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There is a common perception that mathematicians are people who are good with numbers or work with equations all day long. There might be a small amount of truth to this but in general this is a very small part of the life of a mathematician. So what is it that mathematicians really do? In this post I will give a brief introduction to what it means to be a mathematician. 

At the most basic level mathematicians are people who use abstract reasoning to solve complex problems. The closest subject to me in this regard is philosophy. The difference between philosophers and mathematicians is in the type of questions they study. For example, philosophers use pure reasoning to study questions about truth and morality whereas mathematicians use reasoning to study abstract systems. Also, philosophy is more subjective in the sense that philosophers can disagree about  solutions to their questions whereas a mathematical proof is either correct or incorrect.

Mathematicians work with abstract objects. The types of objects varies but some common ones are algebraic, geometric and analytic. Algebraic objects can be thought of as a generalization of the properties of some familiar object like the set of integers (whole numbers). Similarly mathematicians will try to determine all the different types of geometric objects. At the most general level in turns out that this is actually impossible to do. And by impossible I don't mean that nobody has figured out how to do it yet but that nobody ever will be able to figure it out. Analytic objects have to do with things like closeness and limits and are the objects of study in Calculus courses. 

A budding mathematician will spend years in graduate school honing their reasoning abilities and solving mathematics problems and taking rigorous exams to prepare them to write a dissertation. There is no guarantee that the dissertation will ever be completed and somewhere around half of all graduate students in mathematics never graduate with their Ph.D. 

Of those remaining, they choose a problem to work on usually with the help of a professor. This problem must be original and unsolved. That is nobody knows how to solve the problem and nobody knows what the solution will be. Here is where the work of the mathematician begins. Diving in to the unknown guided by their intuition the mathematician begins to try and solve this problem. They might have a guess to what the correct answer will be but they also know in the back of their mind that they could be wrong. 

It is rare at this point for the person to have a moment of insight that leads them to a correct solution. Instead the mathematician works day after day making little to no progress. Usually this process goes on for several years and at then end the person has some sort of partial solution to the problem they were working on. They have made some progress and contributed some original knowledge on their initial question but might not have the complete solution yet. The person then defends the dissertation and graduates.

After this the mathematician may either continue working on this same old problem to try and make further progress or the more adventurous might venture in to new areas of mathematics and try their hand there. Either way their main goal will be to find problems that nobody knows the answer to and continue to solve them throughout their career.   

For the most part the mathematician will use nothing but their brain and pencil and paper to solve these problems. Computers have be used to give proofs for certain types of problems but this is the exception not the rule. I think this is mainly because doing mathematics is a creative pursuit. This is often overlooked because of the way math is taught in schools in a dry and copycat manner. But ultimately a mathematician must come up with original ideas to make any progress.

If you are interested in what a mathematics dissertation looks like you can check mine out at this link: 
https://arxiv.org/pdf/1411.1826.pdf