[HN Gopher] Abel, the Mozart of Mathematics
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Abel, the Mozart of Mathematics
Author : chmaynard
Score : 97 points
Date : 2024-08-16 12:38 UTC (2 days ago)
(HTM) web link (www.privatdozent.co)
(TXT) w3m dump (www.privatdozent.co)
| 082349872349872 wrote:
| The most influential words (in translation) of Abel upon me
| concerned the Pierian Spring:
|
| > "...study the masters, not their students" --NHA
| rramadass wrote:
| 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| -- N.H. Abel (1802-1829), quoted from an unpublished source by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 wrote:
| It seems you prefer to study the masters _and_ the pupils.
| barrenko wrote:
| "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 wrote:
| > 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?
| rramadass wrote:
| You understood it wrong.
|
| It is the _Concepts /Models/Ideas/Intuitions_ as given by
| the "Masters" which we need to focus on and not the
| specific language/phrasing they were expressed in (as long
| as they are not relevant) since these are a artifact of
| their Time/Context/Culture. Think of the differences
| between Transliteration vs. Translation vs. Interpretation
| while maintaining fidelity to the original either in a
| different language or the same language. The early
| scientific papers were written in various European
| languages, but in the absence of reading the original paper
| in the original language ourselves (due to not knowing the
| language or the language being too archaic and unfamiliar)
| most of us nowadays study them only in English trusting to
| the translator/author to do their job faithfully.
|
| Two examples;
|
| 1) In the Principia, Newton uses the phrase "Quantity of
| Motion" to define what we call today as "Momentum". The
| former conveys intuition while the latter is merely a
| formula.
|
| 2) "Imaginary Numbers" were called "Lateral Numbers" by
| Gauss which is intuitive in a geometric sense (without
| generalising too much).
| dxbydt wrote:
| Those are 2 good examples. A lot of the old terminology
| is very powerful because it doesn't try to be slick/pc.
| Saying 'variance' is slick but it doesn't convey as much
| as dispersion. Statisticians also speak in terms of first
| raw moment which has a lot of physical meaning behind it
| - saying 'mean' doesn't convey anything. We frequently
| use coefficient of variation in the old texts. It tells
| you right away that it has something to do with
| variation, so you can dig into it and find you are
| normalizing the dispersion by the expectation so
| basically how much variation for a given amount of
| expectation. Instead of saying useless jargon like sharpe
| of vtsax is 0.7 on a 3 year timeframe, you could as well
| have said a coefficient of variation of 1.4 - that tells
| you right away how much dispersion you can expect in your
| return, whereas by saying sharpe you are simply
| gatekeeping and putting william sharpe on a pedestal for
| what is essentially a very silly ratio that statisticians
| have used for centuries without any fuss. I notice that
| in all the Indian news channels, they have translated the
| word monkey and the word pox, and then combined the two
| translations into a compound word, which right away gives
| much more useful information than calling it mpox because
| of dei reasons. So the people speaking Hindi or Tamil or
| any Indian language know right away what the disease
| refers to, whereas saying mpox gives you zero info. The
| next generation will simply be told mpox and think thats
| an actual word.
| OldGuyInTheClub wrote:
| 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 wrote:
| Also, surprisingly handsome ^^
| BenoitP wrote:
| 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 wrote:
| 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 wrote:
| 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
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