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  1. #1
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    WLC is Wrong: The Argument for Actual Infinities

    Quick Preamble on the Kalaam Cosmological Argument

    As a part of an argument for the existence of God (known as the Kalaam Cosmological Argument), the Christian apologist WLC has argued that an actual infinity as realized in Nature is physically impossible. He argues that the the consequences of physical infinities are illustrated by Hilbert's Hotel. ODN now has many, many other threads on the Kalaam Cosmological argument (e.g. Clive's well-known thread against the KCA), and I encourage readers unfamiliar with the Kalaam Cosmological argument and Hilbert's Hotel to quickly peruse those threads before continuing on this one.

    The purpose of this thread is to conclusively demonstrate the falsity of WLC's assertion by constructing an explicit counter-example. This will be done by creating a physical system that:

    a.) Explicitly obeys the laws of physics.
    b.) Contains an actual infinity, and thus is a physical analog of Hilbert's Hotel.
    c.) In a finite amount of time, fully accommodates an additional "guests" (particle) into it's infinite hotel even though all of its rooms (occupied points) were previously full.

    Note that I am not arguing that such a system exists in the real world, the only point that I am making is that this system could exist in the physical world and if it did, it would break no physical laws and thus cause no physical contradiction. For simplicity, I'm choosing to work with Newtonian mechanics as my model for physical laws, but in my footnote at the bottom, I address why it's unlikely that there's any practical obstructions when incorporating more modern physical principles, such as quantum mechanics or relativity. (Note that I've previously explained that if you understand that quantum gravity might disallow the argument I'm giving here, then you've actually already ceded other premises of the KCA, but I leave this discussion for other threads. I'll ignore the full implications for the KCA, other than noting that if this argument is correct, then at least one central premise of the KCA must be false.)



    Challenge to support a claim.: If you disagree with me that the following system obeys the laws of physics, you need to explicitly explain to me which physical law has been broken in the following system. Note that "it violates common sense" is not a physical law.






    The Argument for the Possibility of Actual Infinities


    Let's begin with definitions for the Newtonian system:

    Def 1: A particle is an object with zero extent (it's a point), but has dynamics (it's a function of time, and so it can move around). At the first instant in time, the only things needed to specify the particle is its starting position, x0, and starting velocity, v0. I'm assuming no forces, so the time evolution of each particle is governed by the equation, x(t) = x0 + v0 t, for the position of the particle at all later times. This uniquely defines the point each particle sits at for all times t, such that t ≥ 0.

    Def 2: The momentum, p, of a particle is defined to be p = m v0, where m is the mass of a particle. The energy, E, of a particle is defined by E = 1/2 m v02.

    Ax 3: In Newtonian mechanics, the only criterion for the system to be physical is that in any finite volume with particles, all of the masses of each particle must be greater than zero, the total energy of the system must be finite, and the the total momentum must be finite. Thus if the system starts off with a well-defined set of initial positions and velocities, and in each finite region of space there's a finite amount of energy and momentum, then the system has not generated any physical contradiction according to classical mechanics. In other words, the system is entirely described with physical laws.

    So that's basic 100 level Newtonian mechanics.


    The Scenario: Hilbert's Hotel in a Finite Amount of Space

    Suppose this hypothetical universe starts out with an infinite number of particles. Again, classical mechanics has no objections to this (in fact, the consistency of infinite particles is used thoroughly in statistical mechanics to approximate systems of large particles), quite literally, the only question is "How many particles, where are they, and how fast are they moving?" As long as you can specify this data whilst keeping finite momentum and energy, there's no physical contradiction. So let's just pick their positions (Remember, I'm not doing this in real time, I'm saying that my initial time, this system happens to be in this configuration).

    Let's start all of them off in a finite interval, say [1,2). For the sake of the argument, let's take some specific length scale, for instance a meter, and just call it "1" in that length scale. To do this, start the particles off in the following positions (known as a geometric series):

    x1 = 1
    x2 = 1 + 1/2 = 1.5
    x3 = 1 + 1/2 + 1/4 = 1.75
    .
    .
    .
    xn = xn-1 + (1/n)2 = 2[1 - (1/2)n ]

    Here's a graphical representation of the positions of the first three particles:

    Click image for larger version. 

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    It's easy to verify that all particles will be at a point greater than or equal to 1 but always less than 2 (i.e. no particle sits at point "2", just arbitrarily close to it, on the interval known as [1,2) by mathematicians). However, each particle is also never lying on top of one another; for each n, each particle is at a distinct point. This is an infinite collection of points where each particle is located, but in a 1 unit distance interval.



    Causing the Hilbert "Paradox" in a Finite Amount of Time


    Now for the alleged paradox. It's been incorrectly conjectured many times now that it would take an infinite amount of time to accommodate the Hilbert Hotel, and this might be resolution to the paradox; it's also been stated that you could never physically accommodate another particle in Hilbert's Hotel (the infinite points on the interval [1,2) which occupy a particle). Let's prove that neither of these assertions are correct. Suppose I give you another particle, let's say at point 0. Let's show that it is physically possible to accommodate the new particle inside of "infinite hotel" in a finite amount of time, and saliently with finite total energy and finite total momentum on the interval. In other words, where no physical contradiction has occurred Specifically, we'll start off with an infinite number of particles occupying an infinite number of points inside the finite interval [1,2) with a single particle lying outside that interval, and then evolve the system forward a unit of time until there's now be "infinity + 1" particles inside of the finite interval, but also with all of the same previous states occupied.

    We also control the initial velocities of these particles, so after a given interval of time, we can uniquely pick each particles next position and that specifies the initial velocity. It's rather simple to solve the set of evolution equations xn(t) = xn vn. Thus, for our setup, we want each particle to move forward by xn -> xn+1 after the allotted finite period of time. So, we require that each particle moves forward by an amount xn+1 - xn = 1/n^2. This means that each velocity needs to be:

    vn = (1/n)2

    So after 1 second, each particle will have moved forward to the next position, particle n moves to particle (n+1)'s spot. This completely defines the system, and the total energy & momentum can trivially be evaluated (let's assume they all have the same mass, m) as

    E = Σ0 1/2 m vn2 = 2/3 m [unit distance]^2/[unit time]^2,

    which is finite (I used the above formula for computing infinite geometric series, which can also be found in the Wikipedia article). The particle at x=0 also has a velocity (which is just 1 [unit distance]/[unit time]), so add 1/2m to get the total energy, including the particle that starts off outside of the "hotel." The total momentum can be more easily calculated with the same formula, and is simply 3 m [unit length]/[unit time].

    So at the instant t=1, the new particle has moved to point 1, particle 1 have moved to point 1.5, particle 2 has moved to 1.75, and so on. Each particle has moved to the next previously occupied point in our set. This means that Hilbert's so-called "paradox" has been realized in a finite amount of time and for a completely physically sensible system.



    In summary: Hilbert's Hotel can be physically realized and everything can happen in a finite amount of time and in a physically sensible manner. Start at t=0, there's an infinite number of points with particles occupying specific points in the interval [1,2), plus one particle outside at x=0. Evolve forward a finite amount of time, i.e. to time t=1, and now an "infinity+1" of particles are sitting inside of the interval [1,2) and occupying the exact same points as were previously occupied. Each particle still has it's own unique point in space, there's no overlaps, and the newest particle was added to the bunch in a finite amount of time, using a finite amount of energy and momentum, not assuming "successive infinities," and the "absurdities" regarding the occupations are as real here as in Hilbert's Hotel. Nevertheless, this is entirely physically possible in classical mechanics and is completely allowed by classical laws of physics. The conclusion? The so-called "paradoxes" that WLC asserts invalidate infinity are essential features, not irresolvable bugs. This is simply how infinite sets work. Your intuition simply has to be modified for how you think about infinite sets.






    N.B. I am also quite certain that I can setup the same system for an infinite number of non-interacting quantum particles (Take it in QFT, where this is easiest to describe and the non-relativistic limit requires it to be valid in QM, too), and where the probability of the transition to including the "infinity + 1" particle should be finite. I may work out this example to make the point really concrete, but it would appear that at first glance, there are zero obstructions to causing this in otherwise completely sensible physical systems. It's also worth pointing out that the Fock space of a free QFT explicitly includes the case of an infinite number of particles (as previously mentioned in other WLC/KCA threads), so literally every assumption there would be unimpeachable (i.e. definitely physically allowed) from the standpoint of QFT, the currently most complete description of Nature until we develop quantum gravity.

    Also, I have notably neglected gravity, but this shouldn't impose any problems. In principle I could add gravity to my evolution equations, only now I'd need to add as a postulate that there's finite mass for every finite region and then setting up the equations to end up at the nth -> (n+1)th positions, for all n. It seems laborious, but not impossible.
    Last edited by GoldPhoenix; July 7th, 2015 at 01:09 PM.
    "Those who can make you believe absurdities, can make you commit atrocities." --Voltaire

  2. #2
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    Re: WLC is Wrong: The Argument for Actual Infinities

    Questions (not rebuttal)

    So the rooms are "Points" ?
    and the guests are "particles"?

    How are you relating to the concept of "all rooms full"?
    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.

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    Re: WLC is Wrong: The Argument for Actual Infinities

    Quote Originally Posted by GoldPhoenix View Post
    Suppose this hypothetical universe starts out with an infinite number of particles...



    Now for the alleged paradox. It's been incorrectly conjectured many times now that it would take an infinite amount of time to accommodate the Hilbert Hotel...

    This is an interesting take GP.


    First a quick question, points are idealizations of particles, not actual particles right? Particles themselves do have extents, correct? I only ask because a major problem with your model would occur if the particles are not points in reality since you would have particles overlapping in space.


    I also have a few objections.


    1) The first is that it seems to beg the question since you are assuming an infinite number of particles and one of Craig's objections is that you cannot form an infinite number of that sort through successive addition. Perhaps I missed it, but did you have a rebuttal to that point?


    2) The second is that you seem to have added a definition (1) which allows you to define away the first half of Craig's objection (by redefining full as not full), and that you don't seem to deal with the second half of the objection.



    Perhaps the best way to bring home the truth of (2.11) is by means of an illustration. Let me use one of my favorites, Hilbert's Hotel, a product of the mind of the great German mathematician, David Hilbert. Let us imagine a hotel with a finite number of rooms. Suppose, furthermore, that all the rooms are full. When a new guest arrives asking for a room, the proprietor apologizes, "Sorry, all the rooms are full." But now let us imagine a hotel with an infinite number of rooms and suppose once more that all the rooms are full. There is not a single vacant room throughout the entire infinite hotel. Now suppose a new guest shows up, asking for a room. "But of course!" says the proprietor, and he immediately shifts the person in room #1 into room #2, the person in room #2 into room #3, the person in room #3 into room #4 and so on, out to infinity. As a result of these room changes, room #1 now becomes vacant and the new guest gratefully checks in. But remember, before he arrived, all the rooms were full! Equally curious, according to the mathematicians, there are now no more persons in the hotel than there were before: the number is just infinite. But how can this be? The proprietor just added the new guest's name to the register and gave him his keys-how can there not be one more person in the hotel than before? But the situation becomes even stranger. For suppose an infinity of new guests show up the desk, asking for a room. "Of course, of course!" says the proprietor, and he proceeds to shift the person in room #1 into room #2, the person in room #2 into room #4, the person in room #3 into room #6, and so on out to infinity, always putting each former occupant into the room number twice his own. As a result, all the odd numbered rooms become vacant, and the infinity of new guests is easily accommodated. And yet, before they came, all the rooms were full! And again, strangely enough, the number of guests in the hotel is the same after the infinity of new guests check in as before, even though there were as many new guests as old guests. In fact, the proprietor could repeat this process infinitely many times and yet there would never be one single person more in the hotel than before.

    http://www.reasonablefaith.org/the-e...f-the-universe



    This the section your OP concerns itself with.

    Your OP deals with the idea of the hotel being able to shift people down a room such that a "full" hotel has a vacancy. Your solution is to argue that the hotel was never really "full" and that despite an infinite number of guests there are always an infinite number of rooms between each guest. This would seem to work fine if we grant particles as points without any extent, such in any dimensional system we could simply use the "space" between the extentless points.

    That possibility falls apart however if we don't assume that definition, which, to my knowledge is more an assumption of convenience than a true necessity of physics. Point Particles are generally only assumed when the spatial dimensions of the particle are irrelevant to the scenario under discussion (https://en.wikipedia.org/wiki/Point_particle), but that doesn't seem to be the case here because once we assume spatial volume for the particle in the above, your conclusion suffers since there would clearly be a limit to how many particles we could fit into any given region of that space.



    But what your OP doesn't deal with, and what is Craig's larger point in this article, is the bolded section of the quote.


    I think that can easily be highlighted with a simple point. Given your assumption that the mass of all particles is greater than 0, the total mass of the universe is unchanged from T=0 to T=N for all N. So we get the problem of adding mass to a universe, but not changing its total mass.







    But, as I noted earlier, you only deal with that one section of one argument. There is quite a bit more in his defense:


    But Hilbert's Hotel is even stranger than the German mathematician gave it out to be. For suppose some of the guests start to check out. Suppose the guest in room #1 departs. Is there not now one less person in the hotel? Not according to the mathematicians-but just ask the woman who makes the beds! Suppose the guests in room numbers 1, 3, 5, . . . check out. In this case an infinite number of people have left the hotel, but according to the mathematicians there are no less people in the hotel-but don't talk to that laundry woman! In fact, we could have every other guest check out of the hotel and repeat this process infinitely many times, and yet there would never be any less people in the hotel. But suppose instead the persons in room number 4, 5, 6, . . . checked out. At a single stroke the hotel would be virtually emptied, the guest register reduced to three names, and the infinite converted to finitude. And yet it would remain true that the same number of guests checked out this time as when the guests in room numbers 1, 3, 5, . . . checked out. Can anyone sincerely believe that such a hotel could exist in reality? These sorts of absurdities illustrate the impossibility of the existence of an actually infinite number of things.

    ...

    At this point, we might find it profitable to consider several objections that might be raised against the argument. First let us consider objections to (2.11). Wallace Matson objects that the premiss must mean that an actually infinite number of things is logically impossible; but it is easy to show that such a collection is logically possible. For example, the series of negative numbers {. . . -3, -2, -1} is an actually infinite collection with no first member.10 Matson's error here lies in thinking that (2.11) means to assert the logical impossibility of an actually infinite number of things. What the premiss expresses is the real or factual impossibility of an actual infinite. To illustrate the difference between real and logical possibility: there is no logical impossibility in something's coming to exist without a cause, but such a circumstance may well be really or metaphysically impossible. In the same way, (2.11) asserts that the absurdities entailed in the real existence of an actual infinite show that such an existence is metaphysically impossible. Hence, one could grant that in the conceptual realm of mathematics one can, given certain conventions and axioms, speak consistently about infinite sets of numbers, but this in no way implies that an actually infinite number of things is really possible. One might also note that the mathematical school of intuitionism denies that even the number series is actually infinite (they take it to be potentially infinite only), so that appeal to number series as examples of actual infinites is a moot procedure.

    The late J.L. Mackie also objected to (2.11), claiming that the absurdities are resolved by noting that for infinite groups the axiom "the whole is greater than its part" does not hold, as it does for finite groups.11 Similarly, Quentin Smith comments that once we understand that an infinite set has a proper subset which has the same number of members as the set itself, the purportedly absurd situations become "perfectly believable."12 But to my mind, it is precisely this feature of infinite set theory which, when translated into the realm of the real, yields results which are perfectly incredible, for example, Hilbert's Hotel. Moreover, not all the absurdities stem from infinite set theory's denial of Euclid's axiom: the absurdities illustrated by guests checking out of the hotel stem from the self-contradictory results when the inverse operations of subtraction or division are performed using transfinite numbers. Here the case against an actually infinite collection of things becomes decisive.

    Finally one might note the objection of Sorabji, who maintains that illustrations such as Hilbert's Hotel involve no absurdity. In order to understand what is wrong with the kalam argument, he asks us to envision two parallel columns beginning at the same point and stretching away into the infinite distance, one the column of past years and the other the column of past days. The sense in which the column of past days is no larger than the column of past years, says Sorabji, is that the column of days will not "stick out" beyond the far end of the other column, since neither column has a far end. Now in the case of Hilbert's Hotel there is the temptation to think that some unfortunate resident at the far end will drop off into space. But there is no far end: the line of residents will not stick out beyond the far end of the line of rooms. Once this is seen, the outcome is just an explicable-even if a surprising and exhilarating-truth about infinity.13 Now Sorabji is certainly correct, as we have seen, that Hilbert's Hotel illustrates an explicable truth about the nature of the actual infinite. If an actually infinite number of things could exist, a Hilbert's Hotel would be possible. But Sorabji seems to fail to understand the heart of the paradox: I, for one, experience no temptation to think of people dropping off the far end of the hotel, for there is none, but I do have difficulty believing that a hotel in which all the rooms are occupied can accommodate more guests. Of course, the line of guests will not stick out beyond the line of rooms, but if all of those infinite rooms already have guests in them, then can moving those guests about really create empty rooms? Sorabji's own illustration of the columns of past years and days I find not a little disquieting: if we divide the columns into foot-long segments and mark one column as the years and the other as the days, then one column is as long as the other and yet for every foot-length segment in the column of years, 365 segments of equal length are found in the column of days! These paradoxical results can be avoided only if such actually infinite collections can exist only in the imagination, not in reality. In any case, the Hilbert's Hotel illustration is not exhausted by dealing only with the addition of new guests, for the subtraction of guests results in absurdities even more intractable. Sorabji's analysis says nothing to resolve these. Hence, it seems to me that the objections to premiss (2.11) are less plausible than the premiss itself.

    With regard to (2.12), the most frequent objection is that the past ought to be regarded as a potential infinite only, not an actual infinite. This was Aquinas's position versus Bonaventure, and the contemporary philosopher Charles Hartshorne seems to side with Thomas on this issue.14 Such a position is, however, untenable. The future is potentially infinite, since it does not exist; but the past is actual in a way the future is not, as evidenced by the fact that we have traces of the past in the present, but no traces of the future. Hence, if the series of past events never began to exist, there must have been an actually infinite number of past events.

    The objections to either premiss therefore seem to be less compelling than the premisses themselves. Together they imply that the universe began to exist. Hence, I conclude that this argument furnishes good grounds for accepting the truth of premiss (2) that the universe began to exist.

    Second Supporting Argument

    The second argument (2.2) for the beginning of the universe is based on the impossibility of forming an actual infinite by successive addition. This argument is distinct from the first in that it does not deny the possibility of the existence of an actual infinite, but the possibility of its being formed by successive addition.

    Premiss (2.21) is the crucial step in the argument. One cannot form an actually infinite collection of things by successively adding one member after another. Since one can always add one more before arriving at infinity, it is impossible to reach actual infinity. Sometimes this is called the impossibility of "counting to infinity" or "traversing the infinite." It is important to understand that this impossibility has nothing to do with the amount of time available: it belongs to the nature of infinity that it cannot be so formed.

    Now someone might say that while an infinite collection cannot be formed by beginning at a point and adding members, nevertheless an infinite collection could be formed by never beginning but ending at a point, that is to say, ending at a point after having added one member after another from eternity. But this method seems even more unbelievable than the first method. If one cannot count to infinity, how can one count down from infinity? If one cannot traverse the infinite by moving in one direction, how can one traverse it by simply moving in the opposite direction?

    Indeed, the idea of a beginningness series ending in the present seems to be absurd. To give just one illustration: suppose we meet a man who claims to have been counting from eternity and is now finishing: . . ., -3, -2, -1, 0. We could ask, why did he not finish counting yesterday or the day before or the year before? By then an infinite time had already elapsed, so that he should already have finished by then. Thus, at no point in the infinite past could we ever find the man finishing his countdown, for by that point he should already be done! In fact, no matter how far back into the past we go, we can never find the man counting at all, for at any point we reach he will have already finished. But if at no point in the past do we find him counting, this contradicts the hypothesis that he has been counting from eternity. This illustrates the fact that the formation of an actual infinite by successive addition is equally impossible whether one proceeds to or from infinity.

    Premiss (2.22) presupposes a dynamical view of time according to which events are actualized in serial fashion, one after another. The series of events is not a sort of timelessly subsisting world-line which appears successively in consciousness. Rather becoming is real and essential to temporal process. Now this view of time is not without its challengers, but to consider their objections in this article would take us too far afield.15 In this piece, we must rest content with the fact that we are arguing on common ground with our ordinary intuitions of temporal becoming and in agreement with a good number of contemporary philosophers of time and space.

    Given the truth of (2.21) and (2.22), the conclusion (2.23) logically follows. If the universe did not begin to exist a finite time ago, then the present moment could never arrive. But obviously, it has arrived. Therefore, we know that the universe is finite in the past and began to exist.

    Again, it would be profitable to consider various objections that have been offered against this reasoning. Against (2.21), Mackie objects that the argument illicitly assumes an infinitely distant starting point in the past and then pronounces it impossible to travel from that point to today. But there would in an infinite past be no starting point, not even an infinitely distant one. Yet from any given point in the infinite past, there is only a finite distance to the present.16 Now it seems to me that Mackie's allegation that the argument presupposes an infinitely distant starting point is entirely groundless. The beginningless character of the series only serves to accentuate the difficulty of its being formed by successive addition. The fact that there is no beginning at all, not even an infinitely distant one, makes the problem more, not less, nettlesome. And the point that from any moment in the infinite past there is only a finite temporal distance to the present may be dismissed as irrelevant. The question is not how any finite portion of the temporal series can be formed, but how the whole infinite series can be formed. If Mackie thinks that because every segment of the series can be formed by successive addition therefore the whole series can be so formed, then he is simply committing the fallacy of composition.

    Sorabji similarly objects that the reason it is impossible to count down from infinity is because counting involves by nature taking a starting number, which is lacking in this case. But completing an infinite lapse of years involves no starting year and is, hence, possible.17 But this response is clearly inadequate, for, as we have seen, the years of an infinite past could be enumerated by the negative numbers, in which case a completed infinity of years would, indeed, entail a beginningless countdown from infinity. Sorabji anticipates this rebuttal, however, and claims that such a backwards countdown is possible in principle and therefore no logical barrier has been exhibited to the elapsing of an infinity of past years. Again, however, the question I am posing is not whether there is a logical contradiction in such a notion, but whether such a countdown is not metaphysically absurd. For we have seen that such a countdown should at any point already have been completed. But Sorabji is again ready with a response: to say the countdown should at any point already be over confuses counting an infinity of numbers with counting all the numbers. At any given point in the past, the eternal counter will have already counted an infinity of negative numbers, but that does not entail that he will have counted all the negative numbers. I do not think the argument makes this alleged equivocation, and this may be made clear by examining the reason why our eternal counter is supposedly able to complete a count of the negative numbers ending at zero. In order to justify the possibility of this intuitively impossible feat, the argument's opponent appeals to the so-called Principle of Correspondence used in set theory to determine whether two sets are equivalent (that is, have the same number of members) by matching the members of one set with the members of the other set and vice versa. On the basis of this principle the objector argues that since the counter has lived, say, an infinite number of years and since the set of past years can be put into a one-to-one correspondence with the set of negative numbers, it follows that by counting one number a year an eternal counter would complete a countdown of the negative numbers by the present year. If we were to ask why the counter would not finish next year or in a hundred years, the objector would respond that prior to the present year an infinite number of years will have already elapsed, so that by the Principle of Correspondence, all the numbers should have been counted by now. But this reasoning backfires on the objector: for, as we have seen, on this account the counter should at any point in the past have already finished counting all the numbers, since a one-to-one correspondence exists between the years of the past and the negative numbers. Thus, there is no equivocation between counting an infinity of numbers and counting all the numbers. But at this point a deeper absurdity bursts in view: for suppose there were another counter who counted at a rate of one negative number per day. According to the Principle of Correspondence, which underlies infinite set theory and transfinite arithmetic, both of our eternal counters will finish their countdowns at the same moment, even though one is counting at a rate 365 times faster than the other! Can anyone believe that such scenarios can actually obtain in reality, but do not rather represent the outcome of an imaginary game being played in a purely conceptual realm according to adopted logical conventions and axioms?

    As for premiss (2.22), many thinkers have objected that we need not regard the past as a beginningless infinite series with an end in the present. Popper, for example, admits that the set of all past events is actually infinite, but holds that the series of past events is potentially infinite. This may be seen by beginning in the present and numbering the events backwards, thus forming a potential infinite. Therefore, the problem of an actual infinite's being formed by successive addition does not arise.18 Similarly, Swinburne muses that it is dubious whether a completed infinite series with no beginning but an end makes sense, but he proposes to solve the problem by beginning in the present and regressing into the past, so that the series of past events would have no end and would therefore not be a completed infinite.19 This objection, however, clearly confuses the mental regress of counting with the real progress of the temporal series of events itself. Numbering the series from the present backwards only shows that if there are an infinite number of past events, then we can denumerate an infinite number of past events. But the problem is, how can this infinite collection of events come to be formed by successive addition? How we mentally conceive the series does not in any way affect the ontological character of the series itself as a series with no beginning but an end, or in other words, as an actual infinite completed by successive addition.

    Once again, then, the objections to (2.21) and (2.22) seem less plausible than the premisses themselves. Together they imply (2.23), or that the universe began to exist.


    ---------- Post added at 05:54 AM ---------- Previous post was at 05:52 AM ----------

    Quote Originally Posted by MindTrap028 View Post
    So the rooms are "Points" ?
    and the guests are "particles"?
    I don't think this is quite what he is offering. To continue the analogy, he is arguing more that there aren't rooms, but a giant hall. And each guest in the hall requires no space in it, so that we could always pack another guest into the hall if needed.
    "Suffering lies not with inequality, but with dependence." -Voltaire
    "Fallacies do not cease to be fallacies because they become fashions. -G.K. Chesterton
    Also, if you think I've overlooked your post please shoot me a PM, I'm not intentionally ignoring you.


  4. #4
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    Re: WLC is Wrong: The Argument for Actual Infinities

    But, as I noted earlier, you only deal with that one section of one argument. There is quite a bit more in his defense:
    This seems like a red herring. A rebuttal to GP's post must establish either that his proposed system is unphysical, or that his proposed system doesn't contain an actual infinity--or that his proposed system is both physical (i.e. violates no laws of physics), contains an actual infinity, but is objectionable on other grounds. Your WLC quote is closest to the last rebuttal, but doesn't seem to establish any grounds on which to object to GP's system. To wit:

    Can anyone believe that such scenarios can actually obtain in reality, but do not rather represent the outcome of an imaginary game being played in a purely conceptual realm according to adopted logical conventions and axioms?
    If there is a physical system in which such scenarios occur, what objection can there be? What would bar belief? Plenty of physical situations violate common sense (see e.g. all of quantum mechanics, or Aristotle's physics, where projectiles don't follow parabolic paths), so the objection that such scenarios would "boggle the mind" doesn't do much to support the claim that any system where such scenarios occur must be unphysical. If such objections would prevail, then they'd similarly prevail against e.g. quantum mechanics, perhaps even GR's description of gravity.
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    Re: WLC is Wrong: The Argument for Actual Infinities

    Quote Originally Posted by MindTrap028 View Post
    Questions (not rebuttal)

    So the rooms are "Points" ?
    and the guests are "particles"?

    How are you relating to the concept of "all rooms full"?
    As stated in the OP, I'm making a analogous system to the Hilbert's hotel. For the analogy, yes, particles analogous to the tenets and the points are analogous to the rooms.

    All of the points inside of the set {1, 1 + 1/2, 1 + 1/2 + 1/4, ..., 2( 1 - (1/2)n ), ...} are the analogies to the set of all rooms identified by {1, 2, 3, ...}. They're the points that the particles are occupying.








    Quote Originally Posted by Squatch347 View Post
    First a quick question, points are idealizations of particles, not actual particles right? Particles themselves do have extents, correct? I only ask because a major problem with your model would occur if the particles are not points in reality since you would have particles overlapping in space.
    1.) In Newtonian mechanics, they are treated as literal points. Rigid bodies, continuum mechanics, and statistical mechanics are all approximations to large collections of particles, so your question is essentially "Why are you using particles from Newtonian mechanics?". I'm ignoring quantum mechanics (where the point-like nature breaks down) in this example for two reasons. The first reason is pedagogical. The form of the physical laws (i.e. the check of consistency) is simple and easy to verify, whereas the quantum mechanical laws are difficult, tedious, and necessarily at a graduate or research level. By choosing Newtonian mechanics, it keeps the math and physics at a 100-level. The simplicity of this example should provide the die-hard fans of the KCA some pause, because coming up with modifications and extensions of this in physical laws more complex than Newtonian mechanics probably isn't going to be very difficult. The second reason is:


    2.) Quantum mechanics is the place where point-like particles begin to not be quite so literally point-like; however, trying to impose the impossibility of actual infinities becomes more openly ludicrous the more quantum you go in physics, not more reasonable. I'll discuss this in more detail at the very bottom of my post.

    Quote Originally Posted by Squatch
    1) The first is that it seems to beg the question since you are assuming an infinite number of particles and one of Craig's objections is that you cannot form an infinite number of that sort through successive addition. Perhaps I missed it, but did you have a rebuttal to that point?
    1.) Firstly, that's not WLC's objection, because if it were then his argument would be trivially fallacious. He isn't saying "I object to having an infinite number of things because it would contradict my statement that you can't have an infinite number of things." That would beg the question. No, his argument is that if you had an infinite number of things, it would necessarily imply a physical contradiction. Note that I do not simply assume that I can consistently have an infinite number of particles. I use an extremely common, logically valid argument known as proof by contradiction: If WLC is correct, then you cannot construct physically consistent system with actual infinities. I construct a system with actual infinities and I check that it is physically consistent. Therefore, by modus tollens, WLC is necessarily wrong.


    2.) Secondly, and I cannot emphasize this enough, there's no successive addition. I explicitly created this example for you, Squatch, because it avoids any kind of successive addition. There's no creation of infinity over time here, Squatch. God simply starts the universe off in this configuration. The only infinity here is a divisible infinity, which you have repeatedly stated in past threads that you have no objection to. God simply says "I wish to begin with this specific universe", and starts the clock running at t=0.


    Quote Originally Posted by Squatch
    2) The second is that you seem to have added a definition (1) which allows you to define away the first half of Craig's objection (by redefining full as not full), and that you don't seem to deal with the second half of the objection.
    I will re-direct to Clive's arguments in post #4.









    To expand on the quantum issues:

    In Quantum Field Theory, you are forced into accepting the existence of Fock space, which is the space that contains all possible particle configurations (The Hilbert space of all possible particle states; a discussion of this can be found in the first few chapters of any book on QFT, so take your pick, but section 2.4 of Tong's intro QFT book is free and discusses this). The dimension of the Fock space is necessarily infinite (In this case, not being infinite would lead to an actual violation of physical law), and this introduces several very simple "actual infinities."

    The first is that I have an actual infinity of different possible particle states. Two examples where this is used, for instance, is in the description of coherent states, for instance in a pulse of light, and the another example is Unruh radiation. Unruh radiation necessarily contains an infinite number of modes in the radiation (or else it violates physical law, see equation 3, which implicitly sums j from minus to plus infinity). Note that even though Unruh radiation hasn't been directly observed a lab, it is a necessary consequence of the current physical laws. So this means if you are accelerating, you see a universe which has a bath containing an infinite number of particles (But all physically finite quantities, like energy, momentum, and temperature), but if you're standing still (or in an inertial frame), you see a universe with zero particles. This highlights the point that trying to demand that infinity is unphysical just doesn't make any sense from a QFT perspective nor does it comport with any known facts of the matter. These theories may break down and render the actual number of particles finite, but as of now, the evidence supporting the assumptions of QFT lead us to these conclusions; and even if so, the only thing that could even conceivably do this is quantum gravity (But quantum gravity also could conceivably lead to past infinities, so this kind of appeal is a double-edged sword for WLC).

    Back to the point of Fock space: This means if I want to start my universe out with an infinite number of point particles, there actually cannot be any principle that forbids me from doing so, because that principle would then by direct implications break the previous physical principles (In this case, Lorentz invariance and unitarity). Thus the laws of QFT + WLC's new "No infinities" principle entails a contradiction. Thus it is "Necessarily possibly true."

    So going to quantum mechanics doesn't provide a way to dispute my overall point about actual infinities, it simply makes WLC's argument even more obviously wrong from the outset. Again, all I'm trying to do here is give a much simpler, much more intuitive example of an "actual infinity." Beyond which, if we accept that we should take quantum gravity into consideration, then his entire argument fails for other reasons, like our lack of knowledge of quantum gravity means we may very well be capable of living in a past-infinite universe.
    Last edited by GoldPhoenix; July 9th, 2015 at 10:34 AM.
    "Those who can make you believe absurdities, can make you commit atrocities." --Voltaire

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    Re: WLC is Wrong: The Argument for Actual Infinities

    Quote Originally Posted by GP
    As stated in the OP, I'm making a analogous system to the Hilbert's hotel. For the analogy, yes, particles analogous to the tenets and the points are analogous to the rooms.

    All of the points inside of the set {1, 1 + 1/2, 1 + 1/2 + 1/4, ..., 2( 1 - (1/2)n ), ...} are the analogies to the set of all rooms identified by {1, 2, 3, ...}. They're the points that the particles are occupying.
    Thanks. Be patient with me here.

    1)So there are no points at "1+.25" or "2" (which of course isn't reached)?
    I'm having difficulty with this, because The rooms are things that stand independent and distinct from the guests, and a Hotel is easy to grasp.
    In your example, looking at the "hotel" of the meter stick, I can't find the first "room", especially if the hotel has the dimension of a meter and the rooms have no such dimension at all.

    2) You called a particle a point in your OP, what is the distinction between the two ideas?
    (I can tell the difference between a room and a guest easy enough

    3) Now,and please correct me if I am wrong, a point in physics has no volume, and particles don't either (If I understand your explanation).
    Is there any problem with using a non volume example to apply to a universe with Volume? (why/why not?)
    - As you can see in #1 I have a problem with finding a dimension less point on a ruler.

    4) All of these elements seem to be very abstract; Numbers, points and Particles. With points being said not to exist (or at least point masses).
    http://scienceworld.wolfram.com/physics/PointMass.html

    How is discussing such abstracts helpful when applied to the real world and what is or isn't possible in the real world? (I guess Re question #3)
    Having an internally consistent idea, doesn't seem to equate to reality.


    5) Any suggestions for reading to answer the question "what is a point or particle"? I'm having difficulty finding anything in google that would directly apply to this thread.
    googling "whats the point" was pretty depressing.
    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.

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    Re: WLC is Wrong: The Argument for Actual Infinities

    Quote Originally Posted by MindTrap028 View Post
    Thanks. Be patient with me here.

    1)So there are no points at "1+.25" or "2" (which of course isn't reached)?
    I'm having difficulty with this, because The rooms are things that stand independent and distinct from the guests, and a Hotel is easy to grasp.
    In your example, looking at the "hotel" of the meter stick, I can't find the first "room", especially if the hotel has the dimension of a meter and the rooms have no such dimension at all.
    Sure, one could say those points exist in space, but there aren't any particles occupying those points.

    Quote Originally Posted by MindTrap
    2) You called a particle a point in your OP, what is the distinction between the two ideas?
    (I can tell the difference between a room and a guest easy enough
    Points are the collections of objects that make up space. In this case, I'm just looking at one dimension of space, so we're just talking about numbers as points and the real number line (all real numbers) is the space.

    Point particles are objects that can traverse this space and occupy one of the points inside of the space.

    So if I have a point particle described by

    x(t) = 1 + 2t

    Then at t=0, it's occupying the point "1." At t=1, it's occupying the point "3." At t=2, it's occupying the point "5."

    Quote Originally Posted by MT
    3) Now,and please correct me if I am wrong, a point in physics has no volume, and particles don't either (If I understand your explanation).
    Is there any problem with using a non volume example to apply to a universe with Volume? (why/why not?)
    - As you can see in #1 I have a problem with finding a dimension less point on a ruler.
    No, there isn't any problem (in classical mechanics). I can't really answer "Why not" unless you give me an example of something you might think is a problem.

    Quote Originally Posted by MT
    4) All of these elements seem to be very abstract; Numbers, points and Particles. With points being said not to exist (or at least point masses).
    http://scienceworld.wolfram.com/physics/PointMass.html

    How is discussing such abstracts helpful when applied to the real world and what is or isn't possible in the real world? (I guess Re question #3)
    Having an internally consistent idea, doesn't seem to equate to reality.
    Because physics uses "abstracts" in order to understand the real world. These "abstracts" are why you can look at a screen right now, and it's not dark, but full of illuminated pixels. The purpose of "abstracts" is that we can conceive of them and they can inform us on conceivable scenarios. In this case, the scenario may not be literally realizable (although 100 years ago, you wouldn't have been able to tell me that this wasn't possible), the purpose is to expose the viewers to a different way of thinking about these issues and make them understand that actual infinities are conceivable in a (simplified) set of physical laws.

    Also, to clarify, I'm not saying points "don't exist." I'm simply telling you where in the space a collection of point particles are located. By saying there isn't a particle at point 1.25, I'm not saying the point doesn't exist, I'm just saying there's no particle occupying that point in this scenario.

    Quote Originally Posted by MT
    5) Any suggestions for reading to answer the question "what is a point or particle"? I'm having difficulty finding anything in google that would directly apply to this thread.
    googling "whats the point" was pretty depressing.
    Honestly, I don't know of any documentation that's not at an upper undergraduate level (And for those cases I suggest David Tong's Classical Dynamics and Statistical Physics free, online books). The problem is that most 100-level or high school courses introduce the point particle as some kind of approximation to macroscopic objects (e.g. a stone, a box, or a planet) and bloviate about this for a long time. They tend never to discuss that for fundamental, massive objects (e.g. an electron), they are the non-relativistic approximation. Thus when you do statistical mechanics of a gas (e.g. like in Tong's Statistical Physics), you see that classical gases are defined as a large collection of point particles (There its approximated as an infinite number of point particles, again illustrating the fact that an infinite number of things does not lead to absurdities, or else, e.g. an important consequence of classical gases, you'd never have learned PV = nRT in your introductory chemistry course, but I digress).
    "Those who can make you believe absurdities, can make you commit atrocities." --Voltaire

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    Re: WLC is Wrong: The Argument for Actual Infinities

    Quote Originally Posted by CliveStaples View Post
    This seems like a red herring.
    Given the thread's title I would argue it is well within scope. GP's argument there (as well as in the OP) is that Craig is wrong about his conclusion on Actual Infinities, not that a specific argument against them fails. It is perfectly valid to note that an opponent did not rebut a major section of an argument without it being a red herring.

    In this case GP did not respond, as part of his OP against 2/3 of that specific argument against actual infinities (the conflicting answer section and the "guests" checking out section), nor against an entirely separate argument against an actual infinity.


    Quote Originally Posted by GP
    1.) In Newtonian mechanics, they are treated as literal points.
    Ok, but I don't think that that really dealt with the nature of my objection. My question was more "do particles occupy a region of space within our universe?" Essentially the issue I had with your example being physical was that it assumes we could fit an infinite number of particles in any sized region of space because none of those particles occupies any actual space.


    As a side note, I didn't see a response to my highlighting that you missed a point in your critique of Craig's defense. Specifically:

    But what your OP doesn't deal with, and what is Craig's larger point in this article, is the bolded section of the quote.


    I think that can easily be highlighted with a simple point. Given your assumption that the mass of all particles is greater than 0, the total mass of the universe is unchanged from T=0 to T=N for all N. So we get the problem of adding mass to a universe, but not changing its total mass.


    Quote Originally Posted by GP
    1.) Firstly, that's not WLC's objection, because if it were then his argument would be trivially fallacious.
    It isn't his only objection, it is clearly one of his objections, I quoted it to you in my last post.

    Second Supporting Argument

    The second argument (2.2) for the beginning of the universe is based on the impossibility of forming an actual infinite by successive addition. This argument is distinct from the first in that it does not deny the possibility of the existence of an actual infinite, but the possibility of its being formed by successive addition.

    Premiss (2.21) is the crucial step in the argument. One cannot form an actually infinite collection of things by successively adding one member after another. Since one can always add one more before arriving at infinity, it is impossible to reach actual infinity. Sometimes this is called the impossibility of "counting to infinity" or "traversing the infinite." It is important to understand that this impossibility has nothing to do with the amount of time available: it belongs to the nature of infinity that it cannot be so formed.

    Now someone might say that while an infinite collection cannot be formed by beginning at a point and adding members, nevertheless an infinite collection could be formed by never beginning but ending at a point, that is to say, ending at a point after having added one member after another from eternity. But this method seems even more unbelievable than the first method. If one cannot count to infinity, how can one count down from infinity? If one cannot traverse the infinite by moving in one direction, how can one traverse it by simply moving in the opposite direction?

    Indeed, the idea of a beginningness series ending in the present seems to be absurd. To give just one illustration: suppose we meet a man who claims to have been counting from eternity and is now finishing: . . ., -3, -2, -1, 0. We could ask, why did he not finish counting yesterday or the day before or the year before? By then an infinite time had already elapsed, so that he should already have finished by then. Thus, at no point in the infinite past could we ever find the man finishing his countdown, for by that point he should already be done! In fact, no matter how far back into the past we go, we can never find the man counting at all, for at any point we reach he will have already finished. But if at no point in the past do we find him counting, this contradicts the hypothesis that he has been counting from eternity. This illustrates the fact that the formation of an actual infinite by successive addition is equally impossible whether one proceeds to or from infinity.

    Premiss (2.22) presupposes a dynamical view of time according to which events are actualized in serial fashion, one after another. The series of events is not a sort of timelessly subsisting world-line which appears successively in consciousness. Rather becoming is real and essential to temporal process. Now this view of time is not without its challengers, but to consider their objections in this article would take us too far afield.15 In this piece, we must rest content with the fact that we are arguing on common ground with our ordinary intuitions of temporal becoming and in agreement with a good number of contemporary philosophers of time and space.

    Given the truth of (2.21) and (2.22), the conclusion (2.23) logically follows. If the universe did not begin to exist a finite time ago, then the present moment could never arrive. But obviously, it has arrived. Therefore, we know that the universe is finite in the past and began to exist.

    Again, it would be profitable to consider various objections that have been offered against this reasoning. Against (2.21), Mackie objects that the argument illicitly assumes an infinitely distant starting point in the past and then pronounces it impossible to travel from that point to today. But there would in an infinite past be no starting point, not even an infinitely distant one. Yet from any given point in the infinite past, there is only a finite distance to the present.16 Now it seems to me that Mackie's allegation that the argument presupposes an infinitely distant starting point is entirely groundless. The beginningless character of the series only serves to accentuate the difficulty of its being formed by successive addition. The fact that there is no beginning at all, not even an infinitely distant one, makes the problem more, not less, nettlesome. And the point that from any moment in the infinite past there is only a finite temporal distance to the present may be dismissed as irrelevant. The question is not how any finite portion of the temporal series can be formed, but how the whole infinite series can be formed. If Mackie thinks that because every segment of the series can be formed by successive addition therefore the whole series can be so formed, then he is simply committing the fallacy of composition.

    Sorabji similarly objects that the reason it is impossible to count down from infinity is because counting involves by nature taking a starting number, which is lacking in this case. But completing an infinite lapse of years involves no starting year and is, hence, possible.17 But this response is clearly inadequate, for, as we have seen, the years of an infinite past could be enumerated by the negative numbers, in which case a completed infinity of years would, indeed, entail a beginningless countdown from infinity. Sorabji anticipates this rebuttal, however, and claims that such a backwards countdown is possible in principle and therefore no logical barrier has been exhibited to the elapsing of an infinity of past years. Again, however, the question I am posing is not whether there is a logical contradiction in such a notion, but whether such a countdown is not metaphysically absurd. For we have seen that such a countdown should at any point already have been completed. But Sorabji is again ready with a response: to say the countdown should at any point already be over confuses counting an infinity of numbers with counting all the numbers. At any given point in the past, the eternal counter will have already counted an infinity of negative numbers, but that does not entail that he will have counted all the negative numbers. I do not think the argument makes this alleged equivocation, and this may be made clear by examining the reason why our eternal counter is supposedly able to complete a count of the negative numbers ending at zero. In order to justify the possibility of this intuitively impossible feat, the argument's opponent appeals to the so-called Principle of Correspondence used in set theory to determine whether two sets are equivalent (that is, have the same number of members) by matching the members of one set with the members of the other set and vice versa. On the basis of this principle the objector argues that since the counter has lived, say, an infinite number of years and since the set of past years can be put into a one-to-one correspondence with the set of negative numbers, it follows that by counting one number a year an eternal counter would complete a countdown of the negative numbers by the present year. If we were to ask why the counter would not finish next year or in a hundred years, the objector would respond that prior to the present year an infinite number of years will have already elapsed, so that by the Principle of Correspondence, all the numbers should have been counted by now. But this reasoning backfires on the objector: for, as we have seen, on this account the counter should at any point in the past have already finished counting all the numbers, since a one-to-one correspondence exists between the years of the past and the negative numbers. Thus, there is no equivocation between counting an infinity of numbers and counting all the numbers. But at this point a deeper absurdity bursts in view: for suppose there were another counter who counted at a rate of one negative number per day. According to the Principle of Correspondence, which underlies infinite set theory and transfinite arithmetic, both of our eternal counters will finish their countdowns at the same moment, even though one is counting at a rate 365 times faster than the other! Can anyone believe that such scenarios can actually obtain in reality, but do not rather represent the outcome of an imaginary game being played in a purely conceptual realm according to adopted logical conventions and axioms?

    As for premiss (2.22), many thinkers have objected that we need not regard the past as a beginningless infinite series with an end in the present. Popper, for example, admits that the set of all past events is actually infinite, but holds that the series of past events is potentially infinite. This may be seen by beginning in the present and numbering the events backwards, thus forming a potential infinite. Therefore, the problem of an actual infinite's being formed by successive addition does not arise.18 Similarly, Swinburne muses that it is dubious whether a completed infinite series with no beginning but an end makes sense, but he proposes to solve the problem by beginning in the present and regressing into the past, so that the series of past events would have no end and would therefore not be a completed infinite.19 This objection, however, clearly confuses the mental regress of counting with the real progress of the temporal series of events itself. Numbering the series from the present backwards only shows that if there are an infinite number of past events, then we can denumerate an infinite number of past events. But the problem is, how can this infinite collection of events come to be formed by successive addition? How we mentally conceive the series does not in any way affect the ontological character of the series itself as a series with no beginning but an end, or in other words, as an actual infinite completed by successive addition.

    Once again, then, the objections to (2.21) and (2.22) seem less plausible than the premisses themselves. Together they imply (2.23), or that the universe began to exist.





    Quote Originally Posted by GP
    He isn't saying "I object to having an infinite number of things because it would contradict my statement that you can't have an infinite number of things."
    Hmm, this response concerns me because it would seem to betray a fundamental misunderstanding of the argument you are attempting to rebut. I would highly encourage you to re-read, or at least skim, the first few paragraphs of the above quoted text.



    Quote Originally Posted by GP
    2.) Secondly, and I cannot emphasize this enough, there's no successive addition. I explicitly created this example for you, Squatch, because it avoids any kind of successive addition.
    Except it doesn't:

    Suppose I give you another particle, let's say at point 0. Let's show that it is physically possible to accommodate the new particle inside of "infinite hotel" in a finite amount of time, and saliently with finite total energy and finite total momentum on the interval. In other words, where no physical contradiction has occurred Specifically, we'll start off with an infinite number of particles occupying an infinite number of points inside the finite interval [1,2) with a single particle lying outside that interval, and then evolve the system forward a unit of time until there's now be "infinity + 1" particles inside of the finite interval, but also with all of the same previous states occupied.

    You are describing successive addition there GP. If I take a quantity, add another to it. Take the new quantity and add another to it. That is successive addition.

    What's more, given the actual nature of Craig's argument that I quote above, you'll note that he is pointing out that successive addition is the only method of adding particles to your space.

    So given that constraint, the natural question arises, how did we get to have an infinite number of particles in that space if I am required to add them successively?


    Hopefully that clarifies a bit of Craig's second argument for you since you seem to have been thinking it was something else entirely.
    "Suffering lies not with inequality, but with dependence." -Voltaire
    "Fallacies do not cease to be fallacies because they become fashions. -G.K. Chesterton
    Also, if you think I've overlooked your post please shoot me a PM, I'm not intentionally ignoring you.


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    Re: WLC is Wrong: The Argument for Actual Infinities

    Squatch, you're really confused here. I'm going to repeat this point elsewhere in this post, but you are very confused about what was presented in the OP. After reading this post (and crucially before responding) please re-read the OP. You have clearly misunderstood what was said. Perhaps I said it poorly, but either way you need to understand what was being presented there before we can have a meaningful discussion about it.

    Quote Originally Posted by Squatch
    GP: "1.) Firstly, that's not WLC's objection, because if it were then his argument would be trivially fallacious."

    It isn't his only objection, it is clearly one of his objections, I quoted it to you in my last post.

    Second Supporting Argument

    The second argument (2.2) for the beginning of the universe is based on the impossibility of forming an actual infinite by successive addition. This argument is distinct from the first in that it does not deny the possibility of the existence of an actual infinite, but the possibility of its being formed by successive addition.

    Premiss (2.21) is the crucial step in the argument. One cannot form an actually infinite collection of things by successively adding one member after another. Since one can always add one more before arriving at infinity, it is impossible to reach actual infinity. Sometimes this is called the impossibility of "counting to infinity" or "traversing the infinite." It is important to understand that this impossibility has nothing to do with the amount of time available: it belongs to the nature of infinity that it cannot be so formed. [...]

    GP: "He isn't saying 'I object to having an infinite number of things because it would contradict my statement that you can't have an infinite number of things.' "


    Hmm, this response concerns me because it would seem to betray a fundamental misunderstanding of the argument you are attempting to rebut. I would highly encourage you to re-read, or at least skim, the first few paragraphs of the above quoted text.

    GP: "2.) Secondly, and I cannot emphasize this enough, there's no successive addition. I explicitly created this example for you, Squatch, because it avoids any kind of successive addition."


    Except it doesn't:

    Suppose I give you another particle, let's say at point 0. Let's show that it is physically possible to accommodate the new particle inside of "infinite hotel" in a finite amount of time, and saliently with finite total energy and finite total momentum on the interval. In other words, where no physical contradiction has occurred Specifically, we'll start off with an infinite number of particles occupying an infinite number of points inside the finite interval [1,2) with a single particle lying outside that interval, and then evolve the system forward a unit of time until there's now be "infinity + 1" particles inside of the finite interval, but also with all of the same previous states occupied.

    You are describing successive addition there GP. If I take a quantity, add another to it. Take the new quantity and add another to it. That is successive addition.

    What's more, given the actual nature of Craig's argument that I quote above, you'll note that he is pointing out that successive addition is the only method of adding particles to your space.

    So given that constraint, the natural question arises, how did we get to have an infinite number of particles in that space if I am required to add them successively?
    There's a lot to say here.

    1.) You need to re-read the OP before responding, because you seem to be operating under the false pretense (in spite of it being repeated multiple times in the OP and in subsequent posts) that I'm building up an infinity of particles over time, one particle at a time. This is crucially, importantly, and explicitly not what I am doing. I did this because I understand that WLC has some bizarre pathological aversion to transfinite induction. Instead, I'm creating a universe with an infinite number of particles inside of an interval (at points 1, 1.5, 1.75, ...), and with a single particle outside of that interval. Then, we start the clock, the particles move in the fashion discussed, and at the end of it, we end up with the initial infinity of particles having shifted and opened up a space for the outside particle to move in. Now every point in the interval [1,2] that was previously occupied is still occupied, but the extra particle from the outside has been "accommodated." And all without breaking any law of physics, and the particle was totally accommodated in a finite amount of time. (Everyone moved for him in the one time interval, and if you're an observer, you could have seen the particles move the finite amount). Please re-read the entire OP with this clarification.

    2.) The entirety of your objections here are centered around the notion that I'm successively adding particles. I'm not adding an infinite number of particles in by hand over time. I'm saying that we play God of this hypothetical universe, and we just will a universe into existence, with the stipulation that the universe is consistent with a set of (simplified) physical laws. If WLC is right, this universe must derive a contradiction or a violation of a physical law. So when I write down the equations that describes the universe, and then I check that it is self-consistent with the physical laws. If WLC is correct then I must derive a contradiction; however, I didn't derive a contradiction, thus WLC is wrong.

    (See later in the post for the objections regarding hypotheticals)

    Quote Originally Posted by Squatch
    Hopefully that clarifies a bit of Craig's second argument for you since you seem to have been thinking it was something else entirely.
    *shakes his head*




    Quote Originally Posted by Squatch
    Ok, but I don't think that that really dealt with the nature of my objection. My question was more "do particles occupy a region of space within our universe?" Essentially the issue I had with your example being physical was that it assumes we could fit an infinite number of particles in any sized region of space because none of those particles occupies any actual space.
    1.) It doesn't even matter if we don't live in a Newtonian universe. The Hilbert's Hotel example that WLC brings up doesn't make use of any quantum mechanics or anything else, so why should that be important in any way for showing why actual infinities are impossible? (In fact, the opposite is true, as shown below) WLC asserts very generically that any physical system capable of realizing the Hilbert's Hotel must be inconsistent; he doesn't appeal to quantum mechanics, relativity, or field theory --and he can't, because if any part of his argument crucially depended on any specific physical law, they might be invalidated by quantum gravity, which would invalidate his argument because that might fail in the complete laws of physics. Thus he is necessarily making a very generic conjecture regarding all physically conceivable universe (And a Newtonian universe is certainly conceivable and a robust set of laws; indeed, if we'd have had this conversation just 100 years ago, you'd have had to concede this entire scenario to me on the assumption that it was all literally true). He simply asserts that if I have a physical model with actual infinities, it will lead to a contradiction in that system. My question is very simple: Where's the contradiction in my model? What law of physics was broken?


    2.) Either way, the specifics were answered in the footnotes of the post you're responding to now (which you apparently did not read):

    Quantum mechanics is the place where point-like particles begin to not be quite so literally point-like; however, trying to impose the impossibility of actual infinities becomes more openly ludicrous the more quantum you go in physics, not more reasonable. [...] In Quantum Field Theory, you are forced into accepting the existence of Fock space, which is the space that contains all possible particle configurations (The Hilbert space of all possible particle states; a discussion of this can be found in the first few chapters of any book on QFT, so take your pick, but section 2.4 of Tong's intro QFT book is free and discusses this). The dimension of the Fock space is necessarily infinite (In this case, not being infinite would lead to an actual violation of physical law), and this introduces several very simple "actual infinities."

    The first is that I have an actual infinity of different possible particle states. Two examples where this is used, for instance, is in the description of coherent states, for instance in a pulse of light, and the another example is Unruh radiation. Unruh radiation necessarily contains an infinite number of modes in the radiation (or else it violates physical law, see equation 3, which implicitly sums j from minus to plus infinity). Note that even though Unruh radiation hasn't been directly observed a lab, it is a necessary consequence of the current physical laws. So this means if you are accelerating, you see a universe which has a bath containing an infinite number of particles (But all physically finite quantities, like energy, momentum, and temperature), but if you're standing still (or in an inertial frame), you see a universe with zero particles. This highlights the point that trying to demand that infinity is unphysical just doesn't make any sense from a QFT perspective nor does it comport with any known facts of the matter. These theories may break down and render the actual number of particles finite, but as of now, the evidence supporting the assumptions of QFT lead us to these conclusions; and even if so, the only thing that could even conceivably do this is quantum gravity (But quantum gravity also could conceivably lead to past infinities, so this kind of appeal is a double-edged sword for WLC).

    Back to the point of Fock space: This means if I want to start my universe out with an infinite number of point particles, there actually cannot be any principle that forbids me from doing so, because that principle would then by direct implications break the previous physical principles (In this case, Lorentz invariance and unitarity). Thus the laws of QFT + WLC's new "No infinities" principle entails a contradiction. Thus it is "Necessarily possibly true."

    So going to quantum mechanics doesn't provide a way to dispute my overall point about actual infinities, it simply makes WLC's argument even more obviously wrong from the outset. Again, all I'm trying to do here is give a much simpler, much more intuitive example of an "actual infinity." Beyond which, if we accept that we should take quantum gravity into consideration, then his entire argument fails for other reasons, like our lack of knowledge of quantum gravity means we may very well be capable of living in a past-infinite universe.


    Quote Originally Posted by Squatch
    As a side note, I didn't see a response to my highlighting that you missed a point in your critique of Craig's defense. Specifically:

    But what your OP doesn't deal with, and what is Craig's larger point in this article, is the bolded section of the quote.


    I think that can easily be highlighted with a simple point. Given your assumption that the mass of all particles is greater than 0, the total mass of the universe is unchanged from T=0 to T=N for all N. So we get the problem of adding mass to a universe, but not changing its total mass.
    Firstly, what on earth are you talking about? My model does not do this under any circumstance. I'm not even coming close to saying that. The mass starts out infinite and ends up being infinite in this scenario. Without having gravity, there's no contradiction, but again this was discussed in my footnotes that you seem not to be reading:

    Also, I have notably neglected gravity, but this shouldn't impose any problems. In principle I could add gravity to my evolution equations, only now I'd need to add as a postulate that there's finite mass for every finite region and then setting up the equations to end up at the nth -> (n+1)th positions, for all n. It seems laborious, but not impossible.

    For instance, I can make each mass of each particle go as mn = m0 (1/n)2. The total interior of the [1,2) will then have a finite 2 m0 amount of mass. Then when the outside particle moves into the interval, there total mass will be 3 m0 (But the crucial point is that it's still finite). Absolutely nothing changes (the energy and momentum simply become even more convergent), because the mass never entered into the problem other than that it's necessary for it to be non-zero and finite.




    To add another quick footnote on the issue of hypothetical universes:

    There's a reason why you can't object to me using point particles; the primary reason is that, for all we know, classical point particles are the fundamental object in Nature.

    Let me take a convoluted example to make this point: You are familiar with the Bohmian interpretation of quantum mechanics, which makes use of a literal point particle and a "pilot wave." Now, is that model likely to be correct? No, not at all. But is it literally impossible? No, not so far as I know. Our universe possibly could be described by this "quantum mechanics" (which isn't quantum mechanics, but simulates the effects of quantum mechanics). But if this model is correct, then everything that I'm describing is literally fine. Yes, there's a pilot wave, which one would need to account for its interactions, but this is essentially no different than setting up the gravity example, because all the wave would do is add forces to the particle. You might even be able to make the pilot wave itself be constructed out of point particles that interact non-locally. But I could deterministically setup the system, and evolve precisely in the manner than I'm describing. The mere fact that this is possible is enough to invalidate everything that WLC is asserting.


    It is precisely because of the fact that we don't understand the fundamental nature of reality that WLC cannot appeal to specific sets of laws if his conjecture about actual infinities is correct, and thus what he is saying must apply very generally to any conceivable, sensible, robust version of physical laws. And the mere fact that every currently known and used theory of physics allows or actively involves various kinds of actual infinities inside of them --this should give anyone who hears WLC's conjecture some serious pause.
    "Those who can make you believe absurdities, can make you commit atrocities." --Voltaire

  10. #10
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    Re: WLC is Wrong: The Argument for Actual Infinities

    Quote Originally Posted by CliveStaples View Post
    You're responding generally to the topic based on a topic sentence that takes up 1% of the OP, and ignoring the other 99% of the OP that presents a specific argument and summarizes it. On-topic responses should respond to the arguments presented in the OP, yes?
    I would encourage you to re-read the OP. It is not only the Topic title statement, it is the opening preamble paragraphs, the structure of the argument (it isn't a rebuttal of an argument, it is an argument for the physical existence of actual infinities), and it is the finding of the summary paragraph. This is hardly me nitpicking a thread title.

    Pointing out a rebuttal conclusion doesn't follow because it fails to address 2/3rds of the initial argument isn't off topic, especially when I'm also addressing the rebuttal presented.


    Quote Originally Posted by CS
    Just because there's "additions" and "infinity" involved doesn't mean there's an infinity being created by successive addition.
    That is true. But we do know a few things given the scenario offered. 1) An actual infinity exists and 2) successive addition is the only offered method of the addition of particles.


    Now, maybe there is another method that allows for that infinite number of particles to have been added to the system, that's fine. But pointing out that the only mechanism presented in the hypothetical is insufficient to bring about the state of the hypothetical is hardly incorrect. What's more, pointing out that the initial assumption "there are an infinite number of particles" needs some robust defense is hardly off topic as well.


    This is even more the case given the context of the discussion being represented. Remember, this is about the passage of time. We are operating within an A-theory assumption, as noted in the argument of Craig's I posted. We can debate it's applicability elsewhere, but given that it is the context GP is trying to justify an actual infinity within, we need to operate within its limits here.

    A-theory posits that temporal passage is, in fact, something real. That we are successively moving through time rather than it all just existing causally concurrent. That kind of passage is represented, in GP's analogy, by the addition of new particles to the system. The problem is, that this kind of temporal passing is the only type of temporal passing that exists within that context.

    So to argue that the actual infinite is created by some other means, is to implicitly move outside of the bounds of the discussion. Now there might be a good reason to do that, but simply assuming it to be the case is insufficient. If we want to posit another method of formation (either of particles or of time) that method needs to be presented by its proponents as part of their analogy, rather than just vaguely appealed to.



    Quote Originally Posted by CS
    Rather, what's going on is that the Hotel (space) already has an infinite number of guests (particles), and GP is explaining how to accommodate an additional guest (particle) in the Hotel (space).
    I fully understand what GP is arguing here. The problem, imo, is that he is arguing the mechanic described in the analogy and completely ignoring the two major objections raised by Craig.


    Craig's two major objections is that we have a hotel that is both "full" and "not full" at the same time. If we were to take that to GP's analogy, GP is implying that there isn't really any such thing as a room and the guest's don't occupy physical space. But GP doesn't address that issue directly, which is why I offered it in my first response.


    He also ignores a) the practical implication of his rebuttal (matter does not occupy space) and b) the primary objection to the process described (we can add mass to a system via non-zero massed particles without changing the overall mass of the system, which was actually what Craig was objecting to).


    What's more, and what gets to the first and second quotes above, is that even if we assume GP is correct in this entire argument and that particles don't occupy space and that the full/not-full issue isn't a big deal, we still don't reach his conclusion, why? Because it misses literally 2/3rds of the objections to the existence of an actual infinity. One being the second argument Craig presents, which is if the process of adding particles to GP's hypothetical system is as he described, we cannot accumulate an actually infinite number of particles.



    Quote Originally Posted by GoldPhoenix View Post
    1.) You need to re-read the OP before responding, because you seem to be operating under the false pretense (in spite of it being repeated multiple times in the OP and in subsequent posts) that I'm building up an infinity of particles over time, one particle at a time.
    Not at all GP. I think it is easy to focus on objections to our comments and not see the other sections. My initial response was essentially, if we ignore the rest of Craig's argument and support, and just focus on this one section, and we assume particles don't actually occupy space, then the process you describe doesn't really have an objection because you've essentially redefined the initial state in Craig's analogy from "every room is full" to "there are no rooms, just a hall." Thus the fully/not full issue drops away. I then pointed out that there was still an objection based on the "adding mass doesn't change the mass of the system" conclusion and that those are some pretty hefty assumptions you've offered, but given them, I don't see an issue.

    The next logical point is to then ask, so what? The assumptions I've asked for a more rigorous defense on, the next question is does the scenario you present have anything to do with both, our universe, and the argument Craig's offering? That is where the quoted text comes in.


    What I said was two fold:

    1) You are invoking successive addition as a process in your hypothetical as part of your explanation of "new guests entering the hotel." I'm not sure how there could be an objection to that, that would be the process described in the indented quote offered.

    2) You are assuming an infinite set of particles exists with no explanation of how it came into existence.


    Now, if we are to tie your hypothetical back to Craig's actual argument, that last point needs to be addressed (along with the issue and assumptions noted).



    Quote Originally Posted by GP
    The Hilbert's Hotel example that WLC brings up doesn't make use of any quantum mechanics or anything else, so why should that be important in any way for showing why actual infinities are impossible?
    The question that prompted this discussion, "do particles in our universe occupy space?" is relevant to this argument because we are asking if your offered exception matches the analogy Craig offers, or, more importantly, the universe we actually live in (since his claim is that an actual infinite cannot exist in reality). That question still remains unanswered (I did read your footnote originally, and again this time, it discussion both a scenario of infinitely possible states or possible particles, not an existent infinite number of particles. And the question would still remain there, can we, under QM treat a particle as a point that occupies no physical space?)

    The question, however still remains, does the assumption you are making comport with our universe? The question could be otherwise stated, "is a point particle a useful idealization, like a frictionless plane, or an actual thing?" I perfectly understand your argument that it would be possible in another universe (which of course would have all kinds of other ramifications), but that misses why this question was asked. Remember, the use of a point particle rather than a particle that occupies space is necessary to avoid the analogy of rooms. IE all possible 'occupiable' points aren't actually occupied in your scenario, which allows you to escape the Hilbert conclusions because the hotel was never full. But the question needs to be asked, does that idealization fit within our universe? Can we actually have an infinitely dense region of space? Black Holes and the initial Singularity of the Big Bang would seem to be examples of this, but generally, as I understand it, those infinitely dense regions are seen as problems to be worked out rather than existent states, right?


    Quote Originally Posted by GP
    Firstly, what on earth are you talking about? My model does not do this under any circumstance.
    Hmm, that wouldn't seem to be the case as I understood your Axiom 3:

    Ax 3: In Newtonian mechanics, the only criterion for the system to be physical is that in any finite volume with particles, all of the masses of each particle must be greater than zero...


    If you have a set mass in the system and you are adding an additional particle of non-zero mass you would seem to be adding net mass to the system. Which is what I was referring to. In this response however, you state:

    I can make each mass of each particle go as mn = m[sub]0[/sub (1/n)2. The total interior of the [1,2) will then have a finite 2 m0 amount of mass. Then when the outside particle moves into the interval, there total mass will be 3 m0 (But the crucial point is that it's still finite). Absolutely nothing changes (the energy and momentum simply become even more convergent), because the mass never entered into the problem other than that it's necessary for it to be non-zero and finite.

    Which initially would seem to avoid the issue because the mass of the particles wouldn't be constant, but would be decreasing as they moved (since the mass seems to be based on their position in line). This would mean that the mass of the particles is a limit approaching 0. From a physical stand point this would seem to be less of adding a new particle, and more of forming a particle from the lost mass of the existing particles. If we physically interpret the new particle as new to the system, where does the mass go as particles are shifted from point n1 to n2?


    Before this gets dismissed as a silly arcane question, I would point out it is critical to the underlying analogy. Either we are really just shifting things around internal to the infinite, in which case none of the "rooms" in the hotel really get empty, we just move bags around, or we need to explain where the mass went that allows for the equivalent masses in the two states.
    "Suffering lies not with inequality, but with dependence." -Voltaire
    "Fallacies do not cease to be fallacies because they become fashions. -G.K. Chesterton
    Also, if you think I've overlooked your post please shoot me a PM, I'm not intentionally ignoring you.


  11. #11
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    Re: WLC is Wrong: The Argument for Actual Infinities

    Quote Originally Posted by Squatch347 View Post
    I would encourage you to re-read the OP.
    Clive clearly understands the OP. You clearly do not, so I would follow your own advice here.

    Quote Originally Posted by Squatch
    2) successive addition is the only offered method of the addition of particles.
    That's completely wrong.

    I Challenge to support a claim. you to quote anything in the OP that suggests that the infinity of particles create was created by successive additions until an infinity was reached.

    Spoiler Alert: You can't.


    Quote Originally Posted by Squatch
    Now, maybe there is another method that allows for that infinite number of particles to have been added to the system, that's fine. But pointing out that the only mechanism presented in the hypothetical is insufficient to bring about the state of the hypothetical is hardly incorrect. What's more, pointing out that the initial assumption "there are an infinite number of particles" needs some robust defense is hardly off topic as well.
    The defense was given, Squatch, you just ignored it, misunderstood it, or otherwise refused to respond to it three times now:

    The OP:

    Suppose this hypothetical universe starts out with an infinite number of particles. Again, classical mechanics has no objections to this (in fact, the consistency of infinite particles is used thoroughly in statistical mechanics to approximate systems of large particles), quite literally, the only question is "How many particles, where are they, and how fast are they moving?" As long as you can specify this data whilst keeping finite momentum and energy, there's no physical contradiction. So let's just pick their positions (Remember, I'm not doing this in real time, I'm saying that my initial time, this system happens to be in this configuration).


    My Second Post:

    Secondly, and I cannot emphasize this enough, there's no successive addition. I explicitly created this example for you, Squatch, because it avoids any kind of successive addition. There's no creation of infinity over time here, Squatch. God simply starts the universe off in this configuration. The only infinity here is a divisible infinity, which you have repeatedly stated in past threads that you have no objection to. God simply says "I wish to begin with this specific universe", and starts the clock running at t=0.


    The Post You're Now Replying to:

    The entirety of your objections here are centered around the notion that I'm successively adding particles. I'm not adding an infinite number of particles in by hand over time. I'm saying that we play God of this hypothetical universe, and we just will a universe into existence, with the stipulation that the universe is consistent with a set of (simplified) physical laws. If WLC is right, this universe must derive a contradiction or a violation of a physical law. So when I write down the equations that describes the universe, and then I check that it is self-consistent with the physical laws. If WLC is correct then I must derive a contradiction; however, I didn't derive a contradiction, thus WLC is wrong.

    I shouldn't need to repeat myself a fourth time here, Squatch.

    Quote Originally Posted by Squatch
    I fully understand what GP is arguing here. The problem, imo, is that he is arguing the mechanic described in the analogy and completely ignoring the two major objections raised by Craig. [...] Craig's two major objections is that we have a hotel that is both "full" and "not full" at the same time. If we were to take that to GP's analogy, GP is implying that there isn't really any such thing as a room and the guest's don't occupy physical space. But GP doesn't address that issue directly, which is why I offered it in my first response. [...] He also ignores a) the practical implication of his rebuttal (matter does not occupy space) and b) the primary objection to the process described (we can add mass to a system via non-zero massed particles without changing the overall mass of the system, which was actually what Craig was objecting to).
    See above. Squatch, you can disagree with a counter argument and you can say that a counter argument doesn't rebut the points on the table, but you can't say it was never given. This idea that I haven't responded to this objection of yours is unacceptable.


    Quote Originally Posted by Squatch
    What's more, and what gets to the first and second quotes above, is that even if we assume GP is correct in this entire argument and that particles don't occupy space and that the full/not-full issue isn't a big deal, we still don't reach his conclusion, why? Because it misses literally 2/3rds of the objections to the existence of an actual infinity. One being the second argument Craig presents, which is if the process of adding particles to GP's hypothetical system is as he described, we cannot accumulate an actually infinite number of particles.
    Except that you raised the 2/3 of his response already, and I responded why it didn't apply, and you've simply ignored it. Ignoring my argument does not mean that you've countered my argument.


    Not at all GP. I think it is easy to focus on objections to our comments and not see the other sections. My initial response was essentially, if we ignore the rest of Craig's argument and support, and just focus on this one section, and we assume particles don't actually occupy space, then the process you describe doesn't really have an objection because you've essentially redefined the initial state in Craig's analogy from "every room is full" to "there are no rooms, just a hall."
    What? In what universe does this represent a response to my argument?

    Quote Originally Posted by Squatch
    Thus the fully/not full issue drops away.
    False. The question "Is there a particle at every point in the set?" corresponds to the question "Is every room full?" It's well-defined.

    Quote Originally Posted by Squatch
    I then pointed out that there was still an objection based on the "adding mass doesn't change the mass of the system" conclusion and that those are some pretty hefty assumptions you've offered, but given them, I don't see an issue.
    The mass of the total system, Squatch, and at t=0 that includes the particles inside of the interval and outside. If you bother to do the calculations yourself, you'll see that the total mass at t=0 is the same mass at t=1.

    Quote Originally Posted by Squatch
    The next logical point is to then ask, so what? The assumptions I've asked for a more rigorous defense on, the next question is does the scenario you present have anything to do with both, our universe, and the argument Craig's offering? That is where the quoted text comes in.
    Asked an answered twice now. You can continue to not read my statements about QM, but it doesn't mean the statements were not responded to.

    Quote Originally Posted by Squatch
    What I said was two fold:

    1) You are invoking successive addition as a process in your hypothetical as part of your explanation of "new guests entering the hotel." I'm not sure how there could be an objection to that, that would be the process described in the indented quote offered.
    I'm adding one new guest, yes. I'm not invoking an infinite successive additions, I'm invoking a single addition of one particle.

    Quote Originally Posted by Squatch
    2) You are assuming an infinite set of particles exists with no explanation of how it came into existence.
    You really need to try reading my posts to you.

    Quote Originally Posted by Squatch
    Now, if we are to tie your hypothetical back to Craig's actual argument, that last point needs to be addressed (along with the issue and assumptions noted).
    Asked and answered.


    Quote Originally Posted by Squatch
    The question that prompted this discussion, "do particles in our universe occupy space?" is relevant to this argument because we are asking if your offered exception matches the analogy Craig offers, or, more importantly, the universe we actually live in (since his claim is that an actual infinite cannot exist in reality). That question still remains unanswered (I did read your footnote originally, and again this time, it discussion both a scenario of infinitely possible states or possible particles, not an existent infinite number of particles. And the question would still remain there, can we, under QM treat a particle as a point that occupies no physical space?)

    The question, however still remains, does the assumption you are making comport with our universe? The question could be otherwise stated, "is a point particle a useful idealization, like a frictionless plane, or an actual thing?" I perfectly understand your argument that it would be possible in another universe (which of course would have all kinds of other ramifications),
    Such as?

    Quote Originally Posted by Squatch
    but that misses why this question was asked. Remember, the use of a point particle rather than a particle that occupies space is necessary to avoid the analogy of rooms. IE all possible 'occupiable' points aren't actually occupied in your scenario, which allows you to escape the Hilbert conclusions because the hotel was never full. But the question needs to be asked, does that idealization fit within our universe? Can we actually have an infinitely dense region of space? Black Holes and the initial Singularity of the Big Bang would seem to be examples of this, but generally, as I understand it, those infinitely dense regions are seen as problems to be worked out rather than existent states, right?
    It's already been explained to you that his argument cannot depend on the very specifics of the laws of our universe. If it did, that would mean that he was appealing to actual laws of reality, which we do not know. That alone would invalidate his argument, wholesale. So this is just bloviating rather than addressing the argument that was forwarded to you.

    Quote Originally Posted by Squatch
    Hmm, that wouldn't seem to be the case as I understood your Axiom 3:

    Ax 3: In Newtonian mechanics, the only criterion for the system to be physical is that in any finite volume with particles, all of the masses of each particle must be greater than zero...


    If you have a set mass in the system and you are adding an additional particle of non-zero mass you would seem to be adding net mass to the system. Which is what I was referring to. In this response however, you state:

    I can make each mass of each particle go as mn = m[sub]0[/sub (1/n)2. The total interior of the [1,2) will then have a finite 2 m0 amount of mass. Then when the outside particle moves into the interval, there total mass will be 3 m0 (But the crucial point is that it's still finite). Absolutely nothing changes (the energy and momentum simply become even more convergent), because the mass never entered into the problem other than that it's necessary for it to be non-zero and finite.

    Which initially would seem to avoid the issue because the mass of the particles wouldn't be constant, but would be decreasing as they moved (since the mass seems to be based on their position in line). This would mean that the mass of the particles is a limit approaching 0. From a physical stand point this would seem to be less of adding a new particle, and more of forming a particle from the lost mass of the existing particles. If we physically interpret the new particle as new to the system, where does the mass go as particles are shifted from point n1 to n2?
    This question betrays a serious lack of comprehension of what is contained in the OP. Squatch, I never said "Let's treat the particle as new to the system." That would lead to the most trivially stupid violation of the conservation of energy. I said, very clearly, in the OP, that we should consider a system with an infinite number of particles lying at the points I specified with the velocities I specified, as well as a particle outside the interval. That's t=0. After we evolve the system forward to t=1, now every particle lies in the old points on the interval. So every old point is still occupied, but now it includes the particle that was previously outside the interval. Yes, the energy inside the interval has increase by 1 m, but the total energy of the system has not changed, there's no discontinuity in any physical observable, and there's no successive additions. The particles all have simply moved, as particles do.

    As for where the particles go, the OP literally tells you. If you ask about the nth particle, you can use the information in the OP to calculate the specifics of the particle at any time, including where it is at t=1.

    More to the point: I'm not saying the mass at every point in [1,2) is the same at t=0 and t=1. I'm only saying the same points have been occupied, just now with one more particle. The mass, energy, and momentum inside of the interval increases with the introduction of the new particle. But everything is still finite and physical.

    Quote Originally Posted by Squatch
    Before this gets dismissed as a silly arcane question, I would point out it is critical to the underlying analogy. Either we are really just shifting things around internal to the infinite, in which case none of the "rooms" in the hotel really get empty, we just move bags around, or we need to explain where the mass went that allows for the equivalent masses in the two states.
    The mass hasn't gone anywhere, Squatch. This is a claim that you're making up from whole clothe.

    I haven't regulated the mass in my example, so if you like you can take my regularization scheme (instead choose the masses to go as 1/n^2) and then check for yourself what the total mass is at t=0 and compare it at t=1. It'll be the same.
    "Those who can make you believe absurdities, can make you commit atrocities." --Voltaire

  12. #12
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    Re: WLC is Wrong: The Argument for Actual Infinities

    Quote Originally Posted by Squatch
    Given the thread's title I would argue it is well within scope. GP's argument there (as well as in the OP) is that Craig is wrong about his conclusion on Actual Infinities, not that a specific argument against them fails. It is perfectly valid to note that an opponent did not rebut a major section of an argument without it being a red herring.

    In this case GP did not respond, as part of his OP against 2/3 of that specific argument against actual infinities (the conflicting answer section and the "guests" checking out section), nor against an entirely separate argument against an actual infinity.
    You're responding generally to the topic based on a topic sentence that takes up 1% of the OP, and ignoring the other 99% of the OP that presents a specific argument and summarizes it. On-topic responses should respond to the arguments presented in the OP, yes?

    ---------- Post added at 02:08 PM ---------- Previous post was at 02:04 PM ----------

    Quote Originally Posted by Squatch
    You are describing successive addition there GP. If I take a quantity, add another to it. Take the new quantity and add another to it. That is successive addition.
    Just because there's "additions" and "infinity" involved doesn't mean there's an infinity being created by successive addition.

    Rather, what's going on is that the Hotel (space) already has an infinite number of guests (particles), and GP is explaining how to accommodate an additional guest (particle) in the Hotel (space).

    An "infinity by successive addition" error would be something like: the Hotel (space) has a finite number of guests (particles), and you keep adding in additional guests (particles) and then claim that the Hotel (space) has an infinite number of guests (particles). But that is not at all what's going on in GP's argument.

    ---------- Post added at 02:29 PM ---------- Previous post was at 02:08 PM ----------

    What's more, given the actual nature of Craig's argument that I quote above, you'll note that he is pointing out that successive addition is the only method of adding particles to your space.

    So given that constraint, the natural question arises, how did we get to have an infinite number of particles in that space if I am required to add them successively?


    You're missing the point of the hypothetical. Note how it beings:

    Suppose this hypothetical universe starts out with an infinite number of particles. Again, classical mechanics has no objections to this (in fact, the consistency of infinite particles is used thoroughly in statistical mechanics to approximate systems of large particles), quite literally, the only question is "How many particles, where are they, and how fast are they moving?" As long as you can specify this data whilst keeping finite momentum and energy, there's no physical contradiction. So let's just pick their positions (Remember, I'm not doing this in real time, I'm saying that my initial time, this system happens to be in this configuration).

    The point of this hypothetical is to exhibit a model with an infinite number of particles and discover what, if any, physical laws it breaks.

    The trivial answer to your question is that it doesn't matter how those infinitely-many particles got to where they were; the points is that the laws of physics tolerates their simultaneous existence as described in the model.

    The less trivial answer to your question is that there might be any number of possible histories that would produce these infinitely-many particles--perhaps there have always been infinitely-many particles, with finite numbers of them coming into or passing out of existence; or perhaps at time t, a particle comes into existence, for t in [-1, 0) (if you intend to argue that these histories would be physically impossible, please cite what physical laws would be broken).
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    Re: WLC is Wrong: The Argument for Actual Infinities

    The following is a discussion, no rebuttal is included nor should be taken to exist.
    The "points" contained here have lack the rebuttal dimension (attempt at physics pun).

    Quote Originally Posted by GP
    Sure, one could say those points exist in space, but there aren't any particles occupying those points.
    Two things I don't understand.
    1) How the point exists, or is said to exist in reality (or possibly in reality).
    2) If the other points exist, and points = rooms, then I don't see how the concept of being full can be applied here.
    I mean your given set may be full, but it's pretty arbitrary.


    Quote Originally Posted by GP
    Points are the collections of objects that make up space. In this case, I'm just looking at one dimension of space, so we're just talking about numbers as points and the real number line (all real numbers) is the space.

    Point particles are objects that can traverse this space and occupy one of the points inside of the space.

    So if I have a point particle described by

    x(t) = 1 + 2t

    Then at t=0, it's occupying the point "1." At t=1, it's occupying the point "3." At t=2, it's occupying the point "5."
    O.K. So one moves, and the other doesn't, but otherwise they are the same, Have I got that right?


    Quote Originally Posted by GP
    No, there isn't any problem (in classical mechanics). I can't really answer "Why not" unless you give me an example of something you might think is a problem.
    I don't really have any objections yet. Your pretty much in God Mode right now, defining and I'm just trying to accuratly grasp what you are saying.
    My forming concern is that, assuming your argument is true and actual infinite can exist, if your example only works with single demnisional things, then it doesn't really support that our universe could be infinity old or counter that an actual infinite universe is not possible.

    I'm not forwarding that, but it plays into a bit of the next few quote & exchange.

    Quote Originally Posted by GP
    Because physics uses "abstracts" in order to understand the real world. These "abstracts" are why you can look at a screen right now, and it's not dark, but full of illuminated pixels. The purpose of "abstracts" is that we can conceive of them and they can inform us on conceivable scenarios. In this case, the scenario may not be literally realizable (although 100 years ago, you wouldn't have been able to tell me that this wasn't possible), the purpose is to expose the viewers to a different way of thinking about these issues and make them understand that actual infinities are conceivable in a (simplified) set of physical laws.
    Certainly I understand the role of imagination, but it is one thing to say "I can imagine X" and another thing to say it is a viable model for reality.
    As it stands, I do not see how some of these abstracts can exist at all, and some seem to be understood as not being able to exist.
    So, I'm unsure of how using something that can't exist in reality can be used to support that Infinite can exist in reality.
    I'm also having difficulty imagining what you are talking about.

    Quote Originally Posted by GP
    Also, to clarify, I'm not saying points "don't exist." I'm simply telling you where in the space a collection of point particles are located. By saying there isn't a particle at point 1.25, I'm not saying the point doesn't exist, I'm just saying there's no particle occupying that point in this scenario.
    Thanks, that is a helpful clarification.
    However, I am still confused by the idea of points being rooms, and some points existing that aren't used, and thus understanding that the hotel is full.

    .... i guess the points not being used, would just be another hotel


    Quote Originally Posted by GP
    Honestly, I don't know of any documentation that's not at an upper undergraduate level (And for those cases I suggest David Tong's Classical Dynamics and Statistical Physics free, online books). The problem is that most 100-level or high school courses introduce the point particle as some kind of approximation to macroscopic objects (e.g. a stone, a box, or a planet) and bloviate about this for a long time. They tend never to discuss that for fundamental, massive objects (e.g. an electron), they are the non-relativistic approximation. Thus when you do statistical mechanics of a gas (e.g. like in Tong's Statistical Physics), you see that classical gases are defined as a large collection of point particles (There its approximated as an infinite number of point particles, again illustrating the fact that an infinite number of things does not lead to absurdities, or else, e.g. an important consequence of classical gases, you'd never have learned PV = nRT in your introductory chemistry course, but I digress).
    Zzzzzz.. O.. sorry, you lost me at the end there *J*


    Thanks for the reference, though if it's that technical it won't make my reading list.. but nice to know anyway

    Now I understand that points are used in continuations, but they do still seem to come up with absurdities. Like the gas thing, if those points were assumed to exist (with mass) they would be black holes (as they would be infinity dense), then of course gas wouldn't act like it does. (from that point mass link I had earlier). It may be handy and extremely precise to treat them that way, but it seems to be admitted that they can't exist.... which confuses me.

    That is one of my blocks in understanding, the things you are talking about (points & Particles) seem to have some pretty clear limits to their ability to exist in reality. Those limits seem to be relevant differences between your examples and objections raised involving rooms and guests.



    ----
    To the point of Abstract.
    If the example you are giving is completely abstract, I think it is exactly like saying.
    "Infinite numbers exist, therefore infinite can exist in reality"

    Is that a fair understanding?
    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.

  14. #14
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    Re: WLC is Wrong: The Argument for Actual Infinities

    Quote Originally Posted by MindTrap028 View Post
    Two things I don't understand.
    1) How the point exists, or is said to exist in reality (or possibly in reality).
    2) If the other points exist, and points = rooms, then I don't see how the concept of being full can be applied here.
    I mean your given set may be full, but it's pretty arbitrary.
    The proper notion of "exist" here should be construed as "existing in space." So you start off with space (here represented as a line, since we're only considering 1-dimensional movement, although you could add more freely). Space, then, is the infinite collection of all points lying on a line. So for a point to exist, we simply mean that it is one of the points making up space itself, which are fixed. Thus this universe is given by a 1-dimensional space, along with a single time dimension.

    When I say "A point particle exists", I mean that it's existing in space and thus must be occupying a single point. Particles move over time, so the particular point that they are occupying can change. In other words, if the particle has a velocity and is moving, then the points that its occupying change over time.


    Quote Originally Posted by MT
    My forming concern is that, assuming your argument is true and actual infinite can exist, if your example only works with single demnisional things, then it doesn't really support that our universe could be infinity old or counter that an actual infinite universe is not possible.
    No, if I wanted I could take a two or three dimensional space, as well. (Actually, I can take as many dimensions as I want.)

    In principle, I could even make the particles rotate inside of a sphere, although the math would be more sophisticated and I would have to check for the finiteness and conservation of angular momentum in addition to momentum, energy, and mass.

    Quote Originally Posted by MT
    Certainly I understand the role of imagination, but it is one thing to say "I can imagine X" and another thing to say it is a viable model for reality.
    As it stands, I do not see how some of these abstracts can exist at all, and some seem to be understood as not being able to exist.
    So, I'm unsure of how using something that can't exist in reality can be used to support that Infinite can exist in reality.
    Well, as I've stated to Squatch, even if Newtonian mechanics doesn't describe our universe (It doesn't) or our universe isn't given by point particles (there's no evidence suggesting now that they do, only evidence to the contrary, but as I stated with Squatch, even that's not technically certain), it really doesn't matter. Our universe very much appears to be described by quantum fields, and quantum fields definitely seem to require an infinite number of quantum particles, at least in principle.

    Thus the falseness of WLC's claim that there can be no infinities seems to fall quite flat. Statistical mechanics makes thorough use of the fact that you can model systems with an infinite number of particles (it's necessary that such a system is physically consistent), the large N expansion of QFT is routinely used as a manner of computing simplified cases (It involves writing down a universe where you copy your fields an infinite number of times, which combined with other aspects of QFT forces the system to be simplified), GR routinely uses infinitely large spacetimes (including the ones used to model the cosmology of our universe, for which there's very strong evidence that our universe should have an infinite volume), and so on. The fact that --even if our universe happens to contain no infinities-- it appears like infinities and all known consistent physical laws have no problems admitting infinities tells me that WLC cannot be correct. He has no idea which universe we live in, but we do not suffer for possible universes and physical laws that consistently admit infinities without problem.

    Quote Originally Posted by MT
    Thanks, that is a helpful clarification.
    However, I am still confused by the idea of points being rooms, and some points existing that aren't used, and thus understanding that the hotel is full.
    The point is that I'm simply giving an example of something imaginable, with simple physical laws, that demonstrates what's going on with Hilbert's Hotel.


    Quote Originally Posted by MT
    Now I understand that points are used in continuations, but they do still seem to come up with absurdities. Like the gas thing, if those points were assumed to exist (with mass) they would be black holes (as they would be infinity dense), then of course gas wouldn't act like it does.

    (from that point mass link I had earlier). It may be handy and extremely precise to treat them that way, but it seems to be admitted that they can't exist.... which confuses me.

    That is one of my blocks in understanding, the things you are talking about (points & Particles) seem to have some pretty clear limits to their ability to exist in reality. Those limits seem to be relevant differences between your examples and objections raised involving rooms and guests.
    No, they've given regulated masses, just like I gave my particles (in my most recent reply to Squatch). Thus the density remains finite and no black holes are generated.

    If this weren't taken care of, then statistical mechanics would not be a consistent approximation to use for systems made out of many bodies (again, e.g., a gas).



    Quote Originally Posted by MT
    To the point of Abstract.
    If the example you are giving is completely abstract, I think it is exactly like saying.
    "Infinite numbers exist, therefore infinite can exist in reality"

    Is that a fair understanding?
    It's not completely abstract because what I'm doing is 100% consistent with the laws of Newtonian mechanics. As I said to Squatch, if we'd have had this conversation 110 years ago, then no one could have argued with me that I was doing something irrelevant and abstract --I would have literally been using what were assumed to be the Laws of Nature. Now, however, we understand that nature is more complicated.

    However, with that said, if we can't even see that infinity imposes problems in simple laws of Nature, then it's difficult for me to imagine making the laws more subtle and complex is going to suddenly prevent actual infinities.

    Again, we don't want for lists of physical laws that describe real systems which use the consistency of infinity in these physical laws (See my examples above). It therefore seems to me that the claim that this should be impossible must either appeal to laws currently unknown (in which case the argument is very weak and necessarily speculative) or else we should already start to see what the kinds of issues might be in simpler versions of the laws of Nature.
    "Those who can make you believe absurdities, can make you commit atrocities." --Voltaire

  15. #15
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    Re: WLC is Wrong: The Argument for Actual Infinities

    Thank You GP for your patient response. I think it was helpful.
    However, before I try to repeat what I understand you to say in completion, I'll focus on this one point.

    Quote Originally Posted by GP
    It's not completely abstract because what I'm doing is 100% consistent with the laws of Newtonian mechanics. As I said to Squatch, if we'd have had this conversation 110 years ago, then no one could have argued with me that I was doing something irrelevant and abstract --I would have literally been using what were assumed to be the Laws of Nature. Now, however, we understand that nature is more complicated.

    However, with that said, if we can't even see that infinity imposes problems in simple laws of Nature, then it's difficult for me to imagine making the laws more subtle and complex is going to suddenly prevent actual infinities.

    Again, we don't want for lists of physical laws that describe real systems which use the consistency of infinity in these physical laws (See my examples above). It therefore seems to me that the claim that this should be impossible must either appeal to laws currently unknown (in which case the argument is very weak and necessarily speculative) or else we should already start to see what the kinds of issues might be in simpler versions of the laws of Nature.
    So I totally understand taking a simplistic view and the reasoning for that.

    So, I am going(read try) to take it even more simply.


    1-In reality we can count things(not necessarily in the physically countable sense), with numbers.
    2-Numbers are infinite
    2-There are no physical laws that would stop the correlation.
    3-Thus Infinites are consistent with physics.
    4-Thus Infinites can exist in reality.


    Now, numbers are abstract, but I think I am applying it to physics in a similar manner to your argument.
    Am I correct in the form of the argument? Is there a specific difference(other than perhaps poor wording, I'm hoping you can see past my poor phrasing)?

    ---edit---
    Explanation of why I think it reflects your argument.

    I think this is basically your first step (numbering locations in space) in a nut shell, and that your argument simply adds layers to this simplified point by repeating it basically.
    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.

  16. #16
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    Re: WLC is Wrong: The Argument for Actual Infinities

    Quote Originally Posted by MindTrap028 View Post
    1-In reality we can count things(not necessarily in the physically countable sense), with numbers.
    2-Numbers are infinite
    2-There are no physical laws that would stop the correlation.
    3-Thus Infinites are consistent with physics.
    4-Thus Infinites can exist in reality.
    Just to clarify a few before agreeing:

    1.) Yes, things are ostensibly countable, even if we can't sit there and physically count them. Counting here, however, specifically means the cardinality of a set, which for infinite sets means the aleph numbers.

    2.) If by "Numbers are infinite", you mean that "There are an infinite amount of natural numbers." (i.e. "The cardinality of the natural numbers, i.e. {1,2,3, ...}, is aleph-null.)

    3.) Thus infinities are in principle consistent with physics. (i.e. we have no reason to assume that they must be inconsistent, due to known counter examples.)


    If you agree to those, then, yes, this is the essence of what I'm arguing.



    EDIT: A note on being able to physically count: Even things aren't infinite, it doesn't mean that we can count them. For instance, you can't count the exact number of atoms in your hand, there's so many it is pragmatically impossible. With that said, you know that there is some finite number of atoms in your hand. Likewise, even if we can't sit there and count the number of particles in the interval [1,2), because here it's infinite, it doesn't mean that they don't or can't exist because you can't personally count them.
    Last edited by GoldPhoenix; July 16th, 2015 at 10:52 AM.
    "Those who can make you believe absurdities, can make you commit atrocities." --Voltaire

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    Re: WLC is Wrong: The Argument for Actual Infinities

    Quote Originally Posted by GP
    I Challenge to support a claim. you to quote anything in the OP that suggests that the infinity of particles create was created by successive additions until an infinity was reached.
    GP, your entire OP is premised on showing that Dr. Craig’s argument is incorrect. Craig’s argument is based on the temporally successive moments created via an a-theory of time. Your objection to his argument is to say, “well lets assume that another mechanism entirely was responsible for creating that infinite history.”

    We can do that, sure, but that doesn’t mean the issue of how that actual infinite was formed is just moot because you assumed it as part of your setup. Questioning that assumption is a perfectly valid exploration of your hypothetical.

    What’s more, saying “let’s just assume they exist” isn’t a mechanism, and you’ll notice my response is that: “2) successive addition is the only offered method of the addition of particles.” You don’t offer another method, you offer their initial creation as an assumption.

    The problem that led to this challenge, imo, is that you broke the response up into several sub-points rather than take it as a single issue. Literally in the next paragraph I point out that we could assume (as you did) that another mechanism could be used to form the actual infinite, but that that mechanism would need to be at least presented as part of the argument if we are going to conclude that Craig is entirely incorrect about his concerns on actual infinites or if we are to conclude that the scenario presented is coherently applicable within our universe.




    Quote Originally Posted by GP
    The defense was given, Squatch, you just ignored it, misunderstood it, or otherwise refused to respond to it three times now:
    Your defense was: “Suppose this hypothetical universe starts out with an infinite number of particles?” That isn’t a defense GP, it is an assumption. My point was that we can’t simply grant this assumption and move along with the hypothetical. To do so, is quite literally, begging the question. You are assuming an actual infinite number of particles exist to show that it is possible for an actual infinite number of particles to exist.


    Or perhaps we could use the next response: “God simply starts the universe off in this configuration.” Fine, then we don’t need the CA, you concede God exists, voila the argument is over. But of course you aren’t doing that, you are assuming He exists to ignore the necessary explanation of the assumption so that you can later conclude the CA doesn’t work and thus God does not exist.

    As I noted in my response, pointing out that we are using the assumption that the initial infinite simply exists as a brute fact is not an unwarranted objection to the argument. I’m objecting to your hypothetical as explanatorily insufficient. There is a big hole in that hypothetical universe that needs to be explained if we are going to conclude that it bears anything like a feasible resemblance to our actual universe.


    Quote Originally Posted by GP
    The entirety of your objections here are centered around the notion that I'm successively adding particles.
    Hmm, this tells me that you haven’t actually read my objections if you think they are centered solely around this one aspect.

    For example, my objection to your use of point particles has nothing to do with this notion.

    My concern about the underlying assumption of an actual infinite to start off the hypothetical isn’t centered around that notion.

    Objections centered around mass aren’t focused on this issue at all either.


    Can you concede that my objection is, at least, a little more robust than you made it out to be?


    Quote Originally Posted by GP
    Squatch, you can disagree with a counter argument and you can say that a counter argument doesn't rebut the points on the table, but you can't say it was never given. This idea that I haven't responded to this objection of yours is unacceptable.
    I will agree that you had a response to the objection made revolving around adding mass to the system, however it isn’t a rebuttal of the point being made (which is that you can add a “guest” to the hotel without adding to the “number of guests in the hotel” category), it was simply a reshuffling of relative values to bypass that point.

    However, to my knowledge no specific rebuttal at all has been offered for the idea that there are practical implications to the idea that matter does not occupy space that are problematic for accepting this hypothetical as representing our universe.


    Quote Originally Posted by GP
    What? In what universe does this represent a response to my argument?
    Well.. Obviously in this one. My point here was that if we ignore all the other objections and reframe the OP as solely a rebuttal of one section of Craig’s argument and we take all the large assumptions of the OP detailed elsewhere as a given, then sure there is a problem with Craig’s argument, and his hotel analogy no longer applies because we’ve begun dealing with particles and space in a way that cannot be treated like guests and a hotel room. But (and it is a big but) those assumptions, and ignoring Craig’s other points, make the hypothetical offered unrelated to the fundamental, underlying argument. That this point needs to represented our universe in order for it to be a reasonable rebuttal.


    Quote Originally Posted by GP
    The question "Is there a particle at every point in the set?" corresponds to the question "Is every room full?"
    Right, and the answer, both before and after the hypothetical additional particle is, “No.” For every two points in your initial state, there are an infinite number of additional points capable of housing a particle between them. That is the crux of your argument, and why a point particle is necessary for it to be coherent.

    Hence why I argued that the structure of your hypothetical is such as to avoid the issue by simply removing the rooms and making a hallway. Perhaps another way to look at it is to say that we could put a screen in every room so that we can fit an additional guest in there. And we can always fit another screen into that room to further sub-divide it and pack another guest in. Which works well and good if we have non-physical guests that don’t occupy space, not so much if those guests need somewhere to sleep.


    Quote Originally Posted by GP
    The mass of the total system, Squatch, and at t=0 that includes the particles inside of the interval and outside. If you bother to do the calculations yourself, you'll see that the total mass at t=0 is the same mass at t=1.



    More to the point: I'm not saying the mass at every point in [1,2) is the same at t=0 and t=1. I'm only saying the same points have been occupied, just now with one more particle. The mass, energy, and momentum inside of the interval increases with the introduction of the new particle. But everything is still finite and physical.



    The mass hasn't gone anywhere, Squatch. This is a claim that you're making up from whole clothe.
    All this does is ignore the objection by focusing on a different system. I asked about the objection that Craig offered, which is that the total mass of the particles in the interval doesn’t change, even though we have added an additional particle with non-zero, positive mass.

    Does the total mass of the interval change from t=0 to t=1? What is that total mass for the interval at both points?

    Quote Originally Posted by GP
    I'm adding one new guest, yes. I'm not invoking an infinite successive additions, I'm invoking a single addition of one particle.
    Fair enough, but you don’t have an objection to the process being carried on successively do you? It wouldn’t cause your scenario any issues to do so, right?


    Quote Originally Posted by GP
    You really need to try reading my posts to you.
    Perhaps you could quote where you explain what process allowed the initial set’s creation? All I’ve seen so far is an appeal to it as an assumption or an invocation of God (which is fine, as long as we are conceding that point fully of course).

    Now if we want to severely limit the scope of your critique to just one issue Craig has with actual infinities I suppose we could assume the above for the sake of argument. The problem I see with that is two fold.

    First, it dramatically reduces the scale of your conclusion in the OP. Perhaps that isn’t a big deal to you, but it is worth pointing out.

    Second, we are still left with a very fundamental question. Is such a set of particles permissible within our universe. We can’t simply assume it is if we have no idea at all how it came into being or the mechanism of its creation. This is a critical burden for the OP to meet if it is really going to overcome Craig’s objections, which relate to an actual infinite being existent in this universe.

    Quote Originally Posted by GP
    Such as?
    This doesn’t really answer any of the questions posed.

    Do particles in our universe occupy space?

    Is a point particle a useful idealization, like a frictionless plane, or an actual thing?

    In QM, do we treat particles as a point that occupies no physical space?
    "Suffering lies not with inequality, but with dependence." -Voltaire
    "Fallacies do not cease to be fallacies because they become fashions. -G.K. Chesterton
    Also, if you think I've overlooked your post please shoot me a PM, I'm not intentionally ignoring you.


  18. #18
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    Re: WLC is Wrong: The Argument for Actual Infinities

    Quote Originally Posted by Squatch347 View Post
    GP, your entire OP is premised on showing that Dr. Craig’s argument is incorrect. Craig’s argument is based on the temporally successive moments created via an a-theory of time. Your objection to his argument is to say, “well lets assume that another mechanism entirely was responsible for creating that infinite history.”

    We can do that, sure, but that doesn’t mean the issue of how that actual infinite was formed is just moot because you assumed it as part of your setup. Questioning that assumption is a perfectly valid exploration of your hypothetical.

    What’s more, saying “let’s just assume they exist” isn’t a mechanism, and you’ll notice my response is that: “2) successive addition is the only offered method of the addition of particles.” You don’t offer another method, you offer their initial creation as an assumption.

    The problem that led to this challenge, imo, is that you broke the response up into several sub-points rather than take it as a single issue. Literally in the next paragraph I point out that we could assume (as you did) that another mechanism could be used to form the actual infinite, but that that mechanism would need to be at least presented as part of the argument if we are going to conclude that Craig is entirely incorrect about his concerns on actual infinites or if we are to conclude that the scenario presented is coherently applicable within our universe.






    Your defense was: “Suppose this hypothetical universe starts out with an infinite number of particles?” That isn’t a defense GP, it is an assumption. My point was that we can’t simply grant this assumption and move along with the hypothetical. To do so, is quite literally, begging the question. You are assuming an actual infinite number of particles exist to show that it is possible for an actual infinite number of particles to exist.


    Or perhaps we could use the next response: “God simply starts the universe off in this configuration.” Fine, then we don’t need the CA, you concede God exists, voila the argument is over. But of course you aren’t doing that, you are assuming He exists to ignore the necessary explanation of the assumption so that you can later conclude the CA doesn’t work and thus God does not exist.

    As I noted in my response, pointing out that we are using the assumption that the initial infinite simply exists as a brute fact is not an unwarranted objection to the argument. I’m objecting to your hypothetical as explanatorily insufficient. There is a big hole in that hypothetical universe that needs to be explained if we are going to conclude that it bears anything like a feasible resemblance to our actual universe.




    Hmm, this tells me that you haven’t actually read my objections if you think they are centered solely around this one aspect.

    For example, my objection to your use of point particles has nothing to do with this notion.

    My concern about the underlying assumption of an actual infinite to start off the hypothetical isn’t centered around that notion.

    Objections centered around mass aren’t focused on this issue at all either.


    Can you concede that my objection is, at least, a little more robust than you made it out to be?




    I will agree that you had a response to the objection made revolving around adding mass to the system, however it isn’t a rebuttal of the point being made (which is that you can add a “guest” to the hotel without adding to the “number of guests in the hotel” category), it was simply a reshuffling of relative values to bypass that point.

    However, to my knowledge no specific rebuttal at all has been offered for the idea that there are practical implications to the idea that matter does not occupy space that are problematic for accepting this hypothetical as representing our universe.




    Well.. Obviously in this one. My point here was that if we ignore all the other objections and reframe the OP as solely a rebuttal of one section of Craig’s argument and we take all the large assumptions of the OP detailed elsewhere as a given, then sure there is a problem with Craig’s argument, and his hotel analogy no longer applies because we’ve begun dealing with particles and space in a way that cannot be treated like guests and a hotel room. But (and it is a big but) those assumptions, and ignoring Craig’s other points, make the hypothetical offered unrelated to the fundamental, underlying argument. That this point needs to represented our universe in order for it to be a reasonable rebuttal.



    Right, and the answer, both before and after the hypothetical additional particle is, “No.” For every two points in your initial state, there are an infinite number of additional points capable of housing a particle between them. That is the crux of your argument, and why a point particle is necessary for it to be coherent.

    Hence why I argued that the structure of your hypothetical is such as to avoid the issue by simply removing the rooms and making a hallway. Perhaps another way to look at it is to say that we could put a screen in every room so that we can fit an additional guest in there. And we can always fit another screen into that room to further sub-divide it and pack another guest in. Which works well and good if we have non-physical guests that don’t occupy space, not so much if those guests need somewhere to sleep.




    All this does is ignore the objection by focusing on a different system. I asked about the objection that Craig offered, which is that the total mass of the particles in the interval doesn’t change, even though we have added an additional particle with non-zero, positive mass.

    Does the total mass of the interval change from t=0 to t=1? What is that total mass for the interval at both points?



    Fair enough, but you don’t have an objection to the process being carried on successively do you? It wouldn’t cause your scenario any issues to do so, right?



    Perhaps you could quote where you explain what process allowed the initial set’s creation? All I’ve seen so far is an appeal to it as an assumption or an invocation of God (which is fine, as long as we are conceding that point fully of course).

    Now if we want to severely limit the scope of your critique to just one issue Craig has with actual infinities I suppose we could assume the above for the sake of argument. The problem I see with that is two fold.

    First, it dramatically reduces the scale of your conclusion in the OP. Perhaps that isn’t a big deal to you, but it is worth pointing out.

    Second, we are still left with a very fundamental question. Is such a set of particles permissible within our universe. We can’t simply assume it is if we have no idea at all how it came into being or the mechanism of its creation. This is a critical burden for the OP to meet if it is really going to overcome Craig’s objections, which relate to an actual infinite being existent in this universe.



    This doesn’t really answer any of the questions posed.

    Do particles in our universe occupy space?

    Is a point particle a useful idealization, like a frictionless plane, or an actual thing?

    In QM, do we treat particles as a point that occupies no physical space?

    Squatch, this is unacceptable. Firstly, it's ridiculous that you're still confused by the technique of proof by contradiction and are continuing to confuse it with the logical fallacy of begging the question. So you're telling me that if I try to assume that I can square the circle, and derive a contradiction, then I'm actually begging the question by assuming I might be able to square the circle? This is a trivial logical error that you're making, Squatch. The only thing that's worse is that I already pointed out the difference to you in the post that you're responding to (and the one before it) and you should have understood the distinction and moved on from this point to actually addressing the argument. Instead, you're back, doubling down on your incorrect understanding.

    Secondly, the closest you've come to addressing the point of the OP is when you keep on bringing up quantum mechanics (Or rather you keep on pontificating about the size of the particles, which was addressed in the OP, and I reminded you that this was synonymous with quantum issues) and how my example (which was explicitly stated to not be realistic, again in the OP) doesn't conform with quantum mechanics. That would be fine, except that again, this is stated clearly in the OP and discussed, then re-discussed, then re-re-discussed, then all three discussions were conveniently quoted for you previously. And you still haven't addressed a single statement that I've made about why you cannot find shelter in quantum mechanics, because you'll just find infinities elsewhere. So instead of addressing anything I've countered with about the nature of quantum mechanics, instead you've bloviated about how I've failed to address WLC's "other claims about successive infinities" without once actually addressing how I'm not addressing them.

    Thirdly, I can understand thinking that the example given in the OP is contrived --it's definitely contrived. But its contrivance doesn't render it's point false, as has been defended by myself at least twice now and, again, ignored twice now by you. This is another point that you've simply deflected rather than addressed, which is the logical nature of the claim that WLC is making about all conceivable universes.


    In short, there's nothing I can address here that I haven't already addressed and hasn't already gone wholly ignored by you. If you want to actually bother responding to my statement with examples of infinity being necessary in quantum mechanics (more specifically quantum field theory) then we can have that discussion. Or you can rebut the point about WLC's "anti-infinity theorem" needing to apply to all conceivable universes --but I'm not repeating myself for a fourth time on these issues only to have you ignore them and pretend like I haven't addressed your point, while you repeat your non-objection, verbatim, for the fifth time. Either you address what's in my posts consistently and wholly with the responses I've given, or there's nothing further to discuss.
    "Those who can make you believe absurdities, can make you commit atrocities." --Voltaire

  19. #19
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    Re: WLC is Wrong: The Argument for Actual Infinities

    Quote Originally Posted by GP
    Just to clarify a few before agreeing:

    1.) Yes, things are ostensibly countable, even if we can't sit there and physically count them. Counting here, however, specifically means the cardinality of a set, which for infinite sets means the aleph numbers.

    2.) If by "Numbers are infinite", you mean that "There are an infinite amount of natural numbers." (i.e. "The cardinality of the natural numbers, i.e. {1,2,3, ...}, is aleph-null.)

    3.) Thus infinities are in principle consistent with physics. (i.e. we have no reason to assume that they must be inconsistent, due to known counter examples.)


    If you agree to those, then, yes, this is the essence of what I'm arguing.
    Thanks for the clarifications they say what I was trying to say much more clearly.

    I'll formulate my objection with the assumption that I have properly and accurately understood your argument. (and pray I don't muck it up anyway)

    Quote Originally Posted by GP
    EDIT: A note on being able to physically count: Even things aren't infinite, it doesn't mean that we can count them. For instance, you can't count the exact number of atoms in your hand, there's so many it is pragmatically impossible. With that said, you know that there is some finite number of atoms in your hand. Likewise, even if we can't sit there and count the number of particles in the interval [1,2), because here it's infinite, it doesn't mean that they don't or can't exist because you can't personally count them.
    Of course I agree . (totally irrelevant fact) As a personal record, I physically counted to 24K (ish) I was very board at work, and had a particularly brainless job (my first.. pushing buggies at the local Grocery store).
    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.

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    Re: WLC is Wrong: The Argument for Actual Infinities

    So, Here I quote the OP header for the section I am addressing.

    Quote Originally Posted by OP
    The Scenario: Hilbert's Hotel in a Finite Amount of Space
    In this section, you have simply repeated with new names the Hilbert Hotel. (as noted in post 2)


    Quote Originally Posted by OP
    Causing the Hilbert "Paradox" in a Finite Amount of Time
    In this section, you have again repeated the Hilbert Paradox portion(with new names) regarding the Clerks solution to admitting a new guest.
    Of course you are giving us the math verbiage equivalent.

    --- First objection ---
    Now regarding both, there is nothing wrong with either of these examples, however you stop there. What you have essentially done is repeated the Hilbert's hotel portion, and declared it possible, and in the process that WLC has been debunked... but you have failed to address the actual objections to that setup laid out by WLC.
    Your set up is fine, just as the Hotel was fine.

    ----First Supporting objection---
    Now, to show that you have not addressed the objection, I will attempt to translate it to your example.

    In the "finite space" portion, you assign each particle a number, now before you can add another particale and do your shuffle, the question should be asked.
    "Did you use a complete list of numbers". IE, in the infinite series were all the numbers used.
    Of course the answer is yes.

    So, then when the Particles move, did you create any new numbers?
    The answer is no, of course.

    So movement doesn't create new numbers, and all the numbers are used. (IE the list is full), how do you manage to find an unused number?
    Or How is the list both sufficient and insufficient to fill the numbers at the same time and in the same sense? Why should we accept such an absurdity as a realizable actuality?

    ---Second Supporting Objection---
    If You subtracted half the Particles, would you have any less particles?
    The answer is no, you would have the same number (IE infinite)

    So you are proposing that I take as a realizable reality, that subtraction doesn't change the total.
    As WLC says, just don't ask the lady who changes the beds.


    Summary of support objections, there is a bit of mathematical slight of hand occurring here. You introduce a mathematical concept that in the end negates some basic mathematical principles (Like subtracting decreases totals) and saying it is logically plausible.


    ---
    Summary of Larger point
    Now the above is to show two objections raised by WLC, that are not addressed in the OP. This is why Squatch brings it up immediately. Your conclusion that "No physical contradictions occur" Is not reached by addressing the Objections WLC brings up, it is reached by being silent on them.
    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.

 

 
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