### THE DEVELOPMENT OF THE FOUNDATIONS OF MATHEMATICS

Summary

What mathematicians have perceived as being the foundations of mathematics has changed over the centuries. Many people still think that maths is based on "numbers".

A more sophisticated belief, particularly after the invention of coordinate geometry, was that all of maths can ultimately be described in terms of geometry, and that hence Euclid's geometry represented the foundations of mathematics. The downfall of this view came with the discovery of non-Euclidean geometry in the 19th century, but this then led to a movement to base each individual branch of maths on its own set of axioms.

### Axiomatics

The following branches of mathematics were among those axiomatized:

- Groups

- Rings

- Fields

- Vector Spaces

- Boolean Algebra (the "laws of thought")

- Natural Numbers (Peano's axioms)

- Integers

- Rationals

- Reals (Dedekind's theory of sections or "cuts")

- Set Theory

The development of the last one in this list was closely connected with a further classification of numbers:

- Algebraic and Transcendental Numbers

- Cantor's Theory

**R** is uncountable Proof

- Paradoxes

Algebraic Transcendental Numbers

Foundations of Mathematics

History of Mathematics Module

Links to other History of Mathematics sites

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These pages are maintained by M.I.Woodcock.