated that not all follow the format you offered and that I have given a specific example where it doesn't.. my position is better supported.

-----
Before we go further, your notion that we should look at the short list of associate editors is challenged.
If you are going to claim THEY are doing the editing four the specific journal at hand you are going to have to support that.
I have here shown another journal (nature) that does not follow the formate described in wiki.

Further from your second link (which appears to be an education text book on the subject)
to the work in hand.
My objective in this phase of the process is to identify
a set of researchers to contact and ask them to provide
a review of the submission. As such, my approach is
quite different to when I am reviewing a paper myself.
As an AE, I do not ?nd it necessary to comprehend every
last detail of the paper, or even to grasp all of the ideas
presented. Rather, my goal is to ?nd experts who can
understand the paper in detail, and provide commentary
on its signi?cance and novelty
In other words, the AE is not reviewing the paper alone, and that is damaging to your critique. Because your dismissal relies on the assumed ignorance and unqualified nature of the reviewer, that
the AE's job is to go out and find the qualified people to submit and opinion is damaging to that case.

So that is two fronts on which your objection fails.

Originally Posted by GP
4.) The Associate Editors are:

a.) Paul Busch (Mathematical Physicist, PhD in Physics; predominant work is on mathematical quantum information theory)
b.) Dennis Dieks (Philosopher of Science, PhD in Philosophy; predominant work is on interpretations of quantum mechanics)
c.) Erik Verlinde (Physicist, PhD in Physics; predominant work on string theory)
d.) Brigitte Falkenburg (Philosopher of Science, PhD in Philosophy; predominant work has been in "Kantian cosmology")

In other words, only 50% of the people who review physics papers are physics experts. More importantly, only one of them is qualified to understand the error made by Shanahan, which is Verlinde. That means 75% of the authors have no training in the physics that Shanahan was getting involved in. (However, it's unclear if the division of labor is even 25% each associate editor. It's possible that Falkenberg and Dieks are the primary associate editors and the other two only occasionally are editors for papers. It would explain a lot.)
see above.
In order to come to the conclusion you have you must first ignore the standard roll of an AE per your own source.
Your objection to them must be that they are too ignorant of the field to even find proper reviewers.

Basically as long as a person knows who to go speak to and is thus a "qualfied" AE, they don't need training in the field in question at all.
This heavily undercuts your criticism based on the qualifications (or lack there of) of the AE's.

Originally Posted by GP
5.) You are right, on paper they should be doing a lot better with the two scientific editors that they have. However, as to how effectual they are at catching these kinds of errors, I need only remind you of the so-called "Einstein-Cartan-Evans theory", which was proven to be crackpot. If you notice, it's not like one of his papers made it through the peer-review process. He was published for years in Foundations of Physics, with over fifteen papers. You think one of their associate editors might have caught it after fifteen papers over the course of nearly four years, wouldn't you? No, I think it is unreasonable to think that they actually engage in scientific peer-review. They are doing exactly what I said they were; in fact, we'll discover soon what the Chief Editors opinions were on this subject.
First of all, I don't think I understand you to say that the paper goes through NO peer review at all. Your challenge is specifically to the scientific peer review.
is that correct?

If so, then you must at least acknowledge my point that the fields overlap significantly so as to make a total dismissal as a source unwarranted.

Originally Posted by GP
The fact that philosophers don't understand these issues isn't surprising.
So.. let me get this strait.
Your saying that unles you have a PHD exclusily in QFT, then you can't comment on it?

Interesting.. Does nature have any QFT PHD's in its advisory panel, has it published any papers on that matter? I couldn't find any, but it's difficult to cover them all.

but they don't do that job anyway

Originally Posted by GP
Notice how he says that ECE theory has been published in Foundations of Physics and then follows it with the statement that ECE has never been published in a peer-review scientific journal? No, 't Hooft (i.e. the Chief Editor of Foundations of Physics) states that it does not conduct scientific peer-review. I don't know who better to say that then he.
O.k. I'm going to say this one more time, and I'll even say it polity and respectfully because I think you will hear it that way.
You are ignoring my point in totality with this response.
You continue to address my point through the goggles of your own point and you are failing to fundamentally grasp the very basics of my point and subsequent argument.
This is a mistake on your part that is preventing you from addressing my point head on and without morphing it or forcing it into the conversation you are fixed in on.

But simply.
I am not defending the source, I'm attacking your counter of it as fallacious and/or flawed (as I have argued).

Originally Posted by GP
8.) Does the fact that they have 50% science trained, associate editors mean that they are a "scientific peer-review journal",
It does support my ACTUAL assertion though. Not that this publication would be classified as a "scientific peer review journal" but that
the field is so closely related to science that it is impossible to be completly ignorant of the science and do valid work in their field.

While you look at 50% "science trained" as basically irrelevant to the scientific claims being made. When it is more reasonable to assume that they are their to make sure no BS gets through.

Originally Posted by GP
You think one of their associate editors might have caught it after fifteen papers over the course of nearly four years, wouldn't you?
More correctly, all the people reading it who have responded in the past, all the people the AE(or board) contacted for referee's.
If it is a reputable journal at all, then your criticism is not just on them but the whole field for not speaking up and pointing it out. The journal is hardly on it's own given the process.

Originally Posted by GP
So, it appears that their track record with Quantum Field Theory is quite poor. And I'm not surprised; to quote Edward Witten, the greatest living field theorist (seconded only possibly by Stephen Weinberg):

"People tend to think that the most difficult thing that physicists grapple with is Einstein's work, but actually that's not true. Since quantum theory --especially since Dirac on relativistic quantum theory-- the most difficult theory in modern physics, by far, is Quantum Field Theory. It's an extremely difficult subject. Well, it was the work of most of the twentieth century to understand it better; we're still [trying to understand] it better. But you could say that the modern understanding took half a century, from Dirac to what we call asymptotic freedom."
By "People" is he referring to those trained in scientific philosophy or just laypeople?
Because that is a very big jump to equate the two... and just assumes your point and doesn't support it.
So I don't think it is as good of support or the "nail in the coffin" that you seem to think it is.

Originally Posted by GP
The fact that philosophers don't understand these issues isn't surprising.
GP you are going to have to do better than that. You have already said somewhere that Physisist have a hard time with it.
That doesn't show that they have a "harder" time or are "less reliable" than others.

All your doing is re-stating what I am challenging. Sure it is a hard topic.. So.. show that they are incompetent as a whole.

Originally Posted by GP
6.) The impact factor of Foundations of Physics is 1.170. Well, I'll quote Chad on this:

"The best indication you can go off of is a journals "impact factor" which is a ratio of the number of citations to number of publications. The higher the impact factor, the more citations a journal receives per paper...which typically indicates the quality and importance of the research. At the very top are journals like Nature, Science, and Cell. These have impact factors in the 20s-30s. Meaning that about every paper published gets on average that many citations. Really good journals specific to fields (in Biology at least) are typically in the range of 7-10. Good solid journals are often in the range 3-7. Lower end journals, which still typically have good peer review and are reputable are in the 1-3 range. Once you get below 1 or even the lower ranges of 1, that means there are almost no citations for the papers published and I would very much question the journal itself."
So.. 1.1 is iether good and reputable, or questionable

Show which one it is in.

Originally Posted by GP
Notice how he says that ECE theory has been published in Foundations of Physics and then follows it with the statement that ECE has never been published in a peer-review scientific journal? No, 't Hooft (i.e. the Chief Editor of Foundations of Physics) states that it does not conduct scientific peer-review. I don't know who better to say that then he.
He doesn't seem to be talking of the journal itself, rather what he is basing his decision on.
He is hardly saying "My journal doesn't do scientific review.
And what would you call researching a paper to see if it is spoken of in any other journals or any other scientific journals?
Isn't that some level of review? I mean, to see what peers are saying about it?

You are literally taking an example where he does his homework on a scientific claim.. and saying it wasn't reviewed for it's scientific claims.

Originally Posted by GP
I think that speaks for itself, as well.
Yea, especially if one is trying to establish a discussion in the scientific community.

===================================

Originally Posted by CS
The fact is that many philosophers of science don't have phds in a scientific field, and yet they still do work in philosophy of science. Big names in philosophy of science too, like David Hume and Daniel Dennett,
Doesn't answer my question.. thanks for playing.

Originally Posted by CS
Someone without a phd in physics should probably not be doing peer-review on physics claims.
Which assumes that it is the only field that covers the subject matter.
You could be saying something along the lines of "A nasa Enganeer shouldn't be reviewing math claims".
Where there is significant over lap (my cousin had taken every math course AT LSU. in persuit of his Engineer)

or,
A plumber really should't be doing peer review on quantum physics.
Where there really is no overlap.

Originally Posted by CS
I never acted or said that philosophy of science and science were unrelated. That's just a strawman.
Then you accept it(the journal) as a relevant source?

Originally Posted by CS
Well, you've said it's false. And you've said that you know about philosophy of science. But since you have no training or experience in philosophy, science, or philosophy of science, your opinion doesn't carry much weight.

And since you're clearly unfamiliar with the CVs of a pretty big swath of philosophers of science (those who lack phds in science fields), I'd say that your statements about the discipline are based on ignorance, not knowledge.
My argument was not based on my personal qualifications. You can address my argument, or take a hike.

Originally Posted by CS
Ah, I see the mistake you're making.

You can make a claim like "This is how you do philosophy of science." If you want people to believe it on your say-so, then your opinion has to mean something. You want people to believe your statements on your say-so; you aren't citing any textbooks on philosophy of science, you aren't citing the statements of any philosophers of science discussing how to do philosophy of science.

You're just some random guy giving his opinion about a field in which you have no training or experience.
Straw-man.
My argument is actually one that the entire field is discredited.
You are free of course to make that assertion by not rebutting my argument, or at this point even acknowledge it's existence.

Originally Posted by CS
Philosophers of science are probably better sources of knowledge for science than, say, someone untrained in science and untrained in philosophy of science.
Thanks for conceding my argument.
AGain, my point was that they are related, and related enough to be a valid source, especially in regards to support for a "discussion in science".

Originally Posted by CS
Rather, the field merely isn't inherently qualified to peer-review physics
Which means an argument must be offered for a positive dismissal of the evidence.
That argument wasn't offered and thus it was fallacious to dismiss them without cause.

Originally Posted by CS
Those weren't questions meant for you to answer, MT. Those were questions that illustrate what kinds of concerns philosophers of science are interested in.

Your answers to them are irrelevant to the point at hand (although if you think you've "answered" the central questions of philosophy of science in the sense of making any other answer untenable, then you're probably beyond hope).
Your perception of what is and is not relevant has been fundamentally flawed.. so please continue to ignore things you don't agree with.

Originally Posted by CS
Sure, but they only have to know things like "In general relativity, the speed of light is constant", or "In special relativity, there are frames of reference, and no absolute frame". Philosophers of science routinely work off of statements like these without understanding their technical, mathematical expressions or basis (see e.g. William Lane Craig).
Thanks for conceding my point, maybe we can make progress.

Originally Posted by CS
What makes you think that philosophers of science generally have that necessary understanding? After all, just because MindTrap thinks that understanding X is necessary to do philosophy of science doesn't mean that all or most philosophers of science understand X.
I have offered a logical argument, you can address it or it stands un-rebutted.

Originally Posted by CS
It discredits the field of philosophy of science to cite the work, CV, and methodology of giants in the field of philosophy of science? Your problem seems to be with philosophy of science, not with me.

I'll again note the irony in talking about the requirements and foundations of a field that you have no training or experience in. When Hume talked about the philosophical foundations of science, Hume was an expert in philosophy. You're not an expert in philosophy, science, or philosophy of science, and yet you think that if philosophers of science don't do it the way you want them to, their entire field is discredited.
I have offered a logical argument, you can address it or it stands un-rebutted.

Originally Posted by CS
Let me make sure I'm understanding your argument correctly.

When you talk about the "viability" of a theory, do you mean physical viability--i.e., the theory's predictions are consistent with all experimental data up to date, etc.? If so, then the kind of equivalence that you're talking about--model equivalence--is a mathematical concept. And mathematical proofs should be peer-reviewed by mathematicians with the appropriate specialization--e.g., model theory. Having a degree in philosophy of science does not endow you with a sufficient understanding of model theory to be able to peer-review statements in model theory.
Are you suggesting that something logically equivilant would have a different mathmatical expression (such as to have a different effect in math equations)?

So lets take it simple.
If I show that X is the logically equivilant to Y
then you are saying that it is possible for X=5 and Y=(4)

Originally Posted by CS
Irrelevant and goalpost-shifting, my point is about the qualifications of philosophers of science. Do you agree that a person doing peer-review for physics should have an understanding of physics at least at the phd level?
No, it's not goal post shifting, because it is the journal being rejected.

*Note, you may have made some other relvant point, but it was surrounded by so much BS that I didn't see it.
Please retract the fallacious appeals to authority and re-state the relevant actual rebuttal (if there even was one).

==================
Originally Posted by GP
So here's what you and MT don't understand very well. Both Clive and I have professional training in mathematics and mathematics & physics, respectively. This is a fact. But you guys don't seem to understand what that means, so let me be clear here:

We're not saying it makes us right about everything. This training does not give us the ability to know the absolute truth and make us beyond reproach. What it does do is give us the ability to know when something is wrong
I'm calling Bulls **** here.
Why? Because above CS is specifically committing the fallacy of appeal to authority. This born of what you are supposedly describing here.

You guys should be the ones to easily point out the problems of reasoning and SUPPORT those assertions. If however you are simply regurgitating things you were told in school that simply will not cut it against an ACTUAL argument.

My argument, for example, doesn't rely on my personal qualifications, and the qualifications of ANYONE else are very much insufficient as a rebuttal to my argument.
Yet, you guys seem more than willing to spew out the same condescending line over and over even though it has been shown to be LOGICALLY FALLACIOUS repeatedly.

Or do you stand by Clives argument? Or is it fallacious and you are simply willing to remain quiet on it?

20. ## Re: WLC's Argument Against an Actual Infinity

Originally Posted by CliveStaples
He's not asking you to refine an argument.
Not exactly. The thrust of his response is, "I can't be bothered to read all of your crackpot sources, you pick one that I can look at." IE I find the form of your argument inconvenient so rephrase it until I find it acceptable and then I'll deal with it.

Originally Posted by CS
Yes, this is essentially my point here
Right, but in both those cases because the world being modeled is hypothetical does not remove the symbolic meaning of the variables or relationships. t is still time in the bacteriological example, even if it is a non-existent species of bacteria. ≺ is still a causal relationship even if the causet discussed is a implied model of our universe (as all the examples used in these papers are).

Originally Posted by CS

Which paper do you think places this limitation on the definition of causets?
As I noted earlier. Causets are part of Causal Set Theory. Causal Set Theory is the theory that events in spacetime can be modeled by sets in which the elements are locally finite and related via a binary causal relationship. So when you argue that these are just mathematical objects and that you are not proposing a physical model you've left the realm of Causal Set Theory and moved into a different discussion. Essentially you are trying to have a different debate, a debate of any partially ordered set whose elements are locally finite, but that is a much broader topic than Causal Set Theory. Operating within that specific framework we cannot ignore the binary relationship and what it is defined as.

Originally Posted by CS
(2) What partial orders don't represent causal structure?

For any set S with any partial order <, isn't there a possible world where there are a set of events corresponding to S whose causal relationships correspond to
This seems to be an incomplete thought. Perhaps you had the same "no spaces around < " problem I've had a few times.

Originally Posted by CS
(1) Yes, events are inherent to spacetime; no, they are not inherent to the choice of coordinates.
I'm not sure where you got the second half of that sentence because I didn't imply anything to that effect. Rather, and you seem to have ceded this point, that the events are both spatial and temporal in nature. We can argue about which coordinate system or which frame of reference to use, it doesn't matter, as long as you hold the frame of reference constant, the temporal and spatial relationship between the points doesn't matter. All that matters is that there is a relationship, they exist within those dimensions and that dimensionality is inherent to their being events. Likewise, the Hasse Diagram relates a different relationship, their causal relationship. But nothing about being in the Hasse Diagram eliminates their spatial/temporal characteristics.

I agree that you cannot derive whether two points share a spatial, a temporal, or both relationship from the Hasse Diagram. That wasn't my point. My point was that you can derive the causal relationship from the Hasse Diagram and the spatial/temporal relationship from the earlier panels.

What you seem to be arguing is that point x is not point x when it is in the Hasse Diagram and so deductions about causality from the last panel cannot be held when reviewing the first two panels. If that is so we'll need a better explanation as to why.

Originally Posted by CS
Hmm, I'm not sure quite what you mean. The person defining the causet dictates it, I suppose.
And if that person decided to dictate a partial order based on say color, how would that conform to the Causal Set Hypothesis?

Originally Posted by CS
And your use of the analogy fits what I'm saying. The child (genetic or otherwise) is still the individual being discussed. Now that child can be a genetic child in relation to family law courts where I am his biological father, or he can be a legal child in relation to say his school where they allow the step-father to be a parental contact.

Originally Posted by CS
(i) and (ii) refer to what information is supplied to the person doing the deducing (the "Deducer").
To rephrase the question, where does the Deducer get this information? "x0"

Whatever that source of information is, how would that be inferred back to a causal situation? IE could one possibly be told the value of a causally future moment?

Originally Posted by CS
First, what specific partial order do they have in mind? Doesn't it depend on the set?
Why would a set that is supposed to model the growth of spacetime events through their causal relationships have a partial order other than the partial order defined under Causal Set Theory?

Originally Posted by CS
Third, how do you know that the poset corresponding to your preference can't be interpreted as a causal order?
Because the order complies with the definition of the word "Causal." What definition of that word are you using in this thread?

You argue that because the order between the set examples given is isomorphic that the relationship is irrelevant which clearly misses the point of modeling something's dynamics. A ball that falls because of gravity and a ball that falls because of magic could be isomorphic in their acceleration profiles, but that doesn't mean the dynamics of why they moved are irrelevant.

Originally Posted by CS
Fourth, what are causal partial orders?
We seem to have covered this ground on several different occasions.

From pg 85 of the axioms paper:
In the special case where M is a directed set, viewed as a model of classical spacetime structure, directed paths are causal paths, in the sense that an event represented by an element x in M influences an event represented by an element y in M if and only if there is a directed path from x to y in M. Such causal paths differ from causal curves in general relativity in two important ways. First, following the causal metric hypothesis (CMH), causal paths represent actual influence, not merely potential influence. In the language of section 1.3 above, causal paths are descriptive, rather than prescriptive.

Or, as quoted earlier in thread:

The elements of a causal set are taken to represent spacetime events, while its binary relation is taken to encode causal relations between pairs of events....
The binary relation < defines an interval finite partial order on C, called the causal order, with the physical interpretation that x < y in C if and only if the event represented by x exerts causal influence on the event represented by y.

Causal partial orders are a partial order on a set that defines the causal influence between elements of the set.

Originally Posted by CS
(2) I don't know how I could agree that causets are a segment of the Causal Set (or metric) Hypothesis, since I don't know what a "segment of the Causal Set (or metric) Hypothesis" is.
It is meant in this way. The idea of a causal set arises from the Causal Metric Hypothesis, correct? It is a concept within that theory (just as the velocity of money is a concept in monetary hypotheses) rather than simply an existing concept (like a set) incorporated to the theory.

Originally Posted by CS
What's the proof that "my definition of the partial order on ℤ" (which I take to mean "the standard order relation on ℤ") fails to "encode causal relations between pairs of events"?
Because ℤ is ordered on its numeric value, not on a expressed relationships based on causal influence.

Originally Posted by CS
As far as I can tell, CMH says that the binary relation is taken to encode causal relations, i.e. when you interpret a model of these axioms, the binary relations are interpreted as causal relations between pairs of events. How does this exclude any partial orders?
Right, the binary relation encodes (meaning to convert information into a symbolic format) a relationship of exerted causal influence into a symbol. If the relationship being encoded is not causal in nature it is not fulfilling the encoding discussed by the CMH.

Originally Posted by CS
It's not a fallacy to say "We don't know whether there is or isn't another mechanism for creating a green item". That's part of the specification of the problem. Calling this an "appeal to ignorance" fallacy is ludicrous; I'm not saying there is such a mechanism because we don't know if there is one. I'm saying we don't know if there is one.
You are arguing the conclusion, "X caused Y" is not warranted because there could be a Z that caused Y, but you offer no evidence or reason as to why Z might be supported. Additionally, Z would be a denial of the first premise and would therefore need to be a rebuttal rather than an appeal.

Originally Posted by CS
Denoting the machine as M, your argument goes like this:

(1) It is unreasonable to assume the existence of a rule for M without evidence of that rule existing.
(2) Every item that is produced by M is produced according to a rule for M.
(3) M does not change the color of green objects.
(4) Suppose G is a green object produced by M.
(5) If it were supposed that G was a non-green object before being output by M, one would assume the existence of a rule for M without evidence of that rule existing.
(6) Therefore, it is unreasonable to suppose that G was a non-green object before being output by M.
(7) Therefore, it is reasonable to suppose that G was a green object before being output by M.

Can you explain the reasoning on (5) and the deduction from (6) to (7)?
Hmm, I'm not sure what explanation of 5 you would like, it is self-evident. If I assume that M only acts according to its rules (2) and that a green item has been produced (4) that was not green when it was put in (5), then by logical necessity there must be a rule for converting things to green objects.

5 also seems internally problematic. Presumably whatever informed us that the object was non-green when input along with the evidence of the output would suffice as evidence that such a rule existed. So premise 5 in internally inconsistent.

Originally Posted by CS
Well, I think a better analogy would be finding a book and knowing about your own journaling and then saying, "I must have written this book myself as part of my journaling!"
The problem with that is we know of other mechanisms for book creation. My objection is that the appeal to an infinitely old universe is an automatic appeal to another form of temporal creation with demands an explanation.

Take your example here. If journaling was the only known source of book creation and you were to say, "well hold on, maybe the book was created by another mechanism?" Fine, then what mechanism would that be?

Originally Posted by CS
Do you have any evidence for this?
First, I have to point out that you didn't highlight the caveat there. We are assuming that we are talking about a gregarious child in relation to all the other elements of the universe. In which case I would point out that it would seem to arise acausally since gregarious children do not have a causal predecessor. That would seem to violate the conditions of causality assumed by this theory, as referenced in the papers:

Together, these axioms represent a straightforward adaptation of familiar notions of causality to the discrete order-theoretic context.
Axioms

The condition of “internal temporality” may be viewed as a very weak type of causality condition. The further causality condition we introduce now is quite strong, being similar to that from which one derives Bell’s inequalities
R/S

Given the conditions of causality assumed in this theory it would seem incoherent to allow for an event to arise acausally.

Originally Posted by CS
This seems to violate the causal relation interpretation of causets. The new maximal element in a gregarious child, x, is such that there is no event y such that y ≺ x. Since all subsequent children of the gregarious child can only add maximal elements, there will never be an element z such that z ≺ x.
It only arises "causlessly" in relation to that particular causet grouping. That certainly doesn't have to be the only causet grouping that element is included in right?

Originally Posted by CS
So we have an embedding of (C,<), which has an infinite chain, in a finite "universe" [0,1].
Aside from my earlier objection to the set of integers as a causet (if we assume each integer is an event here and the ordering is a causal one that objection can be set aside), I think this argument suffers a significant problem. The volume of each event (integer) would need to be zero in order for them to "fit" into the universe as described. However, a zero volume event violates the axiom of local finiteness as well as the measure axiom.

The crucial physical requirement is that local finiteness should coincide with causal local finiteness in the context of classical spacetime structure. Modeling classical spacetime as a directed set D, and applying the independence convention (IC), the definition of causal local finiteness translates to the condition that the relation set R(x) at each element x of D is finite.
Axioms

Measure Axiom: The volume of a spacetime region corresponding to a subset S of C is equal to the cardinality of S in fundamental units, up to Poisson-type fluctuations... Mixing finite and infinite local behavior is dubious even without taking the measure axiom (M) into account, but is particularly problematic when the measure axiom is assumed to hold.
Axioms

4. Local finiteness is compatible with Sorkin's version of the causal metric hypothesis. In particular, local finiteness allows a minimum finite spacetime volume to be assigned to each element in a directed set, without the danger of fatal local behavior. Indeed, since the set D−0 (x) 8 D+0 (x) of direct predecessors and successors of any element x in any locally finite multidirected set (M; h) is finite, this set has a finite volume for any measure mu: P(M) -> R+.
Axioms

Originally Posted by CS
So if X x T = {(x,t) : x ∈ X, t ∈ T} is your set of spacetime coordinates, you're saying that it isn't that X = {x}, i.e. there's only one spatial coordinate, but rather that you're just talking about events that share the same spatial coordinate. Is that right?
Pretty much, assume for the sake of simplicity that events can only arise along that spatial coordinate.

Originally Posted by CS
That only tells me how these four elements relate to each other...If we're going to talk about events of the form (x,t), then I need to know what T is.
If we are to continue to assume that these causal sets exist within a spacetime framework and thus the laws of causality apply, then the causal structure is pretty clear. Each new element (x,t) must be causally related to its predecessor (x,t-1) since no element arises acausally and no two elements can arise at the same t since they could not be discrete (given the restrictions on x).

T is the discrete measure of causal steps in this accretion process.

Originally Posted by CS
Well, every maximal point is causally remote from other maximal points.
But only one maximal point can exist per causal step right?

Originally Posted by CS
A few potential issues with your definition:

(1) Your definition requires only that S not exclude elements that are maximal WRT some chain in C. Your definition does not require that S include only those elements. For example, if S is a set satisfying (ii), and x is an element that is not maximal WRT to any chain in C, then S U {x} also satisfies (ii).

(2) Your definition assumes that every chain in C has a maximal element. How do you know that?
1) You are correct. I'm not sure an alteration to the definition to only include maximal elements really makes much of a difference in what we are talking about, neither does it appear to detract from the discussion. As long as the definition requires that all chains that exist have a maximal element, then it meets my needs.

2) That was part of the definition of there being a "now." Now is the temporal point of maximum accretion. Relativistic objections can be muted via an assumption of a constant observer for the moment, and even with relativistic objections, that kind of change wouldn't allow for the kind of concurrent infinite future that arises from a chain with no maximal element (ie a chain that is infinitely far into the future from now).

Originally Posted by CS
It seems like you're saying that there's some universal poset, say (U,≺), and that your requirement is only that no elements in U be minimax (i.e., both minimal and maximal).

If I've characterized your position accurately, then I have a few questions:

(1) How do you know that this requirement holds, i.e. that (U,≺) has no minimax elements?
(2) Does your concept of a universal poset involve causet dynamics? If (U,≺) is a child (in the R/S sense), then you're requiring that the probability that its parent have a gregarious child is 0, which forces the probability of a non-gregarious transition to also be 0; if (U,≺) is a parent, then I assume that its child must also contain no minimax elements, which again forces the probability of a gregarious child to be 0.

1) I would again appeal to the above defended requirement of causality to argue against no minimax elements in (U,<).

2) It does include causet dynamics, yes. I'm not sure why you hold that the probability of a non-gregarious transition becomes 0 in that case, could you elaborate?

As for your question, in Lemma 2 we see much of this discussion taking place. Let me take an initial stab at a reason why there would need to be a causally prior element somewhere to each gregarious child. On page 13 (if I am reading it correctly) we note that the transition probability of A to B (ax) is the same as C to B (bw). However, if C does not exist (ie there is causally prior causet for that element) then the bw=0 since there is no change of a transition from a non-existent causet. That would also mean that the probability of A to B is zero and hence doesn't exist.

Originally Posted by CS
Do you mean that "for all values x, there exists a y such that y = tn"?
No, to re-iterate, the point under contention was that for all elements (x) in the set there exists an element (y) such that x < y, or y > x.

We are at a point, now, such that there is a causally prior chain to the event (now). (IE we are not in a universe that consists of a single, first event).

Given that latter condition, there does in fact exist a point y that is either causally prior or causally after for each point x.

Again, take a simple example where the universe exists of only 4 events, (1 < 2 < 3 < 4). 4 is now. All points within that universe ( 1, 2, 3 and 4) have a Y that fits the definition given above, right?

But your objection is more to the Cosmological Argument here right? That if I don't allow for the creation of an uncaused element that I can't argue for a beginning right? Without re-hashing the entirety of the CA thread, is it possible to set that objection to the side for the purposes of this example? IE if we come to the conclusion that the universe is temporally finite and has a beginning element, then we can resurrect this objection and argue the question of how such an element would have arisen. I don't think this set aside is too problematic unless one of us is going to maintain that within the confines of our universe that events arise spontaneously.

Originally Posted by CS
What is the evidence for this statement:

[INDENT][COLOR=#333333]Foundations of Physics didn't publish those theories, Foundations of Physic letters, a non-peer reviewed document meant as the equivalent to an editorial section[, did].
I would submit the links from GP's post 386.

Link 1: "The hypothesis was largely published in the journal Foundations of Physics Letters between 2003 and 2005"

Link 2: All papers published by M.W. Evans are in Foundations of Physics Letters. The only papers in Foundations of Physics are An Assessment of Evans’ Unified Field Theory I in which Friedrich W. Hehl critiques Evans' math and On the Non-Lorentz-Invariance of M.W. Evans’ O(3)-Symmetry Law in which Gerhard W. Bruhn shows that Evans' work does not lead to the conclusions he claims.

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