Theorems Versus Computations: Where Is the Mathematical Content?

Continuation of previous post:

There is a striking asymmetry between theorems and computations.

Starting from Peano's axioms, one can generate infinitely many theorems, but perhaps only a relatively small number carry mathematical messages of lasting significance.

Compare this with computation.

There are infinitely many possible computations, but also an enormous number of computations carrying distinct and meaningful information.

A theorem about a differential equation may establish existence of a solution. A computation of that equation can determine what actually happens for particular data. Change the data and a new computation may produce genuinely new information.

The same applies to atomic physics, molecular chemistry, fluid mechanics, weather prediction and engineering design.

In these fields, one general mathematical model can give rise to an unlimited variety of specific problems, each potentially worth computing.

Thus mathematical meaning need not reside primarily in general theorems. It may reside in the specific solutions of mathematical models.

This suggests a fundamental shift in the role of mathematics:

From proving general statements about mathematical objects to computing specific properties of mathematical and physical objects.

AI accelerates both theorem proving and computation, but the consequences are different.

Automated proof production may rapidly exhaust a historically accumulated list of famous open problems. Computational mathematics, on the other hand, faces a virtually inexhaustible supply of new problems arising from science, technology and nature.

Perhaps the future of mathematics lies less in producing more theorems and more in producing reliable computations that tell us something we did not know before.

The essential mathematical question would then be not merely whether something exists, but what it is, how it behaves, and how accurately we can compute it.