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Best mathematician results

  1. Jul 11, 2015 #1
    Hi Everyone,

    There used to be a very interesting topic here about best scientists ever, and I just calculated the results if anyone is interested. Anyway - the top 10 is here:

    01. Newton - 90
    02. Gauss - 87
    03. Euler - 79
    04. Einstein - 77

    05. Riemann - 36
    06. Archimedes - 29
    07. Feynman - 26
    08. Maxwell - 20
    09. Leibniz - 19
    10. Neumann - 16

    And original topic link here: https://www.physicsforums.com/threads/best-all-time-mathematicians-physicists.53558/
     
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  3. Jul 12, 2015 #2

    Svein

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    Some comments:
    • Einstein was not a very good mathematician. He was a theoretical physicist.
    • Archimedes and Feynman are physicists
    So, ranking mathematicians I would put Riemann at the top, followed by Gauss. From there on it is a matter of preference. Great mathematicians are:
    • Weierstrass
    • Fourier
    • Cauchy
    • Gödel
    • Hilbert
    • Andrew Wiles ( the guy who proved Fermat's last theorem)
    • Boole
    • Cantor
     
  4. Jul 12, 2015 #3

    micromass

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    I don't really agree with this. Archimedes was a top mathematician. Of course he lived in an era where "physics" and mathematics were not really separated. But he did a lot of things which should qualify as pure math. If you're going to classify Archimedes as a physicist, then you should classify Newton as one too.
     
  5. Jul 12, 2015 #4

    Svein

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    Yes. I was going to do so, but I did not know exactly how to describe Newton. I also have the same problem with Leibniz. Possibly both should be called polyhistors (jack of all trades)?
     
  6. Jul 12, 2015 #5

    micromass

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    OK, but haven't almost all mathematicians in history been that? Meddling with physics has been very common under mathematicians, and many mathematics has come from physical reasons. This is a lot less the last 100 - 200 years, but great mathematicians such as Euler and Gauss most certainly did a lot in physics too. So what qualifies ass a mathematician to you?
     
  7. Jul 12, 2015 #6

    Svein

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    To be flippant: Having a fundamental theorem or something named after you. Looking through Ahlfors, I note:
    • Cauchy and Riemann
    • Bolzano and Weierstrass
    • Cantor
    • Fibonacci
    • L. Euler (there were several Eulers)
    Royden adds:
    • Banach
    • Fatou
    • Fubini
    • Hausdorff
    • Lebesgue
    Another book on analysis:
    • Fourier
    • Hilbert
    • Zorn
    Have I missed out on any?
     
  8. Jul 12, 2015 #7

    mathwonk

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    I would rank Riemann and Archimedes highly, having read some of their works. I haven't read a lot of Gauss but what I have impresses me (on number theory and surface geometry). I have read some of Hilbert's work on elementary geometry, and it is wonderful, as well as a little of his algebra; oh yes also Euler's algebra and analysis. I have to stop there, as I don't feel qualified to rank people I haven't read closely. I would list also more recent people I have read, like Oscar Zariski, Herman Weyl, David Mumford, Jean Pierre Serre, and Alexandre Grothendieck. Euclid's Elements also impresses me even if it is due to Eudoxus and others. I have read and greatly enjoyed Einstein, but didn't notice any original mathematics in there. On the other hand Feynman and Ed Witten, nominally physicists, seem to have done some very interesting and original mathematics, even if some of it needed further clarification by others. But that is also true of Riemann.
     
  9. Jul 13, 2015 #8

    mathwonk

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    I think it useful if one gives ones reasons for ranking someone highly, so here is my review of Riemann's collected works, in translation:

    Review of Bernhard Riemann, Collected Papers, translated by Roger Baker, Charles Christenson, and Henry Orde,
    published by Kendrick Press, copyright 2004, 555 pages.

    My father's childhood copy of Count of Monte Cristo is inscribed: “this the best book I ever read,” exactly my opinion of this translation of Riemann's works. After the shock of how good and extensive these works are, by a man who died at 39, one is overwhelmed by his succinct, deep insights. It is amazing no English version of these works has appeared before, and this event should be celebrated by all mathematicians and students who read primarily English.

    This translation contains all but one of the papers I-XXXI from the 1892 edition of Riemann’s works, but not the “Nachtrage”. The translation seems faithful, misprints are few, it reads smoothly, and the translators do not edit or revise Riemann's words, in contrast to the selections in "A source book in classical analysis", Harvard University Press.

    I feared Riemann was obscure, and inconsistent with modern terminology, but once one starts reading, the beauty of his ideas begins to flow immediately. There is no wasted motion, computational results are written down with no visible calculation, and their significant consequences simply announced. This is a real treat. Mysterious statements become a pleasant challenge to interpret, in light of what they must mean. Even outmoded language is clear in context.

    This is a concise and understandable source for subjects that paradoxically are harder to learn from books which expend more effort explaining them. That Riemann omits details, and knows just what to emphasize, make it a wonderful introduction to many topics. Even those I thought I understood, are stripped of superfluous facts and shine forth as simple principles.

    Some highlights for me: "Riemann's theorem" and the "Brill - Noether" number, are both derived on page 99. If L(D) = {meromorphic functions f with div(f)+D ≥ 0}, on a curve of genus g, then dimL(D) - 1 = dim ker[S(D)], where S(D) is a (2g) by (g+deg(D)) “period matrix”. Hence (Riemann's theorem) deg(D)-g ≤ dimL(D) -1 ≤ deg(D), and C(r,d) = {divisors D with deg(D) = d and dimL(D) > r} has a determinantal description = {D: rank(S(D)) ≤ (d-r+g)}.

    Hence a generic curve should have a non constant meromorphic function with ≤ d poles only if d ≥ (g/2) + 1, by the intersection inequality (d-1) ≥ (g+1-d) (= codimension of the rank (d-1+g) locus, in (2g) by (g+d) matrices). The similar estimate (d-r) ≥ r(g+r-d) gives the “Brill - Noether” criterion for C(r,d) to be non empty for all curves of genus g, 16 years before Brill and Noether.

    Eventually one realizes Roch's version of Riemann's matrix represents the map H0(O(D))-->H1(O), induced by the sheaf sequence:
    0-->O-->O(D)-->O(D)|D-->0. In particular the ancients understood and used the sheaf cohomology group H1(O) = H1(C)/H0(K).
    The proof of Riemann's theorem for plane curves, although not algebraic, seems not to depend on Dirichlet's principle, since the relevant existence proof follows by writing down rational differentials. Hence later contributions of Brill - Noether and Dedekind - Weber apparently algebraicize, rather than substantiate, his results.

    Riemann's philosophy that a meromorphic function is a global object, associated with its maximal domain, and determined in any subregion, "explains" why the analytic continuation of the zeta function and the Riemann hypothesis help understand primes. I.e. Euler's product formula shows the sequence of primes determines the zeta function, and such functions are understood by their zeroes and poles, so the location of zeroes must be intimately connected with the distribution of primes!

    More precisely, in VII Riemann says Gauss's logarithmic integral Li(x) actually approximates the number π(x) of primes less than x, plus 1/2 the number of prime squares, plus 1/3 the number of prime cubes, etc..., hence over - estimates π(x). He inverts this relation, obtaining a series of terms Li(x^[1/n]) as a better approximation to π(x), whose proof apparently requires settling the famous "hypothesis".

    In XII, Riemann both defines integrable functions, and characterizes them as functions whose points of oscillation at least e > 0, have content zero. I thought this fact depended on measure theory, but it appears rather that measure theory started here, [cf. Watson in Baker’s bilbiography].

    In XIII, Riemann observes that in physics one should not expect large scale metric relations to hold in the infinitesimally small, a lesson I thought taught by physicists writing 50 years later. Elsewhere he hypothesizes that electrical impulses move at the speed of light, another assumption often credited to early 20th century physicists.

    In VI, he proves a maximal set of non bounding curves has constant cardinality by the “Steinitz' exchange” method, 14 years before Steinitz' birth.

    The translator apologizes for Weber’s inclusion of paper XIX on differentiation of order v where v is any real or complex number, written when Riemann was only 21, but I found it interesting: i.e. Cauchy’s theorem shows that differentiation of order v can be expressed as an integral of a vth power, which makes sense for any v, once one has the Gamma function to provide the appropriate constant multiple.

    I hope this sampling from this wonderful book persuades you to read it for your own pleasure.
     
  10. Jul 14, 2015 #9

    lavinia

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    While this is something of a silly question since science proceeds as a communal enterprise and in aggregate is the product of that community, there is no doubt in my mind that modern mathematical thinking arose with Gauss and Riemann.

    After them there were few mathematicians that the mathematics community views as "first rate". For instance I remember one person calling Hilbert the best of the second rate mathematicians. Cantor is a one off and is never mentioned. Godel was a logician as was Boole.

    In the second half of the 20'th century mathematics blossomed into a Renaissance and the discoveries since World War 2 have dwarfed the mathematical knowledge of all of previous human history. There are many geniuses whose contributions are immortal A mathematician recently told me that there are three mathematicians whom the community names as the best mathematicians of this fertile era. They are John Milnor, William Thurston, and Michael Gromov.

    Mathematics is way harder now than it was in the time of Dirichlet or Euler or Archimedes. Just to embark on research can require a couple of years of preliminary study. A breakthrough in today's world is tour de force compared to Weirstrauss's counter examples or Euler's formulas. Many modern second raters have made more difficult discoveries than the first raters that usually pop into people's minds such as Euler or Laplace. The exception is Gauss and Riemann because they not only made discoveries but they changed the way that we think. They are great on a different level, on the level of Einstein and Kepler.

    Also the modern thinkers are just smarter on average than the people of history. Today, there are relative unknowns who are smarter than Newton.

    Here are a few moderns. Thom, Smale, Chern, Simons, Yau, Hamilton, Atiyah, Wiles, Witten, Kervaire, Siefert, Goldman, Cheeger, Sullivan, Singer,Calabi,Nash, De Ligne, Bott, Morse, Pontryagin, Gelfand, Donaldson,Nirenberg,Lax, Perleman. This list doesn't even scratch the surface.

    For those who wish to see the profundity of modern mathematical thinking try reading Milnor's paper on differential structures on the 7 sphere, or Thom's paper on cobordism, or Sullivan's papers on rational homotopy theory, Simons's and Cheeger's papers on differential extensions on homology theories, or read about the Atiyah Singer Patodi index theorem for elliptic operators, or the Bott periodicity theorem, or Hamilton's theory of Ricci flow. And that is just for starters.

    Here is a link to Milnor's paper on the 7 sphere

    http://www.maths.ed.ac.uk/~aar/papers/exotic.pdf
     
    Last edited: Jul 15, 2015
  11. Jul 14, 2015 #10

    lavinia

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    While I know little of physics or other sciences, it would seem obvious that Einstein stands in Physics much the way Gauss and Riemann stand in Mathematics. Earlier one can point to Newton and Kepler. In some sense Kepler was the first real physicist and Newton was the first to mathematically unify the ideas of his time. In this sense he reminds me of Maxwell. Newton's Law of Gravitation and Maxwell's equations have prior to the modern era been the key analytical breakthroughs.

    After Einstein there seems to be the research in Quantum Theory and Elementary Particles. The giants of this field include Dirac, Heisenberg, Shroedinger, Feynmann, de Broglie, Yang and Mills. But my ignorance is surely leaving out many others.

    Also other Sciences have begun to blossom such as Biology and Chemistry. In these one would have to mention Darwin's Theory of Evolution - perhaps the most important idea after Kepler's invention of Physics and Newton's Laws - Linus Pauling's work on the structure of proteins and of course Watson and Crick's discovery of the structure of DNA.

    My mother and father were both biologists and for them Darwin's theory stands out as the most important intellectual achievement of our culture. After that they point to Watson and Crick.

    My mom, who was also an accomplished photographer, felt that science generally was a product of the community and if one person didn't discover something or define a new theory, someone else would. For her, science was not creative in the individual sense but more in the sense of a collegium of coworkers. She felt that creativity by an individual - that is with that person's unique personal mark - happened only in Art and held Leonardo da Vinci, Memling, and Chekov as the the greatest. She felt that Leonardo wasted much of his life in his scientific investigations and that our culture would be richer if he had spent more time painting. I have often thought about her point of view.
     
    Last edited: Jul 14, 2015
  12. Jul 14, 2015 #11

    disregardthat

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    Lists like these are difficult. But in terms of historic influence, we'd have to put people like Descartes, Archimedes, Newton, Euler, Galois, Riemann, Gauss and Poincare very high. In recent times Grothendieck really stands out in my opinion. I may be biased, but he must be the greatest mathematical genius of the 20th century. I would hesitate at putting Einstein high on a list of influential mathematicians. While surely a good mathematician, he did not revolutionize mathematics like the others.
     
  13. Jul 14, 2015 #12

    lavinia

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    This criterion to me does not work. There are many examples of named theorems or named mathematical objects. But a name doesn't necessarily mean that the mathematician was particularly great - just that the object or theorem was important.

    Here are some other examples: Chern-Simons invariants, Milnor invariants of knots, Thom classes, Pontryagin classes, Stiefel-Whitney classes, Chern classes, Bott periodicity theorem, Weil homomorphism, Siefert-Witten invariants, Hopf Index theorem, Atiyah-Singer- Patodi theorem on Index of Elliptic Operators, Dehn invariants, Bieberbach groups, Abelian groups, Weyl transformations, Dirichlet series, Minkowski space, Calabi-Yau manifold, , Eilenberg Maclane space, Hansche-Wendt manifold,Bernoulli numbers ... It is a very long list.
     
    Last edited: Jul 14, 2015
  14. Jul 14, 2015 #13

    mathwonk

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    i always liked the smith normal form, although it seems to be only a cousin of mine. maybe we could start "Lavinia's paradox".
     
  15. Jul 18, 2015 #14
    Quite interestingly, I do not find the name of Al Khwarzimi, the father of Algebra, and not even Aryabhata in above-mentioned list. Modern mathematics would cease to exist without algebra and the concept of zero.
     
  16. Jul 18, 2015 #15

    Svein

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    Here is a list according to Stephen Hawking ("God Created the Integers"). His list is chronological, not ranked.
    • Euclid
    • Archimedes
    • Diophantus
    • Descartes
    • Newton
    • Laplace
    • Fourier
    • Gauss
    • Cauchy
    • Boole
    • Riemann
    • Weierstrass
    • Dedekind
    • Cantor
    • Lebesgue
    • Gödel
    • Turing
     
  17. Jul 18, 2015 #16

    atyy

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    For evolution, one should include Wallace alongside Darwin.

    For DNA, one should include Franklin and Wilkins alongside Watson and Crick. I should also note that Gosling was a co-author on one of Franklin's key papers. In the broader picture, one would include others like Avery, MacLoed and McCarty. My personal favourite is the beautiful experiment of Meselson and Stahl, who demonstrated a key feature of Watson and Crick's hypothesis for DNA replication, which was based on their structure.

    Of course, there is a famous ethical controversy in the DNA story: http://virtuallaboratory.colorado.edu/Biofundamentals/labs/WhatisScience/section_08.html.

    I thought mathematicians were above such things, but I was amazed to learn about Erdos and Selberg: https://2senxcosx.wordpress.com/2009/06/25/117/.
     
    Last edited: Jul 18, 2015
  18. Jul 18, 2015 #17
    I'm a mathematician, and I think I can say that the listings I have read are incorrect. Here is what most mathematicians would agree :

    Top of the top mathematicians (made a large number of first ranked discoveries) :
    Euclides
    Archimedes (his geometric discoveries are incredible, and he is a top of the top physicist)
    Euler
    Gauss
    Hilbert
    Poincaré
    Grothendieck
    probably other more recent like J.P Serres

    and discutably :
    Fermat (first rank discoveries in differential calculus, variational calculus, probability theory, number theory)
    Gödel (despite he proved only one great theorem)

    First rate mathematicians (made a large number of important discoveries, made one or two extremely important discoveries)
    Diophantes,
    Descartes,
    Newton, (but top of the top physicist)
    Lagrange,
    Cauchy,
    Abel,
    Galois,
    Riemann,
    Nash,
    mendelbrot, and many other

    Second rate mathematicians :
    Appolonius,
    Bernoulli
    Jacobi ... many many other

    Einstein is probably the greatest physicist of all times (only Newton can be compared) : he not only established the relativity, but is also the true initiator of quantic mechanics (no one other than Einstein would have dared to say that electromagnetic waves are quanta of energy, an assertion that was presented as a "youthful error" by Max Planck when he defended Einstein position in the university, and made him win the Nobel Price 20 years later). Furthermore, his discoveries in statistical physic are of first importance, and he was the man that sponsored several young savants like Louis de Broglies which made first ranked discoveries, and which would have felt into oblivion without him.
     
  19. Jul 18, 2015 #18

    Ygggdrasil

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    Part of the importance of the discovery of the structure of DNA was that it helped to solidify the idea that information was encoded in DNA. As Sydney Brenner wrote in a piece for Science about the birth of molecular biology:
    Determining how traits of organisms are encoded in DNA remains a major focus of biology, so in addition to those who helped prove that DNA is the genetic material, I'd also recognize those who contributed to thinking about DNA sequences as information, such as Brenner, Francois Jacob, Jaques Monod, Marshall Nirenberg, and Walter Gilbert, as well as the aforementioned Watson and Crick. (For those interested more in the history of molecular biology, a recent essay on the discovery of mRNA was just published in Current Biology, http://www.sciencedirect.com/science/article/pii/S0960982215006065).

    Now to bring this back to the topic at hand, it's clear (for example in the cases discussed above from biology), that scientific principles are often discovered in parallel by many groups and discoveries do not come from individual scientists working in isolation but from many different teams who sometimes compete with each other and sometimes collaborate. Why such a focus, then, on individuals? Perhaps it would be more useful to come up with a list of the most important ideas in mathematics rather than trying to credit singular individuals with advances that built upon much accumulated knowledge that came before them.
     
  20. Jul 18, 2015 #19

    QuantumCurt

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    It is my opinion that this is an almost pointless discussion to even have. This isn't like running a 1000 meter dash or something where there is ONE person who has done it the fastest. Mathematics is very much a communal effort, and there are people who are utterly brilliant in one area of mathematics but perhaps not so knowledgeable in another area.

    Obviously there are some standouts in history like Riemann, Cauchy, Gauss, Newton, Leibniz, Fermat, Archimedes, Euclid, and many others...but I'm not sure that it even makes sense to try and 'rank' them.
     
  21. Jul 19, 2015 #20

    atyy

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    Why is Euler considered so great? Is it because of exp(ix) = cos(x) + isin(x)?

    Presumably not, otherwise shouldn't Napier be rather regularly mentioned too?
     
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