classifying mathematics

See also the nature of mathematics.

Attempts to classify mathematics among other disciplines have always struck me as bizarre. Science is directly about reality, and has habits and procedures that tend to keep it well-grounded. Philosophy attempts to be about reality, but it is at such an abstract level that it is difficult to keep grounded. Math has its roots in reality; it started with counting and measuring and geometry. But math deliberately distanced itself from its applications to gain generality, and now math is the purely formal theoretical part of anything.

Putting math in the sciences, as if it existed only for its applications, doesn’t make sense. Putting math outside the sciences, and acting as if it were a coincidence that it had applications, doesn’t make sense. (People who do this often feel that it is strange and amazing that this unreal math stuff works in the real world, forgetting that that is why it was invented in the first place.) Sometimes you’ll see math as the trunk of the tree of disciplines, which sort of makes sense. But I would say that formal methods are methods, not a subject; it’s not right to put them into a tree with chemistry and biology somewhere nearby. Math is a binding that runs throughout, the veins in the leaves; or a reusable part with infinitely many uses, a programmable camshaft.

Original version, July 2003.
Updated and added here March 2010.