What does a "pure mathematician" do? A shoemaker makes shoes, a musician makes music, an applied mathematician uses mathematics to solve some real-life problems... Each of these job descriptions have some sort of measurable output. What is such output for a pure mathematician?
Some will say that a pure mathematician solves problems in mathematics, i.e., mathematical problems that are not necessarily related to "real life". This does not do justice to the efforts of a pure mathematician: if you are keen to solve problems, rather solve real-life problems! The problem is that the language in which these "pure" mathematical problems are solved is such that it cannot (always) be used to solve the "real-life" problems. A pure mathematician wants to solve only those problems whose solutions are expressed in a pure mathematical language. This does not do justice to the efforts of a pure mathematician either: what a picky attitude! Besides, solve-a-problem style job description applies to every other job. Indeed, any job for which you expect to get paid requires some sort of problem-solving.
The job description of a pure mathematician is actually quite straightforward. A pure mathematician builds "proofs". A proof is a discussion that reaches a certain conclusion with a life-time guarantee of truthfulness of this conclusion. In no other discipline are you able to establish proofs with such a guarantee. Surely having a certainty in a certain fact is a useful thing in any area of life. Unfortunately though, as soon as your conclusions come close to describing how something in "real life" works, their certainty can no longer be guaranteed, i.e., they step out of the reach of pure mathematics. Still, pure mathematics is extremely useful in establishing the real-life-like close-to-certain conclusions, otherwise the disciplines such as applied mathematics, physics, chemistry, and many others, would hardly make any progress (for those who may not be aware of this, these disciplines, as well as many others, rely a lot on conclusions proved in pure mathematics).
The conclusions that a proof proves are called "theorems". Then there are "definitions", which are essentially shortcuts for building complex proofs. Now a proof starts with certain assumptions (always, in fact, for those who may have been deceived that unlike religion, science does not rely on unproved assumptions, but this is a topic for an entirely different discussion...). The universal assumptions, i.e., those that are used over and over in many different proofs, are called "axioms". Part of the task of a pure mathematician is coming up with appropriate definitions and axioms. In the end, they are to be used in a proof, otherwise, they are useless. Solving a pure mathematical problem is all about finding a proof: of a theorem, its negation, or if the theorem has not been precisely stated, finding a precise statement and then its proof. So fair and square, a pure mathematician is someone who builds proofs!
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