Equivocation fallacy. "Probable" =/= "Probability" They are related words, but not of equivalent meaning.
having more evidence for than against...http://dictionary.reference.com/browse/probable
Statistics.http://dictionary.reference.com/browse/probability
a.
the relative possibility that an event will occur, as expressed by the ratio of the number of actual occurrences to the total number of possible occurrences.
b.
the relative frequency with which an event occurs or is likely to occur.
We can see this in a relatively common demonstration. When going to a civil court we are asked to measure the evidence and decide whether or not the event in question "probably" occurred. Clearly they aren't collecting a statistical sample representing a population of possibilities and measuring the ratio of occurrences. The word "probable" does not mean that (neither does probability always mean that either).
In short, you are failing to understand the definitions of the words offered here and how they are being used. You are are attempting to use a term in a manner that it was not implied as meaning in order to question the statement, an equivocation fallacy.
"Suffering lies not with inequality, but with dependence." -Voltaire"Fallacies do not cease to be fallacies because they become fashions.” -G.K. ChestertonAlso, if you think I've overlooked your post please shoot me a PM, I'm not intentionally ignoring you.
Okay, so there's no confusion, there's now two threads having the exact same discussion so I'm going to try to re-route traffic to a single thread, to keep things linear and clear. Here was a very important argument that I made on the other thread. I'll repeated it here, verbatim (other than cstamford's quote which included WLC (somewhat inaccurately) describing Vilenkin's position on just how far the BVG theorem applies).
(EDIT: I modified the conclusion slightly for clarity purposes; I'm going to take this as my official version of this argument and not on the other thread)
The content of this post addresses and rebuts the central claims that Squatch has been making on this thread. Here is the post:
Nope, Vilenkin is definitely a subscriber to the common lore. Vilenkin gave his paper a nice spin, and it's a clever idea, but it's far from a reliable, case-closed argument. Let's not rely on Craig's summary; let's read what Professor Vilenkin has to say to WLC (Note that I've bolded the salient points; I've inserted full names for clarity):
Vilenkin's Response Letter to WLC
"Dear Bill [i.e.William Lane Craig],
"I’m troubled that Lawrence Krauss in some respects misrepresented your views [on your work on the BVG theorem] in our dialogue in Sydney. In an attempt to rebut the evidence for a beginning of the universe, he showed a powerpoint of your letter with the last two sentences of the second paragraph deleted."
My letter was in response to Lawrence’s email asking whether or not I thought the BGV theorem *definitively* rules out a universe with no beginning. The gist of my answer was that there is no such thing as "definitive ruling out" in science. I would say the theorem makes a plausible case that there was a beginning. But there are always caveats."
Pause. A rhetorical point: Knowing my argument, do you care to guess what those caveats are?
I'm going to skip ahead to WLC's question and Vilenkin's response, but you can get the full text here. Here, WLC asks Vilenkin:
"I do have a question about your statement: 'the BGV theorem uses a classical picture of spacetime. In the regime where gravity becomes essentially quantum, we may not even know the right questions to ask.' Elsewhere you’ve written: 'A remarkable thing about this theorem is its sweeping generality. . . . We did not even assume that gravity is described by Einstein’s equations. So, if Einstein’s gravity requires some modification, our conclusion will still hold. The only assumption that we made was that the expansion rate of the universe never gets below some nonzero value' [Vilenkin, 2006, p. 175].
How are these statements compatible? The 2006 statement sounds as if a quantum theory of gravitation would not undo the theorem. But the letter to Krauss sounds as if we are awash in uncertainty.
I have my own idea of how you might understand these statements, but rather than burden you with my surmises, I’d prefer to simply ask you how you understand the situation."
The question of whether or not the universe had a beginning assumes a classical spacetime, in which the notions of time and causality can be defined. On very small time and length scales, quantum fluctuations in the structure of spacetime could be so large that these classical concepts become totally inapplicable. Then we do not really have a language to describe what is happening, because all our physics concepts are deeply rooted in the concepts of space and time. This is what I mean when I say that we do not even know what the right questions are.
But if the fluctuations are not so wild as to invalidate classical spacetime, the BGV theorem is immune to any possible modifications of Einstein's equations which may be caused by quantum effects.
Best regards,
Alex [i.e. Alexander Vilenkin]"
In other words: Assuming (without any reason specified) that quantum physics doesn't matter very much, then the BVG theorem does apply. Assuming then they are large, the BVG theorem doesn't apply. It's worth noting that WLC had actually noticed Vilenkin conceding my exact point to Lawrence Krauss and bothered to make an e-mail asking him to clarify; Vilenkin concedes, clarifies, and explains that the point about quantum effects --which I've been saying here since my initial post-- is correct. Next, I move to the following question:
Are Vilenkin's Assumptions Good Ones?
Vilenkin's paper is good; he's just making one assumption and following the logical conclusions from it. However, and he should really be more clear about this in his e-mail to WLC, there's no reason to suspect that the quantum fluctuations won't become incredibly strong and there's no reason to believe that quantum gravity will (and there's really only evidence to the contrary) preserve the notion of a classical spacetime. We only have three* (currently mediocre) successful ideas about quantum gravity:
1.) The noteworthy example of String theory.
2.) The less noteworthy example of "loop quantum gravity"
3.) The even less noteworthy idea of "causal dynamical triangulation".
Firstly, and most importantly, every single one of them dispenses with the normal idea of a classical spacetime and in each of them we also know that extremely strong quantum gravity effects are present. In other words, if we want to talk about "evidence" in terms of theories that can at least potentially work, then we have 0 for 3 models where the BVG theorem applies. This means that there's no good reason to discuss this theorem as though it's the most relevant piece of information. If you can show me any quantum gravity model which obeys the BVG theorem --and thus has a singularity-- then we could at least compare evidence. But you'd need 4 models before the evidence became 'stronger' that quantum theories of gravity seem to obey the BVG theorem. Because right now, we have no evidence they do and only evidence that it doesn't apply to quantum gravity. At this point, even if one of these models had a past singularity, it wouldn't be because the BVG theorem held; this continues to reenforce the fact that the BVG theorem is not the relevant fact to be discussing.
Secondly, Squatch is fond of saying "We don't have a model where there are quantum effects that lead to an infinite past", I can make an equalently strong statement: "Sure, but we also don't have a model of quantum gravity where we have a finite past, either", which is important to note because Squatch acts as though the default position is "The universe has a singularity", but this is not the correct position. The only correct position is that we cannot make reliable predictions about what the universe looked like when it was younger than when it was about 1.61 10^-35 meters in size. Before that, quantum gravity becomes the dominant force and nothing can be said; maybe extra dimensions come in and change the expansion properties predicted by BVG thereom, maybe quantum corrections cause a bounce in an analogous way to Donoghue's recently proven conjecture, or maybe the universe does collapse to a single point like WLC believes. We really have no idea, so much of no idea that the best we can do is count toy models. And that's hardly convincing evidence.
* There's also a fourth attempt which tries to use the "non-perturbative RG running" of GR to go a "UV fixed point", but it's pretty much just speculation in pure GR; however, even there, too, there's not really a coherent picture of a classical spacetime because the quantum effects completely overwhelm the classical picture, which is why it has to use the "non-perturbative" RG flow. This technically makes it 0 for 4 models, but it is my understanding that this isn't any where near as fleshed out as the other three, so I don't feel like it is worth including.
Last edited by GoldPhoenix; March 26th, 2014 at 08:56 AM. Reason: I edited the conclusion to be a little more readable.
"Those who can make you believe absurdities, can make you commit atrocities." --Voltaire
First, if we're talking about how to compute and compare probabilities, we should probably be looking to technical definitions of probability, not colloquial ones.
Second, you're confusing probability theory with statistics. Statistics is about the behavior of observable random variables; probability theory is about probability measures. The "relative frequency" interpretation of probability theory, often called frequentism, leads to different statistical methods (and interpretations of their results). Conversely, the Bayesian interpretation of probability theory leads to its own set of statistical methods (and interpretations of their results).
Third, if your argument is that the available evidence makes it more likely than not (I'd love a precise definition of what exactly you're using that phrase--or its equivalents--to mean) that the universe has a finite age (or, began to exist a finite length of time in the past), I'd really like to see that evidence. As GP pointed out, applying the BVG result requires certain assumptions with regard to the magnitude of quantum effects; if you are applying the BVG result, you'll need evidence to show (assuming we're relying on the more likely than not requirement) that those assumptions probably hold.
If I am capable of grasping God objectively, I do not believe, but precisely because I cannot do this I must believe. - Soren Kierkegaard
**** you, I won't do what you tell me
HOLY CRAP MY BLOG IS AWESOME
Forgive my ignorance but isn't anything minus itself 0?
On the other hand if I have a never ending number of apples and take away a never ending number of apples I would still have a never ending number of apples unless the number of apples I started with was not never ending.
... ow I just punched myself in the brain
Can anyone explain this to me (please keep in mind that I have virtually no mathematical education past grade 12 physics)?
abc
If I am capable of grasping God objectively, I do not believe, but precisely because I cannot do this I must believe. - Soren Kierkegaard
**** you, I won't do what you tell me
HOLY CRAP MY BLOG IS AWESOME
Or in other words infinity / 2 = infinity. That makes sense.
But shouldn't infinity / infinity = 1?
If I have an infinite number of apples and I divide them up into an infinite number of baskets doesn't each basket contain 1 apple and not an infinite number of apples?
abc
What if you put 3 apples in each basket? What if you numbered the apples, and put every apple of the form 2^k in the first basket, every apple of the form 3^k in the second basket, every apple of the form 5^k in the 3rd basket, etc.? And in general, if an apple is a power of the nth prime, put it in the nth basket.
Then each basket will contain an infinite number of apples, and you'll still have an infinite number of apples left over.
If I am capable of grasping God objectively, I do not believe, but precisely because I cannot do this I must believe. - Soren Kierkegaard
**** you, I won't do what you tell me
HOLY CRAP MY BLOG IS AWESOME
Except that I also did this, so it seems a bit much to argue you are doing it for me.
This kind of restriction only work when there are clear, definitive answers to the question being debated, that isn't the case here. We are talking about two, relatively high level competing theories both of which predict very similar results. Models developed on both those theories arise and fall. Additionally, we have a field, physics, where verificationist philosophy is quite common and as such theories that tend to predict the same things are treated as identical and so discussions on them is usually muted or absent. To reject the higher level concept because someone' attempt to model it failed is hardly a sufficient reason, but you make it sound like because a treatment failed we might as well reject the germ theory of disease. This is a valid alternative theory. You may not like it. It may not be popular. All of those are granted. But hypotheses are not accepted or rejected based on popular opinion.Originally Posted by GP
This also feels a bit like sour grapes to be honest. Trying to unilaterally rewrite the rules of debate in general and this forum in particular because you don’t want to review your opponent’s argument because you arbitrarily assume they didn’t is bad form and disrespectful. If you find the discussion frustrating I’m sorry for that. You are free to pursue the argument as you think is appropriate and I get to do the same within the scope of the rules of this forum. I’ve given you the principle of charity on this forum and in this thread and I would respectfully request the same.
Except I have done all of this GP, I've posted work from those working at research university's with PhDs in Physics, who have multiple peer-reviewed works and are publishing their work in peer reviewed journals. I've offered a non-technical summary of the argument on several occasions. I recognize that you don't accept the argument, but that doesn't mean it hasn't been made.Originally Posted by GP
Reviewing this response, you seem to misunderstand our objection. I'm not saying that being a PhD prevents one's theory from being irrational. I am the first to argue here that technical experts often act irrationally in their field all the time. I said specifically that simply labeling something as "crack pot" is an insufficient rebuttal. Saying that a theory is not the consensus is insufficient to supporting the assertion that it is wrong, especially in the manner you claimed these papers to be wrong.Originally Posted by GP
This is the kind of error one makes when they dismiss their opponent's claims with TLDR initially. If you had read any of my response you would have noted that Prof Székely is writing about the debate in the field, not arguing the underlying physics of the point. Remember, my response here was to show that there is actual debate.Originally Posted by GP
Allegedly. ;-) Just kidding GP, I read your paper and despite it being far beyond my professional training, it was interesting (minus the talk about ghosts ;-) ).Originally Posted by GP
I'm not sure I would call it a no-name university, it is one of the largest universities in Europe. As for his publication number, remember that research gate is only tracking his English language publications. A review of his other work, reveals quite a few more publications in non-english journals.
As for his education, I agree that his website lacks a CV and that is regrettable (I emailed him requesting one, we'll see if I hear back). We do know that he is a tenured professor at a large university and that all of his peers have PhDs in joint programs of philosophy and science (one is in computer science, one has both biology and physics, etc) and so for us to assume he is some untrained hack would require assuming that the university hired someone without credentials and that none of his peers or supervisors seems to have noticed.
Looks like a valid conclusion. I agree, this model for LR seems to be (more or less, obviously we can't rule out a non-zero mass photon, but we can come almost arbitrarily close) incorrect. Conceded, good response.Originally Posted by GP
Nor did I imply there was. I mentioned their membership review process and the moderation of forums process. Your discussion of both where the paper resides and the community of culture of arXiv seems somewhat incorrect and unsupported from arXiv's statements.Originally Posted by GP
Shanahan's paper was published in the Quantum Physics subcategory, is that a "universally recognized crackpot dumping ground?" If so, is Daniel S. Akerib a crackpot? Or Prof. Berezovsky a crackpot? (I think you see where I'm going with this). They both published in the exact same forum as Shanahan. Shanahan's paper resides solely in that sub-section and not at all in the History and Philosophy of Physics sub-section (searching for it there gives you no results). None of the subjects associated with the paper note history or philosophy and there are no moderator actions on it as well, so I'm not sure why you came to that conclusion.
I see under the general physics sub-section that there is a History and Philosophy of Physics sub-section, but nothing seems to indicate that it has looser standards, or is a dumping ground for crackpots. The submissions to this section are moderated in the same manner as any other section. The submission requirements do not appear to differ from any other section. And most importantly, the endorsement system in this section is the same as any other section and requires an endorsement from another user whose credentials have been verified.
I agree with you that none of this means the paper is necessarily correct. But it does make it a bit more serious than I think you are giving it credit for.
Perhaps you were right earlier, perhaps I didn't understand what you meant by crack pot. Both in this specific section and earlier it seems to be more closely correlated to your personal preconceptions than to any objective standard discernible to your readers. You'll forgive me if I don't interpret GP's personal opinion on the philosophy of science as necessarily the objective truth.Originally Posted by GP
This response seems to be a long, "no," in that you don't seem to offer any empirical reason to reject LR. Interestingly, you agree with this statement later.Originally Posted by GP
This is an incoherent statement. Your third sentence is a form of verficationism (a long dead system of thought that is a sub category of positivism) in which you assume that any model, regardless of its inner mechanics is identical if it produces the same results. This statement both says, I'm not making a positivist claim, I just make a form of positivist claim.Originally Posted by GP
Your dismissal aside I don't really have much of an objection here. It can obey Lorentz invariance in an apparent or a material manner, from your point of view it is irrelevant which. It isn't given the subject of this thread because it affects the premise I put forward.Originally Posted by GP
But you haven't offered a reason why it should be incompatible. What about a Lorentz invariance being apparent rather that material would rule out this outcome?Originally Posted by GP
"Suffering lies not with inequality, but with dependence." -Voltaire"Fallacies do not cease to be fallacies because they become fashions.” -G.K. ChestertonAlso, if you think I've overlooked your post please shoot me a PM, I'm not intentionally ignoring you.
1.) This debate is losing focus, so I'm going to go back to what I said in my last post: I would like you to lay out your exact picture of what LR is (i.e. exactly what it is, where it deviates from SR, etc). Then I want you to substantiate it. Present me a source that has passed peer review in the field of physics (After all, you've said the mathematician says there's a dispute so finding the peer-reviewed papers shouldn't be hard), or otherwise explain why you believe LR is true. I apologize if you feel as though you need to repeat yourself in some ways, but, again as per my previous post, I want something clear from you that I can either agree with or dispute. I want a clear claim that I can hold you to as to why LR should be considered a valid alternative to SR, and a clear argument with good supports for why it is a valid contender. Again, you're the one saying that there's a serious discussion. It shouldn't be hard, then, to find a discussion published in a physics journal.
2.) You've basically asserted that my analysis of your experts was done by you (even though you've conceded one case ended up directly working against your argument, one of them you can't provide any peer-reviewed articles for nor any evidence that they have any academic background at all, the other one you've admitted isn't an expert and basically was submitting heresay rather than you providing the actual papers that he was describing, and the fourth one we can't verify actually has submitted a peer-review paper in a physics journal and was just writing about philosophical interpretations of SR --which I've said that I have no problem with, as long as we're still talking about SR and not LR); so independent of who discovered them, they were not vetted as well as they should have been), and the rest of your post appears to be pontificating on points that don't help you. I'm sorry, but I'm not going to let you say that you have substantiated your claim when the best you have is a person whose not in the field saying that there's a dispute inside the field, rather than the actual peer-reviewed articles containing this alleged dispute. Also, I will concede that, although I could swear seeing it in the "General Physics" category, Shanahan's is in the "Quantum" category, but this doesn't make it go from a unreputable source to a reputable one, so while you have my concession on that issue, still nothing in this post is helping support your claims about LR being a viable alternative. I also apologize if my line of inquiry has offended you; I respect you, Squatch, but your sources were far from satisfactory, so I don't think that I'm violating debating etiquette by saying that you didn't vet them properly.
Last edited by GoldPhoenix; March 29th, 2014 at 11:40 AM.
"Those who can make you believe absurdities, can make you commit atrocities." --Voltaire
Believe me, I agree. This was a response to a single line of support for an assumption underlying an argument, we are definitely well down the rabbit hole here ;-).Originally Posted by gp
This request seems a bit inappropriate too. I have offered support for my view of LR. You don’t accept it, granted. You rebutted one paper, completely conceded. But to again ask me to support something that I have already offered support for is only to initiate a cycle that detracts from the thread. I offered LR as a support for an underlying assumption earlier (one of eight supports), you challenged it, I offered support, you find that support unconvincing. Understood. What will happen next? I will offer up another two or three or four peer reviewed papers and you will reject them as “in the wacko section of arXiv” or “I don’t like his layout,” heck you will probably even rebut one of them a la last post, but that doesn’t really get us anywhere. I’ve defended the position, you’ve defended your criticism of it. Some of my evidence still stands (imo) or maybe it doesn’t (in yours), neither response is decisive.
I think you misread my response. I was not implying I did the work for you, I was pointing out that your implication of vetting my sources for me was inappropriate since I had gone through a vetting process too before posting. The remaining implications of this response are likewise inaccurate. Your implication that an environmental scan of the state of the field is “hearsay” is disingenuous at best, you should know better that these papers are quite common in reflect a good source for analyzing the state of work in a field. The idea that Shanahan’s paper is unreputable because ___ (I haven’t seen a real reason beyond the crack pot association) despite showing that it was hardly published in an unmoderated forum with no review is also a bit of a stretch.Originally Posted by gp
Summary, I recognize that you don’t think my argument is up to snuff, got it, but to be honest I don’t see any reason to offer further support given the lack of substantive rebuttal (minus one excellent case) so far. I think post 290 stands relatively untouched as a support of the issue in thread that LR is at least an issue of discussion even if a small one. My take on the subject seems perfectly reasonable that for most physicists the distinction is non-existent given a positivist attitude and its relatively identical predictions thus far, for those that do find the underlying ontology more important, this is a discussion of merit, albeit one that is usually discussed by those outside of model generation (since there really isn’t much of a distinction in the models thus far) and therefore not one I think you are as interested in.
"Suffering lies not with inequality, but with dependence." -Voltaire"Fallacies do not cease to be fallacies because they become fashions.” -G.K. ChestertonAlso, if you think I've overlooked your post please shoot me a PM, I'm not intentionally ignoring you.
I'm moving a private discussion between Squatch and myself into the forums, since the pms were getting too lengthy.
---------- Post added at 10:49 AM ---------- Previous post was at 10:48 AM ----------
I don't know what process you're talking about. Can you give a definition?Originally Posted by Squatch347
I don't know what this means. An infinite union of finite sets can form "actually" infinite sets. A finite union of finite sets forms a finite set. I don't know what union of sets produces a "potential infinite". Can you show your work here? Using definitions, etc.?Given that, x can either be an actual infinite or a finite element.
If it is latter, the best that union can form is a potential infinite since a finite and a finite can only produce a finite, and if continued indefinitely, form a potential infinite rather than an actual one.
...okay, so what's the contradiction?I suspect however you will ask what happens if x is an actual infinite? That only moves the question back a step. Rather than asking, what process formed the fact that we have an infinite history today, we are asking what process formed the infinite history we had yesterday? And so on and so on.
No, you've mistaken your proof obligations. You are obligated to show that an actual infinite is not possible; that is, you have the burden of excluding the possibility of an actually infinite past. If someone were attempting to prove that an actually infinite past were possible, it would indeed be insufficient to simply assume an actual infinite past. The only way that kind of proof would work is in a consistency proof, where you'd show that the system that results from assuming an actually infinite past is either explicitly consistent (these kinds of proofs are rare) or is "as consistent" as some well-regarded system, e.g. arithmetic on real numbers.The only method where x U y (as defined) can produce an actual infinite is if we presume that x is an actual infinite with no explanation of the process that gave rise to it being so. This would seem on the face of it a begging the question fallacy, we are assuming the universe to be infinitely old in order to show that it is infinitely old. It is unappealing further because the mechanism described seems inadequate to elicit the effect without some further assumptions.
Your proof is similar to the following:Let x be an integer. I will show that x is not prime.
If x is not prime, we're done.
If x is prime, then we've merely assumed that x is prime, which is circular/question begging.
Therefore, x is not prime. So all integers are not prime.
What is an "infinite set of finite unions"? If there's infinite unions, an infinite set can result (take e.g. the union of all sets containing precisely one real number; the result of the infinite union is the set of all real numbers).It cannot be an infinite set of finite unions because that produces a potential infinite and we need an actual one, so we would need to adopt a further mechanism to allow for the rise of an actually infinite universe.
Okay. Can you give a definition? The quote you gave doesn't give a formal, precise definition.This is my first experience with them as well. One of my later links offered this:
Causal sets are discrete partially ordered sets, which are postulated to be a discrete substratum to continuum spacetime. The order gives rise to macroscopic causal order, while the discreteness or 'counting' gives rise to macroscopic spacetime volume. Given that causal structure is sufficient to reproduce the conformal metric, and discreteness provides the remaining volume information, it is reasonable to expect that causal sets alone possess sufficient structure to reproduce the entire continuum spacetime geometry.
Take the sequence x_{k} = -k, so {x_{n}} = {-1, -2, -3, ...}Perhaps I don't understand what is meant by a decreasing sequences here. What infinitely decreasing sequence is associated with this set?
Consider the following chain of implications:
n < n+1 (true for all natural numbers n)
-n > -(n+1) (multiplying both sides by -1 and flipping the inequality)
x_{n} > x_{n+1} (by definition of x_{k})
Thus {x_{n}} is a decreasing sequence.
First, what makes it not discrete? Can you give a definition of the terms you're using?Originally Posted by cs
Second, why does it have to be discrete? I assume this has something to do with requiring that the sets in question be causets, but I still don't have a definition for that term, so I don't see how you're reaching your conclusions.
This is particularly frustrating, Squatch. If you would make your claim precise, there would be no confusion on this issue; instead, it seems like you're flipping back and forth on what you mean by "infinite". When you say the past is "finite", do you mean finite by the cardinality measure? Or the "length"/Lebesgue measure? Both? Neither?To rephrase this, we can assume time is either:
Continuous: in which case cardinality loses meaning (as a measure of age because the duration approaches 0) and we must return to the "length" to analyze age. And a point traveling on a line for an infinite amount of "time" is a potential infinite, not an actual one. It approaches an infinite distance traveled, it never attains it.
Quantized: which suffers from the causal chain issue discussed.
You can do a case-by-case claim, e.g. the past must contain either a finite number of events, or the length of the set of events in the past is finite. I'd really like to focus on making your claims precise so that it's clear what support you need and how your conclusion is reached.
---------- Post added at 11:06 AM ---------- Previous post was at 10:49 AM ----------
Okay, nowhere does it say anything like "Sequential growth models cannot have an infinitely large number of prior steps (parents) because no causally prior step can be reached in sequence."Originally Posted by Squatch347
In fact, it does state:
The condition that all past-sets are ﬁnite implies that the partial order is locally ﬁnite. In other contexts one would weaken the condition of past ﬁniteness to local ﬁniteness in the deﬁnition of a causet, but for present purposes there is no harm in using the stronger condition.
The fact that local finiteness is a weaker condition than past finiteness means that there are locally finite sets that have infinite past-sets.
Another source of yours (which you quoted after your claim re: infinitely large number of prior steps) states:
The result of this process, obviously, is a naturally labeled causet (ﬁnite if we stop at some ﬁnite stage, or inﬁnite if we do not) whose labels record the order of succession of the individual births.
The author seems perfectly fine with the notion of an infinite causet.
Erm, what? Can you walk me through how you drew your conclusion? It seems to me that the slide merely states that it's possible to construct a probability measure on the set of infinite, past-finite causets. It doesn't say that all causets have to be past-finite, or that all infinite causets are past-finite, or that every path between elements is finite.In this link: http://igpg.gravity.psu.edu/events/c...s/Bombelli.pdf we have kind of a brief overview of this subject.
On slide 13, we see:If one imposes discrete versions of
I General covariance: The probability of growing a poset c is
independent of the order in which elements are added, and
I Bell causality: Relative probabilities for transitions are not
influenced by what happens in unrelated parts of the poset,
then the possible stochastic evolutions are the Rideout-Sorkin
“generalized percolation” models, each characterized by
non-negative real numbers t0, t1, t2, ..., where tk is related to the
probability that a new “birth” in the causet has some k-element
subset to its past [Rideout & Sorkin 1999].
These conditions imply that every element has a descendant, and
therefore infinitely many, and one obtains a probability measure on
the set of infinite, past-finite causets.
In which the argument posits that given these two assumptions we arrive at an infinite number of possible paths for any specific event, all of which must be finite in length.
Um, it doesn't say that the universe gets arbitrarily large, nor that it happens "quite quickly", or that the universe would have ceased to be dynamic "infinitely long ago". How are you reaching these conclusions?On Slide 15, we discuss the cyclical universe we find that the process propossed descends into an arbitrarily large universe quite quickly and therefore cannot be past infinite or it would have ceased to be dynamic infinitely long ago.• Cosmic renormalization: Each cycle leads to a change in the
effective values of the “coupling constants” tk ; after many cycles,
the universe is typically very large and the tk converge to
asymptotic values [Sorkin 2000, Martin et al 2001].
If I am capable of grasping God objectively, I do not believe, but precisely because I cannot do this I must believe. - Soren Kierkegaard
**** you, I won't do what you tell me
HOLY CRAP MY BLOG IS AWESOME
Honestly, Squatch, I have no idea what your argument even is. I have no idea what you think LR is; I have no idea how the papers you've cited contribute to your argument, since I can't even figure out what your argument is.Originally Posted by Squatch347
It should be relatively quick and simple to just outline your argument and provide citations, right? Something like:
(1) LR is a model / set of axioms that is not identical to SR / a model of SR
(2) If LR, then X,Y,Z
(3) LR is not experimentally disconfirmed + support for this claim
(4) LR is consistent with / predicts currently-available observations/data on relativity.
(5) Therefore, LR is "plausible" (according to some agreed-upon definition of model- or axiom-plausibility)
(6) Therefore, X,Y,Z are plausible
Something like that. It doesn't take a lot of time to do, probably less time than it took to write the post you just authored.
If I am capable of grasping God objectively, I do not believe, but precisely because I cannot do this I must believe. - Soren Kierkegaard
**** you, I won't do what you tell me
HOLY CRAP MY BLOG IS AWESOME
On Ordinal Numbers:
Each ordinal is the well-ordered set of all smaller ordinals.
Each ordinal is a set.
The collection of all ordinals does not form a set.
There is no infinitely-descending sequence of ordinals.
There are infinitely many ordinals.
I bring this up because the implicit, inductive definition of ordinal numbers might intuitively seem like "successive addition", since you're starting at 0 and then combining everything together that went before, but doesn't yield the intuitive results of "successive addition", since there are infinitely many ordinals.
If I am capable of grasping God objectively, I do not believe, but precisely because I cannot do this I must believe. - Soren Kierkegaard
**** you, I won't do what you tell me
HOLY CRAP MY BLOG IS AWESOME
1,3,5,6 have been offered here in thread and, imo, quite clearly. 2 and 4 have been discussed at length and was the major thrust of my argument towards GP. These matters have been offered, the connections have been made, I understand that GP doesn't agree with them (and I don't particularly blame him for that, he is approaching them from a different premise than I am). The dismissal of these connections have taken two forms, 1) counter evidence, which was well done and which I conceded too (that involved one piece of support), 2) dismissal as "crack pot," I understand the temptation to do that rather than spend the energy detailing the problems, but it isn't a valid rebuttal. I also think it is odd that you are asking your opponent here to fully elucidate a more detailed response, give more research, spend more time when his counters were unwilling to do so.
---------- Post added at 07:38 AM ---------- Previous post was at 06:19 AM ----------
I've moved it into this thread. I had made it a PM to help us hash out the difference without burdening the threads, but if we are going to discuss it, might as well be here.
We discussed the creation of sequences of set in PM. Something like:Originally Posted by CS
{ {x_{1}}, {x_{1}, x_{2}}, {x_{1}, x_{2}, x_{3}}}
So for each passing moment, the set defined above as the "past" grows by a finite amount.
3 moments ago: S_{1} = {x_{1}}
2 moments ago: S_{2} = {S_{1}, x_{2}}
1 moment ago: S_{3} = {S_{2}, x_{3}}
So for each moment, the past is defined as the immediately prior set of all past points (x) unioned to the immediately prior "now." For clarity's sake, lets use a simple example. Lets presume, for a minute, that we are only using one year, 2013. And we are only considering days.
Date Past (x) Now (y) 1 Jan {} (null) 1Jan 2 Jan {1 Jan} 2 Jan 3 Jan {1 Jan, 2 Jan} 3 Jan 4 Jan {1 Jan, 2 Jan, 3 Jan} 4 Jan
So we have a growing set of values (the past) that grow by sequentially adding finite values (nows) to that set.
Not here, remember, we have defined a process above, that process can grow infinitely larger, but only in the potential sense. A finite value for x when unioned with another finite y cannot produce a completed infinite set. I think your confusion arises from you perceiving this as a set, completed equation rather than a causally ordered process.Originally Posted by CS
That you are presuming x to be an actual infinite with no explanation of how that would arise. That is simply presuming the answer (the past is infinite) to show that the past is infinite. The mechanism of generating the past (described above) doesn't produce actual infinities on its own, you must add an axiom to show that they exist here (actual infinities exist) which is to beg the question for this debate.Originally Posted by CS
And since the audience has missed some of our back and forth I'll pull back the curtain and show that I have already done so. Lets presume that the argument above holds. I've also offered the premise that the process above is the only mechanism for generating the past (my support being that this is the method of temporal generation in A-theory). If it is the only mechanism and it cannot, on its own produce an actual infinite, then we cannot conclude the past to be past infinite.Originally Posted by CS
I'm not sure why you would conclude that. "Causal sets are discrete partially ordered sets, which are postulated to be a discrete substratum to continuum spacetime." That sounds a lot like a definition to me.Originally Posted by CS
My confusion arose from your statement:Originally Posted by CS
" R, Q, and Z are all totally ordered by their standard orders, and all possess infinitely decreasing sequences."
What is the decreasing sequence for "R" or are you using a short hand implication that R is a set with infinitely decreasing sequence?
You used the term "smooth" implying non-discrete motion, smooth motion would seem to imply that you are ignoring the possible quantized jumps from value to value. If not, what is the meaning you meant to imply with that term?Originally Posted by CS
I can understand your frustration because I feel it. It seems now that this thread is public again you have chosen to forget the explanations offered for these concepts. I offered them to you in PM on two different occasions, only only a few days ago.Originally Posted by CS
Look at my response you are quoting above. Age is always the same thing, the amount of time passed from beginning of x to ending of x (or current time).
To measure that we can use several different techniques. For sets with discrete, non-zero elements, we can use the cardinality times the duration of each element (IE Alex is 10 years old, his lifespan has a cardinality of 10 with each point having a duration of 1 year). That equation becomes meaningless if we consider time as non-discrete. What is the cardinality of a measuring stick? The question loses meaning. In that case, we have to measure the length, in this case of time passed (or to follow your example, distance traveled along the line) and I was pointing out that the distance traveled of that point is likewise a potential infinite. You have described a process (the point moving along the line) which gets larger and larger, approaching infinity, but never reaching it.
That is a pretty unwarranted conclusion, all that statement is actually saying is that they are applying more restrictive category of items to the model. We could be doing a financial analysis and say "it is a weaker condition that banks can print their own money and in other contexts we could consider that" does not mean that banks can print their own money. It means that it would be allowing a wider variance in a condition than it is in this discussion.Originally Posted by CS
Try as one might, the definition offered in that paper was:
The causal set hypothesis is that the continuum spacetime of general relativity is an approximation to a deeper level of discrete structure which is a past finite partial order or causal set (causet). This is a set endowed with a binary relation ≺ such that (x ≺ y) and (y ≺ z) =⇒ (x ≺ z) (transitivity), x 6≺ x (acyclicity), and all “past-sets” {x | x z} are finite.
I'm not sure what mechanism we can use to say that all past sets are finite for causets, but somehow that doesn't mean that past sets are finite.
Again, reviewing this process it seems extremely similar to what I described above:
Each of the dynamical laws in question describes a stochastic birth process in which elements are “born” one by one so that, at stage n, it has produced a causet ˜cn of n elements, within which the most recently born element is maximal. If one employs a genealogical language in which “x ≺ y” can be read as “x is an ancestor of y”, then the nth element (counting from 0) must at birth “choose” its ancestors from the elements of ˜cn, and for consistency it must choose a subset s with the property that x ≺ y ∈ s =⇒ x∈s. (Every ancestor of one of my ancestors is also my ancestor.) Such a subset s (which is necessarily finite) will be called a stem. Stems should be considered causal timelines to any specific event, ie the list of all events that led to some maximal event.
So in that scenario we would have a process that continues to grow on and on forever. Does that sound like a potential or actual infinite?Originally Posted by CS
To be honest, I'm not sure how you read that differently. You seem to be inferring that he is talking about some subset of all cuasets that is finite, but that isn't at all what is being discussed here. He is pointing out that if you apply the two assumptions listed, that all stems (related causets) become potentially infinite in the future (ie that each element will have a descendent) and past finite and that therefore probability measures are obtained on the total set rather than individual stems.Originally Posted by CS
So in a scenario where the universe gets quite large and the constants that govern the strength of interactions approaches zero who does that imply a sustainable infinite cycle?Originally Posted by CS
"Suffering lies not with inequality, but with dependence." -Voltaire"Fallacies do not cease to be fallacies because they become fashions.” -G.K. ChestertonAlso, if you think I've overlooked your post please shoot me a PM, I'm not intentionally ignoring you.
BIG EDIT: I found a good source that gives a mathematical foundation for causal sets. If you'd like, you can read it and see if the definitions are suitable for the argument you'd like to make.
Squatch, you're addressing a hypothetical argument sketch that I gave as an example of what I'd like to see you provide. I need to understand what your argument is. Will you please supply the premises and conclusions of your argument (numbered or indexed would make referencing easy)?
Again, you aren't explicitly giving a definition here, but you've explained your intuition somewhat. I'll try to give a more full definition, and you tell me if it's what you had in mind.We discussed the creation of sequences of set in PM. Something like:
{ {x_{1}}, {x_{1}, x_{2}}, {x_{1}, x_{2}, x_{3}}}
So for each passing moment, the set defined above as the "past" grows by a finite amount.
3 moments ago: S_{1} = {x_{1}}
2 moments ago: S_{2} = {S_{1}, x_{2}}
1 moment ago: S_{3} = {S_{2}, x_{3}}
So for each moment, the past is defined as the immediately prior set of all past points (x) unioned to the immediately prior "now." For clarity's sake, lets use a simple example. Lets presume, for a minute, that we are only using one year, 2013. And we are only considering days.
Date Past (x) Now (y) 1 Jan {} (null) 1Jan 2 Jan {1 Jan} 2 Jan 3 Jan {1 Jan, 2 Jan} 3 Jan 4 Jan {1 Jan, 2 Jan, 3 Jan} 4 Jan
So we have a growing set of values (the past) that grow by sequentially adding finite values (nows) to that set.
(1) Let F = {A_{i}} be a family of subsets of X. F is temporal if and only if Ai = {x_{i}} U (UA_{j} | j < i) for some x_{i} in X.
(2) Let F = {A_{n}} be a countable family of subsets of X. F is temporal if and only if A_{n} = {x_{n}} U A_{n-1} for some x_{n} in X.
So (a) do you understand both of these definitions? (b) Do either of these definitions capture what you're going for, or are different definitions needed?
No, my confusion arises from the ambiguity that comes from doing math without definitions.Not here, remember, we have defined a process above, that process can grow infinitely larger, but only in the potential sense. A finite value for x when unioned with another finite y cannot produce a completed infinite set. I think your confusion arises from you perceiving this as a set, completed equation rather than a causally ordered process.
You misunderstand the nature of your proof. It is incumbent on you to show that the infinite case is impossible. If it is possible to assume an infinite and draw no contradiction, then your argument has failed.That you are presuming x to be an actual infinite with no explanation of how that would arise. That is simply presuming the answer (the past is infinite) to show that the past is infinite. The mechanism of generating the past (described above) doesn't produce actual infinities on its own, you must add an axiom to show that they exist here (actual infinities exist) which is to beg the question for this debate.
And since the audience has missed some of our back and forth I'll pull back the curtain and show that I have already done so. Lets presume that the argument above holds. I've also offered the premise that the process above is the only mechanism for generating the past (my support being that this is the method of temporal generation in A-theory). If it is the only mechanism and it cannot, on its own produce an actual infinite, then we cannot conclude the past to be past infinite.Do you know what each of those words mean? Let's just go through them; since you're supporting this definition, can you please answer the following questions:I'm not sure why you would conclude that. "Causal sets are discrete partially ordered sets, which are postulated to be a discrete substratum to continuum spacetime." That sounds a lot like a definition to me.
(a) What is a discrete set?
(b) What is a partially ordered set?
(c) What is continuum spacetime?
(d) What is a substratum of continuum spacetime?
(e) What is a discrete substratum of continuum spacetime?
R is the set of real numbers. Q is the set of rational numbers. Z is the set of integers.My confusion arose from your statement:
" R, Q, and Z are all totally ordered by their standard orders, and all possess infinitely decreasing sequences."
What is the decreasing sequence for "R" or are you using a short hand implication that R is a set with infinitely decreasing sequence?
Z is a subset of Q is a subset of R (although technically the containments include only isomorphic copies, but that distinction is irrelevant), and the standard order on Z is compatible with the standard order on Q is compatible with the standard order on R.
So any infinite decreasing sequence in Z is an infinite decreasing sequence in Q is an infinite decreasing sequence in R.
Although if you'd like, Q has decreasing sequences that aren't in Z. Take x_{k} = 1/k, so {x_{n}} = {1, 1/2, 1/3, 1/4, ...}. This is an infinite decreasing sequence of rational numbers.
R has infinite decreasing sequences that aren't in Q, either. Take x_{k} = pi/k.
You were talking about causets, and I don't know whether continuous implies non-discrete, because I don't have a definition for discrete (or rather, for the particular use of discrete in the context of causets).You used the term "smooth" implying non-discrete motion, smooth motion would seem to imply that you are ignoring the possible quantized jumps from value to value. If not, what is the meaning you meant to imply with that term?
Is Q discrete?
I don't recall any explanations other than your standard "intuitive" stuff.I can understand your frustration because I feel it. It seems now that this thread is public again you have chosen to forget the explanations offered for these concepts. I offered them to you in PM on two different occasions, only only a few days ago.
Okay, so I guess we're doing Lebesgue measure then, and the "finite additions" to the past are finite in the sense of Lebesgue measure (having finite length). That's the definition I was working with.Look at my response you are quoting above. Age is always the same thing, the amount of time passed from beginning of x to ending of x (or current time).
Uh, for this to work, you'd need the "discrete" (whatever that means) non-zero elements to also have constant Lebesgue measure, i.e. L(x_{i}) = L(x_{j}) for all i,jTo measure that we can use several different techniques. For sets with discrete, non-zero elements, we can use the cardinality times the duration of each element (IE Alex is 10 years old, his lifespan has a cardinality of 10 with each point having a duration of 1 year).
I also think you're not paying very careful attention to "units"/values. Alex's lifespan should be the same regardless of how it is partitioned--whether into years, months, etc. If Alex's lifespan has a cardinality, then Alex's lifespan is a set; in this case, a set with 10 elements, each having a finite length.
Could Alex be 10.5 years old? Could Alex be 10 + pi/3 years old?
Just because you can use a shortcut equation under very specific conditions doesn't mean that the equation should always be valid. We were measuring length in your "cardinality" examples, it's just that the Lebesgue measure of a (countable) union of disjoint sets is countable, so L(X) = sum_{i in I}{L(x_{i})} and under your assumption, L(x_{i}) = L for all i, so L(X) = sum_{i in I}{L} = L*sum_{i in I}{1} = L*|X| when X is finite.That equation becomes meaningless if we consider time as non-discrete. What is the cardinality of a measuring stick? The question loses meaning. In that case, we have to measure the length, in this case of time passed (or to follow your example, distance traveled along the line) and I was pointing out that the distance traveled of that point is likewise a potential infinite. You have described a process (the point moving along the line) which gets larger and larger, approaching infinity, but never reaching it.
No, that's exactly what "weaker" means in this context. If "locally finite" implied "past finite", then all statements that hold for "past finite" sets will hold for "locally finite" sets. Since "past finite" is a stronger condition than "locally finite", it must exclude some sets that "locally finite" would have included.That is a pretty unwarranted conclusion, all that statement is actually saying is that they are applying more restrictive category of items to the model. We could be doing a financial analysis and say "it is a weaker condition that banks can print their own money and in other contexts we could consider that" does not mean that banks can print their own money. It means that it would be allowing a wider variance in a condition than it is in this discussion.
For example, "x is a multiple of 4" is a stronger condition than "x is a multiple of 2".
Well, my first question would be, why are we even talking about causets? But that's the counter-argument stage, and I'd like to get your argument down first before I attempt rebuttal.Try as one might, the definition offered in that paper was:The causal set hypothesis is that the continuum spacetime of general relativity is an approximation to a deeper level of discrete structure which is a past finite partial order or causal set (causet). This is a set endowed with a binary relation ≺ such that (x ≺ y) and (y ≺ z) =⇒ (x ≺ z) (transitivity), x 6≺ x (acyclicity), and all “past-sets” {x | x z} are finite.
I'm not sure what mechanism we can use to say that all past sets are finite for causets, but somehow that doesn't mean that past sets are finite.
It seems like you want to conclude "all past sets are finite for causets", but that's not what they said. They specifically said that in other contexts working with causets, you might weaken the "all past sets are finite" to "past-sets are locally finite", but that the results of the paper hold even if that condition is weakened.
Okay, if you like this definition, let's get into it.Again, reviewing this process it seems extremely similar to what I described above:
Each of the dynamical laws in question describes a stochastic birth process in which elements are “born” one by one so that, at stage n, it has produced a causet ˜cn of n elements, within which the most recently born element is maximal. If one employs a genealogical language in which “x ≺ y” can be read as “x is an ancestor of y”, then the nth element (counting from 0) must at birth “choose” its ancestors from the elements of ˜cn, and for consistency it must choose a subset s with the property that x ≺ y ∈ s =⇒ x∈s. (Every ancestor of one of my ancestors is also my ancestor.) Such a subset s (which is necessarily finite) will be called a stem. Stems should be considered causal timelines to any specific event, ie the list of all events that led to some maximal event.
(a) What is a dynamical law?
(b) What is a stochastic birth process?
If infinite causets are allowed, and the past is a causet, then you'll need to give an argument other than "the past is a causet" in order to conclude that the past is finite.So in that scenario we would have a process that continues to grow on and on forever. Does that sound like a potential or actual infinite?
He doesn't say "potentially infinite in the future". I'd really like it if you'd supply your reasoning rather than just your conclusions.To be honest, I'm not sure how you read that differently. You seem to be inferring that he is talking about some subset of all cuasets that is finite, but that isn't at all what is being discussed here. He is pointing out that if you apply the two assumptions listed, that all stems (related causets) become potentially infinite in the future (ie that each element will have a descendent) and past finite and that therefore probability measures are obtained on the total set rather than individual stems.
Last edited by CliveStaples; April 3rd, 2014 at 11:38 PM.
If I am capable of grasping God objectively, I do not believe, but precisely because I cannot do this I must believe. - Soren Kierkegaard
**** you, I won't do what you tell me
HOLY CRAP MY BLOG IS AWESOME
Great, thanks. I'll give it a look at the first opportunity.
And from my point of view you are asking for me to restate what I've already stated and what clearly seems like a minor point. I see nothing productive about getting dragged down a rat hole of restating my restatements to fit the newest preferable format. I believe the defense has been made adequately and would prefer to spend my response time dealing the with the more practical matters of this thread. It doesn't really matter if I prove LR or concede LR here, in both instances we are going to discuss the below.Originally Posted by CS
Definition 1 seems to be the closest to what I have in mind though with perhaps a few additions.Originally Posted by CS
There is a value x_{i} that is now. It also seems to lack the concept of process which is critical. Perhaps something to the effect of: All values A_{j} exist if and only ifl A_{j-1} has existed. I'm sure you won't find that satisfactory, but we can't substitute an ongoing process for a set equation without an additional
Which has been shown imo.Originally Posted by CS
If the process cannot produce an actual infinite.
and
That is the only process for producing time.
Then it cannot be that time is an actual infinite. Now you are free to debate either comment above (and you are), but that does not mean that the assumption that one just exists, absent any mechanism to create it, is a reasonable objection. In order for that assumption to be an objection it would need some kind of explanation. How did it arise, what kind of mechanism created it? Why is that assumption not a begging the question fallacy? Etc.
First, what warrant do I have to define the words for you? You asked for a definition, I provided it. You seem to be wanting me to do your homework for you here.Originally Posted by CS
Second, I'll be charitable here and offer them:
a) "A set S is discrete in a larger topological space X if every point x element S has a neighborhood U such that S intersection U = {x}. The points of S are then said to be isolated. Typically, a discrete set is either finite or countably infinite. For example, the set of integers is discrete on the real line. Another example of an infinite discrete set is the set {1/n for all integers nɭ}. On any reasonable space, a finite set is discrete. A set is discrete if it has the discrete topology, that is, if every subset is open.
In the case of a subset S, as in the examples above, one uses the relative topology on S. Sometimes a discrete set is also closed. Then there cannot be any accumulation points of a discrete set. On a compact set such as the sphere, a closed discrete set must be finite because of this." http://www.wolframalpha.com/input/?i=discrete+set
b) "A partially ordered set (or poset) is a set taken together with a partial order on it. Formally, a partially ordered set is defined as an ordered pair P = (X, <=), where X is called the ground set of P and <= is the partial order of P.
An element u in a partially ordered set (X, <=) is said to be an upper bound for a subset S of X if for every s element S, we have s<=u. Similarly, a lower bound for a subset S is an element l such that for every s element S, l<=s. If there is an upper bound and a lower bound for X, then the poset (X, <=) is said to be bounded." http://www.wolframalpha.com/input/?i...ly+ordered+set
c) This being a phrase rather than a specific term, he would seem to mean: "a compact connected metric space..." " [of] three spatial coordinates and one temporal coordinate." http://www.wolframalpha.com/input/?i...uum+definition and http://dictionary.reference.com/browse/space-time respectively.
d) "any stratum or layer lying underneath another" http://www.wolframalpha.com/input/?i=substratum
e) "constituting a separate entity or part" http://www.wolframalpha.com/input/?i...ete+definition
IE a duration, yes.Originally Posted by CS
Because it is irrelevant to the question at hand (is the universe infinitely old?). The question is not "how old is the universe?" In which case the precision of the measuring device and the units involved is relevant. Rather, we are simply noting that the units involved in the addition are finite, constant and non-zero in duration which is all the precision necessary to the argument being offered.Originally Posted by CS
You are committing a categorical error here. Local finiteness is not a condition comparable to past finiteness. The former is about the units of measure, the latter is about the total duration of the past. It would be like saying that the phrase "a pound is a pound, the world around" is a stronger condition than the weight of an elephant. The condition being imposed is upon the unit of measure for the elephant (a pound), the weight of an elephant is not a condition here, neither is past finiteness.Originally Posted by CS
No, potential infinites are allowed, I'm not supporting potential infinites here, only actual ones. Because the "past" can continue to grow forever into the future is not a logical reason to conclude that it has already done so.Originally Posted by CS
He says, "each causet will have a descendent." Descendents, by definition are future events (in relation to their parent). By applying the assumption he offered he is explicitly making an assumption about stems and their continuity towards the future. IE if I were to translate his assumption it would read "for all moments there is a next moment." IE a process that for any given moment stretches off into the future. IE a potential infinite growing into the future.Originally Posted by CS
It would be a potential infinite into the past if he had said "each causet has a parent" but you'll notice that wasn't his assumption.
"Suffering lies not with inequality, but with dependence." -Voltaire"Fallacies do not cease to be fallacies because they become fashions.” -G.K. ChestertonAlso, if you think I've overlooked your post please shoot me a PM, I'm not intentionally ignoring you.
The only people arguing against you in this thread have asked you to present your argument in a clear, concise form. I don't think you've stated your argument clearly at all; in fact, I'm not convinced that you've even stated an argument.
There's no "going down the rathole". There's just giving your argument. If you know what your argument is, it takes literally less than a minute to give the argument. I gave an example that would take about 20 seconds to type out.
I don't know what your premises are. I don't know what your reasoning is. All I know is that your conclusion is, "The past had to be finite".
It seems you didn't complete your thought here. I'll refrain from responding until you've completed your thought.Definition 1 seems to be the closest to what I have in mind though with perhaps a few additions.
There is a value x_{i} that is now. It also seems to lack the concept of process which is critical. Perhaps something to the effect of: All values A_{j} exist if and only ifl A_{j-1} has existed. I'm sure you won't find that satisfactory, but we can't substitute an ongoing process for a set equation without an additional
My position is not, "It is reasonable to assume that the past is infinite." My position is, "Your argument must be sufficient to discharge the assumption that the past is infinite."Which has been shown imo.
If the process cannot produce an actual infinite.
and
That is the only process for producing time.
Then it cannot be that time is an actual infinite. Now you are free to debate either comment above (and you are), but that does not mean that the assumption that one just exists, absent any mechanism to create it, is a reasonable objection. In order for that assumption to be an objection it would need some kind of explanation. How did it arise, what kind of mechanism created it? Why is that assumption not a begging the question fallacy? Etc.
I agree that the following argument is valid:
(1) P does not produce infinite sets.
(2) All temporal sets are products of P
(3) Therefore, all temporal sets are finite.
First, you haven't given a precise statement of the "process" you're talking about.
Second, related to the first, you haven't given a proof of (1).
Third, related to the second, you haven't given a proof of (2).
I asked to make sure that you understand what they mean, to ensure that we're both working with the same definitions. Trust me, Squatch, I do not want you doing any math homework for me.First, what warrant do I have to define the words for you? You asked for a definition, I provided it. You seem to be wanting me to do your homework for you here.
Okay, do you know what a topological space is? Do you know what neighborhoods are?Second, I'll be charitable here and offer them:
a) "A set S is discrete in a larger topological space X if every point x element S has a neighborhood U such that S intersection U = {x}. The points of S are then said to be isolated. Typically, a discrete set is either finite or countably infinite. For example, the set of integers is discrete on the real line. Another example of an infinite discrete set is the set {1/n for all integers nɭ}. On any reasonable space, a finite set is discrete. A set is discrete if it has the discrete topology, that is, if every subset is open.
In the case of a subset S, as in the examples above, one uses the relative topology on S. Sometimes a discrete set is also closed. Then there cannot be any accumulation points of a discrete set. On a compact set such as the sphere, a closed discrete set must be finite because of this." http://www.wolframalpha.com/input/?i=discrete+set
Do you understand this definition?b) "A partially ordered set (or poset) is a set taken together with a partial order on it. Formally, a partially ordered set is defined as an ordered pair P = (X, <=), where X is called the ground set of P and <= is the partial order of P.
An element u in a partially ordered set (X, <=) is said to be an upper bound for a subset S of X if for every s element S, we have s<=u. Similarly, a lower bound for a subset S is an element l such that for every s element S, l<=s. If there is an upper bound and a lower bound for X, then the poset (X, <=) is said to be bounded." http://www.wolframalpha.com/input/?i...ly+ordered+set
Do you know what a metric space is? Do you know what a connected space is? Do you know what a compact space is?c) This being a phrase rather than a specific term, he would seem to mean: "a compact connected metric space..." " [of] three spatial coordinates and one temporal coordinate." http://www.wolframalpha.com/input/?i...uum+definition and http://dictionary.reference.com/browse/space-time respectively.
This question is for my own benefit. "Substratum" in this case would refer to something like a "layer" of spacetime, then? An n-dimensional subspace for some n?d) "any stratum or layer lying underneath another" http://www.wolframalpha.com/input/?i=substratum
Hmm. That's not very helpful. "Separate" in what mathematical or physical sense? I suspect they're using the "discrete set" definition that you gave above.e) "constituting a separate entity or part" http://www.wolframalpha.com/input/?i...ete+definition
Okay, then you need to specify that. In general, countable sets have zero lebesgue measure; in particular, finite sets have zero lebesgue measure.IE a duration, yes.
The units are "finite, constant, and non-zero"? I'm not sure what that means. What does it mean for, say, the unit "kilogram" to be finite?Because it is irrelevant to the question at hand (is the universe infinitely old?). The question is not "how old is the universe?" In which case the precision of the measuring device and the units involved is relevant. Rather, we are simply noting that the units involved in the addition are finite, constant and non-zero in duration which is all the precision necessary to the argument being offered.
They're talking about sets. A set is finite if there is a bijection to the set {1,2,..., n} for some positive integer n. A partially-ordered set is locally finite if all of its intervals are finite.You are committing a categorical error here. Local finiteness is not a condition comparable to past finiteness. The former is about the units of measure, the latter is about the total duration of the past. It would be like saying that the phrase "a pound is a pound, the world around" is a stronger condition than the weight of an elephant. The condition being imposed is upon the unit of measure for the elephant (a pound), the weight of an elephant is not a condition here, neither is past finiteness.
"All past-sets are finite" requires that every past-set be a finite set. "All past-sets are locally finite" requires that every past-set be a locally finite set.
Every finite set is locally finite, but not all locally finite sets are finite sets.
I'm not sure what you mean by "potential infinities" in the case of causets. Infinite causets are sets that: a) contain an infinite number of elements (i.e., there is no bijection to a set of the form {1,2,...,n} for any positive integer n; or, alternatively, there is a bijection from the set to one of its proper subsets), and b) meet the critera for being a causetNo, potential infinites are allowed, I'm not supporting potential infinites here, only actual ones. Because the "past" can continue to grow forever into the future is not a logical reason to conclude that it has already done so.
Which infinite causets are "actually finite" and which are only "potentially infinite"?
Each natural number has a "successor". Are there only a potentially infinite number of natural numbers? Or an actually infinite number of natural numbers?He says, "each causet will have a descendent." Descendents, by definition are future events (in relation to their parent). By applying the assumption he offered he is explicitly making an assumption about stems and their continuity towards the future. IE if I were to translate his assumption it would read "for all moments there is a next moment." IE a process that for any given moment stretches off into the future. IE a potential infinite growing into the future.
I'm trying to understand what you mean by "potential infinite" in the context of sets/causets. If you could give a quick, precise definition it would help a lot.
Last edited by CliveStaples; April 8th, 2014 at 11:06 AM.
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You're right, I apologize, I apparently deleted some text. It should read:
Definition 1 seems to be the closest to what I have in mind though with perhaps a few additions.
There is a value x_{i} that is now. It also seems to lack the concept of process which is critical. Perhaps something to the effect of: All values A_{j} exist if and only ifl A_{j-1} has existed. I'm sure you won't find that satisfactory, but we can't substitute an ongoing process for a set equation without an additional axiom defining causal order and subsequent action.
I'll certainly grant the first point to the extent that you don't seem to have a good picture of what I'm talking about. We've continued that discussion above.Originally Posted by CS
Your second point is a requisite of A-theory of time. A-theory holds that time is produced by the process I am attempting to describe. That explanation has been offered on several occasions in thread. A-theory describes a passing now that takes a real moment "now" and iteratively adds those to a past set of nows.
I'm also not clear on your third objection. If the argument is valid, then it is the proof of 3 right?
Clearly then that is a poor analogy for what we are talking about then right? We all recognize that if we had a set representing the units on a ruler {1,2,...,11,12} and that each unit has a duration of 1 Inch, then the length of the ruler is 12 Inches. Likewise if we have a set that represents Timmy's birthdays, {1,2,3,...,12,13,14} with each birthday representing one year of life, then we know Timmy is 14 years old.Originally Posted by CS
Now again you can ask "what if Timmy is actually 14.5 years old?" Then we should argue that birthdays are not an accurate measure of someone's true age at any given moment because they allow for a possible error of 364 days. Which is true, but not germane to the question of whether or not he can drive (14.5 years old is not old enough for a permit). Likewise, we can argue whether centuries, hours, days, minutes, planck moments, etc are accurate measures of the age of the universe. It doesn't matter to the question of whether it is infinitely old or not. As long as the definition of the duration (birthday, inch, etc) is a) finite and b) fixed, the outcome is the same.
Obviously that it does not contain an infinite amount of mass for each kilogram added.Originally Posted by CS
We've already discussed the difference between a potential and actual infinite on several occasions. That we are now talking about causets is irrelevant to that distinction.Originally Posted by CS
There would be if we defined each natural number as causally successive to its predecessor (IE it can only exist causally after the number before it exists.). There are an actual infinite number of natural numbers because they all exist causally independent of each other. 5 can simply be, it does not require that 4 exist prior to its existence in a causal sense which requires 3 to exist prior to 4, which requires 2 to exist prior to 3, etc.Originally Posted by CS
You can create an actual infinite of natural numbers by simply defining the set, bam done. You can't do that in a causally related set because the process of cause/effect must translate through all the necessary steps in order for the set to be "finished" and actual.
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