Economists are not generally mathematicians, any more than physicists are. I draw that analogy intentionally because in both cases there are subsets of the areas where the lines get a little bit fuzzy and individuals are doing a bit of both.
Second to your question. The shortest answer is that mathematicians create mathematics.
There are significant distinctions in how they typically think about world (e.g. discrete vs continuous) or what motivates/justifies the work (i.e. pure vs. applied) but underlying it all at the core is the act of creation, and of understanding those creations and how they relate to other things.
Many non-mathematicians use some mathematics routinely as a means to an end, but for mathematicians it is much of the end itself.
A couple of people have commented on the ways in which it is like art, and aesthetics is important. Hardy said "Beauty is the first test: there is no permanent place in the world for ugly mathematics." , and I can't think of mathematician who would disagree. Applied mathematicians tend to take motivation from a problem from somewhere else, but also value aesthetics of the solution.
Indeed! There is a popular saying, "Arithmetic is to mathematics as spelling is to writing." Of course, a mathematician is often good at arithmetic just like a writer is often good at spelling. However, someone merely good at arithmetic is no more a mathematician than someone good at spelling is a writer.
> There is also a running joke that mathematicians tend to be bad at mental calculations. The story about Grothendieck prime comes to my mind: In a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number. "You mean an actual number?" Grothendieck asked. The other person replied, yes, an actual prime number. Grothendieck suggested, "All right, take 57."
Economists are not generally mathematicians, any more than physicists are. I draw that analogy intentionally because in both cases there are subsets of the areas where the lines get a little bit fuzzy and individuals are doing a bit of both.
Second to your question. The shortest answer is that mathematicians create mathematics.
There are significant distinctions in how they typically think about world (e.g. discrete vs continuous) or what motivates/justifies the work (i.e. pure vs. applied) but underlying it all at the core is the act of creation, and of understanding those creations and how they relate to other things.
Many non-mathematicians use some mathematics routinely as a means to an end, but for mathematicians it is much of the end itself.
A couple of people have commented on the ways in which it is like art, and aesthetics is important. Hardy said "Beauty is the first test: there is no permanent place in the world for ugly mathematics." , and I can't think of mathematician who would disagree. Applied mathematicians tend to take motivation from a problem from somewhere else, but also value aesthetics of the solution.