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Abel's Theorem in Problems and Solutions (2004) [pdf] (ed.ac.uk)
32 points by montrose 9 months ago | hide | past | web | favorite | 7 comments

Note that from the preface these notes are based on a half year high school course. The high school certainly was a specialized math high school. Soviet education was superb.

I’ve perused the notes and they heavily rely on problem solving to learn the material. It seems like a Moore method style of exposition which I really like. Arnold is a master of mathematical writing and teaching.

We no longer teach the formulas for solving cubic and quartic polynomial equations. Nowadays we just teach linear and quadratic equations. For the most part every algebraic equation you can solve by hand that we teach in basic algebra is an equation that can be reduced to a linear equation or a quadratic equation. It’s amazing how many applied problems can be accomplished by this reduction. With the rise of computers and easy numerical computations this isn’t so important but imagine a scenario in which the useful applications all involved degree 5 or higher polynomials. Would we have progressed much?

You ask a trick question. We view and interpret the world through the tools we have (math). Feynman pointed out this absurdity here ( https://www.youtube.com/watch?v=kd0xTfdt6qw&feature=youtu.be... ) The ancients used geometry, classical theory tends to analytical expressions, modern theory is mostly fields, and likely future developments will be computational. The old tools will likely go the way of Greek and Latin and geometry and geography.

>imagine a scenario in which the useful applications all involved degree 5 or higher polynomials. Would we have progressed much?

You can always approximate a polynomial around a point with a lower-degree polynomial. They will only diverge farther out. As a result whenever physics produces a high-degree polynomial we can inspect certain behaviors in a lower degree. This doesn't help with every case, but it does enough for the situation that many questions become answerable.

Oh definitely. Thinking it about it more it’s clear that part of my comment was I’ll thought out. I was attempting to wonder if mathematics would have developed enough theory behind it in a universe in which low degree polynomials were difficult to solve or in which most applicable problems involved high degree polynomials. I think the answer is yes.

What-if questions about the conclusions (as opposed to the axioms) of math being different rarely lead to insight, because they unpin too much. The results mostly depend on whatever else you had to change to keep your primary change from contradicting anything.

If all you changed was physics, the Taylor series thing would put quadratics right back into their position of importance.

I’ll thought out = ill thought out

You must be posting from a phone.


I am! Thanks for catching that mistake.

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