Mathematics consists in deriving consequences (theorems) from a set of assumptions (axioms) by application of given logical rules. The set of axioms mostly used currently is ZFC (Zermelo-Fraenkel-Choice set-theoretical axioms). Axioms and theorems can be formulated in a formal language. ZFC is fairly believable by mathematicians (a typical axiom is ‘there exists an infinite set’). We remind the reader that the consistency of ZFC cannot be proved (this follows from Godel’s incompleteness theorems).This is all true, but it leads people to the conclusion that formal axiomatized math does not really prove what it is supposed to prove, so human math is better.
Human mathematics is based on natural languages (ancient Greek, English, etc.) which can in principle be translated into formal language (but is hardly understandable after translation).
Monday, August 22, 2022
Formal Math v Human Math
David Ruelle wrote and essay on human math, and remarked: