I always wondered how people discovered/conjectured the Pythagorean theorem. Was it by empirically measuring the sides of many right triangles?

Or how can someone conjecture the inscribed square problem “Does every plane simple closed curve contain all four vertices of some square?”? I can’t get how people discover those things…

Also, the axioms themselves are experimental facts, why can’t I start with the Pythagorean theorem as an axiom? Or use other axioms?

Or, are transformational proofs in geometry also considered synthetic geometry? (like the one in http://www.google.ru/books?id=qwyBPybT4oMC&printsec=frontcover&redir_esc=y#v=onepage&q=%22examine%20the%20midpoint%22&f=false ,pg 101)

1. When you first learn mathematics, it’s presented as concepts that exist independent of the people who developed them. You *can* do that, and it shortens the lessons, so why not? If Euclid or Newton is mentioned, it’s in a textbook’s sidebar and won’t be on the test.

2. Over time, you gradually get a sense that *people* were involved in creating all this stuff! There are concepts and theorems named after people throughout mathematics. You might hear a few stories, e.g. about the Pythagoreans or Newton vs. Leibniz.

For many people, that’s where it ends.

3. Some more precocious students might read about the history of mathematics on the side, not because it’s required, but because it’s *interesting*! But anyone who gets into upper-level mathematics inevitably begins meeting other people in their field, seeing the human side of mathematics, and learning the history of mathematics, or at least their corner of it.

In the end, one learns that mathematics is a deeply human endeavor, full of interesting people, many of whom are still alive! Some mathematicians even have blogs!

The sad thing, to me, is that the human element of mathematics, the historical anecdotes, the fun of talking about mathematical ideas with other people… are all but stripped out in the early stages of modern math education. Why?