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Mathematical thinking is actually pretty easy to characterize as thinking that is equipped with a proof checker. Mathematics itself is then just the study of pr
by ouid 5y ago
Mathematical thinking is actually pretty easy to characterize as thinking that is equipped with a proof checker. Mathematics itself is then just the study of proof checkers. Interestingly, this places much of it firmly as a subset of the natural sciences, as proof checkers are physically realizable as compilers (well, certain compilers anyway).
Anything else you bring to the study of mathematics is a heuristic, made admissible by composition with the proof checker. Many of these heuristics seem universal, but they aren't, and it's harmful to assume so.
- Tainnor 5y ago> Interestingly, this places much of it firmly as a subset of the natural sciences, as proof checkers are physically realizable as compilers (well, certain compilers anyway). That's a bold assertion. I think entire books have been written as to what kind of "science" mathematics is, if it even is one, and what the ontological status of mathematical truths is. Regardless of what mathematics truly "is", though, the process of doing mathematics is substantially different from that of doing research in the natural sciences. The arbiter of truth in the latter is experiment, and in particular, inductive reasoning. But experiment and inductive reasoning have no place in mathematical truth (although they may be used to generate hypotheses). And that's leaving aside all of the fundamentally non-scientific questions of mathematical philosophy, such as "is mathematics true/real" or, somewhat more practically, "should we assume the axiom of choice, or the law of excluded middle, or the existence of different kinds of infinities".
- ouid 5y agoEntering a proof into a physical proof checker is mathematics. That's what mathematicians are, after all. That's what publication is, or aspires to be, anyway.