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— N.H. Abel (1802–1829), quoted from an unpublished source by O. Ore in Niels Henrik Abel, Mathematician Extraordinary, p. 138.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;
> 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?
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).
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.
Your username brings to mind Newton's definition as "Fluxion being the instantaneous rate of change of a Fluent" - https://en.wikipedia.org/wiki/Fluxion This too is intuitive since when you consider something "flowing" its "rate of change" becomes easier to think about.
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.
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.
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.
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-innovativ...
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.
I wouldn't apply the term 'underrated' to him. He was treated poorly by Eddington and shamefully by Gale in Chicago who wouldn't let him lecture on the main campus until Hutchins intervened. He's been the subject of a couple of biographies and has a space telescope named in his honor.
His body of work is impressive but, candidly, I am not sure what of that body had the impact of the white dwarf result. His radiative transfer book is still referenced but I am not certain that's what the Nobel Committee considers, at least in physics. Woodward and Corey won in chemistry for bodies of work but I am not sure that would apply in Chandrasekhar's case of if he would have been offended by a "lifetime achievement" award.
> His body of work is impressive but, candidly, I am not sure what of that body had the impact of the white dwarf result.
But this is precisely my point of contention when i said he was "underrated"; Eddington was such a asshole to the young Chandrasekhar (who was just in his early 20s full of ambition, energy and hardwork) that he sent Chandra into a depression, demoralized him and made him rethink his life's future work. See https://en.wikipedia.org/wiki/Chandrasekhar%E2%80%93Eddingto... for reference.
Listen to what Chandra says in the first aeon animated video titled "Shattering Stars" i had linked to above after the 9-min mark;
"I was in my middle twenties; i had to think about my scientific future. Even if i was right, as i thought i was, the idea that one's scientific life has to be motivated by the off chance that we make a great discovery was too risky; too much of a gamble".
Again after 10:35-min mark;
"If you look at my scientific record, how do i judge it? I think one of the motives of Science is to leave some kind of memorial behind oneself. People can make great discoveries and be remembered for that but there is a more modest role a Scientist can play, to assemble material which will be helpful to others and be of some permanent value; I have chosen that approach; __All i think as a consequence of my first shattering experience in Cambridge__".
If he had not moved to the US i believe the scientific world would have lost him. He explicitly toned down his ambitions in order to do Science. He was a stellar mathematician and was able to marry Relativity and Quantum Theory in his work all at a very young age. There is no telling how great he would have become if he had had the right support and encouragement when he needed it.
> if he would have been offended by a "lifetime achievement" award.
He actually was unhappy that the Nobel committee only recognized his stellar structure studies and not the subsequent work that he did in the next 50 years. As wikipedia states; Chandrasekhar accepted this honour, but was upset the citation mentioned only his earliest work, seeing it as a denigration of a lifetime's achievement. I believe he would very much have valued a "lifetime achievement" award.
I agree that he had to come to the US to do science while dealing with a whole lot of different racial BS when he did. The same Wiki articles and his later lectures say that he considered Eddington a friend despite all the sabotage. The academic establishment was derelict in not backing him but that's common.
In a sort-of similar case, Oppenheimer kept shooting Dyson down when the latter was explaining his synthesis of the different approaches to QED. Bethe had to intervene for him to be allowed to speak. I acknowledge that it is an imperfect analogy since Dyson was as British as they came and there was not the racist angle. I think Chandrasekhar's legacy is assured despite his own misgivings. He is remembered and admired and will be for a long time. Regrets? He had a few but he did it his way.
After all, the Nobel Committee only cited Einstein for the photoelectric effect.
Regrets, I've had a few
But then again, too few to mention
Equated what I had to do
And summed them through without exemption
I planned each tensor'd field
Each manifold along the byway
And more, much more than this
I did it my way
-- Albert "Ol' Brown Eyes" Einstein
(whose daughter, Nancy, would go on to have a 1974 hit, "These Roots are Made for Hawking")
I always contrast Eddington's treatment of Chandrasekhar with G.H.Hardy's treatment of Ramanujan decades before. The latter nurtured a younger man's genius which is what every scientist should do; true genius is too scarce and random and so one must be able to recognize it and fan the flames into roaring discoveries.
As Conan Doyle says in the novel, The Valley of Fear; “Mediocrity knows nothing higher than itself; but talent instantly recognizes genius.”.
Thanks for the discussion and the interesting links. It is unfortunate that so few of his lectures were recorded and that there are only clips of him in his own voice.
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
> 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?
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
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.
> "...study the masters, not their students" —NHA