I recently saw a tweet saying most lay people have zero understanding of what high level mathematicians actually do, and would love to break ground on this one before I die. Without having to get a math PhD.
In: Mathematics
Very broadly, you can classify mathematicians as either applied or theoretical.
Applied mathematicians generally start with real-world problems – like determining the optimal shape of an airplane wing, or predicting the path of a hurricane. They start with real-world measurements and observations, look at how those differ from what the existing math predicts, and help come up with better ways to model the real world using math. Sometimes those new models involve new equations or formulas that can’t be solved using existing techniques, so they figure out techniques to solve them.
Theoretical mathematicians generally start with interesting questions – things we don’t understand about math, even if we’re not quite sure if they’re going to be useful or not. One good way to do that is to generalize a concept. For example, take the factorial function n! = n x (n-1) x … x 2 x 1, for example 5! (“5 factorial”) is 5 x 4 x 3 x 2 x 1. It makes sense to take 5! or 29!, but you can’t take 2.7! – but why not? Some mathematicians wondered whether it was possible to generalize factorial to work for any number, not just whole numbers. It started with just curiosity but now their solution (the gamma function) is quite useful in solving some real-world problems.
Sometimes applied math doesn’t lead to new discoveries. Sometimes theoretical math doesn’t have real-world applications. And that’s okay. Also, the line between applied and theoretical isn’t that clear. There are many mathematicians who do some of both, or work on things that are somewhere in-between.
Whether applied or theoretical, essentially all mathematicians try to come up with new theorems with proofs. Basically they come up with a new mathematical solution to a problem that wasn’t solvable before, and they write a proof that their answer is correct. They publish these in journals and present their findings at conferences. Then other mathematicians can build on their solutions to ask new questions and find new answers. So the total knowledge we have in mathematics keeps growing.
There are some great unsolved problems in mathematics. Many of them are easy to state but despite the work of thousands or even millions of brilliant people, no solution has been found yet. Some of these questions are just curiosities, some of them would potentially unlock all sorts of real technological innovations if they could be solved. However, most mathematicians spend most of their time on less ambitious problems. A lot of mathematicians try to focus their career on an area – often an obscure one – that has lots of interesting questions and few answers so far, maximizing their chances they’ll be able to find a lot of answers.
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