♾️ 8 — A Window on Infinity

Whether a mathematical proposition is true or not is indeed independent of physics. But the proof of such a proposition is a matter of physics only. There is no such thing as abstractly proving something, just as there is no such thing as abstractly knowing something. Mathematical truth is absolutely necessary and transcendent, but all knowledge is generated by physical processes, and its scope and limitations are conditioned by the laws of nature. One can define a class of abstract entities and call them ‘proofs’ (or computations), just as one can define abstract entities and call them triangles and have them obey Euclidean geometry. But you cannot infer anything from that theory of ‘triangles’ about what angle you will turn through if you walk around a closed path consisting of three straight lines. Nor can those ‘proofs’ do the job of verifying mathematical statements. A mathematical ‘theory of proofs’ has no bearing on which truths can or cannot be proved in reality, or be known in reality; and similarly a theory of abstract ‘computation’ has no bearing on what can or cannot be computed in reality.

So, a computation or a proof is a physical process in which objects such as computers or brains physically model or instantiate abstract entities like numbers or equations, and mimic their properties. It is our window on the abstract. It works because we use such entities only in situations where we have good explanations saying that the relevant physical variables in those objects do indeed instantiate those abstract properties.

Consequently, the reliability of our knowledge of mathematics remains for ever subsidiary to that of our knowledge of physical reality. Every mathematical proof depends absolutely for its validity on our being right about the rules that govern the behaviour of some physical objects, like computers, or ink and paper, or brains. So, contrary to what Hilbert thought, and contrary to what most mathematicians since antiquity have believed and believe to this day, proof theory can never be made into a branch of mathematics. Proof theory is a science: specifically it is computer science. — page 188