5 ms·

Abel, the Mozart of Mathematics

112 points by chmaynard 2y ago

082349872349872 2y ago
The most influential words (in translation) of Abel upon me concerned the Pierian Spring: > "...study the masters, not their students" —NHA
- rramadass 2y 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 — N.H. Abel (1802–1829), quoted from an unpublished source by O. Ore in Niels Henrik Abel, Mathematician Extraordinary, p. 138.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 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 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 https://archive.org/details/maxwellonelectro0000simp 4) The Annotated Turing by Charles Petzold - http://theannotatedturing.com/ http://theannotatedturing.com/ and https://en.wikipedia.org/wiki/The_Annotated_Turing https://en.wikipedia.org/wiki/The_Annotated_Turing
  - lupire 2y ago
    It seems you prefer to study the masters and the pupils.
    - rramadass 2y ago
      Sure, as long as the "pupil" gives me a simple/insightful/intuitive perspective in to "The Master's" work :-)
  - barrenko 2y 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.
    - rramadass 2y ago
      That is exactly what i point out here - https://news.ycombinator.com/item?id=41283361 https://news.ycombinator.com/item?id=41283361
  - orlp 2y 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?
    - rramadass 2y ago
      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 2y ago
        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 2y 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.
    - rramadass 2y ago
      Ha, Ha! Chandrasekhar being a "Master" himself has a definition of a "Common Reader" which is notches above what the public assumes it to be. It is up to us "the students" to rise to the occasion i.e. there is no royal road to anything worthwhile. His book is also written from a different viewpoint where he gives his proof using modern mathematics notation (hence you are already supposed to know Calculus) to Newton's propositions before showing you how Newton himself worked it out in the Principia. My suggestion is to not let this dissuade you from studying it but to consider it as a lifelong journey towards understanding at your own pace. All of the above texts are non-trivial and hence one should not expect an easy endeavour.
      - OldGuyInTheClub 2y ago
        Maybe in the next life! I never expect anything remotely easy when looking at a Chandrasekhar paper or book. Even his public talks were beyond stratospheric. He gave a couple of them at Berkeley in 1976[1]. I was in middle school at the time and went along with my parents. Large, full auditorium for both sessions and looking back, they were given at such a high level that I think even professional physicists would have struggled to follow all of them. His Ryerson lectures at Chicago are similarly highbrow. I stood in line and shook his hand. I remember the gray hair and the immaculate charcoal-gray suit. I don't think he smiled. [1] https://gradlectures.berkeley.edu/series/hitchcock/page/3/ https://gradlectures.berkeley.edu/series/hitchcock/page/3/ [2] https://press.uchicago.edu/ucp/books/book/chicago/T/bo4432943.html https://press.uchicago.edu/ucp/books/book/chicago/T/bo443294...
        rramadass 2y ago
        Subrahmanyan Chandrasekhar was a very unique and rather underrated Physicist due to the then prevalent colonial racism attitudes. He received his Nobel prize some 50 years after he made his discovery when he was just 19/early-20s. His uniqueness lies in his breadth of study which he consciously adopted after being undermined/betrayed by Arthur Eddington on his stellar structure studies. Here is a nice short film with his own words - https://aeon.co/videos/the-indian-astronomer-whose-innovative-work-on-black-holes-was-mocked-at-cambridge https://aeon.co/videos/the-indian-astronomer-whose-innovativ... A longish biography, Nobel at Ninteen, S. Chandrasekhar here - https://www.youtube.com/watch?v=sXTruZkQ7hI https://www.youtube.com/watch?v=sXTruZkQ7hI A short interview with the man himself - https://www.youtube.com/watch?v=n-lJjR7pM7k https://www.youtube.com/watch?v=n-lJjR7pM7k Finally, from https://en.wikipedia.org/wiki/Subrahmanyan_Chandrasekhar https://en.wikipedia.org/wiki/Subrahmanyan_Chandrasekhar ; He wrote that his scientific research was motivated by his desire to participate in the progress of different subjects in science to the best of his ability, and that the prime motive underlying his work was systematization. "What a scientist tries to do essentially is to select a certain domain, a certain aspect, or a certain detail, and see if that takes its appropriate place in a general scheme which has form and coherence; and, if not, to seek further information which would help him to do that". Chandrasekhar developed a unique style of mastering several fields of physics and astrophysics; consequently, his working life can be divided into distinct periods. He would exhaustively study a specific area, publish several papers in it and then write a book summarizing the major concepts in the field. He would then move on to another field for the next decade and repeat the pattern. Thus he studied stellar structure, including the theory of white dwarfs, during the years 1929 to 1939, and subsequently focused on stellar dynamics, theory of Brownian motion from 1939 to 1943. Next, he concentrated on the theory of radiative transfer and the quantum theory of the negative ion of hydrogen from 1943 to 1950. This was followed by sustained work on turbulence and hydrodynamic and hydromagnetic stability from 1950 to 1961. In the 1960s, he studied both the equilibrium and the stability of ellipsoidal figures of equilibrium, and general relativity. During the period, 1971 to 1983 he studied the mathematical theory of black holes, and, finally, during the late 80s, he worked on the theory of colliding gravitational waves. Chandrasekhar was awarded half of the Nobel Prize in Physics in 1983 for his studies on the physical processes important to the structure and evolution of stars. Chandrasekhar accepted this honour, but was upset the citation mentioned only his earliest work, seeing it as a denigration of a lifetime's achievement.
ngruhn 2y ago
Also, surprisingly handsome ^^
BenoitP 2y 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 2y 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 2y 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_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
data_maan 2y ago
I thought there Terence Tao was the Mozart of maths. So confused https://www.smh.com.au/lifestyle/terence-tao-the-mozart-of-maths-20150216-13fwcv.html https://www.smh.com.au/lifestyle/terence-tao-the-mozart-of-m...
- seanhunter 2y 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/ 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 2y 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-original-source-for-abels-quote-about-gausshe-is-like-the-fox-wh 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 https://archive.org/details/gauss00waltgoog/page/n79 1856
mightyrabbit99 2y 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.

112 points by chmaynard 2y ago

The most influential words (in translation) of Abel upon me concerned the Pierian Spring: > "...study the masters, not their students" —NHA

rramadass 2y 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 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 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 https://archive.org/details/maxwellonelectro0000simp 4) The Annotated Turing by Charles Petzold - http://theannotatedturing.com/ http://theannotatedturing.com/ and https://en.wikipedia.org/wiki/The_Annotated_Turing https://en.wikipedia.org/wiki/The_Annotated_Turing
- lupire 2y ago
  It seems you prefer to study the masters and the pupils.
  - rramadass 2y ago
    Sure, as long as the "pupil" gives me a simple/insightful/intuitive perspective in to "The Master's" work :-)
- barrenko 2y 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.
  - rramadass 2y ago
    That is exactly what i point out here - https://news.ycombinator.com/item?id=41283361 https://news.ycombinator.com/item?id=41283361
- orlp 2y 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?
  - rramadass 2y ago
    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 2y ago
      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 2y 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.
  - rramadass 2y ago
    Ha, Ha! Chandrasekhar being a "Master" himself has a definition of a "Common Reader" which is notches above what the public assumes it to be. It is up to us "the students" to rise to the occasion i.e. there is no royal road to anything worthwhile. His book is also written from a different viewpoint where he gives his proof using modern mathematics notation (hence you are already supposed to know Calculus) to Newton's propositions before showing you how Newton himself worked it out in the Principia. My suggestion is to not let this dissuade you from studying it but to consider it as a lifelong journey towards understanding at your own pace. All of the above texts are non-trivial and hence one should not expect an easy endeavour.
    - OldGuyInTheClub 2y ago
      Maybe in the next life! I never expect anything remotely easy when looking at a Chandrasekhar paper or book. Even his public talks were beyond stratospheric. He gave a couple of them at Berkeley in 1976[1]. I was in middle school at the time and went along with my parents. Large, full auditorium for both sessions and looking back, they were given at such a high level that I think even professional physicists would have struggled to follow all of them. His Ryerson lectures at Chicago are similarly highbrow. I stood in line and shook his hand. I remember the gray hair and the immaculate charcoal-gray suit. I don't think he smiled. [1] https://gradlectures.berkeley.edu/series/hitchcock/page/3/ https://gradlectures.berkeley.edu/series/hitchcock/page/3/ [2] https://press.uchicago.edu/ucp/books/book/chicago/T/bo4432943.html https://press.uchicago.edu/ucp/books/book/chicago/T/bo443294...
      - rramadass 2y ago
        Subrahmanyan Chandrasekhar was a very unique and rather underrated Physicist due to the then prevalent colonial racism attitudes. He received his Nobel prize some 50 years after he made his discovery when he was just 19/early-20s. His uniqueness lies in his breadth of study which he consciously adopted after being undermined/betrayed by Arthur Eddington on his stellar structure studies. Here is a nice short film with his own words - https://aeon.co/videos/the-indian-astronomer-whose-innovative-work-on-black-holes-was-mocked-at-cambridge https://aeon.co/videos/the-indian-astronomer-whose-innovativ... A longish biography, Nobel at Ninteen, S. Chandrasekhar here - https://www.youtube.com/watch?v=sXTruZkQ7hI https://www.youtube.com/watch?v=sXTruZkQ7hI A short interview with the man himself - https://www.youtube.com/watch?v=n-lJjR7pM7k https://www.youtube.com/watch?v=n-lJjR7pM7k Finally, from https://en.wikipedia.org/wiki/Subrahmanyan_Chandrasekhar https://en.wikipedia.org/wiki/Subrahmanyan_Chandrasekhar ; He wrote that his scientific research was motivated by his desire to participate in the progress of different subjects in science to the best of his ability, and that the prime motive underlying his work was systematization. "What a scientist tries to do essentially is to select a certain domain, a certain aspect, or a certain detail, and see if that takes its appropriate place in a general scheme which has form and coherence; and, if not, to seek further information which would help him to do that". Chandrasekhar developed a unique style of mastering several fields of physics and astrophysics; consequently, his working life can be divided into distinct periods. He would exhaustively study a specific area, publish several papers in it and then write a book summarizing the major concepts in the field. He would then move on to another field for the next decade and repeat the pattern. Thus he studied stellar structure, including the theory of white dwarfs, during the years 1929 to 1939, and subsequently focused on stellar dynamics, theory of Brownian motion from 1939 to 1943. Next, he concentrated on the theory of radiative transfer and the quantum theory of the negative ion of hydrogen from 1943 to 1950. This was followed by sustained work on turbulence and hydrodynamic and hydromagnetic stability from 1950 to 1961. In the 1960s, he studied both the equilibrium and the stability of ellipsoidal figures of equilibrium, and general relativity. During the period, 1971 to 1983 he studied the mathematical theory of black holes, and, finally, during the late 80s, he worked on the theory of colliding gravitational waves. Chandrasekhar was awarded half of the Nobel Prize in Physics in 1983 for his studies on the physical processes important to the structure and evolution of stars. Chandrasekhar accepted this honour, but was upset the citation mentioned only his earliest work, seeing it as a denigration of a lifetime's achievement.

ngruhn 2y ago

Also, surprisingly handsome ^^

BenoitP 2y 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 2y 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 2y 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_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

data_maan 2y ago

I thought there Terence Tao was the Mozart of maths. So confused https://www.smh.com.au/lifestyle/terence-tao-the-mozart-of-maths-20150216-13fwcv.html https://www.smh.com.au/lifestyle/terence-tao-the-mozart-of-m...

seanhunter 2y 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/ 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 2y 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-original-source-for-abels-quote-about-gausshe-is-like-the-fox-wh 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 https://archive.org/details/gauss00waltgoog/page/n79 1856

mightyrabbit99 2y 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.