Wednesday, October 07, 2009

The last Universal Mathematician was ???

The book Prime Obsessions has the following on Page 159:
You will see it written that Hadamard was the last of the universal mathematicians--- the last, that is, to encompass the whole of the subject, before it became so large that this was impossible. However, you will also see this said of Hilbert, Poincare, Klein, and perhaps of one or two other mathematicians of the period. I don't know to whom the title most properly belongs, though I suspect the answer is actually Gauss.
  1. I have never heard Hadamard or Klein mentioned as such, though this does not mean it has not been said.
  2. I would add Kolmogorov to the list of candidates.
  3. My wife would add Aristotle and Archimedes to the list.
  4. Gauss seems like a good choice to me.
  5. Do any current mathematicians qualify? One stumbling block--- I do not know of anyone lately who has made serious contributions to logic and some non-logic branch of mathematics.
  6. It is, of course, an ill defined question. Is there a way to better define it and answer it?
~

28 comments:

  1. Bill -- Do you type up your posts in vi and copy/past them onto Blogger? Guess how I guessed? :-)

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  2. John von Neumann. Serious contributions to both logic and nonlogic branches.

    The mathematician Jean Dieudonné called von Neumann "the last of the great mathematicians." -from wiki.

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  3. To vifan's question: by the ~ at the end.

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  4. BTW: "Prime Obsession" is only one of several Riemann Hypothesis books out there. Du Sautoy's "The Music of the Primes" is both more entertaining and much better at giving a feel for the flow of the mathematics involved, including discussions of recent work that touches on TCS.

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  5. Bill, notice the quote is asking for the last universal mathematician - this is the 'title' in the last sentence. The author is saying nobody after Gauss was universal. Aristotle seems like an odd choice to me - and I am not aware of any mathematical contribution he made outside of logic.

    In any case, Weyl and Kolmogorov seem like good later universalists (at least they both contributed to logic and other fields).

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  6. Gelfand (RIP)

    Drinfeld

    Manin

    Kontsevich

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  7. Serre also comes to mind

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  8. Stephen Smale? Robert Solovay?

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  9. Shaharon Shelah, who is still active, made contributions to both logic and non-logic branches of mathematics.

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  10. They say Thomas Young was the last man to know *everything*.

    Some people who did make serious contributions to logic and non-logic fields: Alfred Tarski, Paul Cohen, John von Neumann.

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  11. You could count Grothendieck's work in topos theory as a serious contribution to logic. He began his career with important work in functional analysis (as a student of Laurent Schwartz), proceeded to revolutionize homological algebra, algebraic geometry and number theory, and during his brief return in the 80's also contributed (among other things) to homotopy theory. He's perhaps as close as anyone to being a modern universal mathematician.

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  12. I'll second von Neumann.

    I think the list "Gelfand, Drinfeld, Manin, Kontsevich" indicates more about the anonymous commenter's interests than truly universal mathematicians. Surely, these were/are great mathematicians, but they are mostly focused on algebraic geometry and the related physics, with the possible exception of Manin.

    As a side note: Hrushovski has done work both in logic and algebraic geometry, and I've heard from several people that he's given the (only?) most interesting application of model theory to algebraic geometry. I have no idea if he qualifies as universal, I just thought it was interesting that model theory can be applied to algebraic geometry that algebraic geometers actually care about.

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  13. GASARCH: "It is, of course, an ill defined question. Is there a way to better define it and answer it?"

    LOL ... in the immortal last words of Monty Python's Sir Robin ... thhhaaatt'ss eeasssyyy!

    That answer to GASARCH's question is another question:

    "Who is the *next* Universal Mathematician?"

    And (more interesting) what new insights into mathematics will she create?

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  14. Back in the day (the 1700s), you could actually know everything. Now we can hardly find someone who understands all of mathematics.
    Some day in the distant future, a blogger will be asking his readers, "Do you think there's anyone today who understands the entire field of planar graph algorithms?"

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  15. come on, we all love von neumann due to the legacies left behind in terms of anecdotes.

    If I had the choice of having the brain capacity of either:
    1)Euler
    2)Gauss
    3)Von Neumann

    I'd still pick von neumann.
    But i don't want to die of bone cancer .... he probably experienced the most painful death eva.

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  16. Remark: Smith is 100% right about the agonizing pain of metastatic bone cancer.

    Also, in von Neumann's era, cancer therapies and pain management techniques were tragically primitive and ineffective .... and regrettably, today they *still* tragically primitive and ineffective.

    ---------------------

    On a more cheerful note, regarding the question "The *next* Universal Mathematician", IMHO it is perfectly feasible to conceive a reasonably specific (albeit highly optimistic) vision of what such a mathematician might work on.

    For starters, this mathematician might embrace Mac Lane and Grothendieck as role models. In Colin McLarty's memorable phrase, these work of these mathematicians "cleared away tangled multitudes of
    individually trivial problems ... puts the hard problems in clear relief and makes their solution possible."

    This is an immense service. The "tangled multitudes of individually trivial problems" can then be attacked in brush-clearing books like Petkovsek, Wilf, and Zeilberger's A=B.

    As for the "hard problems in clear relief", they can be attacked via the Imperial Message of Grothendiek's rising sea: "the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it. . . yet it finally surrounds the resistant substance."

    Of course, for us engineers, the brush-clearing work of Petkovsek, Wilf, and Zeilberger is by far the most important! So from our point of view, the next Universal Mathematician will be a person who thinks in the style of Mac Lane and Grothendieck ... and publishes in the style of Petkovsek, Wilf, and Zeilberger!

    For this to make sense in a broader context, science and engineering have to change greatly. And fortunately, science and engineering *are* changing greatly ... and compatibly with the above changes in mathematics.

    The two traditional branches of science---theory and experiment---are being augmented by two *new* branches of science. One new branch is observational science (long an impoverished Cinderella, now becoming a queen in her own right).

    The other branch is so new that it doesn't even have a name ... some folks call it "simulation-based science and engineering" (SBES) ... but this is a misnomer ... it really should be called "narrative-based science and engineering" ... the point being that conceiving the enterprise narrative is the relatively hard part ... whereas simulating the physics and chemistry is the "A=B" part.

    Is it really the role of mathematicians, scientists and engineers to conceive narratives?

    Reflection on history suggests: "Oh yes ... every past Universal Mathematician has created new narratives ... and this will be true for every future one too."

    From this (exceedingly optimistic) point-of-view, there never will be a "last" Universal Mathematician ... instead we will be inspired and sustained by the narratives that past, present, and future Universal Mathematicians create.

    ---------------------
    ... *Say* ... that must've been a good cup of coffee! :)

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  17. I'd say that Terry Tao is looking very good as a candidate for the next univeral mathematician. His range is far wider than most distinguished mathematicians and he has shown interest in a wide variety of problems.

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  18. Since no less than five people mentioned John von Neumann, perhaps I'll contribute a personal anecdote about him.

    Back around 1976 or so, at the University of Chicago, there was a rather shy person, often to be encountered in the evening hours, who was known to graduate students as Mike Newman.

    Mike ran a small laboratory (in the attic of the Fermi Institute!) focusing on medical imaging and microscopy.

    Mike and I got to be pretty good friends, and one day he mentioned that John von Neumann had taken a keen interest in biomedical imaging and microscopy. "That's news to me," I said "Wherever did you read *that*?"

    "I don't need to read it," said Mike quietly "because I am John von Neumann's brother." This was true, and known to some of the senior UC faculty, who respected Mike's preference for privacy.

    Mike went on to say, passionately, that none of John von Neumann's biographers had ever captured more than a tiny portion of von Neumann's interests, and that furthermore, von Neumann's personality had been far warmer and more humanitarian than the ultra-conservative and militaristic caricatures that some biographers had projected onto him.

    Well, I let the matter drop---it didn't seem all that interesting at the time---and Mike and my relationship continued as before.

    Now I wish that I had asked Mike some questions about his brother!

    Years later, I was able to confirm that von Neumann *had* been interested in biomedical imaging and microscopy (see BibTeX), and this episode largely accounts for my own research interests relating to quantum spin microscopy.

    If there is a point, it is this ... if in the twenty first century we are lucky enough to grow *more* mathematicians like John von Neumann---which IMHO is pretty sure to happen---then nonetheless, only those privileged to work closely with her/him, are likely to have any reliable appreciation of them ... because what appears in mathematical biographies is only largely a fog onto which we project various mythic stories.

    I append, also, a BibTeX reference to Neil Sheehan's recent A Hot Peace in a Cold War, in which von Neumann figures prominently as a "cold warrior" (see chapter 34).

    I have often felt---and even written obliquely per the "Heritage" article also cited below---that if von Neumann had found support to direct his genius toward his 1946 interests in biomedical imaging and microscopy---rather than missiles and thermonuclear bombs---then the history of the twentieth century might have been radically different, and the prospects of our twentieth century, correspondingly more hopeful.

    ---- BibTeX ----

    @book{, Author = {Neil Sheehan}, Publisher = {Random House}, Title = {A Fiery Peace in a Cold War: Bernard Schriever and the Ultimate Weapon}, Year = {2009}}

    @inproceedings{, Author = {J. Von Neumann}, Booktitle = {Proceedings of the Norbert Wiener Centenary Congress, 1994}, Editor = {V. Mandrekar and P. R. Masani}, Pages = {506--512}, Publisher = {American Mathematical Society, Providence, {RI}}, Series = {Proceedings of Symposia in Applied Mathematics}, Title = {Letter to {N}orbert {W}iener from {J}ohn von {N}eumann}, Volume = 52, Year = 1997}

    @article{, Author = {J. A. Sidles}, Journal = {Proc. Nat. Acad. Sci. USA}, Number = {8}, Pages = {2477--2478}, Title = {Spin microscopy's heritage, achievements, and prospects}, Volume = {106}, Year = {2009}}

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  19. There are a number of good candidates (for breadth of accomplishment in Mathematics) who are probably more recent than Von Neumann - although perhaps not as brilliant.

    I nominate for a start: Penrose

    There are probably a few others to also be considered, like Zeeman.

    I would be pleased to think that we have not reached the point where there will be no more last universal mathematicians.

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  20. oh boy, i wished i had known von neumann personally. he must have had an amazing mind. (superior than euler's and gauss's). I have no idea why people still believe gauss tops everyone else ... clearly von neumann.

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  21. Anonymous writes: [von Neumann] must have had an amazing mind. (superior than Euler's and Gauss's)

    Appreciation of von Neumann need not diminish our appreciation of Euler and Gauss!

    An instructive exercise for students struggling to learn scientific writing is to deconstruct Gauss' Disquisitiones generales circa superficies curvas. The full text of Gauss' work is available on Google books both in Gauss' original Latin and in English translation (General Investivations of curved surfaces of 1827 and 1825).

    These articles contain the proof of the theorem named by Gauss himself the Theorema Egregium (wonderful theorem).

    Just as remarkably (IMHO) this work illustrates how the generation of Gauss (and Euler too) invented the narrative style of the modern mathematical literature ("by combining these equations", "it can be shown", etc.).

    Many researchers acquire this now-classical mathematical narrative style of writing by osmosis; for slower students (uhhh ... me?) it is a good learning experience to explicitly deconstruct a dozen or so pages of Gauss' nonpareil example. `Cuz heck, *someone* had to invent that style!

    Do you want to write in the classical style of Gauss? The attached LaTeX style will at cause your typography to appear (broadly) in the classical style of Gauss ... by selecting one of the oldest mathematical fonts (Garamond).

    The resulting typography will perhaps seem familiar for another reason ... a well-known series of books about a boy wizard were set using these fonts! :)

    ---- the LaTeX style "Hogwarts" ---

    % ***********************************
    \RequirePackage{pifont}% loads dingbat fonts, see "psnfss2e.pdf" on how to access these
    \RequirePackage[garamond]{mathdesign} % requires that "garamond" fonts be installed
    \RequirePackage[small,euler-digits]{eulervm} % works well with garamond at "small" size
    \RequirePackage{beramono} % nicer \ttfamily
    \renewcommand{\boldsymbol}[1]{\mathbold{#1}} % to ensure that eulervm fonts are used
    \linespread{1.04} % make the typography more open, Hogwarts style
    % ***********************************

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  22. I'll be sorry to see this delightful thread drop off the front page ... and yet that's what happens eventually to all threads ... and so in closing (on my part) here are a couple of *audio* clips of Universal Mathemicians:

    Hilbert: Wir mussen wissen, wir wollen wissen ("We must know, we will know").

    And von Neumann, rejoicing that his new computer "yesterday ran for four hours without making a mistake!"

    Happy history, everyone! And GASARCH/Lance, my thanks for a wonderfully fun topic! :)

    --- audio links ---

    "http://courses.washington.edu/goodall/MRFM_movies/Hilbert_we_must_know.mp3"

    "http://courses.washington.edu/goodall/MRFM_movies/vonNeumann_short.mov"

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  23. I have to agree with your wife and say Archimedes. After reading the Archimedes Codex it became clear to me how great he really was. I always was taught that the Greeks didn't really understand the concept of infinity and therefore modern ideas of calculus were not figured out till Newtown and Liebnez. Well, it appears that that statement is no longer going to be true.

    Of course, that is a particular bias at this point in my Math career (College Sophmore) so it'll probably change as I learn more in Math and those that helped developed its proofs.

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  24. Thanks for the Hilbert and von Neumann audio clips. What Hilbert actually says is, "Wir müssen wissen, wir werden wissen".

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  25. i have not studied all of his works, but one biography of von neumann wrote that he contributed to all the mathematical fields of his time, except for number theory.

    i think that in many ways his lightning virtuoso style of research is comparable to euler as they both thought 'computationally'. gauss's style was more like a philosopher, i think.

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