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Why was the idea of writing, formalising and creating new proofs brought about? For example, why even though we have found hundreds of proofs of the Pythagorean Theorem are we still trying to find more and refine the current proofs that we have?

The motivation for this question was basically to feed my own curiosity, as writing proofs for Trig and Calculus are my favourite parts of math.

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One day in my introductory calculus course, a professor observed that "there are two kinds of obvious statements: those that are true and those that are false".

In physical theories, the golden standard for "truth" is agreement with reality. In math it's not so easy: a million right-angled triangles that satisfy the Pythagorean Theorem do not rule out a right-angled triangle that violates it. Somewhere in the sands of time must have evolved the realization that mathematical statements needed to be proved starting from statements that were either obvious or simply stipulated to be true. This may have originated as a way of settling arguments, as a way of building knowledge (you often learn a lot when you try to prove something, whether it's obvious or non-obvious), as a guard against error, or as an end in itself.

There are of course two kinds of proof: those that are valid and those that are not. Mathematical proofs during history have often been less than rigorous, or had subtle flaws. Sometimes the theorem to be proved was actually false. Sometimes the proof was faulty. Formalization became increasingly necessary so that no obvious step or assumption was left unscrutinized.

As far as the apparent drive to "find more proofs of the Pythagorean Theorem", I think that's the wrong way of looking at it. More often than not, I expect that most of these proofs come about as bi-product of noticing that other proofs, theorems and techniques can lead to new proofs of the Pythagorean Theorem. Often prized are proofs that are short or or beautiful or intuitive or expose unexpected connections with other mathematical ideas.

Circling back to your current favourite parts of math, one interesting category of Trig proofs is the so-called Proof Without Words. And good example of an "obvious" Calculus theorem is the Intermediate Value Theorem, which is what I recall my prof was referring to.

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Proof has ancient antecendents lost to the modern world.

Lets say I said to someone, I can throw a stone fifty yards.

And she goes, prove it.

And so I pick up a stone and throw it.

Then we measure the distance, and its fifty four yards.

So assertion proved.

This is proof by demonstration. And probably the earliest kind of proof developed.

But what has this got to do with mathematical proof? Everything. The notion of proof wasn't developed from nothing in mathematics, but imported from elsewhere. This is why it is wrong to think that the ancient mathematicians of Egypt and Babylonia had no notion of proof. They did. It was simply different from todays. Once an idea is newly imported into a new domain, it evolves in ways that is consonant and peculiar to this new domain. Thus the ancient Greeks did not invent proof by inventing the notion of axioms, they were merely thinking through what proof - an idea already understood - meant in mathematics.

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