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  1. #101
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by CliveStaples View Post
    I don't see anything in your post that contradicts anything I've said. Yes, by the law of excluded middle, "x is possible OR x is not possible" is true. If it is known that "x is impossible" is necessarily false (i.e., that "x is possible" is necessarily true), then one can deduce from the impossibility of "x is impossible" that x must be possible.
    There are a million ways you can put the same proposition rhetorically, Clive, which is the point here. MT's question remains unanswered by you. These are the facts as they stand at present. You should try living within their constraints on the UOD in making your arguments.

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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by cstamford View Post
    There are a million ways you can put the same proposition rhetorically, Clive, which is the point here. MT's question remains unanswered by you. These are the facts as they stand at present. You should try living within their constraints on the UOD in making your arguments.
    I think your claim here is absolutely false, and I think you should retract it if you have either good manners or intellectual integrity.

    First, the question. The question was, as MT put it:

    Is it o.k. to appeal to known impossibilities in order to demonstrate possibility?

    This question led me to think of what kind of counterexamples there might be. So I offered the following line of reasoning:

    (i) p is possible or q is possible [premise]
    (ii) q is impossible [premise]
    (iii) Therefore, p is possible.

    If (ii) holds, then q is an impossibility. So the question is, does (i) through (iii) involve "appealing to known impossibilities in order to demonstrate possibility"? It depends entirely on what the working definition of "appealing" is. It seems to me that you are, in some sense, appealing to the impossibility of q in order to demonstrate the possibility of p; you're referencing the impossibility of q, and then from that impossibility you deduce the possibility of p.

    So the answer to MT's question, based on the above reasoning, was "Sometimes, yes."

    But I didn't just leave it there. I further offered a more narrowly-defined sense of "appeals", in order to capture the intuition that a reasonable explanation can't require that impossible things obtain. So I wrote:


    "If p is impossible, then the truth of p cannot be a premise of nor deducible from any reasonable explanation".


    So not only did I answer MT's question, but I attempted to help clarify what precisely the features of an explanation or demonstration are, with regard to impossible propositions.
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  3. #103
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by CliveStaples View Post
    That argument doesn't make much sense to me. What's wrong with an infinite amount of hydrogen having been consumed?
    Besides that it doesn't match observational data? There would be an infinite amount of helium now and an infinite amount of hydrogen. That would also mean that our universe has an infinite amount of matter right? That would be a very different gravitational picture than we see now I think.

    Quote Originally Posted by CS
    I'm not quite sure what your question is about.
    It is about actually referencing Craig's argument. He doesn't argue that infinities are impossible. He argues that actual infinities formed by successive addition are impossible. IE You add one to one, then add one to that total, then one to that total ad infinitum.

    ---------- Post added at 04:04 PM ---------- Previous post was at 03:31 PM ----------

    http://www.reasonablefaith.org/hilbert-and-kalam

    I would like to add a bit more context to this debate on what Craig's position is:

    I have not at any time made the claim that Hilbert offered a proof that an infinite regress of causes is mathematically impossible. Rather I cite Hilbert as an example of a great mathematician who, though enthusiastic about the mathematical existence of the infinite, denied that the actual infinite exists in reality.




    Quote Originally Posted by GP
    Either way though, I'm going to need you to give me a slightly more descriptive definition of "infinities caused by successive addition."
    Please see my response to Clive. Time is the obvious example being used by Craig here, time would be a set formed by successive addition.


    Quote Originally Posted by GP
    Not to my knowledge. I'm not an expert on the ekpyrotic model (though I do know one of the authors of the original paper), but so far as I know it's only been observationally constrained (Which has nothing to do with an a priori argument like Craig's) but it doesn't have any mathematical inconsistencies. It's pretty much completely removed from string theory, but again, this is technically not a problem. In other words, there's nothing a priori wrong with cyclic universes.
    Well I would point out the obvious that I am even less an expert on these models than you, but to my knowledge they have two major problems. 1) Entropy is not returned to its initial level between cycles and as such builds up in the models and the 2) Borde-Guth-Vilenkin Theorem states that the universe must, on average been in a state of inflation, which wouldn't be true of an infinite cyclical universe.

    Quote Originally Posted by GP
    Yes. There's an uncountably infinite number of successive infinities contained in spacetime.
    Making the infinite uncountable as opposed to countable is simply a useful fiction. Because we can imagine an infinite breakdown of each point does not mean that it exists in actuality. In fact the point system itself doesn't even fit the limitations of Craig's argument. Point systems are idealizations, not actualities.

    Really it would seem we are re-living Zeno's paradox here with you affirming Zeno's tongue in cheek conclusion. These mathematical subdivisions represent a mathematical reality, which is fine, no one is arguing that a mathematical set formed by successive additions is problematic.

    I would also point out that in your example, we still have point 0, which is the point Craig is making. We don't get rid of the beginning in either example you point out.
    Last edited by Squatch347; September 9th, 2013 at 08:50 AM.
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  4. #104
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by Squatch347
    It is about actually referencing Craig's argument. He doesn't argue that infinities are impossible. He argues that actual infinities formed by successive addition are impossible. IE You add one to one, then add one to that total, then one to that total ad infinitum.
    What does "ad infinitum" mean?

    Consider the sequence: {{1}, {1,2}, {1,2,3}, {1,2,3,4}, ... , {1,2,3,...,k}, ...}
    It is true that all of the sets in this sequence are finite.

    That is, a finite union of finite sets is necessarily finite.

    But an infinite union of finite sets can indeed be infinite. So what does it mean to form a set "by successive addition"? Is the following set formed "by successive addition":

    {1} U {2} U {3} U ...
    = U{n | n a positive integer}
    ={1,2,3,...}

    ---------- Post added at 06:41 PM ---------- Previous post was at 06:37 PM ----------

    Quote Originally Posted by cstamford View Post
    There are a million ways you can put the same proposition rhetorically, Clive, which is the point here. MT's question remains unanswered by you. These are the facts as they stand at present. You should try living within their constraints on the UOD in making your arguments.
    As I said before, I answered MT's question in the affirmative: I gave a case where an "appeal to impossibility" can be used in an explanation. Specifically, one can appeal to the impossibility of ~p to explain the possibility of p.

    ---------- Post added at 06:42 PM ---------- Previous post was at 06:41 PM ----------

    Really it would seem we are re-living Zeno's paradox here with you affirming Zeno's tongue in cheek conclusion. These mathematical subdivisions represent a mathematical reality, which is fine, no one is arguing that a mathematical set formed by successive additions is problematic.
    ...how is he affirming Zeno's conclusion? Zeno's "paradox" is simply a failure to recognize that a sum of infinitely many elements can itself be finite.
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  5. #105
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by CLIVE
    What does "ad infinitum" mean?
    It means "insert the idea of infinity".

    Quote Originally Posted by CLIVE
    But an infinite union of finite sets can indeed be infinite. So what does it mean to form a set "by successive addition"? Is the following set formed "by successive addition":

    {1} U {2} U {3} U ...
    = U{n | n a positive integer}
    ={1,2,3,...}
    No.
    "...." is the addition of the concept of infinity. Any set that adds infinity to it will by definition be infinite. Thus dong so is circular reasoning for those trying to support an actual infinite.

    What it isn't is "actual" list.

    (1, 2, 3) Is a an actual list up to 3. IE all of them have actual individual representation.

    (1,2,3 ....) is a set of an actual 1,2,3, and a purely conceptual representation of infinity added onto it.

    -----Analogy discard as seen fit----
    It is as though I said I can count to 100, so I counted thusly.
    1,2, skip a few, 99, 100. .. There I counted to 100.

    Now you are not saying you can "count" to an infinite, but your holding that the infinite can exist in reality. However the support you are offering is
    the equivalent of me saying I counted to 100, even though I didn't "actually" "count" in a successive manner.

    So, using non-successive counting to support the possibility of successive addition, is a categorical error of support.
    I apologize to anyone waiting on a response from me. I am experiencing a time warp, suddenly their are not enough hours in a day. As soon as I find a replacement part to my flux capacitor regulator, time should resume it's normal flow.

  6. #106
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by MindTrap028 View Post
    It means "insert the idea of infinity".
    Okay. What does that mean? "Insert the idea of infinity" into successive addition? That's nonsense until you define what it means to "insert the idea of infinity" into successive addition.


    No.
    "...." is the addition of the concept of infinity. Any set that adds infinity to it will by definition be infinite. Thus dong so is circular reasoning for those trying to support an actual infinite.

    What it isn't is "actual" list.

    (1, 2, 3) Is a an actual list up to 3. IE all of them have actual individual representation.

    (1,2,3 ....) is a set of an actual 1,2,3, and a purely conceptual representation of infinity added onto it.

    -----Analogy discard as seen fit----
    It is as though I said I can count to 100, so I counted thusly.
    1,2, skip a few, 99, 100. .. There I counted to 100.

    Now you are not saying you can "count" to an infinite, but your holding that the infinite can exist in reality. However the support you are offering is
    the equivalent of me saying I counted to 100, even though I didn't "actually" "count" in a successive manner.

    So, using non-successive counting to support the possibility of successive addition, is a categorical error of support.
    Uh, no. Infinity is not an element of {1,2,3, ...}. The set of natural numbers, N, contains all and only the "actual" natural numbers. There is no representation of "infinity" in the set {1,2,3, ...}, "conceptual" or otherwise. Another way to denote the set {1,2,3, ...} is {n, such that n is a positive integer}. The "..." means something like "all the next elements follow the sequence as given", so in that sense it represents an infinite collection of numbers, but not infinity itself.

    And how are the representations "1", "2", and "3" different from "purely conceptual representations"?
    If I am capable of grasping God objectively, I do not believe, but precisely because I cannot do this I must believe. - Soren Kierkegaard
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  7. #107
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by CliveStaples View Post
    What does "ad infinitum" mean?
    Its latin for "to infinity" meaning, usually, that you repeat a process successively forever.

    Quote Originally Posted by CS
    Is the following set formed "by successive addition":

    {1} U {2} U {3} U ...
    = U{n | n a positive integer}
    ={1,2,3,...}[COLOR="Silver"]
    I would say that it is not. 1+1+1... is not formed by successive addition, it is all done simultaneously, right? Rather, 1+1=2, 2+1=3, 3+1=4... is formed by successive addition.

    Quote Originally Posted by CS
    ...how is he affirming Zeno's conclusion? Zeno's "paradox" is simply a failure to recognize that a sum of infinitely many elements can itself be finite.
    Isn't that what GP's position was (certainly possible I was misunderstanding), but that a finite set can be infinitely divided to be an uncountable infinity.
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  8. #108
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by Squatch347
    http://www.reasonablefaith.org/hilbert-and-kalam

    I would like to add a bit more context to this debate on what Craig's position is:

    [indent] I have not at any time made the claim that Hilbert offered a proof that an infinite regress of causes is mathematically impossible. Rather I cite Hilbert as an example of a great mathematician who, though enthusiastic about the mathematical existence of the infinite, denied that the actual infinite exists in reality.[indent]
    I don't think that I've claimed otherwise, my claim has more or less been that his statement that "There are no actual infinities" is just an unsubstantiated claim and goes against everything that we know about science.


    Quote Originally Posted by Squatch
    Well I would point out the obvious that I am even less an expert on these models than you, but to my knowledge they have two major problems. 1) Entropy is not returned to its initial level between cycles and as such builds up in the models and the 2) Borde-Guth-Vilenkin Theorem states that the universe must, on average been in a state of inflation, which wouldn't be true of an infinite cyclical universe.
    1.) Only in early cyclic universes was a problem, it has been known for over 10 years now how to evade the second law of thermodynamics (specifically in the Ekpyrotic model that I'd previously referenced). Although, it's worth commenting that thermodynamics has many non-intuitive features in the presence of General Relativity (Bekenstein-Hawking entropy, for one thing) than it does in ordinary statistical mechanics (And it's not necessarily terribly intuitive there, either). To the question though, of why it isn't a problem in the Ekpyrotic model, I would refer to Wikipedia:

    "As Richard C. Tolman showed, the earlier cyclic model failed because the universe would undergo inevitable thermodynamic heat death.[1] However, the newer cyclic model evades this by having a net expansion each cycle, preventing entropy from building up. However, there remain major open issues in the model." -- Wikipedia

    There's a lot of stringy questions about the model, but the physics that we do understand is consistent with this, and moreover even if string theory is wrong this provides by itself a technically correct model of cosmology. Personally, I don't find it to necessarily be a believable one, but that's not the question. The question is "If it's is possible", because that provides the necessary counter example to Craig's statement that it is "impossible."

    2.) Skimming through the Borde, et al, paper, it seems like they're missing out on an important detail --quantum gravity. Before I comment on that, there's a website that does a decent job refuting Craig's interpretation of the Borde, et al paper, here. But my problem with all of these arguments about cosmology is that no one knows what the UV-complete theory of gravity is. This means that at the small scales near the Big Bang, we simply cannot say what came --if anything-- before the Big Bang. People who work in one area of physics (Particularly in cosmology) really like to make strong claims using General Relativity about these issues, but there's no reason to trust GR at all if you go far enough back. And importantly, if you go far enough back to where the beginning of the Big Bang occurred, this is precisely where you trust GR the least. You can't, therefore, make any statement about the case where some quantum gravity effects come into play and create a stable way --independent of the ekpryotic model-- of cycles of the universe. We simply don't know, so Craig cannot, in good conscience, make these claims.

    3.) Also, the point about hydrogen consumption in stars is concretely ruled out as a valid argument. This is why I was trying to say before that having a big bang before star formation is a crucially important piece of information that invalidates that line of reasoning. If there were an infinite number of big bang cycles, then consuming an infinite number of hydrogen over the cycles is completely consistent because an infinite amount of hydrogen given an infinite amount of cycles. Each cycle just burns through the amount made in that cycle's nucleosynthesis.

    4.) Honestly, Craig's lost the argument at this point. His statement that the universe cannot be past-infinite is just plainly wrong. He didn't need to make a statement about all infinities in the universe, but since he did, I'll point out that that is wrong, too. My example of "points in spacetime" works perfectly well for that. The article that I linked you to, the one on why the Borde, et al paper was less convincing than Craig had wanted, also points out that even if the universe is past-finite, there's nothing that prohibits a purely natural explanation --as the authors of the Borde, et al paper explicitly say when asked about the paper. Or really any physicist for that matter.

    Quote Originally Posted by Squatch
    Making the infinite uncountable as opposed to countable is simply a useful fiction.
    Support or retract.

    Quote Originally Posted by Squatch
    Because we can imagine an infinite breakdown of each point does not mean that it exists in actuality. In fact the point system itself doesn't even fit the limitations of Craig's argument. Point systems are idealizations, not actualities.

    Really it would seem we are re-living Zeno's paradox here with you affirming Zeno's tongue in cheek conclusion. These mathematical subdivisions represent a mathematical reality, which is fine, no one is arguing that a mathematical set formed by successive additions is problematic.
    1.) Points, to this day, represent one of the only physical things that human's know about that has never changed. They are in literally every theoretical explanation of the universe.

    2.) Zeno's paradox is just an example of radically misunderstanding infinity. Zeno simply misunderstands that you can sum an infinite number of quantities into a single finite quantity. In fact, if you parametrize time in the manner which Zeno suggests in his thought experiment, you get the correct, finite answer. If you request, I'll even do it for you; it's not hard to demonstrate. (Or you can read Wikipedia, they do it there).

    This is honestly a really basic fact of mathematics and physics that is taught to every single first year student of mathematics and physics. I'm not trying to attack you personally, Squatch, or say that you're stupid (because you're not), but to people who have even a small amount of training in the field of either mathematics or physics, this is really basic stuff. This is honestly why Clive and I are so flabbergasted at people's responses. We're not trying to be rude or mean, we simply cannot understand how anyone could not understand this after hearing the correct arguments or having taken these classes. Again, I emphasize, I'm really not trying to be mean or rude.

    Quote Originally Posted by Squatch
    I would also point out that in your example, we still have point 0, which is the point Craig is making. We don't get rid of the beginning in either example you point out.
    I'm confused, we're not talking about any "beginnings" in this part of the debate. You had asked me, at the end of post #64, for an example of set of "infinite successions" that were realized in Nature:

    "Can you point out which of these countable infinites [used in science] is formed by successive addition? "


    And I provided that example to rebut only that point. Everything that we know about the cosmos tells me that I can keep moving forward 1 meter ad infinitum. The collection of these points is an actual infinity and moreover its an actual infinity that is used in science. More strongly, there's an uncountably infinite number of these countable infinities!
    Last edited by GoldPhoenix; September 3rd, 2013 at 03:28 PM.
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  9. #109
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by Squatch347 View Post
    I would say that it is not. 1+1+1... is not formed by successive addition, it is all done simultaneously, right? Rather, 1+1=2, 2+1=3, 3+1=4... is formed by successive addition.
    See, this is what's bizarre to me. When you "form sets", you don't do it in a time-based process. The sets exist on their own. What you're doing with something like {n|n is a positive integer} is identifying which set you're talking about.

    Take the expression "1+2+3". Are all of these addition operations performed "simultaneously"?
    Take the expression (1/2) + (1/4) + (1/8) + ... + (1/2)^k + (1/2)^(k+1) + ... Are all of these addition operations performed "simultaneously"? I assure you that this sum is just as well-defined as the previous sum. I simply don't understand how "simultaneous" operations differ from "non-simultaneous" ones.



    Isn't that what GP's position was (certainly possible I was misunderstanding), but that a finite set can be infinitely divided to be an uncountable infinity.
    ...I don't see GP affirming anything like that. I think you're confusing two measures: length versus cardinality. A line segment of length 1 has infinitely many points on it. I don't think GP is at all committed to the proposition that a finite set can be "infinitely divided".
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by CliveStaples View Post
    See, this is what's bizarre to me.
    I think this confusion arises because you are only dealing with the mathematical representations, Craig's arguments is about that process in a physical and temporal universe. The sets you described are not performed successively in the mathematical sense right? There is no order of operations that says I add 1 and 2 together then add 3 and 4 together. In the process of actually doing the set, in the real universe, yes the mechanics are successive by necessity, but then we also never complete the set and never actually get an infinite result right?

    Quote Originally Posted by GoldPhoenix View Post
    "As Richard C. Tolman showed, the earlier cyclic model failed because the universe would undergo inevitable thermodynamic heat death.[1] However, the newer cyclic model evades this by having a net expansion each cycle, preventing entropy from building up. However, there remain major open issues in the model." -- Wikipedia
    This too then removes the possibility of having a cyclical model that is infinitely old in the past right? Because in that model we would have had an infinite number of cycles in the past, each with net expansion, which means that we should already be infinitely large.

    Quote Originally Posted by GP
    Before I comment on that, there's a website that does a decent job refuting Craig's interpretation of the Borde, et al paper, here.
    As is so common with internet debates, both sides end up re-hashing old objections. The linked post has several problems. First, its reference to the discussion with Valenkin supports Craig's point more than the poster's. Valenkin says, sure you can avoid a beginning, if you appeal to some pretty extreme assumptions. In a separate published paper he calls this approach "metaphysical cosmology." (A. Vilenkin, "Birth of Inflationary Universes," Physical Review D 27 (1983): 2854. See J. Hartle and S. Hawking, "Wave Function of the Universe," Physical Review D 28 (1983): 2960-75; A. Vilenkin, "Creation of the Universe from Nothing," Physical Letters 117B (1982): 25-28.)

    Due diligence comment here, I am relying heavily on Craig's response to these points in his academic work in formulating my reply.

    I think the appeal to the Hawking Hartle model is more interesting, though I think he misstates it a bit. He is correct that the HH model "rounds off" the starting point to avoid a definitive starting point, a T=0 moment as it were. This definitely implies there is not a beginning point , not that there isn't a beginning. This type of universe comes into being just as it has in the classical model, it just avoids an infinitely dense singularity in doing so. To quote the author's themselves: "the amplitude for the Universe to appear from nothing... [it] would quite literally be created out of nothing: not just out of the vacuum, but out of absolutely nothing at all, because there is nothing outside the universe."
    Hartle and Hawking, "Wave Function of the Universe," p. 2961; Hawking and Penrose, Nature of Space and Time, p. 85.

    Going from a state of "not existing" to "existing" is by definition a beginning. Now it is certainly possible, and unclear, if that happens at a T=0 point or if it happens over a fuzzy or imaginary construct around that. But nevertheless, there is a finite number of planck instants between now and that state change, that is what is meant by the universe being finitely old.

    I won't pretend to fully understand the imaginary time implication, but I'll rely on Hawking's discussion of the subject when he says: "Only if we could picture the universe in terms of imaginary time would there be no singularities . . . . When one goes back to the real time in which we live, however, there will still appear to be singularities." Hawking, Brief History of Time, pp. 138-39.

    The use of imaginary time is more of a non-realistic thought experiment than a real application. Hawking again: "I take the positivist viewpoint that a physical theory is just a mathematical model and that it is meaningless to ask whether it corresponds to reality." (Stephen Hawking, "The Objections of an Unashamed Positivist," in The Large, the Small, and the Human, by Roger Penrose [Cambridge: Cambridge University Press, 1997], p. 169).)

    Craig goes on to state that all Quantum Gravity models rely on imaginary time and as such are more thought experiment than actual reality. He compares this to electron quantum tunneling in Euclidean space where imaginary time is used to clarify the effect, but that no one actually maintains that that device accurately describes reality. I'll leave this analysis to you, since I am not qualified to fully defend it.

    Quote Originally Posted by GP
    3.) Also, the point about hydrogen consumption in stars is concretely ruled out as a valid argument. This is why I was trying to say before that having a big bang before star formation is a crucially important piece of information that invalidates that line of reasoning.
    My apologies if it came across that I disagree with this point. I think this is correct. Allow me to rephrase it and I think we'll have an agreement.

    In any given cycle of the universe, a finite amount of hydrogen and observed star formation indicates that there is a finite amount of time between the observation and the cycle "reset."

    The question is whether or not we can assume an infinite number of cycles.

    Quote Originally Posted by GP
    Support or retract.
    Perhaps it was inaccurate for me to say that it is a "useful fiction," I'll retract that comment and say rather that an uncountable infinite in the manner you describe is materially different than the argument Craig is making.

    I would say this for a couple of reasons.

    1) Uncountable infinites are formed not by successive addition, but by successive division and thus not relevant to the point.

    2) This type of set still has a beginning, which is Craig's point, there is still an X=0.

    Quote Originally Posted by GP
    1.) Points, to this day, represent one of the only physical things that human's know about that has never changed. They are in literally every theoretical explanation of the universe.
    I agree, but they are assumptions underlying those theories and models of an ideal rather than a material thing right?

    Quote Originally Posted by GP
    2.) Zeno's paradox is just an example of radically misunderstanding infinity. Zeno simply misunderstands that you can sum an infinite number of quantities into a single finite quantity.
    No need, you are absolutely correct on this point and I'll retract the Zeno comment above.

    Quote Originally Posted by GP
    This is honestly why Clive and I are so flabbergasted at people's responses.
    I understand, I think the problem we are having is that the arguments being posited by you and Clive either seem to assume an infinite to prove an infinite or rely on a bit of a strawman of Craig's actual argument, imo.

    Quote Originally Posted by GP
    Everything that we know about the cosmos tells me that I can keep moving forward 1 meter ad infinitum. The collection of these points is an actual infinity and moreover its an actual infinity that is used in science.
    By "collecting" these points you are removing the successive addition of the claim though. If you actually move forward 1 meter at a time, you never actually cross an infinite distance, the distance crossed at any point is still finite. It approaches infinity as T approaches infinity, but never actually equals infinity.
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by Squatch347 View Post
    I think this confusion arises because you are only dealing with the mathematical representations, Craig's arguments is about that process in a physical and temporal universe. The sets you described are not performed successively in the mathematical sense right? There is no order of operations that says I add 1 and 2 together then add 3 and 4 together. In the process of actually doing the set, in the real universe, yes the mechanics are successive by necessity, but then we also never complete the set and never actually get an infinite result right?
    The "then" doesn't make sense to me. 1+2 is another way of writing "3". You could consider addition as an operator, something like (1+2)+3 means "add 1 and 2 together, then add the result to 3". But (1+2)+3 doesn't "first" equal 3 and "then" equal 6 "after" the second addition; it's just 6.

    I'll say this: if you start at time t, and every k seconds you "add" an element to a finite set S, there is no t' such that at t', S is infinite.

    But where's the contradiction? If the universe had no beginning, then there is no "t" at which you being adding elements to S. So ultimately this argument relies on already accepting that the universe had a beginning in the finite past.


    I don't know what "the process of actually doing the set" means. Sets are mathematical objects, not real objects. You don't ever "actually" "do" anything to a set. When you add 1 and 2 together, you aren't "doing" anything to 2. You aren't changing or affecting "1" and "2". You're just describing a different number, specifically the number 1+2 (also written "3").
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by CliveStaples View Post
    I'll say this: if you start at time t, and every k seconds you "add" an element to a finite set S, there is no t' such that at t', S is infinite.
    This is really the point of the argument, so why don't we stick with this agreement?

    Quote Originally Posted by CS
    But where's the contradiction? If the universe had no beginning, then there is no "t" at which you being adding elements to S. So ultimately this argument relies on already accepting that the universe had a beginning in the finite past.
    Approach it from the other direction then. We have a T=Now and we can successively count backwards from this point and lets apply your point above (I'll change the variables).

    If you start at time [now], and every [1] seconds you "add" an element to a finite set [length of the past], there is no [past point] such that at [past point], [length of the past] is infinite.

    Variable substitution.

    t --> Now as the starting point to establish our set.

    S ----> The set is "temporal length of the past."

    t'-----> any point in the past.

    k ---> arbitrary division of time, 1 second.
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by Squatch347 View Post
    This is really the point of the argument, so why don't we stick with this agreement?

    Approach it from the other direction then. We have a T=Now and we can successively count backwards from this point and lets apply your point above (I'll change the variables).

    If you start at time [now], and every [1] seconds you "add" an element to a finite set [length of the past], there is no [past point] such that at [past point], [length of the past] is infinite.

    Variable substitution.

    t --> Now as the starting point to establish our set.

    S ----> The set is "temporal length of the past."

    t'-----> any point in the past.

    k ---> arbitrary division of time, 1 second.
    None of this supports Craig's assertion--unless he's merely claiming that a finite union of finite sets is finite. But that would be insufficient to show that the past cannot be infinite.

    Consider the set defined as S = {x| x is a point in spacetime in the actual universe, where x is 1 second previous to another element in S}, with the stipulation that the "now" point in spacetime is included in S. This is essentially the set containing "now", and every second previous to now. The question is whether this set is infinite. You're answering a different question, which is whether "now" along with a finite number of elements in S forms an infinite set. Those are not the same question.
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by Squatch347 View Post
    This is really the point of the argument, so why don't we stick with this agreement?

    Approach it from the other direction then. We have a T=Now and we can successively count backwards from this point and lets apply your point above (I'll change the variables).

    If you start at time [now], and every [1] seconds you "add" an element to a finite set [length of the past], there is no [past point] such that at [past point], [length of the past] is infinite.

    Variable substitution.

    t --> Now as the starting point to establish our set.

    S ----> The set is "temporal length of the past."

    t'-----> any point in the past.

    k ---> arbitrary division of time, 1 second.
    Not weigh into your guys debate too heavily, but Clive already countered this very well with the question he asked in post #111:

    But where's the contradiction? If the universe had no beginning, then there is no "t" at which you being adding elements to S. So ultimately this argument relies on already accepting that the universe had a beginning in the finite past.


    I really cannot overstate or overemphasize the importance of his question. Before you guys get lost in a discussion about unions of finite sets (Technically, it's integrally related to this, but I doubt that people without mathematical training have the intuition for that kind of argument), you should really address that. Or, to put it another way:

    In a universe which is past-infinite, there is no such thing as a "first moment". Therefore, it is categorically erroneous/conceptually confused/ill-defined/illogical to make the statement 'It would have taken an infinite amount of time to traverse from the initial moment to now'. There was no initial moment. It is quite literally like talking about the 'number blue' or the 'color one'.

    I really cannot overstate that point. I'll also comment that you already accept this as true, Squatch, just in a different context:

    Quote Originally Posted by Squatch
    By "collecting" these points you are removing the successive addition of the claim though. If you actually move forward 1 meter at a time, you never actually cross an infinite distance, the distance crossed at any point is still finite. It approaches infinity as T approaches infinity, but never actually equals infinity.
    Yes. Exactly. Let's separate this from the issue of having these points (Which I still contend is an infinite set of points formed by successive addition, but that's not important for the following). There's no "end-point" to something is infinitely long (either as a time span or a spatial length). Now just flip directions.

    I can keep on going backwards, and I will never reach infinity. There's no end-point; I can keep on going backwards for an arbitrarily long time. Now just change the context; a past-infinity (an infinite amount of time in the past) means that I can keep on going backwards in time ad infinitum. I can go backwards 10 seconds, then another 10 seconds, again and again. And I'll never reach the beginning, because there's no beginning to reach. Just like there's no end in a spatial infinity.


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    Back to Squatch and I's debate:

    Quote Originally Posted by Squatch347 View Post
    I think this confusion arises because you are only dealing with the mathematical representations, Craig's arguments is about that process in a physical and temporal universe. The sets you described are not performed successively in the mathematical sense right? There is no order of operations that says I add 1 and 2 together then add 3 and 4 together. In the process of actually doing the set, in the real universe, yes the mechanics are successive by necessity, but then we also never complete the set and never actually get an infinite result right?
    I wouldn't think of it that way. I would request you to really think about the following thought experiment that I'm going to provide.

    I think the best way to understand "infinity" is that it means you can do something an arbitrary number of times. It doesn't matter if it cannot reach infinity; the entire point is that it cannot reach infinity because being infinitely long means that it has no boundary or ending point, including infinity (which is not a part of space, there is no point in space that is infinitely far away from you). I think, from what you're telling me and Clive, that you're worried about there being no boundary point (either an "initial time" or a "ending point" in space), but the whole point of infinity is the definitive lack of such a thing. So you can travel arbitrarily far and never reach reach the end.


    1.) The mere fact that "I can go arbitrarily far away from where I am currently" is the statement that "Space is infinite in length" (And, btw, I can take an infinite number of integer 1 meter points in space and that forms an actual infinity). Try not to think about actually traversing the distance; it cannot reach the end-point anyways, the point is that it can keep on going.

    2.) Not being able to tell me that "There's a specific point that a light beam (Or pick any moving object you like) cannot reach at some point in time." === You are admitting that "Space is infinite".

    Proof: If there's no point that the light beam has to stop at, then space just keeps on going forever. Not having an ending point == spatially infinite (In the direction it's moving in, anyways).

    If space is infinitely long, I'm free to take collections of an infinite number of points (because you've now admitted that they must exist) which can be derived by successive addition {1 meter, 2 meters, 3 meters, ..., N meters, ....} just like time can, {1 second ago, 2 seconds ago, 3 seconds ago, ...}.


    There's some subtleties to this with this in an expanding spacetime, but using every piece of empirical data that we currently have, it all strongly indicates that we live in an infinitely large universe.

    Quote Originally Posted by Squatch
    This too then removes the possibility of having a cyclical model that is infinitely old in the past right? Because in that model we would have had an infinite number of cycles in the past, each with net expansion, which means that we should already be infinitely large.
    No, it's infinitely long and cyclic. The net expansion I think just asymptotes to a constant, although to be fair I haven't read the details of the model, but no, as a mathematical details if the net expansion asymptotes to a constant then everything would work out fine. There are also other possible outs that I don't have off the top of my head, but it's not necessarily true that a net expansion causes any problem.

    Again, there's also no "already" because, again, there's no starting point from which to compare. There's no "beginning" in a past-infinite universe. That's why Craig was trying to rule it out from the outset, because it would disagree with a creationist model.

    Quote Originally Posted by Squatch
    Due diligence comment here, I am relying heavily on Craig's response to these points in his academic work in formulating my reply.

    I think the appeal to the Hawking Hartle model is more interesting, though I think he misstates it a bit. He is correct that the HH model "rounds off" the starting point to avoid a definitive starting point, a T=0 moment as it were. This definitely implies there is not a beginning point , not that there isn't a beginning. This type of universe comes into being just as it has in the classical model, it just avoids an infinitely dense singularity in doing so. To quote the author's themselves: "the amplitude for the Universe to appear from nothing... [it] would quite literally be created out of nothing: not just out of the vacuum, but out of absolutely nothing at all, because there is nothing outside the universe."
    Hartle and Hawking, "Wave Function of the Universe," p. 2961; Hawking and Penrose, Nature of Space and Time, p. 85.

    Going from a state of "not existing" to "existing" is by definition a beginning. Now it is certainly possible, and unclear, if that happens at a T=0 point or if it happens over a fuzzy or imaginary construct around that. But nevertheless, there is a finite number of planck instants between now and that state change, that is what is meant by the universe being finitely old.

    I won't pretend to fully understand the imaginary time implication, but I'll rely on Hawking's discussion of the subject when he says: "Only if we could picture the universe in terms of imaginary time would there be no singularities . . . . When one goes back to the real time in which we live, however, there will still appear to be singularities." Hawking, Brief History of Time, pp. 138-39.
    This is all fine and well, and Hawking is a smart guy to say the least, but the Hawking-Hartle calculation is strictly done in terms of General Relativity. If we run the clock backwards, we shrink our universe as we go back to that initial instant; however, between now and where the singularity actually occurs, we go to sizes that are too small and General Relativity no longer remains a valid descriptor of our universe. This means that we cannot make predictions that we have any right to claim are accurate. While it is true that if you take GR as a literally true theory, you can get something like the Hawking-Hartle universe and have a universe with a finite past. But, unfortunately, we live in a universe where quantum mechanics is true and you cannot ignore quantum corrections to General Relativity, which invalidate General Relativity before you can reach the singularity that they're speaking of.

    Quote Originally Posted by Squatch
    The use of imaginary time is more of a non-realistic thought experiment than a real application. [...] Craig goes on to state that all Quantum Gravity models rely on imaginary time and as such are more thought experiment than actual reality. He compares this to electron quantum tunneling in Euclidean space where imaginary time is used to clarify the effect, but that no one actually maintains that that device accurately describes reality. I'll leave this analysis to you, since I am not qualified to fully defend it.
    The short story is that Craig is misunderstanding physics. I understand why Craig would think this; it's a subtle distinction that I wouldn't expect anyone but an expert in the field to understand. If you want more than that, then here's the long story:

    1.) We use imaginary time all of the time (we call it "Wick rotation"); it doesn't render our work to a "thought experiment". It's literally just a very powerful mathematical procedure (Using a concept called "analytic continuation") which allows us to do a calculation in imaginary time (Which improves the mathematical properties of integrals) and then translate the answer, consistently, back into "real" time. The best example that I can give for why this is absolutely not "a thought experiment" is that physicists used imaginary time tricks in computing from first principles --correctly to all known decimal places-- the mass of a proton from the fundamental theoretical physics of quarks and gluons. It was one of the first groundbreaking results in the field of physics that numerically solves quantum field theory about 10 to 20 years ago.

    2.) When physicists use the word "quantum gravity", we actually mean two different things. Most of the time, people don't ask us about these things and don't need to understand this at all so being lazy about what we mean causes no confusion. There are two quantum gravities:

    Quantum Gravity A: The quantum description of gravity that is below the Planck scale (The scale at which quantum effects on gravity become too large to consistently neglect).

    Quantum Gravity B: The quantum description of gravity that is at and above the Planck scale.


    Contrary to what you might have heard on Nova or read in books on String Theory, we know pretty much everything that we could possibly want to know about "Quantum Gravity" --so long as we're sufficiently below the Planck scale (Distances larger than about .00000000000000000000000000000001 cm).

    It's actually quite remarkable what we know, in fact. We know that Quantum Field Theory predicts General Relativity as the unique candidate of gravity (Unless you want to add stuff on top of GR, which is allowed but so far not experimentally observed, but it doesn't change the fundamental features of General Relativity). As it's often said in the field, "If Einstein hadn't discovered General Relativity, then particle physicists would have discovered it 50 years later out of necessity." We know how to use Quantum Field Theory to generate quantum corrections to the predictions made by General Relativity (This is how we know that GR is absolutely no good above the Planck scale, because these corrections explode). We use the "imaginary time trick", as we usually do in QFT, to compute quantum mechanical corrections to GR. This is actually most of what Hawking is known for doing in physics: Hawking-Bekenstein entropy, the Hawking-Moss instanton, Hawking radiation, etc. However:

    A.) Imaginary time here is perfectly fine, it's just a mathematical trick like when physicists computed the mass of a proton.
    B.) Even supposing that it weren't, this quantum theory of gravity isn't the correct one once we go up to the Planck scale (Which is the one that matters when we talk about the beginning of the universe). We don't know what that theory is, but it's certainly not just a "thought experiment", as it will fundamentally describe Nature.


    Quote Originally Posted by Squatch
    My apologies if it came across that I disagree with this point. I think this is correct. Allow me to rephrase it and I think we'll have an agreement.

    In any given cycle of the universe, a finite amount of hydrogen and observed star formation indicates that there is a finite amount of time between the observation and the cycle "reset."

    The question is whether or not we can assume an infinite number of cycles.
    I agree, but I'd like to emphasize an issue. The question is whether or not it is possible to assume that there is an infinite number of cycles. It is certainly possible. Therefore, Craig is wrong --independent of any philosophic argument he'd like to state that infinities cannot be real.

    Quote Originally Posted by Squatch
    Perhaps it was inaccurate for me to say that it is a "useful fiction," I'll retract that comment and say rather that an uncountable infinite in the manner you describe is materially different than the argument Craig is making.

    I would say this for a couple of reasons.

    1) Uncountable infinites are formed not by successive addition, but by successive division and thus not relevant to the point.

    2) This type of set still has a beginning, which is Craig's point, there is still an X=0.
    Not exactly how I'd have phrased it, but additive or divided, there's an uncountable number of them.

    Quote Originally Posted by Squatch
    I agree, but they are assumptions underlying those theories and models of an ideal rather than a material thing right?
    Personally, I would argue that points have to be physical, even if for some reason there's a finite number of them, countably finite number of them, or an uncountably infinite number of them. They're the building blocks of geometry. Basically, "geometry" + "quantum fields" == the fundamental building blocks of physics. The only part we properly don't understand is the "quantum gravity" bits, which is (Because gravity and geometry are tied up together as one) basically the issue of "quantum geometry", but either way they are quite intrinsic to physics.

    Quote Originally Posted by Squatch
    No need, you are absolutely correct on this point and I'll retract the Zeno comment above.
    Cheers.

    Quote Originally Posted by Squatch
    I understand, I think the problem we are having is that the arguments being posited by you and Clive either seem to assume an infinite to prove an infinite or rely on a bit of a strawman of Craig's actual argument, imo.
    That's not the issue, in my opinion. Craig is arguing that it is impossible for infinity to be realized in Nature. Since he cannot mean that it is "logically impossible", he means that it is "nomologically impossible" (Contradicts the laws of Nature). And that's a claim which simply wrong; it is certainly possible within the known laws of physics (albeit with modifications, but consistent modifications nevertheless) to have a cyclic universe. A cyclic universe is both possible within the known laws of physics and has no beginning. The mere possibility obviously refutes his claim that it is impossible.

    There's no issue of "assuming" that there's an infinity. By Craig saying that it is impossible, he means that it is necessarily inconsistent. True or not, if I assume that it is possible and this does not necessarily entail a contradiction, it means that it is not impossible precisely because it is possible. Providing a counter example that shows that it is even possible --ignoring if it's actually true or false-- is sufficient to render Craig's claim false. Saying something is "impossible" is a strong, strong claim.
    "Those who can make you believe absurdities, can make you commit atrocities." --Voltaire

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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by GoldPhoenix View Post
    There's some subtleties to this with this in an expanding spacetime, but using every piece of empirical data that we currently have, it all strongly indicates that we live in an infinitely large universe.
    Tangent, but what precisely is meant by "expanding universe"? Is there a center to the expansion? My thinking goes like this: if space is expanding, then points in spacetime are moving. But moving in reference to what?

    [/quote]The short story is that Craig is misunderstanding physics. I understand why Craig would think this; it's a subtle distinction that I wouldn't expect anyone but an expert in the field to understand. If you want more than that, then here's the long story:

    1.) We use imaginary time all of the time (we call it "Wick rotation"); it doesn't render our work to a "thought experiment". It's literally just a very powerful mathematical procedure (Using a concept called "analytic continuation") which allows us to do a calculation in imaginary time (Which improves the mathematical properties of integrals) and then translate the answer, consistently, back into "real" time. The best example that I can give for why this is absolutely not "a thought experiment" is that physicists used imaginary time tricks in computing from first principles --correctly to all known decimal places-- the mass of a proton from the fundamental theoretical physics of quarks and gluons. It was one of the first groundbreaking results in the field of physics that numerically solves quantum field theory about 10 to 20 years ago.[/quote]

    From what I understand, Wick rotation relies on a correspondence between real-valued equations in Minkowski space and complex-valued equations (at least in one variable) in Euclidean space. If this is true, then using "imaginary time" to solve physical problems in reality is no different than using "integer arithmetic" to count the number of apples in a basket: they rely on a correspondence between a mathematical object/process and an actual object/process.

    2.) When physicists use the word "quantum gravity", we actually mean two different things. Most of the time, people don't ask us about these things and don't need to understand this at all so being lazy about what we mean causes no confusion. There are two quantum gravities:

    Quantum Gravity A: The quantum description of gravity that is below the Planck scale (The scale at which quantum effects on gravity become too large to consistently neglect).

    Quantum Gravity B: The quantum description of gravity that is at and above the Planck scale.


    Contrary to what you might have heard on Nova or read in books on String Theory, we know pretty much everything that we could possibly want to know about "Quantum Gravity" --so long as we're sufficiently below the Planck scale (Distances larger than about .00000000000000000000000000000001 cm).
    Reading your post, I got a little confused about whether you were talking about measurements above vs. below the Planck scale, so if you'll indulge me, I'm going to try to keep track of that now.

    Here, you're saying, "If we're below the Planck scale, we are really familiar with how quantum gravity works." Right?

    It's actually quite remarkable what we know, in fact. We know that Quantum Field Theory predicts General Relativity as the unique candidate of gravity (Unless you want to add stuff on top of GR, which is allowed but so far not experimentally observed, but it doesn't change the fundamental features of General Relativity). As it's often said in the field, "If Einstein hadn't discovered General Relativity, then particle physicists would have discovered it 50 years later out of necessity." We know how to use Quantum Field Theory to generate quantum corrections to the predictions made by General Relativity (This is how we know that GR is absolutely no good above the Planck scale, because these corrections explode). We use the "imaginary time trick", as we usually do in QFT, to compute quantum mechanical corrections to GR. This is actually most of what Hawking is known for doing in physics: Hawking-Bekenstein entropy, the Hawking-Moss instanton, Hawking radiation, etc.
    This confused me. If quantum effects become larger as the scale becomes smaller (especially once you go below the Planck scale), shouldn't the quantum corrections to GR grow large in magnitude when you're below, not above, the Planck scale?

    However:

    A.) Imaginary time here is perfectly fine, it's just a mathematical trick like when physicists computed the mass of a proton.
    B.) Even supposing that it weren't, this quantum theory of gravity isn't the correct one once we go up to the Planck scale (Which is the one that matters when we talk about the beginning of the universe). We don't know what that theory is, but it's certainly not just a "thought experiment", as it will fundamentally describe Nature.
    Oooooooooooooooooooooooh. I think I actually understand now. When you say "up to" or "above", you're using upward movement in reference to going to smaller scales. I think. Or something.

    And your (B) is basically what I said above. The only reason physicists use math at all is because the math is useful to describe something physical/real. Sometimes the math will only involve integers (say, when counting chromosomes), sometimes it will involve things like rotations and reflections (which aren't "numbers" at all, in the typical sense), and sometimes it will involve algebraically-closed fields--e.g., the complex numbers.
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by CliveStaples View Post
    Tangent, but what precisely is meant by "expanding universe"? Is there a center to the expansion? My thinking goes like this: if space is expanding, then points in spacetime are moving. But moving in reference to what?
    1.) The definition of length (The infinitesimal Pythagorean theorem/the Arclength Integral) at a given point in time is a function of time (A monotonically increasing function of time, as it were). This means that objects' distance grows with time. Put into a formula, when you're at an instant in time:

    dl^2 = a(t)^2 [ dx^2 + dy^2 + dz^2 ]

    is the simple case with uniform matter distribution, no radiation, no spatial curvature, and no comsological constant. With those you can't use rectangular coordinates anymore (You have to use spherical), but you catch my drift. In that case, a(t) in General Relativity is predicted to be a(t) = (t/T)^(2/3), where T is, up to numerical factors that I don't recall, is related to one over the Hubble parameter.


    2.) There's no center. Take any uniformly expanding space (like a sheet of rubber being stretched), and each point thinks that all the other points are moving away from it at increasingly faster speeds the farther the other points are away from them. But if you move to any other point on the surface of the rubber sheet, then they'll say the same thing. This means there's no center because everyone thinks they're the center away from which everything is expanding. The salient point is that this is possible when everywhere is expanding equally (Well, I should be careful, galaxies are expanding away from each other at approximately equal amounts up to relatively small corrections do to some galaxies being more massive, etc). This is also an empirical fact of our universe.

    Quote Originally Posted by Clive
    From what I understand, Wick rotation relies on a correspondence between real-valued equations in Minkowski space and complex-valued equations (at least in one variable) in Euclidean space. If this is true, then using "imaginary time" to solve physical problems in reality is no different than using "integer arithmetic" to count the number of apples in a basket: they rely on a correspondence between a mathematical object/process and an actual object/process.
    It does, yeah, more or less. It's a bit subtle because it uses analytic continuation which says we're allowed to do it for non-trivial reasons, unlike {1} <--> 1 Apple. Although I guess that depends one what one thinks is trivial.

    Quote Originally Posted by Clive
    Reading your post, I got a little confused about whether you were talking about measurements above vs. below the Planck scale, so if you'll indulge me, I'm going to try to keep track of that now.

    Here, you're saying, "If we're below the Planck scale, we are really familiar with how quantum gravity works." Right?
    I was internally debating with myself to say this or not, but the Planck scale is an energy scale. Due to the ideas most idealistically captured by de Broglie, energy scales correspond with inverse length scales. Therefore, "Large energy scales correspond to small length scales" and "Low energy scales correspond to large length scales."

    More mathematically, "10^N [Length Scale]" <==> "[Energy Scale] == 10^-N { Planck's Constant * Speed of Light / [Length Scale] }". The factor is quite important, because it's what allows the scales to be related. In classical physics, there's no real correspondence because you're required to use Planck's constant to make the equivalence of energy and inverse length scales, but Planck's constant only appears in quantum corrections. (Except in strongly coupled quantum systems, but in that case the classical description is 100% wrong)

    Anyways, this is a long way of saying: "The Planck scale is tremendously huge <==> The associated Planck length scale is tiny."

    Quote Originally Posted by Clive
    This confused me. If quantum effects become larger as the scale becomes smaller (especially once you go below the Planck scale), shouldn't the quantum corrections to GR grow large in magnitude when you're below, not above, the Planck scale?
    They grow with energy. Energy scales go as inverse length scales. In reverse, this means smaller lengths means bigger quantum corrections.


    Quote Originally Posted by Clive
    And your (B) is basically what I said above. The only reason physicists use math at all is because the math is useful to describe something physical/real. Sometimes the math will only involve integers (say, when counting chromosomes), sometimes it will involve things like rotations and reflections (which aren't "numbers" at all, in the typical sense), and sometimes it will involve algebraically-closed fields--e.g., the complex numbers.
    Well, yeah. Math is a language; physicists use it to describe Nature.
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    Re: WLC's Argument Against an Actual Infinity

    I'm coming in late, I'm butting into the middle of the convo, I've not read most of this discussion, and I'm certainly not staying for a full on, verbose debate.

    That being said, this caught my attention:

    Quote Originally Posted by GP
    Or, to put it another way:

    In a universe which is past-infinite, there is no such thing as a "first moment". Therefore, it is categorically erroneous/conceptually confused/ill-defined/illogical to make the statement 'It would have taken an infinite amount of time to traverse from the initial moment to now'. There was no initial moment. It is quite literally like talking about the 'number blue' or the 'color one'.
    Agreed 100%.

    However, this only applies if the universe actually is past-infinite. And it only applies as a correction to something someone said if they argued that it was past-infinite. Else, it's circular reasoning as "the other team" has been arguing against a past-infinite. That's the whole issue here really: "Does the universe have a beginning or not?" If it does, then it would seem that actual infinities cannot exist. If it doesn't, then this is problematic (in as far as the context that WLC argues against sequentially measured entities as he defines time as such).

    Did I miss the context of the passage in question or something?
    Last edited by Apokalupsis; September 6th, 2013 at 07:27 AM.
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  20. #118
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by Apokalupsis View Post
    I'm coming in late, I'm butting into the middle of the convo, I've not read most of this discussion, and I'm certainly not staying for a full on, verbose debate.

    That being said, this caught my attention:


    Agreed 100%.

    However, this only applies if the universe actually is past-infinite. And it only applies as a correction to something someone said if they argued that it was past-infinite. Else, it's circular reasoning as "the other team" has been arguing against a past-infinite. That's the whole issue here really: "Does the universe have a beginning or not?" If it does, then it would seem that actual infinities can exist. If it doesn't, then this is problematic (in as far as the context that WLC argues against sequentially measured entities as he defines time as such).

    Did I miss the context of the passage in question or something?
    I'll assume you meant "If it does, then i would seem that actual infinities can't exist." I don't see how that follows; a universe could be infinitely large but have a finite beginning. A universe could have finite volume but have infinite divisibility, like a fractal with structure at arbitrarily small scales: the Koch snowflake encloses a finite area but has infinitely many "peaks", and an infinitely-long perimeter. A finitely-large universe could contain an infinitely-long string.

    Could you go through WLC's arguments about how "The universe does not have a beginning" is problematic? The "sequentially measured" arguments are irrelevant, so far as I can tell. Yes, if you start adding finitely many elements to a finite set, there is no point at which the set is infinite. But that doesn't contradict any claim; nobody is claiming, "If you start adding finitely many elements to a finite set, there is indeed a point at which that set is infinite."

    (This of course is making certain assumptions about what is meant by "there is no point at which that set is infinite"; technically, I'm interpreting this to mean, "In any sequence {Sn}, where |S0| is finite and Sk+1 = Sk Tk, where |
    Tk| is finite, there is no m such that Sm is infinite." This statement is provably true.)




    Hmm. I have two thoughts. GP has claimed something along the lines, "If we form a set that includes all points in spacetime that are 1-meter increments in a single direction away from me, that set is actually infinite and contains only 'things' that actually exist in the universe--specifically, points in spacetime." So, two things:

    1. What about the set of all points in spacetime that are 1 meter away from you? This should also be infinite, if for every 0 <= θ <= 2π and 0<= φ<= 2π there is a point in spacetime (x0 + (cos θ)(sin φ), y0 + (sin θ)(sin φ), z0 + cos φ, t0) where (x0,y0,z0,t0) is your coordinate. In fact, any region of spacetime with non-zero length/area/volume/Lebesgue measure will contain an infinite number of points in spacetime.

    2. Another way of applying your idea, using the convention I described above in my response to Apok, might be the following:

    Nobody is claiming that some Sk is infinite. Rather, the claim is that the set n Sn is infinite [technical note: this statement is provably true if and only if there does not exist a k such that for all n > k, Tn is empty] and all of its elements actually exist.

    The point in (2) gets at part of my frustration in this argument. The question is whether an "actual infinite" exists; one way of showing that an "actual infinite" exists would be to exhibit a set that is: (i) infinite (in the sense of cardinality); and (ii) contains only elements that "actually" exist. Showing that some particular set satisfies (ii) but not (i) (such as arguing that no set in the sequence {Sn} is infinite) is insufficient to show that every set that satisfies (ii) cannot satisfy (i).

    Imagine you were trying to figure out whether there are finitely many positive integers, or infinitely many. You'd be looking for a set S that satisfies: (i) S is infinite; (ii) S contains only elements that are positive integers.
    You then prove that no set in the sequence defined by Sk = {1,2,3,...,k}--that is, the sequence {Sn} = {{1}, {1,2}, {1,2,3}, ... , {1,2,3, ..., k}, ...}--is infinite. Yes, all of the sets in that sequence are finite. But none of them are the set of all positive integers! The set of positive integers is equal to
    n Sn, and it is indeed infinite.

    To relate this to the current discussion: Proving that the sequence defined by Sk = {the kth year previous to now} contains only finite sets doesn't show that the set S = {all years previous to now} is finite.
    Last edited by CliveStaples; September 6th, 2013 at 07:02 PM.
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  22. #119
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    Re: WLC's Argument Against an Actual Infinity

    Quote Originally Posted by CliveStaples View Post
    I'll assume you meant "If it does, then i would seem that actual infinities can't exist."
    Correct. I've since edited the post, thank you.

    I don't see how that follows; a universe could be infinitely large but have a finite beginning. A universe could have finite volume but have infinite divisibility ,like a fractal with structure at arbitrarily small scales; the Koch snowflake encloses a finite area but has infinitely many "peaks", and an infinitely-long perimeter; a finitely-large universe could contain an infinitely-long string.

    Could you go through WLC's arguments about how "The universe does not have a beginning" is problematic?
    Well, that was explained in the last statement I made in the post.

    If it doesn't, then this is problematic (in as far as the context that WLC argues against sequentially measured entities* as he defines time as such).


    or rather "entities formed by successive addition" or "measured by equal temporal intervals".

    He argues that counting down from infinity can't work (yet this is a consequential position of having an infinite past).


    (P) IF (i) the temporal series of all past events is actually infinite in its duration (as measured by equal temporal intervals), THEN (ii) there COULD be some mind/clock/counting machine/computer/angel/god which would SUCCESSIVELY pair all the past equal intervals (say, seconds) to all negative whole numbers in the corresponding order.

    The series of past events could have the order type ω* + ω*, the order type exemplified by . . . , -3, -2, -1, . . . , -3, -2, -1. But it's obvious why proponents of the infinite past don't adopt this view: for then there are past events which lie at an infinite distance from the present. But then one could never traverse the infinite distance from, say, the first -3 to the second -3. That's why proponents of the infinite past always insist that the existence of an infinite past doesn't entail an infinitely distant starting point. The Tristram Shandy paradox about a man writing his autobiography so slowly that it takes him a year to record the events of a single day challenges the assumption that an infinite past would have the order type ω*. The only hope for proponents of the infinite past is to insist that the series of past events has the order type ω* so that every event lies at only a finite distance from the present. In that way, forming an infinite past by successive addition doesn't involve, they claim, traversing an infinite distance.

    http://www.reasonablefaith.org/count...#ixzz2e7xJNaZb
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    Re: WLC's Argument Against an Actual Infinity

    The series of past events could have the order type ω* + ω*, the order type exemplified by . . . , -3, -2, -1, . . . , -3, -2, -1. But it's obvious why proponents of the infinite past don't adopt this view: for then there are past events which lie at an infinite distance from the present. But then one could never traverse the infinite distance from, say, the first -3 to the second -3. That's why proponents of the infinite past always insist that the existence of an infinite past doesn't entail an infinitely distant starting point.
    My guess is that his argument here relies on the idea that for every interval of time, there's a "next most recent" interval of time, which this order type would contradict. What's the "next most recent" interval of time after the interval encoded by the first -1? Well, there isn't one.

    But of course that's true; the assumption that there's always a "next most recent" interval of time is an assumption about the order type of time, one that explicitly rejects the order type WLC is supposedly analyzing. So if you assume that the universe has order type
    ω*, it can't possibly have order type ω* + ω*.

    The Tristram Shandy paradox about a man writing his autobiography so slowly that it takes him a year to record the events of a single day challenges the assumption that an infinite past would have the order type ω*. The only hope for proponents of the infinite past is to insist that the series of past events has the order type ω* so that every event lies at only a finite distance from the present. In that way, forming an infinite past by successive addition doesn't involve, they claim, traversing an infinite distance.
    Okay, so I guess he wasn't using that assumption. I guess his point is that it's impossible to traverse an "infinite" time-distance. I don't see any reason to think this is true--certainly it's true for humans, who have finite lifespans. But why would it have to be true for God? What's the problem with the notion of an eternally-existing God who exists at all points in time in a past-infinite universe?


    ---------- Post added at 10:47 AM ---------- Previous post was at 10:32 AM ----------

    2.) There's no center. Take any uniformly expanding space (like a sheet of rubber being stretched), and each point thinks that all the other points are moving away from it at increasingly faster speeds the farther the other points are away from them. But if you move to any other point on the surface of the rubber sheet, then they'll say the same thing. This means there's no center because everyone thinks they're the center away from which everything is expanding. The salient point is that this is possible when everywhere is expanding equally (Well, I should be careful, galaxies are expanding away from each other at approximately equal amounts up to relatively small corrections do to some galaxies being more massive, etc). This is also an empirical fact of our universe.
    Let's say you've got a rubber pizza-shaped object, and you somehow uniformly pull out its edges. Isn't it true that the center point might not move (with respect to, say, the ground) but that every point on the rubber "thinks" every other point is moving away?

    I guess I'm having trouble thinking of what "center" even means in this context. Suppose your rubber pizza was rotating at uniform acceleration. Wouldn't there be a way to measure which points on the rubber were undergoing to least acceleration? If there's one point that always undergoes no acceleration, wouldn't that be the "center" of the pizza, even if it was uniformly expanding?

    How is this kind of expansion different from something like a simple dilation transformation on R2? Clearly something like f(x) = cx is an expansion/contraction (c>1 or 0
    I'm sure this is all super basic stuff that I should just read an undergrad text on, but whatever reading is gay.
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