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.”

]]>I think stages 2 and 3 correspond to the conscious competence and unconscious competence stages.

]]>This post was pointed out to me by a reader of my blog, Mr Peter Munro, as a comment to a post about my ongoing troubles (http://www.alexcolovic.com/2016/08/before-olympiad.html#comment-form). Even though I am a chess grandmaster for quite some time, and I can safely put myself in the “post-rigorous” stage, I still find that I am very prone to formal mistakes. This has affected my performances of late, which has also affected my confidence. I think mathematicians are lucky not to have to live in a competitive environment!

]]>I contrast this with an IBL (independent based learning) class I took in number theory. In that course, there was motivation for the problem. There were explanations and excerpts about history that allowed me to gain insight into why someone would prove something a certain way, or define a thing that way.

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