Many years ago, when dinosaurs roamed and naked-mouthed humans could mingle with impunity, I was a wide-eyed student working on a bachelor’s degree in physics. Sometime during my final year, I had trouble sleeping. Now, there are many, curricular reasons that a physics student might have a bout of insomnia (many are practically tradition) but this was the only instance in my career (so far) where mathematics alone was responsible for it. This was the day I read about the so-called Banach-Tarski paradox.
I won’t explain the whole thing here for two reasons – it’s already been done well in the Wikipedia page linked above and I still don’t understand enough of the underlying mathematics to do it justice. That said, the crux of the “paradox” is the following: one can take a mathematical object (frequently a sphere) which can be disassembled into parts that make up the whole. These parts can then be rotated and translated in such a way that, when reassembled, they yield two identical copies of the original object. The way this is achieved has to do with certain quirks of set theory and concepts of whether something can be measured.
As far as paradoxes go, it’s not particularly exciting (technically, it’s actually not even a paradox, check out the article). However, as a blossoming physics student, the concepts of conservation of mass and energy had been hammered into my brain, and this paradox had lodged in my brain like a piece of popcorn in the teeth. I actually lost sleep over it; in my view, mathematics was self-inconsistent and broken. How could we rely on mathematics for logical exactness when there was clear proof that it was broken? If math is inconsistent, how can it reliably describe the universe? I now can say that my understanding was incomplete, but at the time I was pretty distraught.
I eventually wound up in my advisor’s office, telling him of my concerns: that it wasn’t worth it to study physics if the descriptions we use are internally flawed, that the logic on which I built all my understanding was flawed. After thinking for a moment, he leaned back in his chair and allayed my fears as only a physicist could. As he put it, while mathematics can deal with infinity, physics takes place within a universe that we know to be granular with objects that are necessarily finite, which we extend to get our concepts of conservation of matter and energy. While I was not completely satisfied that mathematics wasn’t broken, my fears about physics collapsing about our ears had been sufficiently put to rest.
Now, years later, I have come to a slightly more complete understanding of the problem. The so-called paradox is not indicative of a fundamental flaw of mathematics, but rather an interesting quirk that arises from the application of the axiom of choice. Again, I won’t go into the details here (since I don’t understand them well enough to explain them simply), suffice it to say that the axiom of choice is used to cast the mathematical problem of disassembling and reassembling a ball in such a way that uses non-measurable sets. This is significant because it allows the pieces of the ball to be described without volume, and as soon as we abandon the constraint of volume, we get the “paradox”.
The thing that intrigues me most is that the simple choice of how to describe the problem can create or resolve such a glaring paradox. It is a stark reminder that, for all it’s elegance, mathematics is still a human construction that we use to try to understand the universe, and far from perfect or complete. To view it in this way really begs the question of whether something like mathematics can really achieve a truthful description of the universe, but that is a discussion for another post. In my opinion, this sentiment applies to all branches of science and is a beautiful explanation to why we will always need to keep learning and researching.