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).
ZFC is not supposed to prove the consistency of ZFC. It doesn't make sense. Consistency is only proved in a larger system. Goedel's theorems are widely misunderstood.
To hell with this 'human' math nonsense, who else is using it that we know of but us woeful lowly humans? USEFUL not 'beautiful' math is always best.
ReplyDeleteLogic (and math which is a subset of logic) is always at the absolute mercy of the limits of what we know, crippled by what we assume, and truncated by the finite amount of time we have to do it in.
No logic can not produce a damn thing Ex Nihilo beyond these limits. There are no logical process or calculation (casuality) outside of time, as cause and effect are only possible if there is a before and after.