Mathematical Proof and the Principles of Mathematics/History

Missing from many books on mathematics is material on the history and development of subject covered. But not only does this do a disservice to the subject, leaving the impression that it's dry and academic, but it does a disservice to the reader in that it removes some of the potential avenues of understanding for a subject that may already be difficult.

Mathematics did not spring into being complete with proofs for every theorem and free of logical errors. The road to reach that stage was long, difficult, and full of wrong turns and surprises. Modern standards of mathematical proof mostly developed in the last 300 years, but the journey really started on the Greek peninsula some two and a half millennia ago.

Chapter summary

A history of mathematics in general would take up many volumes, but what what we're mainly concerned with here is are the events that to the standards of mathematical rigor currently in common use. We start with Euclid who set a standard of mathematical proof that was not greatly improved upon for around two thousand years. During that time mathematics as a whole increased tremendously in scope, and eventually it became clear that a new standard would be needed if mathematics was to continue to grow while maintaining its prestige as the most logical of the sciences. The result was a new subject, the mathematics of mathematics itself, or metamathematics. This new subject was developed extensively in the 20th century, but yielded some results which seem to go against common sense, so the journey continues on.