Mathematicians fall into two categories, or stereotypes, depending on your point of view. We are either pure or applied mathematicians. These two types of mathematicians are inherently different from one another. The pure mathematicians are pleased that their work is totally abstract; they are not worried about how or why a particular theorem is helpful in the real world. Think Plato’s perfect forms. These abstract ideals are shared by pure mathematicians. For example Godfrey Harold Hardy once said,
“No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.”
As it turns out, his work was later used for practical applications, but in his heart he was a pure mathematician. I have known some purist who are particularly upset of the later use of their work in applications. A friend of mine is pleased to know her math will never help make a bomb. Although, applied mathematicians don’t all want to make bombs, we all want to solve a real life problem.
Applied mathematicians discover new mathematics to solve a particular real life dilemma. One such applied mathematician was Jacques Salomon Hadamard. Although Hadamard did much work with Number Theory (which at that time had no great applications) he was an applied mathematician. He is quoted to saying:
“Practical application is found by not looking for it, and one can say that the whole progress of civilization rests on that principle”
Applied mathematician suffered through a rigorous 5-6 year Mathematics PhD program which consisted of proofs and logic and rigor just like the pure mathematicians. The purists are no more mathematicians than the applied mathematicians. I happen to be an applied mathematician. And a found a quote which really makes me laugh. This is from Pafnuty Lvovich Chebyshev (1821-1894) who is famous in the world of probability and statistics.
“To isolate mathematics from the practical demands of the sciences is to invite the sterility of a cow shut away from the bulls.”