Counting forever

0. Introduction

Here I just want to explain a simple point which comes up in the discussion of whether it is possible to ‘count to infinity’, and what that tells us about whether time must have had a beginning. Wes Morriston deserves the credit for explaining this to me properly. All I’m doing is showing two places where his point applies.

  1. The targets

I have in mind two contemporary bits of philosophical literature. One is found in Andrew Loke’s work, specifically his 2014 paper, p. 74-75, and his 2017 book, p. 68. The other is found in Jacobus Erasmus’ 2018 book, p. 114. In each case, the authors are arguing that it is not possible to count to infinity because no matter how high one counts, no matter which number one counts to, there are always more numbers left to count. Here is how they express this point.

Firstly, here is Loke in his paper:

“If someone (say, George) begins with 0 at t0 and counting 1, 2, 3, 4, … at t1, t2, t3, t4, … would he count an actual infinite at any point in time? … The answer to the question is ‘No’, for no matter what number George counts to, there is still more elements of an actual infinite set to be counted: if George counts 100,000 at t100,000, he can still count one more (100,001); if he counts 100,000,000 at t100,000,000, he can still count one more (100,000,001).” (Loke, 2014, p. 74-75)

Secondly, here is Loke in his book:

“Suppose George begins to exist at t0, he has a child at t1 who is the first generation of his descendants, a grandchild at t2 who is the second generation, a great-grandchild at t3 who is the third generation, and so on. The number of generations and durations can increase with time, but there can never be an actual infinite number of them at any time, for no matter how many of these there are at any time, there can still be more: If there are 1000 generations at t1000, there can still be more (say 1001 at t1001); If there are 100,000 generations at t100,000, there can still be more (100,001 at t100,001), etc.” (Loke, 2017, p. 68)

Finally, here is Erasmus in his book:

“Consider, for example, someone trying to count through all the natural numbers one per second (i.e. 1, 2, 3, . . . ). Can the person count through the entire collection of numbers one per second? Clearly not, for no matter how many numbers the person has counted, there will always be an infinite number of numbers still to be counted (i.e. for any number n that one counts, there will always be another number n + 1 yet to be counted). Therefore, it is impossible to traverse an actually infinite sequence of congruent events and, thus, if the universe did not come into existence, the present event could not occur.” (Erasmus, 2018, p. 114)

What each is saying is that if someone starts counting now, they will never finish counting. And this is true, of course.

Think about the following mountain: it has a base camp at the bottom, but it is infinitely tall and has no highest point (for each point on the mountain, there is another one which is higher than that point). Can one start at the bottom and climb to the top of such a mountain? No, because there is no top of such a mountain.

(My dispute with Craig was not over exactly this point, but on something slightly more subtle. That was whether the following is false:

A) It is possible that George will count infinitely many numbers.

I say that A is true. All Loke and Erasmus’ sorts of considerations get you is to say that the following is false:

B) It is possible that George will have counted infinitely many numbers.

But we can leave this point here for now.)

2. My point

All I want to highlight today is that the fact that there are ‘always more numbers left to count’ only applies to certain types of infinite count. Imagine the following three scenarios:

i) George is trying to count the positive integers, in this order: (1, 2, 3, …)

ii) George is trying to count the negative integers, in this order: (…, -3, -2, -1)

iii) George is trying to count all the negative and all the positive integers, in this order: (…, -3, -2, -1, 0, 1, 2, 3, …)

We can think of the scenarios like this:

  • Scenario i) is like climbing up an infinitely tall mountain that has a bottom but no top;
  • Scenario ii) is like climbing up an infinitely tall mountain that has no bottom but does have a top;
  • Scenario iii) is like climbing up an infinitely tall mountain with no bottom and no top.

In each case, due to the nature of infinite sets, the tasks involve counting the same amount of numbers. Simple intuition tells you that the third involves counting more numbers than the first two (and should be the sum of the first two). However, it is actually the case that each scenario involves counting sets with the same cardinality (that is, ℵ0). Put another way, each mountain is the same height as the other two.

The key point I want to make is this. It is obviously true that the Loke-Erasmus observation (that “there will always be an infinite number of numbers still to be counted”) only applies to scenarios i) and iii). It just doesn’t apply to scenario ii).

When George is starting at 0 and counting up, he always has the same amount of numbers left to count (always ℵ0-many). The same is true for when he is in scenario iii), no matter where George is along his task.

But if he is in scenario ii) he is in a very different situation. No matter where George is in his task in this scenario, he only ever has finitely many numbers left to count. He doesn’t have infinitely more mountain to climb in scenario ii). It has a top, and no matter where he is, George is only finitely far away from the top of the mountain. Clearly, he can reach the top of such a mountain if he is already some way along his climb.

And here is where the rubber meets the road. If there is a problem with scenario ii), it is not that “there will always be an infinite number of numbers still to be counted”, because that is just false. And scenario ii) is the one that Loke and Erasmus ultimately have in their sights. That’s because it is the one where George ‘completes’ an infinite task by counting an actually infinite amount of numbers.

Put simply, the argument they are making looks like this:

  1. George cannot count up to infinity, because there would always be more numbers left for him to count.
  2. Therefore George cannot count down from infinity.

Put like this, the fact that the argument is invalid is plain to see.

The problem, if there is one, is not about completing, or finishing, an infinite task. It might be that the problem has something to do with starting such a task. But if so, there is really no point in talking about problems that involve the impossibility of finishing, as that is a different point.

3. Conclusion

There might be other reasons to think that one cannot count down from infinity, of course. Indeed, Loke and Erasmus both have more to say about this issue. But what one often finds in discussions like this is the following sort of move: they make an observation that applies to an endless series, and try to apply that to a beginningless series. As is simple to see, sometimes the initial observation (such as that there will always be more numbers left to count) just doesn’t apply to both. And the switch from one to the other is thereby not valid.

35 thoughts on “Counting forever”

  1. Another great post. I enjoyed this — and, of course, we agree.

    Here’s something I’ve told you about before, but that is relevant to the argument you’ve offered here. Most of the friends of the Kalam argument — certainly Craig and Loke — take big bang cosmology to have empirically confirmed their idea that the past is finite. They think that this is so because some General Relativistic cosmological models retrodict that the curvature of space-time diverges at a finite time in the past. To understand what this means, consider the following analogy. The function f(x) = 1/x is not defined for x=0 because f(x) diverges at x=0. So, x=0 represents a kind of boundary to the positive real numbers. Likewise, space-time becomes undefined when the curvature diverges, so that the divergence in the curvature marks a boundary to time. There is no time t=0, just as there was no point x=0 for f(x).*

    However, if we believe that General Relativistic cosmological models retrodict a beginning, then we must also believe that the universe has counted down from infinity. Consider, again, the function f(x) = 1/x. If we start at, say, x=10 and move backwards, no matter how large f(x) becomes, I can find another value of x, closer to x=0, where f(x) is larger. Likewise, if we start at the present day and look backwards, we find that the space-time curvature increases. No matter how large the curvature becomes, we can find another time, closer to t=0, where the curvature is larger. Just as there was no upper bound for f(x), so, too, there is no upper bound for the curvature. **

    So, now consider integer values of the curvature. No matter how large the integer, we can find a past time where the curvature was equal to that integer. So, according to the cosmological models that supposedly confirm a beginning of the universe, the universe has counted down from infinity.

    All of this requires that space-time is continuous and not discrete. Space-time may well be discrete, or have some other substructure, but, if space-time is discrete, then General Relativity is not a correct theory of space-time. In fact, if General Relativity turned out to be false, but empirically adequate within a particular domain of inquiry, then the early universe is precisely the place where we’d expect General Relativity to break down. In that case, friends of the Kalam argument cannot use big bang cosmology to defend their view that the universe is finite into the past. So, friends of the Kalam have two choices: either accept that counting down from infinity is metaphysically possible or accept that Big Bang cosmology does not confirm a beginning of the universe.

    ————–

    * For those who might visit this site and know some General Relativity, I mean the Ricci scalar curvature diverges at the so-called “big bang” singularity in some FLRW models. The variable t represents the coordinate time in ds^2 = -dt^2 + a(t)^2 dOmega^2, where dOmega is the spatial part of the metric and a(t) is the scale factor.

    ** Again, for readers who may know General Relativity, the technical way of saying what I have in mind is that if you take an FLRW model, foliate the model into space-like surfaces of constant extrinsic curvature, and label the surfaces with the cosmic time, then you will find that the Ricci scalar diverges as one approaches the singularity.

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    1. Daniel, the point-like nature of the Big Bang singularity, if it existed, would entail a beginning of time in the local spacetime region (“local” meaning the spacetime region temporally connected to “us” observers on this side). This is because it would “pinch off” (disconnect) any prior region by virtue of the infinitesimal size of the singularity. But it is almost a physical certainty that singularities cannot exist. This is because (1) GR predicts them right at the point where GR itself gives out (sort of like division by zero), and so GR’s own prediction cannot be trusted in that limit, and more importantly (2) While GR is very accurate at the large scales, it is a classical (non-QM) theory, we know that QM must apply at the small scales, and QM prohibits singularities. (Stephen Hawking, one of the co-authors of the singularity theorem spent a great deal of time disabusing people of the actual existence of singularities because they are prohibited by QM, and the singularity theorem does not take QM into account).

      While there was a transition some 13.8 million years ago, a huge rate of expansion of the observable universe, etc., it is likely that our local spacetime region is temporally connected to prior events, absent a singularity. Past finitude is still an open question, though, because there is, at present, no satisfactory quantization of gravity (the marriage of QM and GR). There are some candidates, but it has proven to be a very difficult question to answer and it remains one of the last loose ends remaining in the immensely successful description of the world afforded by the Standard Model. I think it’s fair to say, though, that the “smart money” in present Cosmology is on the side of past non-finitude (an infinite past).

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      1. Miguel — Much of what you wrote is correct, though with some fairly heavy caveats. The tone of your comment suggests that you don’t think I understand GR. I offered a succinct explanation of the Big Bang singularity in the post that you are responding to, so I’m not sure why you re-explained the Big Bang singularity in your own post. (In any case, I compelled to say that I have studied GR through the graduate level and have two publications on philosophy of cosmology and a third article, under review, dealing with General Relativity in particular.)

        You mentioned Hawking’s singularity theorem. Hawking and Penrose proved a variety of singularity theorems through the 1970s, though all such theorems make an explicit assumption about the universe’s energy density (e.g., that the energy density is positive for all observers). Quantum fields can have negative energy density and this remains true in semi-classical quantum gravity. The inflaton field that many folks think dominated the early universe has a negative energy density. Historically, this was part of the motivation for Borde, Guth, and Vilenkin’s theorem, though that theorem also fails to demonstrate a singular origin for the universe. Consequently, we can add a third reason as to why we should think there was no big bang singularity.

        You are right that singularities are a feature of classical space-times and are not expected to survive in whatever quantum gravity theory will supplant GR. In my post, I assumed GR because many friends of the Kalam argument assume that GR is a good theory of space-time. As I argued, there is tension between their view that the universe began with a singularity and their view that one cannot count down from infinity, since, at least acccording to GR, the Ricci scalar curvature has counted down through all of the integers! So, according to the GR models that Craig, etc, appeal to, nature *did* count down from infinity and that undercuts their claim that counting down from infinity is impossible.

        There’s an old argument, originally due to Milne, related to this issue. As you may know, one can slice an FLRW space-time into space-like surfaces of constant mean extrinsic curvature and then label those surfaces with the cosmic time. And then the claim that the universe is finitely old amounts to the claim that the cosmic time has a finite value on the “present” space-like surface. Unfortunately for Craig, et al, one can re-label those space-like surfaces with a variety of functions; for example, any function that monotonically increases with the scale factor can label the surfaces. On Milne’s view, on the “correct” labeling, the Big Bang singularity occured infinitely far in the past. Milne’s cosmology is now outdated, but a number of folks — including a number of A-theorists, like Craig — have argued that one should label the surfaces with the York time. According to York time, supposing GR were right, the Big Bang singularity occured infinitely far in the past.

        Here’s the relationship with the argument that I offered. One can label the surfaces of constant mean extrinsic curvature with the negative Ricci scalar curvature. If one does so, then, according to that “clock”, the singularity happened infinitely long ago. As Milne put the point, there’s been enough time for an infinity of things to happen –there’s a bijection between the past cosmic time and infinite time. The General Relativistic model that Craig endorses not only entails that the universe can count down from infinity in finite time, but also entails — like in Craig’s thought experiment — that the universe can count down from infinity over infinite time!

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    2. Daniel – A cosmologist goes to a conference and asks the security guard: “Excuse me sir, do you know what room the General Relativity seminar is in?” Security guard: “No, I wouldn’t know that, sir.” Cosmologist: “It’s where a bunch of guys say ‘gee-mu-nu, gee-mu-nu’ over and over.” Security guard: “Oh, it’s right over there, sir.” Lame, I know, sorry. 🙂

      Congratulations on the publications in the philosophy of cosmology and GR, references would be appreciated, if you have them handy. I too had a year of GR in grad school–that Einstein summation convention sure was convenient when solving the Field Equation over and over under different boundary conditions… Much midnight oil was burned deep into those weeds, to mangle two clichés together. It was one of my favorite courses and made me consider going into gravity, but I ended up specializing in something else instead. So I don’t claim to be an expert, only to know enough to be gently dangerous (and that probably holds for all my comments here).

      I was not responding to your main point about how those who rely on the Big Bang singularity must commit to a countdown from some sort of infinity because I don’t find it particularly compelling. I was speaking to what I think is a more direct approach. Either (1) Point out that GR is at best incomplete and at worst incorrect and thus cannot be recruited to reach sweeping conclusions like past finality; or (2) Just point directly to the infinities inherent in singularities (infinite density, infinite temperature, infinite curvature, etc.). If infinities cannot be “actual,” then neither can singularities, no countdown required.

      But I don’t think that past finitists are concerned so much with countdown methods as they are with the alleged impossibility of “traversing” an infinite. You may find a way to relabel time with a re-scaled (fast-enough decreasing) index such that you transform a count from a finite past into a count from infinity via that re-scaled counting index. But that strikes me as a Zeno-like move, and doesn’t really entail that the singularity “occurred” an infinite time ago, which is what finite-past proponents really have a problem with.

      Likewise, I don’t think it is particularly impressive either, to find a bijection between past cosmic time and infinite time (as long as you’re dealing with a classical spacetime that is, since QM wouldn’t allow it). For example, there is a bijection between the open interval (-1,1) and the Real Line. Yet one is “traversable” and the other isn’t. Likewise there’s a bijection between (-1,0] and (-&#8734,0]. There’s nothing particularly deep here, that I can see.

      In the same vein but more directly, one could, I suppose, present a dilemma to past finitists who subscribe to both (1) infinities cannot be “actual” and (2) classical (continuous) spacetimes are “actual.” Since any “traversable” interval of time has embedded in it an infinity of infinitesimal intervals (bijectible to the Real Line and thus to an infinity), either there are no “actually” traversable time intervals (because their embedded infinities cannot be “actual”), or classical (continuous) spacetimes cannot be “actual” and bye-bye singularities. I’m not sure how convincing this would be either, though.

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      1. Miguel — Here are pre-prints for the two publications:

        https://arxiv.org/abs/2006.08890
        https://arxiv.org/abs/2006.07748

        “Either (1) Point out that GR is at best incomplete and at worst incorrect and thus cannot be recruited to reach sweeping conclusions like past finality”

        Of course, I agree with this point. Singularities are features of classical space-times and we know enough about quantum gravity to know that space-time is not classical. Unfortunately, our theistic interlocutors appear to disagree, for they understand the various singularity theorems to be strong evidence for a beginning of the universe.

        “or (2) Just point directly to the infinities inherent in singularities (infinite density, infinite temperature, infinite curvature, etc.). If infinities cannot be ‘actual,’ then neither can singularities, no countdown required.”

        I don’t think the argument that you have described suffices. The space-time manifold does not exist at the singular point. For example, consider the standard radiation dominated universe in which a(t)=t^1/2. In that case, there is a singularity at t=0. But the space-time manifold is well-defined *ONLY* for t > 0. This is a point that I made in my first post: in FLRW models, the boundary to space-time is open. So, there is no part of the manifold at which the density, etc, is actually infinite. Craig is well aware of this and comments on this fact in the book he co-authored with Quentin Smith. One needs a more subtle argument for showing that the singularity is inconsistent with the arguments put forward by Kalam proponents. The countdown in the Ricci scalar makes the more subtle move required and directly parallels the argument that Alex considered in his post.

        “But I don’t think that past finitists are concerned so much with countdown methods as they are with the alleged impossibility of ‘traversing’ an infinite. You may find a way to relabel time with a re-scaled (fast-enough decreasing) index such that you transform a count from a finite past into a count from infinity via that re-scaled counting index. But that strikes me as a Zeno-like move, and doesn’t really entail that the singularity ‘occurred’ an infinite time ago, which is what finite-past proponents really have a problem with.”

        Perhaps I didn’t express myself well enough in my post! In any case, I don’t think your comments reflect the argument that I offered.

        FLRW space-times are globally hyperbolic and, for that reason, can be foliated into space-like surfaces. Of course, at least on an orthodox understanding of relativity, there is no uniquely best way to foliate a given space-time. But, as it turns out, there is a unique foliation of an FLRW space-time into surfaces of Constant Mean (extrinsic) Curvature. This is called the CMC construction. On the CMC construction, one can label the CMC surfaces with the cosmic time, where the cosmic time is the proper time of observers who are locally at rest with respect to the universe’s expansion. (To put that another — perhaps more correct — way, the cosmic time is a coordinate time that locally corresponds with the proper time of observers who are co-moving with the universe’s expansion.) When cosmologists say that the Big Bang happened at a finite time in the past, they mean that the lapse in cosmic time between the Big Bang and now is finite.

        We can ask: is the cosmic time the *right* labeling of the CMC surfaces? For most physicists, this question doesn’t really make sense to ask; after all, there is no absolute time that one can use to label the CMC surfaces. Instead, one may choose a variety of conventions. If this is one’s view, then it is adequate for my purposes to point out that there are coordinate conventions — such as the York time — that are standard choices and in which an infinite amount of time lapses between any observer and a past (or future) singularity. In other words, if the choice of coordinate time is merely conventional, then, in FLRW models, the lapse in coordinate time between any observer and the BB singularity is a matter of convention. In that case, there is no fact, at least as traditionally understood, as to whether there was a beginning, at least with respect to coordinate time.

        On the other hand, this is not the view that is adopted by most friends of the Kalam argument. For most friends of the Kalam argument, including Craig, there is an absolute time and so a correct way to label the CMC surfaces. On Craig’s view, the absolute time is time as kept by God. And now the question becomes: is the absolute time the York time, the cosmic time, or something else? How one answers this question will determine whether FLRW models depict the universe as having had an absolute beginning.

        Unfortunately for Craig, many authors who have argued for an absolute time have argued that the absolute time is York time. For example, Bohmian mechanics requires an absolute reference frame. Antony Valentini, who has perhaps done more work than anyone else in articulating Bohmian cosmology, has argued that the absolute time that one should use for Bohmian mechanics is the York time. There’s even a few passages in one of Craig’s books where *Craig* argues for the adoption of York time as the absolute time. (Elsewhere, Craig asserts, without much argumentation, that the absolute time is the cosmic time.)

        I think the right conclusion to draw is that Craig, and other friends of the Kalam argument, have much more work to do. Setting aside worries about quantum gravity, Craig and other friends of the Kalam need to provide a convincing argument for a specific labeling of the CMC surfaces and then show that, according to that labeling, the singularity happened at a finite time in the past.

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    3. Daniel – Regarding your response, I wasn’t able to catch the “fairly heavy caveats” to my points that you mentioned. As far as I can tell, you agreed with everything I said? I am aware of the history. Yes I realize that there are several singularity theorems, but I was referring to the singularity theorem pertinent to the Big Bang, which is more specific to the topic at hand than other singularity theorems, as I thought that’s what was relevant in this thread about the beginning of the universe and time, etc. Incidentally, neither Hawking nor Penrose embrace(d) past finality, so they don’t(didn’t) really “buy” into the classical singularity, and that’s probably for good reasons. And, yes negative energy density is a non-classical feature of QFT (not sure how that’s relevant). The positive energy density assumptions in the Field Equation (in the BB Hawking-Penrose case) is a type of weak pseudo-boundary condition to enforce physicality, absent specific knowledge of what should actually go into the energy tensor (as in a start-of-the-universe scenario), otherwise you get a whole host of unphysical solutions.

      I have to take issue with your statement that the purported inflaton can have negative energy densities? Would this occur only in rare fluctuations? How could this be true on average? I can see that a low tunneling amplitude could be overcome by waiting a very long time, but how would a decay to the vacuum occur without a massive input of energy at the start of the process in the negative energy density case? (Maybe God? 🙂 If you meant negative energy, I’m not sure that the required negative energy necessarily derives from a quantum effect in semi-classical quantum gravity (inflatons or not). Classical gravity can provide negative energy in droves, I think, to counterbalance the spontaneous positive energy and produce the ungodly free-lunch. A negative energy density would not make sense to me, but there again, my intuition may well be relying on outdated models. If you’re right, that would be very interesting to me, as I can’t see how that would work. Maybe you meant negative pressure instead of negative energy density? I was also not aware that the BGV theorem was motivated by negative energy density? I’m also not entirely sure what the BGV has to do with the BB singularity?

      Again, I don’t know that you disagreed with anything I said (although I’m not sure I agree with what you said regarding negative energy densities).

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      1. Miguel — A number of great (and difficult!) questions.

        “I have to take issue with your statement that the purported inflaton can have negative energy densities?”

        You are right that I misspoke here and that my phrasing was rather sloppy. The Hawking-Penrose theorems assume different conditions on the energy depending on the theorem, such as the dominant energy condition. The vacuum expectation value of a scalar field is known to be able to violate the dominant energy condition. The inflaton field is hypothesized to have been a scalar field and so the possibility arises that the dominant energy condition was violated in the early universe. If so, perhaps a singularity can be avoided. This gave rise to the thought that the inflationary mechanism might prevent a singularity.

        Borde, Guth, and Vilenkin wrote their paper in reply to the idea that the inflationary mechanism could prevent a singularity. They develop a generalization of the Hubble parameter that applies to non-homogeneous and non-isotropic space-time. And then they consider the average of their generalized Hubble parameter along a time-like geodesic congruence. As they point out, if the average of the generalized Hubble parameter is greater than zero, then the congruence must be past-incomplete. Geodesic incompleteness is often considered an indication that a space-time is singular, so one may conclude that the BGV theorem tells us that inflationary space-times do not avoid a past singularity. (Of course, at best, one has only that the geodesic congruence is bounded by a singularity and not that the whole space-time is bounded to the past.) Borde, Guth, and Vilenkin reach a more modest conclusion in their original paper — that their result indicates a geodesic congruence of the sort previously described must exit classical space-time at a finite past time.

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    4. Daniel – Thank you for the arxiv papers, I’ll take a look.

      Yes, exactly, the classical theory gives out right at t=0 and it’s defined only for t>0 precisely to avoid the infinities involved at the singularity. But all that means is that the singularity itself is unphysical. At the singularity there would be infinities and the classical theory breaks down.

      Maybe I’m not understanding, but I still don’t get why the re-scaled countdown must entail an “untraversable” infinite past, even counting from a time t0 arbitrarily close to the singularity. Is this the reverse of, for example, an observer (obs1) falling into a singularity? From an outside observer (obs2)’s perspective, the original observer (obs1) never quite makes it into the singularity (it takes an infinite amount of time), but from obs1’s perspective, he/she does (it does not take an infinite amount of time, even assuming a point observer so tidal forces can be ignored). However, I’m not sure it’s meaningful to conceive of an observer outside the universe itself watching the unfolding of the BB singularity (i.e., stuff “unfalling” from it). From the perspective of an observer within the universe, the time since he/she and all the other stuff started “unfalling” out of the singularity (i.e., the BB event), is past finite, regardless of counting re-scalings. While observer perspectives can be different, counting re-scalings or conventions don’t change the physics. If they do, something went wrong somewhere.

      I agree that there can be different coordinate conventions, but whatever conclusion you reach should not be coordinate-convention-dependent! That is verboten/meaningless in physics.

      I also don’t think Alex’s arguments are dependent on counting re-scalings or conventions.

      Incidentally, Bohmian mechanics is about as “fringe” as one can get in physics and are not taken very seriously at all, I don’t think. There’s been a great deal of empirical research on Bell’s inequalities, even recently, which have increasingly constrained any such “interpretations” to practically non-existent.

      Anyway, that’s not particularly relevant. I’m not sure that past finitists (including Kalam proponents) would accept this, but who knows, maybe they would.

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      1. “Yes, exactly, the classical theory gives out right at t=0 and it’s defined only for t>0 precisely to avoid the infinities involved at the singularity. But all that means is that the singularity itself is unphysical. At the singularity there would be infinities and the classical theory breaks down.”

        What you say here is close to being right, but not quite. You are right that the singularity is an indication that the theory has broken down and needs to be supplanted by a quantum gravity theory. But suppose we take the classical theory literally, as our theistic interlocutors insist. If we take the classical theory literally, then the classical theory states that there is an open boundary such the manifold includes only those points which are such that t > 0.

        In other words, on the classical theory, there is no t=0 point and so no actual infinities. Again, to demonstrate an incompatibility between the classical theory and Kalam-style arguments, one needs to do more than point out that there is a sense in which singularities are infinite. Singularities do not, for example, instantiate an actual infinity.

        “Maybe I’m not understanding, but I still don’t get why the re-scaled countdown must entail an ‘untraversable’ infinite past, even counting from a time t0 arbitrarily close to the singularity. Is this the reverse of, for example, an observer (obs1) falling into a singularity?”

        No. In the case of obs1 falling into a black hole and obs2 watching from outside, we think about the issue in terms of the proper time of either observer. The statement that obs1 sees themselves as falling in for finite time means that obs1 measures themselves as taking a finite time to do so. The statement that obs2 sees obs1 take an infinite amount of time means that, according to obs2’s clock, obs1 takes an infinite amount of time.

        Notice two things. First, if we ask which of the two observers has the objectively right answer, we will be disappointed; on an orthodox understanding of relativity, both answers are equally correct. Second, the black hole scenario does not mention coordinate time, whereas the arguments that I gave were about coordinate time and not proper time.

        “While observer perspectives can be different, counting re-scalings or conventions don’t change the physics. If they do, something went wrong somewhere. I agree that there can be different coordinate conventions, but whatever conclusion you reach should not be coordinate-convention-dependent! That is verboten/meaningless in physics.”

        I want to put a word of caution here. Part of what I’ve been trying to do is to explain the perspectives adopted by Craig, et al, and another part is to explain some of my replies. So, I agree with you that we should not understand anything that is coordinate-dependent as physical; there’s another question here about whether Craig agrees.

        The view that anything which is coordinate dependent is not physical has an important implication for FLRW models, in that, even in singular FLRW models, whether the singularity is at finite or infinite coordinate time is a matter of convention.

        Absolute time, if such a thing exists, is not coordinate independent. You might take this to be a reason to reject absolute time. Unfortunately, there are philosophers who think that there is such a thing as absolute time. In fact, Craig, and many other friends of the Kalam argument, insist that the Kalam argument cannot be formulated without absolute time. Perhaps you’d want to say that this is already good enough reason to reject the Kalam argument.

        In any case, once Craig, et al, have placed absolute time on the table and have insisted that the universe began to exist because there’s been a finite absolute time since the beginning, it’s fair game to ask which coordinate convention corresponds to absolute time.

        “I also don’t think Alex’s arguments are dependent on counting re-scalings or conventions.”

        Good! I never said that Alex’s arguments did depend on those things! My claim was that Alex’s interlocutor — that is, Craig, Loke, etc — have arguments that depend on those things.

        “Incidentally, Bohmian mechanics is about as ‘fringe’ as one can get in physics and are not taken very seriously at all, I don’t think. There’s been a great deal of empirical research on Bell’s inequalities, even recently, which have increasingly constrained any such ‘interpretations’ to practically non-existent.”

        A couple things here.

        First, I want to note that my mention of Bohmian mechanics was only by way of explaining Craig’s view. I am not a Bohmian, myself. Craig is a Bohmian and Bohmians accept a preferred coordinate system; moreover, on independent grounds, Craig accepts absolute time. In some places, Craig uses Bohmian mechanics to argue for absolute time. In any case, given that friends of the Kalam argument typically endorse absolute time and insist that the Kalam argument requires absolute time, in offering a reply, we might examine the most sophisticated articulation of absolute time. One such articulation has been offered by Bohmian cosmologists.

        Second, the Bell inequalities do not provide us with any ammunition at all against Bohmian mechanics. In fact, Bell *was* a Bohmian and argued for Bohmian mechanics. The Bell inequalities say only that if there are hidden variables, then they are non-local. Bohmian mechanics postulates non-local hidden variables from the get-go and does not postulate local hidden variables. All the same, this provides us with a different reason for rejecting Bohmian mechanics — i.e., we think that Lorentz invariance is an actual symmetry of space-time, whereas Bohmian mechanics, in virtue of being non-local, violates Lorentz invariance. Unfortunately, Bohmian mechanics is remarkably good at hidding violations of Lorentz invariance, so that, if there are any such violations, Bohmian mechanics predicts we’d never observe them.

        Third, while Bohmian mechanics might be considered “fringe” among practicing physicists, Bohmian mechanics is not fringe among philosophers of physics. But perhaps you’ll take that as a reason to think there’s something wrong with philosophers!

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      2. Daniel – “You are right that the singularity is an indication that the theory has broken down and needs to be supplanted by a quantum gravity theory. But suppose we take the classical theory literally…” We (they) shouldn’t because it has broken down. Nor am I saying anything about what it needs to be supplanted with. The theory’s failure is independent of what may or may not supplant it (though I agree it will likely be some sort of quantization of gravity).

        “the classical theory states that there is an open boundary such the manifold includes only those points which are such that t > 0… there is no t=0 point and so no actual infinities.” Either there is a singularity which gives rise to the open-boundary definition of time, or there isn’t. Either the singularity is physically possible or it is not. What is the open boundary (0,Inf) referring to? A singularity at t=0. Just because infinities can be avoided by requiring t to be arbitrarily close to but always > 0, doesn’t mean that infinities are not lurking in the singularity itself. They are. If I drive just to the edge of a deep precipice and stop just short of falling into it, that doesn’t mean that therefore there is no precipice with a large drop. It means that I have avoided falling into it while driving arbitrarily close to it, but not quite into it. I shouldn’t get to assume that there really is no precipice and no deep fall only because I can drive just short of it if I want to live to measure how close/far away I am to/from it.

        ”Singularities do not, for example, instantiate an actual infinity.” I beg to differ. That is, in fact, what defines them and where their name comes from. That the manifold is undefined right at the singularity is another way of saying that the manifold avoids the intractable infinities concomitant with the singularity, not that the singularity itself does not contain infinities.

        ”both answers are equally correct” Not in this case. Yes, as I mentioned, each observer has equally valid proper times in their respective reference frames. However, obs2’s frame is not meaningful when the singularity is the start of the universe itself, because obs2 cannot be outside the universe, as he could for example, stand outside a black hole singularity or something like that within the universe.

        ”there’s another question here about whether Craig agrees… Unfortunately, there are philosophers who think that there is such a thing as absolute time. In fact, Craig, and many other friends of the Kalam argument, insist that the Kalam argument cannot be formulated without absolute time.” Then Craig and others have no business relying on GR, irrespective of the singularity problems, because GR is precisely the opposite of “absolute” time. Hence the “R” in “GR”.

        ”once Craig, et al, have placed absolute time on the table and have insisted that the universe began to exist because there’s been a finite absolute time since the beginning, it’s fair game to ask which coordinate convention corresponds to absolute time.” I beg to differ on both counts. That a finite time has elapsed since the BB singularity (whether you start counting really, really close to 0 or from 0 itself), does not mean that one needs to commit to a notion of absolute time. It is possible for all (non-massless) observers in the universe to agree that an interval of time is not infinite without having time be “absolute.” A finite time interval is not a committal to “absolute” time. Coordinate conventions are, again, irrelevant in physics. They are a convention used to describe phenomena. They do not affect the phenomena being described nor do the phenomena depend on the convention used to describe them. If you’re referring to reference frames that’s a different matter. Yet again, all physical models (thanks to GR) are constructed from the ground up so that they are reference-frame invariant.

        ”I never said that Alex’s arguments did depend on those things! My claim was that Alex’s interlocutor — that is, Craig, Loke, etc — have arguments that depend on those things.” OK, but I thought that you were using a transformation to represent a finite interval as obtained by “counting from infinity,” which you then equated with Alex’s count of a bona fide infinite interval. I think those are very different things. You are taking a finite interval of time, superimposing a time grid which changes with time (re-scales), and end up with an infinite “count.” That can be done, sure. As I said you can biject (-1,0] to (-Inf,0]. That doesn’t mean that (-1,0] is infinite in the same way that (-Inf,0] is. But in Alex’s case, he does not re-scale the time “ticks”, the ticks represent constant (fixed) intervals. I believe this is the same “construction” used by Craig et al. and Alex and Craig et al. disagree on whether incremental addition can achieve infinities, but they don’t disagree on the constancy of the counter “ticks”.

        ”Craig is a Bohmian and Bohmians accept a preferred coordinate system; moreover, on independent grounds, Craig accepts absolute time. In some places, Craig uses Bohmian mechanics to argue for absolute time.” If you mean a preferred reference frame, then in that case, Craig should eschew GR if he is to be consistent, and toss singularities etc. (If you do mean a preferred coordinate system, then that’s even worse, as that is not coherent.)

        You are right about Bohmian mechanics not having been ruled out by Bell’s inequality experiments (I just looked it up briefly). I remembered Bohmian mechanics as a hidden-variables theory and conflated hidden variables with local hidden variables, which is what the Bell’s inequalities’ experiments have severely constrained. There are, I believe, other reasons to reject non-local hidden variables, though. Also, I don’t think it matters much if Bell himself was a “Bohmian” or not, what matters are the empirical verifications of Bell’s theorem. Anyway, this is not particularly relevant, I don’t think.

        ”while Bohmian mechanics might be considered “fringe” among practicing physicists, Bohmian mechanics is not fringe among philosophers of physics. But perhaps you’ll take that as a reason to think there’s something wrong with philosophers!” No, not at all! My thinking is that physics (and science more generally) is a sub-branch of philosophy (“natural philosophy” maybe out of fashion, but is a dead-on accurate descriptor). I think of science as “philosophy plus data” as others have said. Science has split off and struck out on its own, has been vastly successful and has progressed at dizzying speeds in the relatively recent past. (Think about how a century ago, we thought our galaxy was all there was, how our phones are more powerful than the supercomputers of 25 years ago, how we can do fMRIs, etc.) Perhaps the speed of science’s success derives from the fact that it is tethered to reality via empirical confirmation and less so to opinion trends. Bad ideas tend to be weeded out quickly in science because of brutal confrontation with observation. The results speak for themselvs. I don’t think there’s “something wrong with philosophers” but I do think that sometimes there can be abuse of science or abuse of philosophy by both scientists and philosophers. Few things irritate me more than physicists like Krauss (whom I otherwise admire) dismissing or putting down philosophy, though I think he’s changed this attitude more recently. Not only is this ignorant, but any scientist that does this is shooting themselves in the foot because science is a branch of philosophy. The converse also holds, of course as there are philosophers who are dismissive of science, despite its undeniable and impressive success.

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      3. “Maybe I’m not understanding, but I still don’t get why the re-scaled countdown must entail an ‘untraversable’ infinite past, even counting from a time t0 arbitrarily close to the singularity.”

        I’ve already replied to this comment, but I’ll offer another reply because I think that, at certain points, we may be talking past each other!

        I have discussed *two* objections to Kalam-style arguments. And neither of my objections involves the view that a “re-scaled countdown” entails an “untraversible infinite past”.

        First, a reply to the notion that counting down from infinity is impossible. Craig, et al, have insisted that we should take the singularity results from classical GR literally, that is, as evidence that the universe had a beginning. And Craig, et al, have offered an a priori argument for a beginning of the universe, namely, that counting down from infinity is impossible. In reply, I argued that on the results from classical GR that Craig accepts, nature counts down from infinity by going through all integer values of the Ricci scalar (or of the energy density or of a variety of other parameters).

        Second, I have argued that while Craig endorses absolute time, Craig does not tell us which labeling of the CMC surfaces corresponds to absolute time. If, for example, the negative Ricci scalar were the absolute time, then absolute time would not have a beginning. So, this is a challenge to Craig: tell us which of several possible labelings corresponds to absolute time and offer us an argument for choosing that labeling.

        The two arguments are related in that they both depend upon the fact that, in some classical models, the Ricci scalar (or a variety of other parameters) has traversed an infinity of values.

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      4. Daniel – Yes, I think we may be talking past each other (not on purpose, though, I’m sure!).

        ”Craig, et al, have offered an a priori argument for a beginning of the universe, namely, that counting down from infinity is impossible.” Indeed, but let me offer two comments. First, the “counting” presumes a straightforward identification of constant intervals with the counting index. In other words, each count corresponds to a fixed proper time interval, a second, or a day, or a year… Craig et al. are not objecting to a finite period of (cosmic) time where the intervals being counted change as the count progresses, in such a way as to make the count infinite. They are objecting to the interval itself being infinite. You mentioned York time, which changes (re-scales) as the spacetime curvature changes. The steeper the curvature, the shorter the tick time. The closer you get to a singularity, the steeper the curvature, the smaller the tick size and the faster the counter increases. This will, of course, go to infinity as one approaches the singularity. That’s what I meant by “re-scaling.” But the proper time interval is still finite, even though the count approaches infinity.

        Second, when you look at Craig et al.’s defense of the “absurdity” of past infinitude, it’s not so much the counting specifics that matter as it is the alleged inability to get to the present from an infinite past. It’s about the “traversal” of an infinite, not the details of counting. Again, I can concoct many counting regimes that will go to infinity at midnight on New Year’s Eve 2019 by decreasing the tick time accordingly, as I approach that time. Yet, New Year’s Eve happened, and here we are, we have “traversed” that interval, regardless of how we counted it. The interval itself is finite, even if the count can be concocted to be infinite.

        ”In reply, I argued that on the results from classical GR that Craig accepts, nature counts down from infinity by going through all integer values of the Ricci scalar (or of the energy density or of a variety of other parameters).” I don’t think nature is counting. I think the count depends on the definition of the counter’s “tick” size. While there may be more or less theoretically convenient ways to count, the question is whether the (cosmic) time elapsed since the BB is finite, not whether it can be counted to be infinite using one metric or other.

        ”So, this is a challenge to Craig: tell us which of several possible labelings corresponds to absolute time and offer us an argument for choosing that labeling.” Sure. But that doesn’t give us license to concoct or select a counting scheme that produces an infinite count and declare victory, because of the dependence on the choice of counter re-scaling. Labelings are one thing, and the actual size of the interval is another.

        ”in some classical models, the Ricci scalar (or a variety of other parameters) has traversed an infinity of values.” Yes, but that’s not what Craig et al. are referring to when they talk about an “actual” infinity. The cosmic time since the singularity is not infinite, even if there are ways to count which yield an infinite count for some definition of the counting procedure.

        Personally, I wouldn’t be impressed with this re-scaling of counters to yield an infinite count and represent that as an actually infinite interval of time. The interval of time is itself a sum of the individual “ticks”. If the “tick” sizes are decreasing fast enough, the sum can be finite. I don’t find anything profound here. It’s possible that you’re right and I’m not understanding this. By all means, give it a try. I’d be curious to see how they’d respond. Hopefully the discussion has been somewhat useful and helps you anticipate some objections.

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      5. Daniel, Miguel — this was a very thoughtful discussion, and personally I find it very gratifying to see GR discussed at this kind of level on a public forum. I don’t have much to add regarding the main substance of the discussion, but I do have a couple of points to add that you might both find interesting.

        Daniel is right when he states that, within classical GR, singularities are not actually a part of the solution. So, for a big bang solution, there is no “t=0”, just an open boundary. Miguel is slightly wrong when to suggest that, in some sense, we are just excluding t=0 because the theory breaks down there, and that is why we are left with an open interval of time. For theories defined on fixed manifolds you can make sense of a statement like this — after all, you can point to the place where some function “would” become infinite. It’s like saying that the function 1/x is singular at x=0. But GR is not defined on a fixed manifold — the manifold itself is the solution, and so there is no place that you can point to and say “that would be t=0, if only we allowed various quantities to be undefined”. It’s more like considering the function 1/x but defined only on the open interval (0, infinity). Since “x=0” is not a part of the space up for discussion, we cannot point to the point x=0 and say that that is where the function becomes singular.

        This actually raises a very interesting question, and I think Miguel has some good intuition here with the analogy of driving to the edge of a cliff. The question that is raised is the following: given that the singularity is not actually a part of the manifold, how are we justified in saying that there is, in fact, a singularity present? After all, all physical quantities take finite values everywhere on the manifold!

        One obvious answer would be to point out that various quantities (like the Kretschmann scalar) are unbounded as you move along certain paths in the manifold (i.e. those paths which “approach the singularity”). This doesn’t quite capture what we mean by a singularity though, for reasons that I’ll explain later. Instead, a better solution is to point out that the manifold cannot be extended (within some regularity class). One can again think of the function f(x) = 1/x, defined only on the open interval (0, infinity) — you can say that this function is singular because there is no way to extend the open interval (0, infinity), and to also extend the function f(x) (i.e. by defining some value for f(0)) in such a way that the function remains, say, continuous. This is really the correct way to think about singularities in GR: a spacetime is singular if there are inextendable, finite-length (or finite proper-time) curves, and if the spacetime cannot itself be extended within some regularity class. Precisely what that regularity class “should be” is still up for debate — we probably require something a bit more that just continuity of the metric, since it is likely impossible to make sense of the Einstein equations on spacetimes which are only continuous (and not, say, differentiable).

        I mentioned earlier that simply pointing out that some quantities are unbounded along certain paths doesn’t quite capture what is meant by a singularity, and the reason for that is that there are solutions to the Einstein equations where certain “physical” quantities relating to curvature are actually unbounded at spacetime points which are genuinely a part of the manifold. These are admittedly obscure, but there are things called “impulsive gravitational waves” which are genuine solutions to the Einstein equations where the curvature is undefined at certain points on the manifold. The idea is something like an impulse in Newtonian mechanics, which can be thought of as an infinite force acting over an infinitesimal time. Similarly, these impulsive gravitational waves are solutions to the Einstein equations where the curvature is infinite but in an appropriately small region of spacetime (and the singularity is suitably “weak”). Mathematically, these can be thought of as “weak solutions”, if you’re familiar with that concept (if not, see https://en.wikipedia.org/wiki/Weak_solution). Physically, if you were in such a spacetime then you could move right through the place where the curvature becomes singular — you would experience an infinite tidal force, but only for an infinitesimal period of time. By the way, in my opinion these spacetimes are much better candidates than either black holes or cosmological solutions, if you’re looking for a solution to the Einstein equations that instantiates an “actual infinity”.

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      6. Joe —

        I agree with everything that you said and have little else to add.

        You asked, “given that the singularity is not actually a part of the manifold, how are we justified in saying that there is, in fact, a singularity present? After all, all physical quantities take finite values everywhere on the manifold!” As you rightly point out, folks who defend a beginning of the universe on the basis of classical GR do not need to maintain that there is literally a point on the manifold where the curvature becomes infinite. Instead, they appeal to geodesic incompleteness. Friends of the Kalam argument — and, in particular, Craig — deny that there is any point on the manifold where the curvature becomes actually infinite and, for that reason, do not face an immediate conflict between their denial of actual infinities and their use of GR. At least on this point, I think Craig is right; in the context of General Relativity, curvature singularities are not actually part of the manifolds that they infect.

        I will also reiterate that for Craig, et al, the notion that past time is infinite is not a reference to any observer’s proper time. Instead, Craig maintains that what he calls “absolute time” is finite into the past and denies an orthodox understanding of relativity. It’s not clear that Craig’s heterodox understanding of relativity is consistent with GR (let alone relativistic cosmological models) in the first place, but, supposing that they can be made consistent, one can then rightfully ask Craig which time coordinate corresponds to his absolute time. This is crucial because other authors who have adopted heterodox views about relativity that resemble Craig’s have maintained, as their absolute time, a time coordinate that would place the BB singularity infinitely far into the past. This is a somewhat well known issue in philosophy of relativity, but is not an issue that Craig has ever addressed.

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      7. Daniel —

        “At least on this point, I think Craig is right; in the context of General Relativity, curvature singularities are not actually part of the manifolds that they infect.” This is not quite true if, by a “curvature singularity”, you mean a point on the manifold at which the curvature is infinite. It is true that there are no such points in black holes or the usual cosmological spacetimes that people consider. But there are other spacetimes — such as those containing “impulsive gravitational waves” — on which the curvature is not defined at every point in the manifold (and so, in some sense, is “infinite” at some spacetime points). Despite this fact, these spacetimes are genuine solutions to the Einstein equations. This might sound odd — after all, the Einstein equations involve the curvature, so how can we say that a spacetime solves the Einstein equations if the curvature is undefined?! The answer is that they are “weak solutions”. Suppose we want to solve an equation f(x) = 0. Then f is a weak solution if, for any suitable function h(x), the integral (over any region) of h(x) f(x) is zero.

        One can, of course, just forbid these kinds of solutions and only consider “classical solutions” where the curvature is defined at every point on the manifold, but this begs the question if we are trying to decide whether the curvature is “actually infinite” in any solutions to GR. To put it another way, you can ban these kinds of solutions if you want, but there is nothing about the theory of GR which means that you have to. One doesn’t need the curvature to be defined at every point in order to make rigorous the concept of a solution to the Einstein equations: It is enough, for example, to demand that the components of the curvature (in some coordinate system) are locally square integrable. If this is true, then you can prove local existence and uniqueness of solutions etc. Consider the function x^(-1/4). This is square integrable near x=0, but it is not finite at x=0! The curvature could look something like this, and you could perfectly well include the point corresponding to x=0 in your manifold.

        This is all slightly besides the point, because none of these issues arise in the usual cosmological spacetimes, but hopefully you at least find it interesting.

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  2. Thank you, Alex, for sharing your thoughts on this. However, I wonder whether the relevant question with respect to the Kalam is not whether it is possible that George will count infinitely many numbers, but whether the value of the count can be infinite in the actual world. Otherwise, I don’t see how an infinite would have been instantiated in the actual (and present-time) world.

    With regard to that question, it seems to me that if the actual world contains only a finite number of discrete, coexisting elements, then the value of the count must always be finite. For in order to know the next number to count, there must exist some way in the actual world to store the current number. But in a world with a finite number of coexisting elements it is possible to store only finite numbers.

    On the other hand, it’s easy to see how an eternal system with a finite number of coexisting elements could create a sequence that would extend infinitely into the past and the future. An eternally-spinning odometer with a finite number of symbols on the wheels would do the trick.

    Loke’s argument for building the Hilbert hotel also seems to be begging the questions, since there would be no way to build an infinite number of hotel rooms in the actual world without an infinite supply of beds (or material to make the beds) in the actual world.

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    1. Your comment is fairly perceptive! I enjoyed reading this.

      You might take your objection to Loke’s argument about Hilbert’s Hotel a step further. An infinite hotel would presumably occupy an infinite amount of space. So, unless space is infinite in extent, a Hilbert Hotel is not possible. Perhaps what Loke has actually shown is that space must be finite in extent and not that time must be finite.

      I am a bit worried, though, as to whether your objection to Loke works. Loke might say that he’s imagining that God builds the Hilbert Hotel and pops each room — including the material the room is constructed from and the space the room occupies — into existence ex nihilo every hour. In that case, one wouldn’t need an infinite supply of material or an infinitely large universe. You might reply that creation ex nihilo is impossible, but that’s a different objection.

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      1. Thank you, Daniel. About Loke, I was addressing only the non-supernatural case, because, as I understand it, theists object to eternal time in a non-supernatural world because that would create an instantiated infinite in the actual world, which then leads to paradoxes, like the HH. These paradoxes seem to pop up only for an infinity of things existing together at the same time, so that seems to be the only infinite to worry about, which I assume is what they call an instantiated infinite. However, arguments like Loke’s, or George counting, must first assume that the actual, non-supernatural world has a pre-existing instantiated infinite to show that eternal time would create an instantiated infinite in the actual world. That seems like the textbook example of begging the question.

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      2. Alexander —

        Friends of the Kalam argument maintain that a Hibert Hotel is impossible not only in the actual world but in any metaphysically possible world, including worlds where there are supernatural beings. Loke explicitly phrases his argument in terms of the im/possibility that God could create a Hilbert Hotel. For Loke, the claim that not even God could make a Hilbert Hotel amounts to the claim that even a being that could do any metaphysically possible task cannot construct a Hilbert Hotel — and, presumably, this is because Hilbert Hotels are not metaphysically possible.

        In any case, I don’t think we need God to formulate Loke’s argument in a way that seems to make trouble for your objection. Supposing, as many friends of the Kalam argument do, that presentism is true, all we need is that it is metaphysically possible for objects to come into existence ex nihilo, since all that is needed is the following counterpossible conditional:

        (C) If an infinite past is metaphysically possible then one can construct a Hilbert Hotel.

        Suppose that an infinite past is metaphysically possible and that it is metaphysically possible for objects to come into existence ex nihilo. If it is metaphysically possible for objects to come into existence ex nihilo, there would presumably be a metaphysically possible world where, every hour, a new patch of space comes into existence ex nihilo and a hotel room comes into existence ex nihilo in the same hour. If the past could be infinitely long, then there would be a metaphysically possible world where a Hilbert Hotel has been constructed.

        In a world where each room of the hotel is generated ex nihilo, there wouldn’t need to be an infinite resource of materials to draw upon to construct the hotel. This pushes the issue back on to whether creation ex nihilo is possible, but, as I said, that’s a distinct worry about the Kalam argument.

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      3. Thank you for those comments, Daniel. I think I have two comments about your objection:

        1) I wonder whether a non-supernatural world that allows for an infinite number of objects to pop into it ex nihilo is not, in some objectionable (paradox-causing) way, already infinite.

        2) If a theist allows objects to come into existence ex nihilo in a non-supernatural world in order to get the instantiated infinite, mustn’t he also concede that all of the world could have just have popped into existence ex nihilo, implying that god is not necessary? Are there any scenarios in which eternal time would create an instantiated infinite in the actual non-supernatural world that would not also make god unnecessary? While the immediate conclusion of the Kalam is mildly interesting, I think it is the broader implication of god being necessary that gives the Kalam its real weight. A defense that undermines the real goal of the Kalam seems self-defeating.

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      4. Alexander — Great questions!

        “1) I wonder whether a non-supernatural world that allows for an infinite number of objects to pop into it ex nihilo is not, in some objectionable (paradox-causing) way, already infinite.”

        Hm. I’m not sure what to say about this. I suspect that what the Kalam proponent would need to say here is just that for any possibly existent object x, x could pop into existence. But from this, it does not follow that all possibly existent objects can pop into existence — that would be the fallacy of composition (i.e., it would involve a for-each to for-all inference).

        On the other hand, as I think Alex has done a great job explaining, for-each to for-all inferences are not always fallacious. If we are told that any possibly existent object x could pop into existence, but that it’s false that all possibly existent objects can do so, I’m left wondering what prevents all possibly existent objects from doing so. For example, are there some objects whose popping into existence is not compossible? That could block some combinations from popping into existence, but, of course, would not show that there is no compossible combination that is infinitely large. If there’s no reason to rule out an infinitely large compossible combination of objects that pop into existence, then the Kalam proponent has not met their burden of proof. Perhaps that’s a way to save your argument and show that there is something question begging about the Kalam proponent’s argument.

        “2) If a theist allows objects to come into existence ex nihilo in a non-supernatural world in order to get the instantiated infinite, mustn’t he also concede that all of the world could have just have popped into existence ex nihilo, implying that god is not necessary? Are there any scenarios in which eternal time would create an instantiated infinite in the actual non-supernatural world that would not also make god unnecessary? While the immediate conclusion of the Kalam is mildly interesting, I think it is the broader implication of god being necessary that gives the Kalam its real weight. A defense that undermines the real goal of the Kalam seems self-defeating.”

        Well, the Kalam proponent needs the premise that everything that comes into existence comes into existence by way of a cause. So, the Kalam proponent cannot consistently say that objects can pop into existence, ex nihilo, without a cause. There’s a further debate about what the attributes of that cause have to be.

        We have to remember that when we talk about worlds sans supernatural entities, we are talking about counterpossible states of affairs. For most proponents of the Kalam argument, God is not only necessary for the creation of the universe, but metaphysically necessarily existent. Nonetheless, the usual strategy in defending the view that God is the cause of the universe coming into existence ex nihilo is not to invoke necessity but, instead, to appeal to an inference to the best explanation, i.e., the best explanation of an ex nihilo beginning is the spontaneous creation of a powerful mind with libertarian agency.

        Incidentally, while a “defense that undermines the real goal of the Kalam” may be a phyrric victory for most Kalam proponents, I think that all of these issues are much more interesting than the debate over God’s existence. I’m interested in what we can (or cannot) say about the origin of the totality of the physical world and about fundamental reality. If God’s existence could be conclusively demonstrated to be metaphysically impossible, I think we could still ask interesting questions about the Kalam argument.

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  3. Interesting post as usual Alex, thank you. I don’t think that you have invalidated Premise 1, though:

    P1. A collection formed by successive addition cannot be an actual infinite.

    P1 is about the impossibility of obtaining an infinity (an infinite collection) from successive addition. As far as I can tell, P1 does not make temporal references, and is not dependent on tenses (what will have been counted vs. what will be counted, etc.). P1’s mention of “successive” refers to incremental addition which, unlike Premise 2, is not temporal nor refers to tenses in any way that I can see.

    I don’t think that it is possible to “form” an infinite collection by incremental addition without starting with the presumption of an already-existing infinite collection. It would not be particularly impressive to assert that an infinite collection can be “formed” by starting with an infinite collection to begin with. One might say that’s question begging. A limiting process like “in the limit as a counter approaches infinity” does not resolve the problem either because (a) it presumes that the collection of counters is infinite to begin with, and (b) “approaching” infinity is not the same as arriving at infinity, and in fact, no matter how large the counter becomes, it will always be infinitely far away from finishing the count.

    But suppose one does not start with the presumption of an infinite collection of numbers, and consider “A) It is possible that George will count infinitely many numbers.” But notice that any process of counting that does not already include an infinite collection in its running count, will not arrive at an infinite collection at any specified time. If the process of counting will not arrive at an infinite collection at any specified time, then there is no time at which the process of counting will arrive at an infinite collection. If there is no time at which the process of counting will arrive at an infinite collection, then the process of counting will never arrive at an infinite collection. If the process of counting will never arrive at an infinite collection, then it cannot arrive at an infinite collection. Since infinitely many numbers constitutes an infinite collection, it follows that “not-A) it is not possible that George will count infinitely many numbers.”

    I mentioned in a comment to your Successive Addition article over a year ago that early in the 20th Century number theorists axiomatized “infinity” (in Zermelo-Fraenkel and its derivatives) because it could not be “constructed” by incremental addition, but was too useful to give up. While the natural numbers can be “constructed,” essentially, from incremental addition (counting) and other axioms, “infinity” cannot. I don’t think it’s an accident that mathematicians gave up on “constructing” infinity through successive addition. Nor is it likely that they were careless or not thoughtful enough. Most likely it is because P1 is correct.

    Natural language is a fluid and rather imprecise affair that tends to cloak problems or inconsistencies in vagueness. I don’t think this is particularly controversial. For example, it’s quite possible that there is a problem lurking in my move: “If the process of counting will not arrive at an infinite collection at any specified time, then there is no time at which the process of counting will arrive at an infinite collection.” But it may be difficult to detect such a problem (quantifier shift, equivocation…?) because of inherent vagueness in natural language. Math is every bit as much a language, but unlike natural language, it is formalized and usually much more precise and perhaps better suited to concepts like infinity.

    I would agree that the argument you attribute to Loke and Erasmus is invalid:

    1. George cannot count up to infinity, because there would always be more numbers left for him to count.
    2. Therefore George cannot count down from infinity.

    However, this neither refutes P1, nor does it establish that 2. is false. In fact, I’m not sure that “counting down from infinity” is even coherent, let alone possible. After all, infinity is neither an origin (to start counting from), nor a destination (to arrive at by counting). Infinity refers to a collection being unbounded.

    The problem with Loke’s (and others’) argument is not with P1 (or with Loke’s reframing of P1 to strip the entire argument of A-Theory baggage). The rub is with Premise 2 and their insistence that the past must be “traversed,” or “crossed” in its entirety. Of course, it is not possible to entirely “traverse” infinities. Infinities may be “traversable” but cannot be “traversed” in their entirety for the same reason that they cannot be “constructed” by incremental addition only. I.e., precisely because of P1. That one cannot traverse to an infinitely faraway destination or from an infinitely faraway origin does not mean that infinities cannot exist.

    Introducing distinctions such as “actualized” vs. “potential” to break the symmetry between past and future shouldn’t have any currency either, I don’t think. As I understand it, the contention is that, unlike the future, the past is “actual” because all past events have “actually happened” at some time in the past and thus must be traversable either as a destination going backwards from the present, or as an arbitrary origin which will eventually arrive at the present. As to symmetry-breaking to preserve the “potential” infinitude of the future, I don’t see how the same could not be said about the future simply by shifting tenses, (“will actually happen” and the like) but that’s a separate matter. Even if one defines “actualizable” or “actual” as traversable from the present, there is no event in the past that is not traversable (i.e., can be arrived at) from the present, so there is no event in the past that is not “actual” in that sense.

    It is quite another matter, to insist that the past in its entirety be traversable from some origin (beginning) to now, which, of course makes the past finite, perforce, because (a) infinities cannot be entirely traversed, and (b) there cannot be such a thing as an origin (beginning) if the past is boundless. This is not a demonstration that the past is finite, but rather, it is forcing a definition of the past to make it finite.

    To say that the past must be traversed from some infinitely-long-ago origin is to beg the question, because an origin (i.e., a beginning) entails that the past cannot be boundless. Infinitely-long-ago beginnings and traversability are the wrong way to go about demonstrating past finitude because of their circularity.

    To demonstrate past finitude, its proponents must demonstrate that the past is somehow bounded. Starting from, say, the present, if one goes back into the past some finite amount of time, one can always go back further and further, which seems to indicate that the past is boundless, and therefore infinite, absent any demonstration of a finite bound in the past. (This, it seems to me, is entirely symmetrical to going forward into the future instead but, again, is a separate matter.) Past-finitude proponents must demonstrate that there has to be a stopping point. The impossibility of complete traversability, while a property of infinities, is neither here nor there, and neither is “actualizability.” Also, none of this seems to hinge on the apparent clash between the A-Theory and modern physics, as far as I can tell.

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    1. Hi Miguel. Insightful comments, as usual.

      “I don’t think that you have invalidated Premise 1”. Agreed; I’m not trying to do that in this post. However, I have a paper coming out in Mind where I do argue that this premise is false. I will share that when the final proofs have been agreed.

      As a preview though, here is how I see the argument going:

      1. It is possible that George starts counting now and will never stop.
      2. If George starts counting now and will never stop, then for each natural number, n, George will count n.
      3. If George will count each natural number, then George will count ℵ0-many numbers.
      4. Therefore, it is possible that George will count ℵ0-many numbers.

      I think that the conclusion is an example of an actual infinite (each natural number will be counted) that is formed by successive addition (each one is counted successively).

      Your comments, to the effect that George’s counting “will not arrive at an infinite collection at any specified time”, are right. But to me this just means that there is no final counting event, at which the job is complete. And that is to say that there is no point at which the infinite collection ‘has been’ formed. So I think this is an example where ‘it will be that p’ is true without ‘it will have been that p’. That inference (from the simple future to the future perfect) sounds like it is plausible, but we can construct other counterexamples, ones that have nothing to do with infinity. Imagine that time ends and at the final moment p is true for the first (and last) time. At all earlier times, ‘it will be that p’ is true, but there is no point where ‘it has been that p’ is true, so ‘it will have been that p’ is always false. This shows that one cannot logically derive a future perfect from a simple future tense. The counting forever case is just another counterexample. The key thing to notice, the common property to both counterexamples, is that there is no temporal standpoint from which one can look back on p and say that it has happened. In the case where p happens at the point where time ends, that’s because there just is no subsequent point in time to look back at it from. In the counting case, there is no such vantage point just because the counting events ‘fill the future’. But in both cases, we have the same result.

      So I am not bothered by the fact that there is no point at which George has finished his counting. I think that is true, but that it doesn’t settle the question. I also think this point is very subtle, and that many philosophers content themselves by answering this close-by question when the salient issue is just next door, so to speak.

      Sometimes you also use the language of ‘arriving at’ infinity. This makes it sound like what is required is counting ℵ0 as a member of the series of natural numbers. Of course, it’s not a member of that series. It is not the immediate successor of any natural number, and if counting is just stating immediate successors, then one will never count that number. And again, I think this is right but not strictly relevant. In my argument from above, it’s a question of the cardinality of the numbers that George will count. There is no ‘infinity-eth’ natural number, but if George will count every natural number, then he will count infinitely many numbers. That’s all one needs. The natural numbers are an example of an actually infinite collection, even though each one is only finitely greater than 0. So this sort of point, about ‘arriving’ at infinity, though clearly closely related, doesn’t settle the question in my view.

      I think it is helpful to consider the reverse operation; George counting down forever and finishing now. Whatever one wants to say about why that is not possible, clearly it’s not got anything to do with George counting a number with no immediate successor. Obviously, he doesn’t ‘start’ at ℵ0 and have to say the number that comes ‘next’. Nobody thinks *that* is what is wrong with the infinite countdown (they have other things to say, but not that). Yet somehow it is very often what people think is wrong with the infinite count up. That attitudinal asymmetry alone is very revealing, in my opinion.

      The last thing you say that I wanted to respond to is about ‘traversing’ the infinite; “Of course, it is not possible to entirely “traverse” infinities”. If we picture an infinitely long path, and stipulate that George starts walking down it and never stops, then he will traverse each part of the path. How many parts is that? Infinitely many. So in a sense it is an infinite path that can be traversed. Will he arrive at the end? No, there is no end. Will he ever ‘have’ traversed infinitely many parts? No. But we are back to a perfect tense again. So in some ways, this just sounds like the same thing in different language.

      But there is something I find objectionable for a different reason here, and it is my biggest issue with the A-theory. The concept of ‘traversal’ seems to me to involve both spatial and temporal components. It is basically just another word that means ‘movement’, or ‘motion’, etc. In the final analysis, it is some kind of ratio between distance and time. But whenever we get an argument that involves reference to ‘the present moment’ having to traverse the infinite past, I feel like we are double-counting the same thing at some point. That’s because the infinite past is usually conceived as an infinitely long line, along which there is a ‘moving-now’ that is rolling along it like a train car on a rail track. So some versions of this argument insist that the train car would still be on its way to this point if the track was infinitely long. Something like that. But the problem is that the ‘distance’ here is time. If we appeal to something like a clock that we can measure how long it takes the moving now to cover a bit of the timeline, we have introduced a second time dimension into the picture. Fun though it might be to speculate about there being more than one time dimension, the point here is that we were just trying to cash out the A-theory, which is not a ‘two-times’ theory. Plus we can iterate this conjuring trick, and think about a new timeline that plots out where the moving now was at different points on the original timeline. An A-theorist should want there to be a new moving now that rolls along this timeline, and we are back at the start of this process of conceptualising time as both a space and as a ticking clock again.

      So, to me, the lesson of this is that the metaphor of the ‘moving now’ shouldn’t be taken literally. Whenever anyone says things along the lines of ‘the past couldn’t be infinite because it would take infinitely long to get to the present’, they are appealing to a metaphor that breaks down at exactly the point where they need it to be sustained. No infinite distance is actually traversed by anything. There is not distance and time here; just time. So the objection, when spelled out without the misleading metaphor there is just something like ‘the past couldn’t be infinite because otherwise there would be an infinite amount of time in the past’.

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      1. Excellent! Can’t wait for the Mind paper. And thank you for the response. I’ll respond to it after I’ve read it more carefully–there’s a lot to absorb.

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      2. Alex, I read your response in more detail, thank you again. This is turning out to be more interesting than I’d thought. I hope my response below is at least somewhat helpful to anticipate objections. I remain unconvinced that you have demonstrated that Premise 1 is false. I think that 2 and 3 in your argument against Premise 1 make a quantifier scope assumption which may well be assuming what you’re trying to prove. Also, a better argument can be made that reaches the opposite conclusion, I think.

        Before addressing that, let me first point out the reason why I responded as if you had been trying to address (falsify) Premise 1 in your post. Premise 1 claims that your Scenario i) is impossible and your post aims to show that, regardless of Scenario i)’s possibility, Scenario ii) is possible. But I don’t think there is a meaningful distinction between Scenario i) ‘counting up/down from a starting point’ and Scenario ii) ‘counting up/down to an end point’. Scenario i) is essentially what Premise 1 refers to. In their argument, Craig et al. make the implicit identification of the impossibility of Scenario i), which is essentially Premise 1, with the impossibility of counting down from an infinite past, which is essentially Scenario ii). I.e., Craig et al. implicitly assume that your Scenarios i) and ii) are equivalent so that if Scenario i), via Premise 1, is impossible, then something like Scenario ii) must also be impossible. I believe that this move by Craig et al. is permissible because there is no meaningful distinction between the possibilities of Scenarios i) and ii), as far as I can tell.

        When I say that there is no meaningful distinction between i) and ii), I mean that one is possible iff the other one is possible. The count direction under addition/subtraction and labeling the finite side as a beginning/end, respectively, is entirely symmetrical and, in my view, an inconsequential difference. After all, the cardinality is the same in both cases, the counting procedures are mirror images of each other, and the beginning vs. end labels are, in some sense, arbitrary. The contention is that you cannot get to an infinity from a finite start by successive finite additions any more than you can get from an infinity to a finite end by successive finite removals. You either have to add an infinity in the first case, or remove an infinity in the second. Both cases require the existence of infinities which either cannot be generated by successorship (successive addition), or has to be assumed to be formed by successorship in the first place (but which your arguments do not demonstrate, so far as I can tell).

        In your mountain-climbing illustration of Scenario ii), the reason why George’s count is only finitely far away from the top of the mountain is that there is no meaningful sense in which he could have started at the bottom of a bottomless mountain. So every possible count is only finitely far away from the top. But this is just like Scenario i), since any possible count is only finitely far away from the bottom. None of this says that it is possible to form an infinity by successorship alone.

        Now to your refutation of Premise 1. I’ve taken the liberty of expanding two of the premises in your argument (italicized) to elucidate what I think are implicit statements (please let me know if I’m not representing your argument accurately):

        1. It is possible that George starts successively counting now and will never stop.
        2. If George starts counting now and will never stop, then for each natural number, n, George will count n.
        2a. If George will count each natural number n, then George will count all the natural numbers.
        3. If George will count all the natural numbers, then George will count ℵ0-many numbers.
        4. Therefore, it is possible that George will count ℵ0-many numbers.
        4a. Therefore, it is possible that George will form an ℵ0-infinity by successive counting.

        So 4a refutes Premise 1, according to the argument. To be clear, we agree that ‘all the natural numbers’ refers to the infinite set of all positive integers with strong-limit cardinality ℵ0. I believe that 2 and 2a have not been substantiated and are simply restating the conclusion. You are saying that George will complete a task that he will never stop completing, which doesn’t seem possible to me. To be more precise, there is an unwarranted “jump to infinity” via a quantifier scope assumption which is equivalent to assuming that the set of natural numbers can be formed by the process of successive counting, which in turn is what is supposed to be demonstrated in the first place. It seems to me that 2 and 2a have not demonstrated that George will count all the numbers or that he will count ℵ0-many numbers, they have simply assumed this.

        Your argument, as I understand it, hinges on a quantifier logic of the form ‘there does not exist an n which will not be counted, therefore all n will be counted’. (Or, if you prefer to remove the passive voice you can say ‘there does not exist an n that George will not count, therefore George will count all n’.) This is, of course, valid for the usual quantifiers with finite scope: ~∃n ~C(n) ∀n C(n), where C(x) stands for ‘x will be counted’ (or ‘George will count x’). The problem here is with the scope of the quantifiers, which is assumed to be infinite by successorship to begin with. To wit, the following are equivalent ways of stating 2 and 2a:

        ~∃n ~C(n) ~[~C(1) V ~C(2) V … V ~C(n)] [C(1) & C(2) & … & C(n)] ∀n C(n).

        And this is fine for the usual scope of the existential and universal quantifiers when n is finite. Assuming that there’s no problem with an infinite quantifier scope definition, this would also be valid if the scope of the quantifiers is assumed to be infinite in the first place, but that’s the rub. The problem comes in when you allow n to grow unbounded, in which case you have to assume that the scope of the quantifiers also grows unbounded, and allows you to ‘cross the ellipsis’, as it were, and make the jump to infinity (to ℵ0 cardinality). But notice how this is the same as assuming that one can form an ℵ0-infinity by successive counting, which is what you set out to prove in the first place. In other words, assuming that the universal quantifier can be applied to the counting predicate C(x) as

        ∀n C(n) [C(1) & C(2) & … & C(i) & …]

        under the assumption that the successively-labeled set of the {C(i)} has ℵ0 cardinality to begin with, is equivalent to assuming that it is possible to form an ℵ0-infinity by successive counting. (This would work for the existential quantifier in exactly the same way.) Put another way, if you do not start out assuming that successorship will produce a set with ℵ0 cardinality, then you cannot assume that the successively-labeled {C(i)} set will have ℵ0 cardinality either and the quantifier scope cannot be relied upon to claim that it is possible to arrive at an ℵ0 cardinality by successorship without that presumption. The quantifier’s infinite-cardinality scope presumed to be arrived at by successive labeling, does not demonstrate that infinite cardinality can be arrived at by successorship; it presumes it.

        A better way to look at this, I think, is to note that if George will never stop counting, this means that he will never finish counting, and there will always be numbers left to count. Since there will always be numbers left to count, this means that George will never count all the natural numbers. In argument form:

        1’. It is possible that George starts successively counting natural numbers now and will never stop counting.
        2’. If George starts counting natural numbers now and will never stop counting, then there will never be a natural number n that George will count such that there will not be uncounted natural numbers left to count.
        3’. Therefore, for all natural numbers n that George will count, there will always be uncounted natural numbers left to count.
        4’. If there will always be uncounted natural numbers left to count, then George will never count all the natural numbers.
        5’. Therefore, it is impossible that George will count ℵ0-many numbers.
        6’.Therefore, it is impossible that George will form an ℵ0-infinity by successive counting.

        So 6’ agrees with (substantiates) Premise 1, according to this argument. But it’s even more dramatic than that. Another way to look at this is that I am issuing a ‘promissory note’ that George will, in fact, count all the natural numbers. All of them. Bar none. But of course, a promissory note is not the same as fulfilling the promise. I’m stating that he will, in fact, count them all, but will he? George starts at t=1 and I issue a promissory note that he will count all the numbers (trust me, he will). George counts for a long time, and he gets to t=n. How much progress has George made towards fulfilling the promise? He’s counted 1,2,…n numbers, so surely he’s made some progress towards fulfilling the promissory note that he will count all the numbers? To determine how much progress George has made, I can simply relabel the counter to t’=t-n such that at the next increment of the count, t=n+1, we have t’=1, followed by n+2 which gives t’=2, etc. But this looks like the count 1,2,… all over again. Actually, it is the same count 1,2,… all over again, only relabeled. This means that George has exactly the same task ahead of him that he had when he started at t=1. George has exactly the same amount of counting left to do at n as he did at 1. He has made no progress towards fulfilling my promissory note, and in some sense, George is always starting the task, and thus never finishing it. So why should you believe me that George will, in fact, count ℵ0-many numbers, ever? It seems that the best I can do is issue the exact same promissory note for every n and never, ever, demonstrate one iota of progress towards fulfilling the promise, no matter the value of n. A long sequence of identical promissory notes of a future promise fulfilment, no matter how large, is not the same as fulfilling the promise in the future, particularly when no progress between promissory notes can be demonstrated for any n. An even stronger argument can thus be made by substituting 2’ with:

        2’’. If George starts counting natural numbers now and will never stop counting, then there will never be a natural number n that George will count such that there will not be ℵ0-many uncounted natural numbers left to count.
        Etc…

        The conclusion, then, is that George will never even come close to counting an ℵ0-infinity by successive counting.

        While English is not my native language, I think I understand the nuanced contrast that you draw between the simple future vs. the future perfect indicative tenses. I’m not convinced, however, that this is a valid objection to Premise 1 (nor a defense of your argument against it, as the case might be). I’ll address this below, but in the meantime, note that the argument can be framed, as I did above, without a future perfect reference. It can be framed in a simple future tense, thereby defusing the future perfect objection, I think. Instead of saying that there is no time at which the task will have been completed, one can say that the task will always be incomplete, or will never be complete. No appeal to the future perfect is needed. One may quibble that saying that a task will never be complete is the same as saying that the task will never have been completed. But that would be conceding the point that, for a boundless future at least, the tense distinction is not particularly meaningful. Maybe that’s because of the vagaries of language and the stretch to our intuitions that infinities exact.

        Even assuming that the ‘simple future does not require a future perfect’ objection is valid, at best we have two different but valid ways of arriving at contradictory conclusions via the simple future indicative tense: George will form an ℵ0-infinite set by counting vs. George will never form an ℵ0-infinite set by counting. In that case this would likely mean that it is time for a new axiom (per Gödel’s famed incompleteness theorems) if we want to remove this contradiction. This could be something like the Axiom of Infinity (or alternatively some sort of ‘Axiom of Finitude’, if we wanted to go that route, which some have). What it would not mean, however, is that your argument has established that an ℵ0-infinity is reachable, or can be formed, by successorship only. You can assert it by way of an axiom, but you have not demonstrated it via successorship only, in my view.

        In that case, Premise 1 can either be supported or not supported depending on which axiom you choose, but has not been demonstrated either way. However, I think that my argument concluding (6’) does not suffer from a self-referential quantifier scope assumption. It does not rely on the quantifier scope being finite or infinite, and remains in full force in either case. That’s why I think it is more reasonable to say that Premise 1 is supported vs. claiming that it is not.

        It seems to me, that the counterexample you offer against the ‘inference from the simple future to the future perfect’ is disanalogous and thus inapplicable to the infinite case precisely because p becomes true at a specific time in the future. I.e., in your example, p becomes true only at the end of time which occurs at a specific time, whereas in the infinite case there is no end of time, and no clear way that p must be true at any specific time. Both cases share the common property that there is no time at which p will have been true. Agreed. But in one case this is so because p is true precisely when time ends and in the other it is so for precisely the opposite reason, namely because time does not end. The analogy breaks down precisely where it counts (no pun intended), namely in deciding whether p will ever be true. In one case it certainly will, while in the other it arguably either never will or it cannot be determined that it ever will. I.e., just because p will not have been true in both cases and p will be true in one case (finite future), does not entail that p will be true in the other case (boundless future). The future truth of p needs to be demonstrated in the second case, not concluded just because of the commonality of the future perfect falsehood in both cases. I could be wrong, of course, but this strikes me as a sort of undistributed middle leap, or something like that.

        You mentioned that ‘we can construct other counterexamples that have nothing to do with infinity’, but you only offered the one where p is true only at the end of time, which I don’t see as being relevant to the infinite (boundless future) case. In fact, I don’t see any way in which the finite-future case applies to the infinite set of natural numbers, which is boundless. I can’t think of other counterexamples (in the indicative), which of course doesn’t mean there are none. But in general, it seems to me that, given a boundless future, saying that ‘it will be that p’ does not seem consistent with saying that ‘there will never be a time when it will have been that p’. If it will be that p, then it will be that p at some time, even if this time is an indeterminate time. Then under a boundless future, there will always be a later time when it will have been that p. But if there is no time (even an indeterminate time) when it will have been that p, then how can one say that it will ever be that p? To be more precise, it seems to me that I can always help myself to the following definitions of ‘it will be that p’, ‘boundless future’, etc:

        Define t0 as the present time, and let q = ‘it will be that p’ for some proposition p with a well-defined truth value. We say that q is true only if there is a time t > t0 at which p is true (where t is not required to be known at t0). Now define r = ‘it will have been that p’. We say that r is true only if q is true at t0 and there exists a time t’ > t. But in a boundless future, there will always exist a t’ such that t’ > t for any t (whether t is known at t0 or not). Therefore, in a boundless future, q –> r, or in other words, in a boundless future ‘it will be that p’ entails ‘it will have been that p’. Equivalently, ‘it is not the case that it will have been that p’ entails ‘it is not the case that it will be that p’. (In fact, for a boundless future, the stronger statements q r and equivalently ~q ~r can be made by replacing the ‘only ifs’ in the definitions with ‘iffs’.)

        Because, in a boundless future, there will always exist a t’ > t, I don’t see how, in the indicative, other counterexamples to q –> r can be found, unless any of these definitions are inconsistent or somehow untenable. Allowing t to increase boundlessly does not invalidate this, because even then a t’ can always be found such that t’ > t.

        Be that as it may, I don’t think that the argument in support of Premise 1 that I presented above (6’) is dependent on a future perfect tense.

        Barring your example of a bounded future, which I would argue is disanalogous to the boundless future case, we have an ancillary point of agreement. I think we would agree that, if p has not yet become true, and if, sometime in the future, ‘it will have been that p’, then ‘it will be that p’ (that seems unassailable). It follows (by contraposition, or modus tollens etc.) that if ‘it is not the case that it will be that p’, then there will be no time at which ‘it will have been that p’. Of course, this does not mean, on its own, that if there will be no time at which it will have been that p, then it is not the case that it will be that p, necessarily. More succinctly:

        (a) It will have been that p –> It will be that p. (True)
        (b) ~(It will be that p) –> ~(It will have been that p). (Contraposition/m.t.)
        (c) It will be that p –> it will have been that p. (Asserting (a)’s consequent)
        (d) ~(it will have been that p) –> ~(it will be that p). (Denying (a)’s antecedent)

        But I am not arguing (b) nor am I concluding (d) from (a). And I would have to agree that (c) and (d) do not follow from (a), of course. (Although a case for (c) and thus (d) could be argued independently from (a), as I did above, but I’m not arguing that because I don’t think it’s necessary in order to defend Premise 1.) However, I am saying that there are reasons to accept the consequent in (d) independent of (d)’s antecedent and independent of (a). It seems to me that you are arguing that ‘it will be that p’. I am saying (1) that you have not demonstrated that ‘it will be that p’, and (2) that ‘~(it will be that p)’ is more reasonable. I think the tense differences, while interesting and illuminating, are ultimately extraneous.

        It’s not about there being ‘no point at which George will have finished his counting’, it’s about the fact that there is no demonstration that George will ever finish his counting (actually, it is stipulated that he will never stop) and in fact there is an argument to the contrary. And if George will never finish his counting, how can one say that George will count all that there is to be counted?

        The remainder takes us a bit off topic, but the time ‘traversal’ or ‘crossing’ language is one that Craig et al. use when defending the notion of the ‘absurdity’ of an infinite past. I don’t think that it has to do with the tense differences, for the reasons I offered earlier. Despite metaphors or visual analogies (‘a train car on a rail track’ or ‘climbing a mountain’, etc.), I don’t think that time ‘traversal’ has, necessarily, a spatial component, nor that it refers to ‘movement’. While spatial ‘traversal’ does entail time ‘traversal’ (lest one should be able to travel a spacelike interval instantaneously), this does not mean that time ‘traversal’ must involve spatial movement. It is possible for me to be stationary with respect to some reference frame or other and for an interval of time to ‘elapse’ while I am stationary. I think when Craig et al. refer to ‘traversal’ in time they are alluding to an interval of time elapsing, not to spatial movement or spatial traversal. I see nothing wrong with this, nor does the notion of time intervals elapsing vitiate Premise 1, in my view.

        I think that, if there is any reference to a train car on a rail track, it might be alluding to a causal chain or a sequence of events, or something like that, not necessarily to spatial movement. To the extent that there is spatial movement, it is incidental (or perhaps it is used only as a metaphorical visualization aid, much like your mountain climbing examples).

        I accept Premise 1, but I think the problem is with Premise 2 as I’ve mentioned before. Just because infinities cannot be constructed by successive addition only, does not mean that an infinite past is impossible, even if we cannot count it or ‘traverse’ it in its entirety. As Wes Morriston said in your discussion video, if Craig et al. have a problem with infinite regressions of causes, they should state that up front, rather than leaping from Premise 1 to past finality. For Craig et al. to demonstrate past finality, they, like Truman hitting the horizon here, would have to find a justification for a hard stop. Short of that, past finitude is not justified either, regardless of Premise 1.

        As far as the A-Theory, to my mind, it is likely wrong on physical grounds, but that is quite apart from the a priori arguments discussed here, I think. I’d have to reflect a bit more on your thoughts regarding the A-theory and how it applies to the arguments here.

        I’m not an expert in any of this, nor have I done my homework nearly as much as you probably have, so I don’t have to tell you to take all of this with a grain of salt (you do anyway). But hopefully the discussion has been somewhat useful and will help you anticipate some objections. Thanks again.

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      3. Thanks for the lengthy thoughts. I’m glad it got you thinking!

        On the first quick point, you say basically that endless counts up and beginningless counts down should be equivalent, in the sense that one is possible iff the other is. So I see where you are coming from on that. My issue, and really this was the point of the blog post, is that if that biconditional held, then we would expect the reasons given for why one was impossible would be symmetric with respect to time. Yet the reason given – namely that there is always more numbers left to count – isn’t symmetric. It doesn’t apply to the counting down case, but it does apply to the counting up case.

        As an analogy, it might be that Jack can ride a bicycle iff Jill can, and that Jack cannot because he only has one leg. But because Jill has two legs, this asymmetric fact doesn’t really *explain* the biconditional. How come Jill can’t ride a bicycle, we might think. Sure, we have a biconditional, but the actual reason given for thinking that one side of that is true doesn’t apply to the other. The biconditional itself doesn’t prevent Jill from riding a bike. This is a bit like how I see the discussion about counting. It might be that counting up and counting down are equivalent, but if so I would expect that the actual explanation of why I can’t finish a downwards count would actually apply to that case. What we have is someone saying ‘Jill can’t ride a bike because Jack only has one leg’. That won’t do.

        Moving on, your version of my argument adds in a new premise, 2a, “If George will count each natural number n, then George will count all the natural numbers.” I think it’s very important not to go this route. I deliberately did not want to involve ‘each-to-all’ inferences anywhere in the argument. Part of the problem is that these quantifiers are not well modelled in classical (first-order) logic. Sure, we have the nice existential/universal quantifier distinction. But the difference between each and all is surprisingly hard to capture using this machinery alone. It’s the sort of distinction medieval logicians were much better at making than modern ones. What we need to do is consider the difference between distributive and collective predication. ‘Each’ naturally comes across as distributive, but ‘all’ (and ‘every’) are somewhat equivocal in natural English and can be used in both senses. Here is the difference. Consider the following:

        A) Each item in the shop costs £5
        B) All items in the shop cost £5

        If I take three items to the shop, then according to A, the bill will be £15. That means that the predicate ‘x costs £5’ has been used distributively. It is possible to hear B having the same effect (that is, to read ‘all’ distributively). But an equally natural way to read this is collectively. If I did, then no matter how many items I bring to the till in one go, my total bill would be £5. The predicate ‘x costs £5’ applies to the totality, rather than to each element in the totality.

        Now, at least sometimes, we cannot infer collective predication from distributive. The above case is a good example (as the prices show). So I don’t want to infer from ‘x will count each number’ to ‘x will count all the numbers’, if we mean by the latter something like collective predication. On the other hand, if all we mean by the latter is distributive prediction, then it muddies the water to introduce the word ‘all’ rather than sticking to the less ambiguous ‘each’. So I would recommend not adding in premise 2a.

        You also changed the third premise so that it uses the word ‘all’. I wouldn’t want to do that either. Rather, just use ‘each’ (so that it is clear that we are talking distributive predication). Then the inference is:

        If George will count each natural number, then George will count ℵ0-many numbers.

        And that seems solid to me. In fact, I think that is an instance of a more general rule, which is something like:

        R) For all collections, S, if *each* element of S is P, and the number of elements in S is x, then the number of elements in S that are P is x.

        To me, R sounds almost trivially true. Note that it is a principle concerning distributive predication, not collective.

        You also change the conclusion, 4a, in such a way that it seems less clear to me exactly what the scope is supposed to be. True, it mirrors Craig’s premise more closely, but I actually don’t think that’s too important here. The point is that Craig says that an actual infinite cannot be ‘formed by successive addition’ because one cannot count ‘to infinity’. I want to say ‘hold on a minute here, there is a sense in which I can: namely, it is possible that I will count ℵ0-many numbers.’ That is ‘counting to infinity’ in a relevant sense. What I am not doing is saying ‘it is possible that I will complete an infinite count up’. That’s not possible, and I’m not trying to say that it is. I’m merely pointing out that ‘it is possible that I will count ℵ0-many numbers’ is true anyway, and constructing an argument for it. If you change the argument such that it makes me saying that ‘it is possible that I will complete an infinite count up’, then you are misrepresenting my point. And, to me, saying “it is possible that George will form an ℵ0-infinity by successive counting” comes close to implying the latter. I read it as saying that he has completed an infinite count at some future point. So I’m not sure how helpful it is as the conclusion of my argument, because it seems to be expressing something I take to be false.

        You make a few comments like “You are saying that George will complete a task that he will never stop completing, which doesn’t seem possible to me” which aren’t right. I’m not saying anything about him ‘completing’ the count up. I’m not saying ‘when he completes his count, he will have counted infinitely many numbers’ I’m just saying ‘if he never stops, how many future counting events are there?’ The answer is infinitely many. I’m not evaluating the claim as of some future point when they are all behind me (because there is no such point for one thing). I’m evaluating it as of now, when they are all future. I think this is a subtle point, and basically key to my whole argument. While you have obviously thought long and hard about this, I’m still not quite sure you are seeing this. That’s no criticism, of course, as the view you are expressing is a very entrenched way of viewing things. But I’m trying to enact a gestalt shift here 🙂

        Although I really liked the mirror image argument you gave, where you turned the tables on (your version of) my argument, I think that it is still operating under the wrong way of thinking about what I am up to. I mean, when you see my point properly, it doesn’t matter that he always has the same number of numbers left to count. No point of my argument requires that he completes this task (indeed, I think he cannot). I just think that if he is going to count each number (i.e. not stop), then because there are infinitely many numbers, distributive predication and R tells us that he will count infinitely many – even if he never stops or completes his task.

        I’m going to stop here. Not because there aren’t other valuable comments in your post worth responding to, but just because I need to get on with things today. Hopefully I have said something that is fruitful for you though. It is very useful for me to think about objections like this, anyway. So thanks for the engagement 🙂

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      4. You’re welcome, it’s all very interesting. On the counting up vs. down symmetry, to make it fully symmetric with respect to time, you’d have to reverse the arrow of time as well as exchange subtraction for addition and ‘beginning’ for ‘end’. If you want full symmetry, you’d have to start at -1 and continue counting backwards. I think the key point is that in one case you have an infinity left to be traversed by counting and in the other you have traversed an infinity by counting. Both need to be demonstrated with equal force, and I would claim they are equivalent demonstrations.

        You’re claiming that you have been counting from infinity past and now you have a finite count left: …,-3,-2,-1. Sure, once you’ve arrived at -3, you only have a finite amount left to count. But how did you manage to arrive at -3? To arrive at -3, you have presumably successively counted an infinity, and that is what needs to be demonstrated. When you say that you have somehow arrived at -n and now you have a finite amount left to count, the question is how you could have arrived at -n from infinity past in the first place, not that you have a finite amount left to count. This is a claim that needs to be demonstrated, not just asserted to be possible. To see the ‘numbers-left-to-count’ symmetry, note that if your count really comes from infinity past, then you would’ve always had numbers left to count to arrive at any finite -n. In fact, you would’ve always had an infinite number left to count in order to arrive at a finite -n from now. Otherwise, you would not have been counting from infinity past. In some sense, -n would always be infinitely far away from infinity past. It’s not enough, I think, to say ‘well I am at -n now, regardless of how I got here’. Unless, that is, you can demonstrate that an infinity can be successively counted, and we’re back to the same demonstration.

        On the collective vs. distributive predication, your example is right, of course. In fact, this was one of the most interesting sections of your video with Morriston and Hedrick on the Craig discussion (I remember rewinding it and watching it again—really enjoyed it). With this, I go back to natural language being a bit squishy and perhaps inadequate for these tasks, at times—those medievals were on to something. To ensure there’s no funky collective-from-distributive inference due to language imprecision (interpretive ambiguities in what is possible or natural to ‘hear’), it’s sometimes helpful to write things out precisely (it’s not mathematical pedantry, or anything, it’s just for clarity). While it is possible to incur such natural-language ambiguities, it is also possible to avoid them by clearly and precisely representing the intended interpretation of the expressions at hand. I won’t go through the exercise, but in the shop items example, one can remove the ambiguity by labeling the predicate appropriately (Wes Morriston, I think, alluded to something similar in your video). I don’t think that my representation of your argument incurred such ambiguities.

        In the finite case there is no getting around the fact that ~∃n ~C(n) is logically equivalent to ∀n C(n). What you say about one, you necessarily say about the other. You are right that it is not clear how this equivalence translates in the infinite-scope case (I put this caveat in my other comment). But I’m not the one presuming infinite scope for these quantifiers. I’m only saying that the assumption that they can have infinite scope (even if this assumption were warranted), is tantamount to assuming that ℵ0 infinities can be counted, which is what is at issue to begin with. If infinite scope is not warranted, then the argument, without the assumption, fails to even get off the ground.

        But OK, let’s throw away 2a (apologies if I misrepresented anything), and let’s stick to the original distributive predication in 2 and 3. In passing, I mentioned in my previous comment that the same would hold for the existential quantifier, so I’ll explain what I meant by that. To say that George will count each n (distributive predication) is equivalent to saying that there is no n that he will not count, or ~∃n ~C(n). Without resorting to the universal quantifier this is the same as saying:

        ~∃n ~C(n) = ~[~C(1) V ~C(2),…,~C(n)].

        (I used ‘=’ instead of the biconditional because the double arrow doesn’t seem to print in these comments.) There’s no getting around this logical equivalence for finite n. Next, the move is to say that if George will count each n (or equivalently, if there exists no n that he will not count), then he will count ℵ0 many numbers. But you have now ‘crossed the ellipsis’ in the square brackets, and in the process ‘jumped to infinity’, and assumed what you were trying to prove. If the quantifier scope is not extendable to infinity, then your conclusion is not warranted either.

        On the other hand, the argument that I presented does not rely on assuming infinite quantifier scope (nor does it fail upon that assumption either). Letting the predicate U(x) stand for ‘when George counts x there will be ℵ0 many uncounted numbers’, allows me to conclude that there will be ℵ0 many numbers left to count for any n, regardless of whether the quantifier scope is infinite or not. No presumption of finitude or infinitude is required and the argument remains in full force either way. My argument does not make the ‘jump to infinity’ nor does it rely on the finitude or infinitude of the quantifier scope. That’s why I believe its conclusion is more defensible than your argument’s conclusion.

        I agree that R is almost trivially true. I’m just not sure what relevance it has to counting an infinite x, though. Again, I point back to the existential quantifier problem above. If each element, s, of the natural number set, S, is countable, P(s), and there are x elements in S, then the number of elements in S that are countable, P, is x. But if x can be counted, (is P) how does that get us to x being ℵ0? By assuming that incrementing s will get us to ℵ0. But we haven’t demonstrated this, we have assumed it.

        I agree that you can ‘count to infinity’ if by that you mean a process that starts somewhere and will extend indefinitely. I think it would be more accurate, in that case, to say ‘I will count towards infinity.’ But that is different from saying that it is possible that you will count ℵ0-many numbers. If the ‘possibility of counting to infinity’ is simply the ability to start a count and continue indefinitely towards infinity, sure, that’s possible. But that doesn’t mean that you will count ℵ0-many numbers unless such a process can be demonstrated to be possible. If I hop on a treadmill and aim it to London, it will never get me there from NY. In some sense I can say ‘I will go towards London’ but in another I cannot say ‘I will go to London’.

        When Craig says that ‘an infinite cannot be formed by successorship because one cannot count to infinity’, I don’t think he’s arguing that the process of counting toward infinity cannot happen. He’s arguing that you will not ‘form’ the natural number set because you will never count ℵ0-many numbers.

        I think Craig can always, just as easily, help himself to saying that if George will never stop counting, then he will never count ℵ0 many numbers and that, therefore, it is impossible that he ever will.

        At this point, I’m repeating myself, so I’ll stop. Maybe you’re right that I’m not seeing some subtlety in your arguments. Barring such a blind spot on my end (which is very possible), I think it might be more likely that these types of discussions bring to the fore how ambiguities in natural language can lead to different, and even contradictory interpretations.

        And, yes, my view is the ‘entrenched’ orthodoxy, but perhaps that is for a good reason? Maybe the world is not ready for your ‘gestalt shift’ quite yet 🙂 . All the more reason why this is a worthwhile discussion. Thanks again.

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      5. Sorry, the post missed the ‘is logically equivalent to’ double arrows, (biconditional connective) so, alternative, i’ll substitute iff instead:

        ~∃n ~C(n) iff ∀n C(n)

        ~∃n ~C(n) iff ~[~C(1) V ~C(2) V … V ~C(n)] iff [C(1) & C(2) & … & C(n)] iff ∀n C(n)

        ∀n C(n) iff [C(1) & C(2) & … & C(i) & …]

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  4. First, let me say that I’ve really enjoyed reading this thread and I agree with Alex’s main point. When I first read the quotes from Loke and Erasmus in the post, their reasoning seemed so patently absurd that I assumed that I must be missing something. After following this thread, I still have the same feeling so I think it’s time to ask for help.

    For example, in the quote from Erasmus, he wrote:

    “…Therefore, it is impossible to traverse an actually infinite sequence of congruent events and, thus, if the universe did not come into existence, the present event could not occur.”

    If you will accept that argument, it seems to me that you should also accept this:

    “Therefore, it is impossible to traverse an actually infinite sequence of congruent events and, thus, if the integers did not have a beginning, the number “3” could not occur.”

    Where am I going wrong?

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    1. Well, it includes a cute version of the grim reaper paradox, and I am writing a paper on that. Seems more productive than directly responding to his blog, but the points will apply to it. I’ll post some stuff on here as I progress.

      Liked by 1 person

      1. allright then…a paper on the grimm reaper paradox sounds nice
        (maybe I even understand it 🙂 this time)

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  5. Btw will you make a blog post on The premises of the kalam that try to show thqata the Cause has to be personal…I think Alexander pruss made some New and some older Arguments for this on Capturing xchristianitys channel I can send it to you if you want to.
    but Anyway some posts on these Premises would be cool

    Liked by 1 person

  6. But if we keep in mind that they both have the kalam in Mind.
    Doesnt we have to say that scenario 2 is Impossible too since, what they have in mind is that George will have counted to 0 and above which then would be a problem again.
    Or am I getting something wrong here

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