Math is a really useful subject—at least, that's what your parents and teachers told you. But math also leads to scenarios, like Zeno's paradoxes, that seem to inspire skepticism.
Is math a realm of timeless, universal truths? Or are mathematicians just making it up as they go? If equations are made up, why are they so useful? We’ll be discussing these questions, and more on this week’s episode, “The Mysterious Timelessness of Math.”
Math is obviously good for many things: we use it for everything from building bridges to designing lasers to predicting the motions of planets to explaining why snowflakes have that odd six-pointed shape. But why is it good for so many things?
Maybe it describes the fundamental structure of the universe. But that makes its methodology look puzzling, on the face of it: how can anyone learn about the fundamental structure of the universe just by scribbling symbols on a whiteboard or a piece of paper? Mathematicians don’t run experiments. They don’t even write down observations about the physical world.
Some would say that there’s no real puzzle here: if math is latching on to deep truths (in particular, the kind that philosophers call necessary truths) then maybe we don’t need evidence to trust it. A claim like “I’m wearing purple socks” isn’t a necessary truth. It’s contingent: even if it happens to be true, it could easily have been false. In order to know whether it’s true, you have to actually check. But for a necessary truth, like 2+2=4, there’s no need to check, because no circumstance could possibly make it false.
But this reply isn’t entirely adequate. We do have ways of checking whether something is a truth of mathematics; mathematicians develop proofs, and try to generate counterexamples to important conjectures. It’s just that those methods typically don’t involve any reliance on physical experiments (although the rise of computers has changed this, making some branches of mathematics increasingly reliant on Monte Carlo methods and computer-assisted proofs).
Another possible answer to the question “how do we know that math describes the fundamental structure of the universe?” is that math works. Math is indispensable for the electronics that brought this blog post to you: if you want to build a computer, you need to understand electromagnetism, and you can’t do that without equations. But this answer doesn’t seem totally adequate either: the indispensability of math might be a good reason to believe that it’s latching on to important truths, but it still doesn’t explain how math is capable of latching onto such important truths.
Another response to the challenge is to say that math doesn’t give us factual information at all. Instead, it’s a useful filing system for organizing what you know—one where you still have to add the information yourself. This raises some worries about arbitrariness (why not pick an organizing system where 2+2= 5, if that’s convenient?), but we can cut down on some arbitrariness by requiring that mathematical systems be internally consistent. Regular arithmetic, where 2+2=4, is not inherently better than mod-3 arithmetic, where 2+2=5; they're both internally consistent systems but with different uses.
I think that reconciling the usefulness of math with its methods is going to take some more philosophical work! I’m looking forward to exploring possible answers with our guest this week, philosopher Arezoo Islami from San Francisco State University.