Hmm... question for thought:
Without using modern techniques, other than definitions, how could one make the "method of exhaustion" (used by Euclid and Archimedes) rigorous?
I haven't yet answered this question myself, but I was just thinking about it and thought you peeps might enjoy it.
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i think a lot of it comes down to the well ordering principle
actually scratch that, that is for the method of exhaustion used to prove for example that the square root of two is irrational. the method of exhaustion for finding areas is integration
We could try... and you mean like showing that a process terminates using it? But it's more like how does one define an "infinitesimal"? Like how can we make it rigorous?
What does it mean to repeatedly inscribe?
There's an easy definition for a limit (epsilon-delta), but how can we make it useful and rigorous? I think that's the hardest question.
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I mean, the concept is much easier, now that we know calculus, but the case just seems so massively complicated when dealing with it using only elementary operators.
"To those who ask what the infinitely small quantity in mathematics is, we answer that it is actually zero. Hence there are not so many mysteries hidden in this concept as they are usually believed to be. "
I'm no stranger to calculus. I'm looking for a way to *prove* or show the case of the method of exhaustion to be mathematically rigorous using axiomatic, rather than intuitive, principles and definitions.