Abel, the Mozart of Mathematics

https://www.privatdozent.co/p/abel-the-mozart-of-mathematics

by chmaynard • 2 years ago

112 30 2 years ago

privatdozent.co

0823498723498722• 2 years ago

The most influential words (in translation) of Abel upon me concerned the Pierian Spring:

"...study the masters, not their students" —NHA

rramadass10• 2 years ago

The actual quote is;

It appears to me that if one wants to make progress in mathematics one should study the masters and not the pupils.

— N.H. Abel (1802–1829), quoted from an unpublished source by O. Ore in Niels Henrik Abel, Mathematician Extraordinary, p. 138.

This is my absolute favourite quote too and i always keep this in mind. I think this is very relevant today since it seems every "Tom, Dick and Harry" wants to write a book most of which is mere parroting from other sources with no insight/simplification/intuition whatsoever. It is only the "Masters" who have a way of directly getting to the heart of the matter in the simplest manner possible. To this end i try and collect some of the original texts with detailed explanations so that i can learn from them. Some interesting ones are;

1) A Source Book in Mathematics by David Eugene Smith (Dover Publications) - https://archive.org/details/sourcebookinmath00smit

2) Newton's Principia for the Common Reader by Nobel-prize winning Physicist S.Chandrasekhar - https://archive.org/details/newtonsprincipia0000chan

3) Maxwell on the Electromagnetic Field: A Guided Study (Masterworks of Discovery series) by Thomas Simpson - https://archive.org/details/maxwellonelectro0000simp

4) The Annotated Turing by Charles Petzold - http://theannotatedturing.com/ and https://en.wikipedia.org/wiki/The_Annotated_Turing

lupire• 2 years ago

It seems you prefer to study the masters and the pupils.

barrenko• 2 years ago

"Do not seek to follow in the footsteps of the wise; seek what they sought."

Or from 'Enter the Dragon' - "It's like a finger pointing to the moon. If you focus on the finger, you'll miss all of it's heavenly glory.".

But great essay about Abel, wonderful and somewhat tragic life.

orlp• 2 years ago

It is only the "Masters" who have a way of directly getting to the heart of the matter in the simplest manner possible.

If this is true, why are three of your four references secondary references? Surely if the master said it in the 'simplest manner possible' you would be reading Newton, Maxwell and Turing directly, rather than the paraphrasing/annotating by Chandrasekhar, Simpson and Petzold?

OldGuyInTheClub5• 2 years ago

I am a great admirer of Chandrasekhar but his style of writing is very difficult to learn from. I thought I was common enough to read his Principia exposition but learned I was a few levels below standard.

ngruhn• 2 years ago

Also, surprisingly handsome ^^

BenoitP• 2 years ago

Can someone knowledgeable about this chime in:

The invention of group theory. In proving that there are no general algebraic solutions for the roots of quintic equations, Abel invented (independently of Galois) what later became known as group theory. In addition to Galois, the topic was also studied in the same period by Joseph-Louis Lagrange (1736–1813).

How are quintic equations related to group theory?

cloudvertigo• 2 years ago

You may want to look up Galois Theory. The core idea is to study the equations' roots via their permutation group. If and only if the permutation group of the roots (Galois group) is a Solvable group, the equation has algebraic solutions.

eigenket• 2 years ago

The connection is a field of study today called Galois theory, and especially the "fundamental theorem of Galois theory"

https://en.wikipedia.org/wiki/Fundamental_theorem_of_Galois_...

Roughly speaking what you do is you start with a polynomial over some field, for example over the rational numbers, then you see what you need to add to the rational numbers to get to a field in which you can fully factor that polynomial into linear factors.

For example say we have the polynomial x^2 - 2, we know there isn't any solution to this in the rationals, so we can't factor the polynomial. We then consider the expanded field you get when you add the square root of 2 to the rationals. This expanded field includes root 2, and all products and sums of root 2 with rational numbers. You can check that the elements of this new field look like

a + b sqrt(2)

where a and b are rational. In this new field you can factor the above polynomial as (x + sqrt(2))(x - sqrt(2)).

The connection with group theory comes when you realise that the central object of your study is the bijective (invertable) functions which map this new extended field to itself, while mapping the rationals to themselves. For example for the field formed from the rationals by adding root 2 there are two such bijective functions: the identity function which maps everything to itself, and a second one which swaps root 2 with minus root 2, and leaves everything else the same.

This jump to having to think about these groups of functions (automorphism groups) is a big imaginative leap, but let's you turn hard problems about polynomials into easier problems about groups

vismit2000• 2 years ago

https://news.ycombinator.com/item?id=41255456

rramadass10• 2 years ago

https://math.stackexchange.com/questions/2067201/need-help-u...

data_maan• 2 years ago

I thought there Terence Tao was the Mozart of maths. So confused

https://www.smh.com.au/lifestyle/terence-tao-the-mozart-of-m...

seanhunter• 2 years ago

Yeah. Or maybe Grothendiek. Oh sorry I forgot he’s the Einstein of maths[1]. In general calling people the Mozart of some field is pretty lazy.

[1] https://www.spectator.co.uk/article/the-einstein-of-maths/ in spite of the fact that Einstein probably would have considered himself the Einstein of maths.

gwern• 2 years ago

"Most notably was Gauss at Göttingen, of whose style Abel once wrote:

    “He is like the fox, who effaces his tracks in the sand with his tail.”"

This is incorrect. Abel was reporting someone else's comment, who remains anonymous: https://hsm.stackexchange.com/questions/3610/what-is-the-ori...

Gauss was also not responding to this specific criticism (which he probably never heard); it was a general statement of his, reported in an obituary afterwards:

"Gauss's aim was always to give his investigations the form of perfect works of art. He would not rest sooner and never gave a piece of work to the public until he had given it the perfection of form he desired for it. A good building should not show its scaffolding when completed, he used to say. In his demonstrations he used almost entirely the synthetic method, which he had come to prize through his studies of Archimedes and Newton. It is distinguished from the analytic method by its brevity and comprehensiveness. But the road leading to the discovery remains veiled; and indeed it often seems that Gauss frequently and intentionally turned aside from the road that led to mere instruction." https://archive.org/details/gauss00waltgoog/page/n79 1856

mightyrabbit99• 2 years ago

Comparing him with Mozart isn't really fair. At the very least Mozart was definitely not self taught. Mozart's father is an established musician.

Abel, the Mozart of Mathematics

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