foundations of mathematics The axiomatic method

Perhaps the most important contribution to the foundations of mathematics made by the ancient Greeks was the axiomatic method and the notion of proof. This was insisted upon in Plato’s Academy and reached its high point in Alexandria about 300 bc with Euclid’s Elements. This notion survives today, except for some cosmetic changes.

The idea is this: there are a number of basic mathematical truths, called axioms or postulates, from which other true statements may be derived in a finite number of steps. It may take considerable ingenuity to discover a proof; but it is now held that it must be possible to check mechanically, step by step, whether a purported proof is indeed correct, and nowadays a computer should be able to do this. The mathematical statements that can be proved are called theorems, and it follows that, in principle, a mechanical device, such as a modern computer, can generate all theorems.

Two questions about the axiomatic method were left unanswered by the ancients: are all mathematical truths axioms or theorems (this is referred to as completeness), and can it be determined mechanically whether a given statement is a theorem (this is called decidability)? These questions were raised implicitly by David Hilbert (1862–1943) about 1900 and were resolved later in the negative, completeness by the Austrian-American logician Kurt Gödel (1906–78) and decidability by the American logician Alonzo Church (1903–95).

Euclid’s work dealt with number theory and geometry, essentially all the mathematics then known. Since the middle of the 20th century a gradually changing group of mostly French mathematicians under the pseudonym Nicolas Bourbaki has tried to emulate Euclid in writing a new Elements of Mathematics based on their theory of structures. Unfortunately, they just missed out on the new ideas from category theory.