Just like medical doctors there are several different disciplines of high level math. Some of them are more abstract than others. It would be hard to truly describe them all in a simple manner. However the broadest generalization I can make is high level mathematicians use complex math equations and expressions to describe both things that exist physically and things that exist in theory alone.

An example would be, One of the most abstract fields of mathmetics is “number theory” or looking for patterns and constants in numbers. Someone working in number theory might be looking to see if they can find a definable pattern in when primes occur (so far it has been more or less impossible to put an equation to when a prime number occurs).

Now you may ask, “why work on something so abstract and purely theoretical” well sometimes that work becomes used to describe something real. For instance for hundreds of years mathematicians worked on a problem they found in the founding document of math “the elements” by Euclid. One part of it seemed to mostly apply, but their intuition told them something was wrong. Generations worked on this problem without being able to prove Euclid wrong. Eventually they realized the issue. Euclid was describing geometry on a perfectly flat surface. If we curve that surface and create spherical and hyperbolic geometry the assumption Euclid made was wrong, and our Intuition was right. Later we learned we can apply that geometry to how gravity warps space and time. Thus the theoretical came to describe reality.

Solve problems, that’s the essence of it. Some of them can be stated simply, like the Collatz Conjecture (iterate a function: on even numbers, divide by 2; on odd numbers, multiply by 3 then add 1; for any positive integer starting point, does it eventually reach the loop 4-2–1-4-2-1-etc.?), and some of them require more advanced knowledge, like the Reimann hypothesis (do all the non-trivial zeros of the analytic extended Reimann function satisfy Re(z)=-1/2?).

It might not be apparent why these problems are important, but their applications can be hidden in the real world and not known for years or decades or centuries. Fermat’s Little Theorem, for example, is why encryption on your computer works. Or, finding solutions of the Navier-Stokes equation is useful for fluid dynamics, which affects engineering of planes, cars, etc. On the flip side, we might never know if there’s a practical use for the Goldbach Conjecture or the Twin Primes Conjecture, but even if there isn’t there’s still the pursuit of knowledge, applying those methods to other problems.

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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