Gradually the level of mathematics that we analyse very carefully gets simpler and ever more elementary, say from continuous functions, to real numbers, to (because of Dedekind cuts) infinite sets, to … and that’s where things more or less stopped in the 1920’s and 30’s.

But with the advent of modern computers, it is gradually becoming clearer that our pure mathematics is actually not that rigorous at all. Terry’s categories, while perhaps useful, hide a painful truth: most of pure mathematics does not work logically if we had to explain it to a smart computer, even though it may seem reasonable to some of us equipped with the correct “intuition”. Unfortunately pretty soon we will have to adjust to the new kid(s) on the block, with zero intuition but loads of relentless computing power. Watch out pure mathematicians: Google and Deep Mind’s Alpha is coming, and it is going to walk all over our intuition, just as it has for the modern Go and Chess players.

]]>“The Methodology of Mathematics”

available as a download from

http://education.lms.ac.uk/the-journal/ ]]>

http://math.huji.ac.il/~ehud/MH/Gauss-HarelCain.pdf

Regarding the first, the author writes:

“This outline of the proof shows that Gauss’s first proof of the FTA

is based on assumptions about the branches of algebraic curves, which might appear plausible to geometric intuition, but are left without any rigorous proof by Gauss. It took until 1920 for Alexander Ostrowski to show that all assumptions made by Gauss can be fully justified.”

Regarding the second, the author writes:

“This proof was purely algebraic and very technical in nature.”

This proof is valid if some extension field of the real numbers is algebraically complete.

Regarding the third, the author writes:

“Gauss’s third proof is easily followed step by step. But how could it be obtained? How was the complicated function y constructed? One can make some reasonable guesses.”

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