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 , right?” Some will remember even more, that it discusses a property of right triangles which gives a relation between the lengths of the sides of such triangles. Even the few people who remember the name but can’t remember anything about the Theorem still remember it to be a meaningful, true statement of mathematics.
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.