Banach-Tarski paradox, א Infinities, Infinitesimals, and the A-theory

I will offer an analysis of what is going wrong with the Banach-Tarski paradox suggesting that points, construed as infinitesimal surface areas, are nothing more than mathematically useful fictions. I will suggest that infinitesimals raise the same kinds of modally-prohibitive paradoxes in metaphysics as positing actually infinite quantities does (and for the same or similar reasons), and then consider an argument against the A-theory (in most of its forms) which can be purchased from these insights. I will then scout out some philosophical avenues available to the A-theorist.

The Banach-Tarski paradox is a famous mathematical paradox according to which it can be proved that if you divide the surface area of a sphere into little bits, and simply rearrange the bits appropriately, you can reconstruct two spheres each with the same surface area as the original sphere. In layman’s terms, you can prove (something just a shocking as) that 1=2.[1] To explain how it works, it may be worth calling to mind the various paradoxes associated with actual infinities.

Consider what it would be like to count upwards from -7 to infinity and stop only once you’ve arrived. Even if given an infinite amount of time you would never arrive, because no finite additions can sum up to a transfinite quantity. Subtract infinity from infinity, and what do you have? You have zero, but you also have infinity, and you also have 18.9801 (and every other real number); all of these are not just legitimate answers, they are mathematically correct answers. However, clearly 18.9801 is not equal to either zero, infinity, or anything else! Have a (Hilbert) hotel with an infinite number of rooms, all of which are occupied, and you want to check in an infinite number of new guests? No problem, just move every person from the room they are in (n) to the room with a room number equivalent to two times the original room’s room number (2n). Done; you’ve managed to move people around in such a way as to create an infinite number of vacant rooms without asking anyone to leave. Most of us (who are interested in this sort of thing) know the myriad paradoxes which arise from postulating even the possibility of an actual infinity. It seems relatively philosophically secure that there cannot be an א number of things (where א represents the first transfinite number, not to be confused with ∞ which symbolizes infinity taken as a limit rather than a quantity). If there are philosophically sophisticated caveats then so be it, but the point will remain that there are plenty of examples of things for which having an א number of them is clearly (broadly logically) impossible.

Let’s return, for a moment, to Hilbert’s Hotel, because it’s a particularly useful illustration. Suppose that the guest in room 3 checks out, while all the (infinitely many) other rooms remain occupied. The desk clerk decides that they want every room occupied, so they ask each person in room n (where n>3) to move one room over; that is, from room n to room n-1. That will fill up room 3, but the process will also leave no room empty because there is no room number n for which there is not an occupied room n+1. This works equally well for two dimensional shapes, such as circles; remove one ‘point’ from the circumference of a circle and you may have an infinitesimal gap, but simply move every other point along the circumference over (uniformly) by an infinitesimal amount and, voila, the gap is plugged and there will be no new gap. The trick in the case of the Banach-Tarski paradox is to apply the same reasoning to three-dimensional objects. For the best explanation of this paradox I’ve ever seen, (especially for readers who aren’t familiar with it, please make your life better and) check out Vsauce.

Alexander Pruss has noted on his blog that this result “is taken by some to be an argument against the Axiom of Choice.”[2] However, he argues that you can get the same paradoxical result in similar cases (and even in the same case) without the axiom of choice, so that the axiom of choice should be cleared of all suspicions. I agree (though I’m certainly no expert). Richard Feynman is purported to have said, upon being shown the proof, that “it’s fine you can do it with ‘continuous spheres’, since there’s no such thing. The important thing is you can’t do it with oranges, because oranges are made of a finite number of indivisible parts.” I think he is wrong about oranges (being actually comprised of indivisible finite parts, at least if the ‘parts’ are extended in three spatial dimensions), but his sentiment is appreciably insightful nonetheless.

The problem with the paradox, in my submission, is that it divides the surface of the sphere up into points. However, points on a sphere, like points on a line segment, are infinitesimals. This is precisely why (Aristotelians) say that line segments are not composed of points the way walls are composed of bricks, but, instead, points act as the limits between which a line segment is continuously extended. An infinitesimal is a quantity which is infinitely small. It is non-zero, but it is also smaller than any finite quantity. Sure infinitesimals are useful for doing things like infinitesimal calculus, developed by one of my all time favorite philosophers Gottfried Wilhelm Leibniz, but they remain, I believe, nothing more than useful fictions. To borrow a phrase from W.L. Craig;

“They are akin to ideal gases, frictionless planes, points at infinity, and other useful fictions employed in scientific theories.”[3]

If we are to accept the possibility of infinitesimal quantities in reality, then we will quickly run into paradoxes like the Banach-Tarski paradox (which, quite apart from being obnoxious to the rational intellect, seems to violate the law of conservation of matter and energy). Positing infinitesimals is just as paradoxical as positing sets of actually infinitely many discrete things (where ‘things’ is an ontologically loaded term). I am suggesting that infinitesimals are just as paradoxical as actual infinities, and, at bottom, for the same reason(s). In fact, I have this intuition that every argument for thinking that there cannot be any actual infinities (as opposed to potential infinities, where ‘infinity’ merely acts as a limit), admits of a parody for an argument against the existence of infinitesimals. I’m not sure I can rigorously prove it, but I think it’s very plausible.

It seems to me that there’s something conceptually parasitic about infinitesimals relative to infinities. They each conceptually supervene on each other symmetrically. To visualize this symmetry, consider plotting the function ƒ(x)=  1/x which will look like this:

[http://mathworld.wolfram.com/images/eps-gif/AsymptotesOneOverX_1000.gif]

The distance between the curved line and the x-axis (i.e., y=0) as x approaches (positive or negative) infinity is shrinking (or, at least, its absolute value is shrinking), and approaching an infinitely small non-zero measure. When X is infinite, the absolute value of the y-axis coordinate of the curved line (i.e., the distance between the curved line and it’s asymptote, here being the x axis) is infinitesimally small. This example helps to illustrate the point that the concept of an infinitesimal is bound up with the concept of infinity, so that in the absence of one the other would be inconceivable. That at least motivates the suspicion that if one turns out to be metaphysically impossible, so will the other.

What relevance does this have for the philosophy of time? Well, consider that on the A-theory there is such a time as the present. How long does the present last? What, precisely, is its magnitude, its duration? Let’s consider the following argument:

  1. If the A-theory is true, then the present is either infinitesimal in duration, or it is finite in duration.
  2. The present cannot be infinitesimal in duration.
  3. The present cannot be finite in duration.
  4. Therefore, the A-theory is false

Premise 3 can be established with Leibniz’ argument against the (logical) possibility of a physically indivisible element, or ‘atom’ (in the etymologically literal sense). For anything extended in three-dimensional space, however small, it will always be logically possible for me to divine it in two, even if I am physically incapable of doing so (due to some constraint, such as not having the appropriate equipment for the job, or maybe not even being able to develop any tool which could do the job). Physical impossibilities are not (all) logical impossibilities, and logically there is no constraint on how many times I could divide an object extended in space. To say that there is an object extended in space which is not logically possibly divided up into smaller constituent pieces is, according to Leibniz, incoherent. The exact same argument, mutatis mutandis, works against there being chronons (i.e., atomic chunks of time).

The denial of premise 2 is absurd given our observations that positing infinitesimals leads to modally unconscionable paradoxes like Banach-Tarski.

Ways out: I see four ways, not all of them equally viable, for an A-theorist to escape the conclusion of this argument.

First, they could challenge premise 3 on the grounds that, if there are chronons, then by definition they are entities which cannot be physically divided. The suggestion would be that the prima facie absurdity of a Chronon de dicto doesn’t entail the impossibility of a chronon de re. This dangerously dislocates rational intuition from epistemic reliability, but I can imagine extreme empiricists embracing this response.

Second, they could challenge premise 2 by arguing that positing any more than one real infinitesimal of any kind might be problematic, but that there’s no way to derive similar paradoxes from positing a maximum of one infinitesimal. In other words, perhaps paradoxes involving infinitesimals only arise when there are n infinitesimals, where n ∈ ℕ, and n>1. Multiply an infinitesimal by any natural number, or even a transfinite number, and you will still get an infinitesimal result, so it seems harder to show that from one infinitesimal you could derive some kind of contradiction.

Quick thought: Perhaps if there are rules/axioms such as (i) no infinitesimal can be larger or smaller than any other infinitesimal, but (ii) anything (other than 1) to the power of itself is larger than itself, you could derive a contradiction by taking an infinitesimal X, running it through Xx=Y, and then asking whether Y is larger than X, or the same size (it appears to be both). However, I don’t have the kind of facility in mathematics to be able to produce a rigorous proof that even a single infinitesimal would lead to some kind of contradiction or unconscionable paradox. Moreover, it isn’t entirely clear to me what relevance that kind of mathematical paradox would have for the metaphysical consideration at hand. In any case, the second challenge to premise 2 cannot be lightly dismissed.

Third, one could adopt a really wild philosophy of time, such as the Apresentism I wrote about in the last post (thus denying the first premise).

Fourth, one could deny the first premise by adopting what has been called a non-metric view of the present. This is the view preferred by William Lane Craig.[4] I have more than expended my allotted time for blogging and casual writing today, so I will leave this post here for now. I may return to the idea of non-metric present in the (near) future in another post.

[Ha, I don’t presently have time to write more. Get it?]

[1] For fun, check out and try to find the mistake in the following mathematical proof that 1=2 here: https://www.math.toronto.edu/mathnet/falseProofs/first1eq2.html

[2] http://alexanderpruss.blogspot.ca/2013/06/the-banach-tarski-paradox-and-axiom-of.html

[3] William Lane Craig, “Response to Greg Welty,” in Beyond the Control of God: Six Views on the Problem of God and Abstract Objects, ed. Paul Gould (A&C Black, 2014), 102.

[4] See: Craig, William. “The extent of the present.” International Studies in the Philosophy of Science 14, no. 2 (2000): 165-185.

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3 thoughts on “Banach-Tarski paradox, א Infinities, Infinitesimals, and the A-theory

  1. Halloo Tyler!
    I just discovered your very interesting blog/ website.
    I am a Moslem physicist from Jordan, and am very interested in theological discussions, in search of the truth that gives our life meaning. I am somewhat older than you (52 years old) and follow new-ashaari-sunni confession. By new-Ashaari I mean the essence of the theology of Al-Ghazali but believing in the B-theory of time and accepting actual infinities in the creation. I am arguing here exactly against your thesis. If you believe in the B-Theory of time and believe in the eternity of the life in paradise (as all followers of the 3 Abrahamic religions do) the you should believe in actual infinities:

    1. There are either potentially infinite or actually infinite events in paradise
    2. B-Theory is true
    3. Then the events in paradise are actually infinite.

    And if you believe that actual infinities are impossible then you should believe in the A-theory of time:
    1. Actual infinities are impossible
    2. Then God’s creation (the number of created entities) is at most potentially infinite
    3. x is potentially infinite implies there exists a moving measure according to which x is finite at any point of the measure
    4. Then there is a moving measure t, for which the number of created entities is n(t)
    5. This measure, t (call it universal time), has all properties of time according to the A-theory
    6. Then A-theory is correct

    Sorry for this quick input and excuse my bad English.
    I read that you are (were) interested in Islam. I welcome any comment or question.

    Best regards

    Ismail Hammoudeh

    • Thank you Ismail for your perspicuously clear thoughts. In fact, I share your sentiments here, and I’ve been thinking about that particular problem for several years now. Best I can do to answer it is appeal to distinctions: perhaps it is not logically possible to have an actually infinite number of concrete entities, or perhaps it is not logically possible to have a causal chain which is infinitely long. Clearly, there are infinitely many true propositions (indeed, the number of true propositions isn’t even infinite, it is literally indefinitely large, which can’t be quantified at all – set theory literally can’t even deal with it). I’m inclined to think that events aren’t real things in a robust enough sense for them to pose a problem in the same way that penguins and planets would pose a problem. So, perhaps some species of actual infinities can exist, but there cannot be an actually infinite number of concrete objects. However, I’m not sure how much this helps. After all, you may say that if the past were infinitely stretched out, and at any moment God can create a new concrete entity, then it would be logically possible for an actually infinite number of concrete beings to exist if only the past were infinite. That reductio ad absurdum of the past, however, works just as well, on the B-theory, for an infinite future. So, I think we should have to say that there can’t be an actually infinite number of members in a single causal chain. Here, however, the infinite future may pose a problem just in case there can possibly be (temporally) backwards-causation, but perhaps there is still this advantage: if there is an infinite past, then it seems plausible that it follows relatively uncontroversially that the causal story of the world involves an infinitely long causal chain. It certainly wouldn’t follow as uncontroversially from positing an infinite future, even admitting that backwards-causation exists.

      Pruss has addressed concerns in this vein on his blog as well. You may want to check that out: http://alexanderpruss.blogspot.ca/search?q=Actual+infinite

      Thank you for your thoughts again, and I’m sorry that this response came so late. I haven’t been blogging regularly lately (I really should).

  2. Thanks Taylor for your prompt reply and for the Alexander-Pruss-Collection-Link. A lot of interesting material (that seem to be coming in a book soon). I admire Pruss and his line of thinking.
    I also found a paper that you wrote defending the B-theory of time (https://thirdmillennialtemplar.files.wordpress.com/2013/06/the-b-theory-is-true-arguments-from-natural-theology-set-theory-and-epistemology.pdf). I had a quick view and seem to like it a lot. I must study it. But it leaves me somewhat confused as it seemed here that you accept the A-theory.
    But let’s return to your thoughts and my comments:
    1. You wrote: ” I’m inclined to think that events aren’t real things in a robust enough sense for them to pose a problem in the same way that penguins and planets would pose a problem”. By events I mean all actions that take place in the afterlife (e.g. prayers 1 on day 1, prayers 2 on day 2, …). They constitute an actual infinity in the same sense that proponents of the “standard”-Kalaam-argument claim to be impossible. They do exist in the B-theoretic “block-universe” and they do exist in the knowledge of God.
    2. You wrote: ” I think we should have to say that there can’t be an actually infinite number of members in a single causal chain”. I totally agree on the impossibility of this form of actual infinities on one condition: “the chain should be without beginning”. Until one finds a proof that real-number-analysis is logically incoherent I can’t deny the possibility of a continuous causal chain, which in the context of differential equations, given the relevant initial values, has a unique solution (i.e. is fully causally explained). I believe that denying this special form of actual infinities (a beginningless causal chain) is all one needs for the cosmological Kalaam argument (if one accepts the A-theory of time, of course).
    3. About the possibility of backwards-causation I have this to say. I believe that the whole world is the creation of God according to his will. The world may be modelled as the block-universe (space-time). It is fixed (in reality=according to God’s view). You may regard it as a book in which there is a written novel. Some characters of the novel “are conscious” of the world of the novel they live in. They see a dynamic world which is “apparently” causally connected (that is the way the novel is constructed). These apparent causes can be blind “natural laws” (e.g. fire burns paper) or more complicated “supernatural” phenomena (e.g. the force of praying with strong belief). If you see fire burns paper, the casual view (the view of the characters in the novel) is: fire causes the burning (oxidization) of the paper. This is the apparent view. The real view is: God creates the fire, God creates the paper and God creates the burning. It is his will that in most cases fire comes with burning. But in the case of Abraham e.g. fire came with peaceful coolness (apparently a “miracle”). Sometimes one calls this theory of causality “occasionalism”. In any case, according to this view of apparent causes, the causal relation is a correlation that the conscious observer according to some subjective factors (habit, expected action, explanatory power, what is known and what is unknown …), understands as between cause and effect. Normally, this apparent causal relation respects time (or defines time?): the earlier KNOWN influences (“determines”) the later UNKNOWN. But I believe, there are other possible scenarios. Suppose my friend is being operated now in China and that his situation is very critical. Suppose that the operation has just ended now at 3:00 AM but I wake uo only 4 hours later at 7:00 AM. Knowing nothing about the result of the operation I begin to pray sincerely for my friend. I believe this act at 7:00 AM is an apparent “partial” cause for the success of the operation of my friend. I should still add that in the novel world “apparent” causality is for all practical purposes the known CAUSALITY according to which conscious characters with “free will” act, define moral code, … and live (so is the structure of the novel).
    Sorry for the lengthy comment

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