Believe it or not, a mathematician has needs similar to yours. He/she needs to discover a problem connected to the existing mathematical culture. Then she needs reassurance and encouragement as she struggles with it. And in the end when she proposes a solution she needs agreement or criticism. No matter how isolated and self-sufficient a mathematician may be, the source and verification of his work goes back to the community of mathematicians.
Sometimes new theories seem to spin out of your head and the heads of your predecessors. Sometimes they’re suggested by real-world subjects, like physics. Today the infinite-dimensional spaces of higher geometry are models for the elementary particles of physics.
Mathematical discovery rests on a validation called “proof,” the analogue of experiment in physical science. A proof is a conclusive argument that a proposed result follows from accepted theory. “Follows” means the argument convinces qualified, skeptical mathematicians. Here I am giving an overtly social definition of “proof.” Such a definition is unconventional, yet it is plainly true to life.
In logic texts and modern philosophy, “follows” is often given a much stricter sense, the sense of mechanical computation. No one says the proofs that mathematicians write actually are checkable by machine. But it’s conventional to insist that there be no doubt they could be checked that way.
Such lofty rigor isn’t found in all mathematics. From one specialty to another, from one mathematician to another, there’s variation in strictness of proof and applicability of results. Mathematics that stresses results above proof is often called “applied mathematics.” Mathematics that stresses proof above results is sometimes called “pure mathematics,” more often just “mathematics.” (Outsiders sometimes say “theoretical mathematics.”)
A naive non-mathematician—perhaps a neo-Fregean analytic philosopher—looks into Euclid, or a more modern math text of formalist stripe, and observes that axioms come first. They’re right on page one. He or she understandably concludes that in mathematics, axioms come first. First your assumptions, then your conclusions, no?
But anyone who has done mathematics knows what comes first—a problem. Mathematics is a vast network of interconnected problems and solutions. Sometimes a problem is called “a conjecture.” Sometimes a solution is a set of axioms!
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In developing and understanding a subject, axioms come late. Then in the formal presentations, they come early.
Sometimes someone tries to invent a new branch of mathematics by making up some axioms and going from there. Such efforts rarely achieve recognition or permanence. Examples, problems, and solutions come first. Later come axiom sets on which the already existing theory can be “based.”
The view that mathematics is in essence derivations from axioms is backward. In fact, it’s wrong.
