To get away from my take-home finals for a bit, and to get this back into a mathy-type of mood I would like to talk about something I’ve been thinking about recently. Does the Banach-Tarski paradox have philosophical significance in philosophy of math?
Recently I gave a talk on a proof of the Banach-Tarski Paradox. What does it say? Well, formally: any bounded subset of Euclidean space is equidecomposable with two copies of itself. More famously, a corollary to this is that the unit sphere can be broken into five pieces that can be rearranged (using rigid motions) into two unit spheres.
Lots of people reject that this is true (I assure you it is, though) due to the fact that it is absurd. You could never actually do this in truth. You can’t break something into a finite number of pieces, then rearrange those pieces to get two of the thing you started with back. My question is: what type of grounds is that to reject something? Who ever said math had to reflect real life? You can’t make a true circle in the mathematical sense, but we still use them. On to the philosophy…
I guess my response to these people is a little troublesome since it puts a slight damper on my view that empiricism should be a perfectly acceptable practice in math (in fact, it secretly is, but don’t tell mathematicians that). Let’s say we wanted to try to prove or disprove the B-T paradox empirically. Well, since we cannot actually do this in practice, we would without a doubt come to the wrong answer that it is false.
What is actually going on then? Why does this work? I think most people reading this already know the answer. The proof of BT uses the Axiom of Choice. Hmm…are we now just converting one philosophical debate to another? Probably. Philosophically BT is only philosophically interesting because of the AC. So now we should ask, should the AC be used or not? That’s for another day. (P.S. the answer is yes, since without it all sets would be measurable).