A Gift from the Greeks: The Postulational Method

[BaalChatzaf]

The Greeks invented the axiomatic or postulational method for mathematics. This was first done by Thales (624 - 546 b.c.e) and Pythagoras (572 - 500 b.c.e). Both sets of dates are approximate. The idea of basing a mathematical system on a set of undefined terms and postulates which gave meaning (" the semantics " ) to those terms was perfected by Euclid for his famous system of plane and solid geometry (along with some number theory and theory of proportions derived from the work of Eudoxus). Euclid (323 - 283 b.c.e.) systematized geometry and derived 486 theorems from a small set of axioms (common notions) and postulates (geometric assumptions). His method of proving theorems became the pattern for all subsequent rigorous mathematics even unto our own day.

See

http://en.wikipedia.org/wiki/Euclid%27s_Elements

His system had several gaps and would not be regarded as truly rigorous by contemporary standards but it was version 1.0 of the methodology. Euclid's system was purified of its defects by David Hilbert in 1899 c.e. in Hlibert's -Grudlagen der Geometrie-.

See http://en.wikipedia.org/wiki/Hilbert%27s_axioms

The important thing about the postulate method is that intuition and philosophy do NOT enter into proofs directly. Only the postulates, and definitions based (ultimately) on the undefined terms of the theory are used and theorems are proved by logic, generality conditional (as opposed to syllogistic) logic which is adequately captured by Natural Deduction (see See http://en.wikipedia.org/wiki/Natural_deduction_calculus ) if one were to formalized the logic. Generally the logic is not formalized in most mathematical discourse. Philosophical discourse is external to and separate from proofs which are only allowed to use the postulates of the system and logic. This is not to say the philosophy does not inform the system. A philosophical view is part of means by which the undefined terms are identified and the postulates governing them are formulated. A combination of philosophy and intuition are an integral part of mathematical discovery, but intuition and philosophy are kept out of the process of justification, that is to say proof of theorems. It is the proof process that keeps the mathematics honest.

The result is that a flea circus such as we witnessed on the thread The Opposite of Nothing Is/Isn't Everything, should not occur.