Mathematics as Just a Game

There is a quote attributed to Hilbert that " Mathematics is a game played according to certain rules with meaningless marks on paper." It is perhaps the most cynical characterization of the whole field I have ever encountered.

Pure mathematics, in its modern form, is a race to find prove theorems on the basis of some set of axioms, some set of definitions and a given system of logic. Unlike a science or medicine in which a result must be repeatedly checked against experience and observation (and there is a good career to be had in doing so), in mathematics once a theorem is proven there are only three ways to work with it:

1. To prove another theorem.
2. To get some applied work done.
3. Find another, better way to prove it.

While this last looks on its face like verification, note the term "better"--the new proof must add some new insight, not simply verify as one would accept in physics or medicine. On first glance, Hilbert has a point.

On the other hand, those theorems can describe all manor of objects (and in fact describe anything which meet the hypotheses). One simple example, set theory, works fine for piles of sand or stacks of money. Geometry we see around us all the time. The Navier-Stokes Equations govern, on the applicable scales, the flow of every fluid known to man.

The point here is simple: given a system of logic, a set of axioms, and a set of definitions power sets of theoretical tools for describing not only other theoretical objects but myriad real world systems can be effectively analyzed. While, yes, mathematics for its own sake does resemble a game on paper with meaningless symbols, it is those powerful tools which set it apart.