Santy asked:

Were the Ancient Greeks correct in thinking that mathematics is an absolute truth?

Answer by Helier Robinson

Yes, definitely. But you have to understand that the truth that is absolute is mathematical truth, not empirical truth. All the mathematical truths that the ancient Greeks discovered are as true today as they were then — Pythagoras’ Theorem, for example. It does not matter what language a mathematics book might be written in, the mathematical statements proved in it will be true in any other language.

It does not matter what mathematician discovers a new mathematical truth, if he proves it then all other mathematicians will agree that it is true. It may be modified under special circumstances, but is otherwise absolutely true: for example, a right-angled equiangular triangle is impossible in Euclidean geometry, and that is absolutely true; but it is possible in spherical geometry. In spherical geometry, which is geometry on the surface of a sphere, the shortest distance between two points is a great circle; lines of longitude are great circles, as is the equator. One equilateral triangle is formed by the lines of zero longitude, longitude 90 west, and the equator — and this is a right-angled equiangular triangle, possible in this geometry because all triangles in this geometry have their interior angles sum to more than 180 degrees, unlike Euclidean triangles which sum to exactly 180 degrees.

The history of logic is interesting in this context. Formal logic was invented by Aristotle and developed by the medieval philosophers. It is a logic of classes (hence our word ‘classification’). Modern symbolic logic is a logic of truth-functions and, in quantificational logic, a logic of sets — and sets, basically, are classes. All of this is derivative from the logic used by mathematicians, which has never been formalised but which includes sets and functions and valid argument forms such as modus ponens and modus tollens and all the other valid argument forms of formal logic — as well as purely mathematical argument forms such as mathematical induction, and concepts not appearing in formal logic, such as numbers, shapes, equations, and algebraic and topological structures. So if you are interested in reasoning, study mathematics, not logic.