Wildberger says that Banach-Tarski is Nonsense
Dr. N.J. Wildberger has added a new video to his “Sociology and Mathematics” series in which he discusses the Banach-Tarski Paradox. If you are unfamiliar with this particular concept, it suffices to say that Banach-Tarski illustrates some very peculiar and counterintuitive properties of infinite sets. Fairly unsurprisingly for anyone familiar with Dr. Wildberger’s work, he considers the entire discussion undertaken by Banach-Tarski to be nothing but nonsense. In the video, Dr. Wildberger explicitly notes that he rejects the Axiom of Choice (one of the major axioms upon which Banach-Tarski relies) and I have discussed previously that he also rejects the Axiom of Infinity (which is similarly necessary for Banach-Tarski). Thus, Dr. Wildberger’s video (and his original blog post which inspired the video) seemed fairly curious to me.
Yes, of course the Banach-Tarski Paradox is nonsense if you reject the axioms upon which it depends. Any and every mathematical theorem in existence would be nonsensical to a person who rejected the axioms underlying that theorem.
For example, let’s look at one of the most famous theorems in all of mathematics– one which is so well-known that even people who hated math in school still recognize it by name. Let’s discuss the Pythagorean Theorem. The Pythagorean Theorem is, after all, such a favorite of this author that I literally have it tattooed on my chest. Over my heart. In the original Greek. When I tell people about it, most reply with something like, “Oh, yeah, that’s 
But what if we were to reject the Parallel Postulate?
If you are unaware, the Parallel Postulate is one of the primary axioms of the geometry which everyone learns in High School. As Euclid phrased it, the Parallel Postulate is:
And that if a straight-line falling across two (other) straight-lines makes internal angles on the same side (of itself whose sum is) less than two right-angles, then the two (other) straight-lines, being produced to infinity, meet on that side (of the original straight-line) that the (sum of the internal angles) is less than two right-angles (and do not meet on the other side).
Euclid’s Elements, Book I, Postulate 5. Translation by Richard Fitzpatrick
More simply, this just says that two infinitely long straight-lines which cross each other at some point are not parallel. However, axioms and postulates are supposed to be very simple, self-evident concepts. For example, two of Euclid’s other postulates are “you can draw a straight-line between any two points” and “all right angles are equal to one another.” The Parallel Postulate, notably, is rather complex and seems more like a theorem to be proved from other postulates than a postulate, itself. In fact, many brilliant generations of mathematicians over the course of a couple thousand years were plagued by that exact thought. They struggled, century after century, building one upon another’s work, trying to find some way to prove a Parallel Theorem which could take the place of the Parallel Postulate.
Then, in the 1800’s, we learned that this was impossible. It is entirely logically possible to create an entire, coherent system of geometry in which we reject the Parallel Postulate but maintain Euclid’s other axioms. And in such non-Euclidean geometries, we find out that the Pythagorean Theorem is not true.
So by simply rejecting one axiom of geometry, the Parallel Postulate, we find that one of the most well-known and lauded results in all of mathematical history, the Pythagorean Theorem, is nonsense.
This is absolutely and clearly a direct analog to Dr. Wildberger’s claims about Banach-Tarski. Many, if not all, of the reasons which Dr. Wildberger gives for rejecting the Axiom of Choice and the Axiom of Infinity are just as applicable for someone who wants to reject the Parallel Postulate. They are all impossible to demonstrate empirically. They are all far more complex than a self-evident axiom ought to be. None can be proven from the other assumed axioms in their respective systems. All can be abandoned in other mathematical systems which are still perfectly coherent.
So, yes, if one rejects the axioms upon which Banach-Tarski was built, then it is clear that person will view Banach-Tarski as nonsense. However, if one doesn’t reject those axioms, then there is nothing about Banach-Tarski which is nonsensical or incoherent or self-contradictory. For someone like me, who completely accepts both the Axiom of Infinity and the Axiom of Choice, Banach-Tarski is a perfectly rational theorem, though its implications might have been surprising or counterintuitive at first glance.
With all that said, the reason I decided to comment on Dr. Wildberger’s video isn’t to challenge his view that Banach-Tarski is nonsense. Again, I completely agree that if one rejects Choice and Infinity, then this paradox is nonsense. Rather, I wanted to comment because Dr. Wildberger seems to be inferring that the nonsensicality of Banach-Tarski supports his rejection of Choice and Infinity. However, that would quite clearly be problematic as it would entrench him in a completely circular argument. Rejecting the axioms makes Banach-Tarski nonsensical; and we should reject the axioms because Banach-Tarski is nonsensical. Such a position is clearly fallacious.
So we are left with one of two possible scenarios: either Dr. Wildberger is simply saying that if we reject the axioms on which the theorem is built then the theorem is nonsensical, which is a fairly trivial and irrelevant observation; or else he is arguing that we should reject the axioms because Banach-Tarski is nonsensical without them, which is a fallacious and circular argument. Either way, one is left to wonder at the real point which he is attempting to make with this video.
Boxing Pythagoras 
I think the point is that someone who otherwise finds the Axiom of Choice and other ZFC axioms plausible, yet finds the conclusion of the Banach-Tarski Paradox nonsensical, should call the ZFC axioms into question. If from hypotheses A, B, C, … one derives an absurdity, at least one of the hypotheses must be false.
I believe the fundamental problem with the Banach-Tarski Paradox is not solely that the conclusion is counterintuitive, but that the sets claimed to exist can never be exhibited in any sort of constructive way. As an analogy, suppose some set of hypotheses proved that there exists a root of the Riemann Zeta function with real part greater than 0.51, but the proof gives no possibility of an explicit bound on the imaginary part. Would it not be reasonable to question the hypotheses, given the implausibility of the conclusion and the fact that the actual thing claimed to exist is not possible to exhibit from the argument?
Thank you for taking the time to read and reply! Though, I’m not quite sure I understand what you’re trying to say, so I’m going to ask for some clarification.
What reason would a person have for finding Banach-Tarski to be nonsensical if that person accepts ZFC? Insofar as I can see, Banach-Tarski is perfectly coherent on ZFC. No contradictions are inherent or can be logically inferred from its hypotheses. Are you perhaps using a more colloquial understanding of the word “nonsensical?” In Mathematical Logic, “nonsensical” usually means that something is either undefined or else incoherent, neither of which is the case with Banach-Tarski.
Well, yes, if you take a Constructivist perspective, the Banach-Tarski paradox becomes nonsensical because– again– this involves a rejection of the axioms upon which it relies. To view Banach-Tarski in such a manner, one must reject either the Axiom of Infinity or the Axiom of Choice or both.
I honestly don’t see how this would be analogous. It seems you’re saying that such a proof claims that a root both exists and is undefined, which would be inherently contradictory. In mathematics, for a root to exist, it must be well-defined. If it is undefined, it cannot be the root of a function. What part of Banach-Tarski implies a contradiction on ZFC?
You ask:
What reason would a person have for finding Banach-Tarski to be nonsensical if that person accepts ZFC?
My own point of view is that I view ZFC as probably logically consistent, and so things which it proves to be true which are accessible to computation probably ARE true–but it makes all kinds of statements about things that are not computationally accessible: undefinable real numbers, for example (which, despite the vast majority of real numbers being undefinable, a single one of these can never be exhibited explicitly).
When something like Banach-Tarski comes along making an outrageously counterintuitive existence statement, yet the thing whose existence is claimed can not possibly be exhibited in a way accessible to computation or concrete representation, one might wonder whether the objects in the domain of discourse of ZFC really do have any meaning.
In contrast, the theorem that every positive integer can be represented as the sum of four perfect squares, while counterintuitive, makes a claim that can be verified in as many cases one cares to check. If the theorem were false, it could be disproved by a specific, concrete example. The contrast with the Banach-Tarski decomposition of a sphere is, to my mind, very stark, and this was probably was one factor leading me to question whether ZFC is the ‘right’ foundation for math.
If one is content to be told by a theory that something exists without being able to write it down, calculate with it, or in any way probe its internal structure, then probably one is fine with ZFC, Banach-Tarski, and so on. But what I myself always found most interesting in math are ‘concrete’ objects, things which can be written down and calculated with: numbers, polynomials, integrals, finite graphs, finite groups, explicitly defined functions…and the general theorems about their relationships that can be computationally verified in specific instances. I end up wondering whether Banach-Tarski, the Continuum Hypotheses, and other things of this sort, really belong in mathematics, and whether a theory more grounded in computational meaning, rather than nonconstructive existence statements, can serve as the foundation for mathematics.
So you agree that ZFC is logically consistent, but you are disturbed by the fact that formal systems of mathematics are limited in their ability to describe all the numbers which arise in that system? Why would the limitations of a descriptive language imply that a logically consistent system is inconsistent with reality?
I hardly consider it to be “outrageously counterintuitive,” though I will certainly admit that’s a fairly subjective qualifier. Generally, by the time one encounters Banach-Tarski, that person has already gotten a grip on Hilbert’s Hotel; and Banach-Tarski seems to be a fairly expectable follow-up to that earlier thought experiment, discussing uncountably infinite sets rather than countably infinite ones.
Can it be? Assuming you mean that you can verify the theorem by computationally finding the four perfect squares which add up to any given positive integer, I’m quite certain that this is impossible. For an easy example, Graham’s Number is a finite, positive integer, and yet I will assert that it is computationally impossible for anyone to determine the four perfect squares which sum up to it.
That’s cool. The thing I find most interesting in math is the mathematics of infinities: infinitesimals, cardinalities, non-standard analysis, the peculiarities of operating on infinite sets…
Such a position seems strangely self-contradictory, to me. If what you truly care about is computation, then ZFC already serves as a well-tested and thoroughly successful foundation for applied mathematics. Do you think that a Constructivist foundation to mathematics will somehow do a better job of supporting computational mathematics? It seems to me that any computation which could be performed on such a foundation could just as easily be performed on ZFC, and using the latter doesn’t force us to re-invent the wheel for the vast majority of mathematics.