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Disclaimer:

I am asking this question as an improvement to this question, which should be community wiki. This is in line with the actions taken by Andy Putman in a similar case (cf. meta).

See the relevant meta thread about the previous question.

Edit: If it wasn't already obvious, I only asked this question to prevent the other one (which was not made community wiki) from being reopened.

Question:

The scope of mathematics has grown immensely since ancient times. At what point in time did it become impossible for a single person to understand the majority of mathematics enough to keep current with contemporary research?

Edit: Clarified the wording.

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I have no idea. That's why I put a disclaimer and made it community wiki. –  Harry Gindi Jun 13 '10 at 9:50
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Aren't you ask about who was the last universal mathematician? This is a very subjective question which is asked quite often (see, for example, blog.computationalcomplexity.org/2009/10/…). I would say that Hilbert is officially recognised as such, although I wonder about his serious contributions to geometry. –  Wadim Zudilin Jun 13 '10 at 9:58
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$ \dddot\smile $ –  Wadim Zudilin Jun 13 '10 at 10:20
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Doesn't Terry Tao understand most of mathematics? –  Gerry Myerson Jun 13 '10 at 12:51
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I very much doubt that Professor Tao would object if I disagreed with you =). –  Harry Gindi Jun 13 '10 at 12:59
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2 Answers

up vote 8 down vote accepted

The world's output of scientific papers increased exponentially from 1700 to 1950.

One online source is this article (which is concerned with what has happened since then). The author displays a graph (whose source is a 1961 book entitled "Science since Babylon" by Derek da Solla Price) showing exponential increase in the cumulative number of scientific journals founded; an increase by a factor of 10 every 50 years or so, with around 10 journals recorded in 1750.

Perhaps someone can locate similar statistics specific to mathematics, but it's reasonable to expect the same pattern. If so, it is a long time since any individual could follow the primary mathematical literature in anything close to its entirety.

But then, gobbling papers is not how leading mathematicians (or scientists) actually operate.

By making judicious choices of what to pursue when, and with sufficient brilliance and vision, it is possible even today to make decisive contributions to many fields. Serre has done so in, and between, algebraic topology, complex analytic geometry, algebraic geometry, commutative algebra and group theory, and continues to do so in algebraic number theory/representation theory/modular forms.

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Indeed, if you want general background, reading the research literature would be a last resort. The sheer number of papers has a great deal to do with library budgets, also. The remark that growth is exponential does support the idea that the change might come rather suddenly. (By the way Cartier once said something like "Serre has no idea what a Laplacian is", which is not to be taken literally but an indication of non-universality in a very versatile algebraist.) –  Charles Matthews Jun 13 '10 at 17:57
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I don't think we need complete statistics to see that it became impossible to keep up with the mathematical literature by the mid 18th century. Euler alone produced work that took nearly a century for others to notice. For example, his 1751 discovery of addition theorems for elliptic integrals did not bear fruit until Jacobi picked the idea in the 1820s. And by then the mathematical community had to catch up with the work of Lagrange, Gauss, Abel, Jacobi ... (and they hadn't even noticed Galois yet). –  John Stillwell Jun 13 '10 at 23:29
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At some point between Harald Bohr's foundation of the theory of almost periodic functions, and the major paper of van der Corput that J. E. Littlewood regarded as the most technical paper in the whole of mathematics so far. In other words some time round 1915, or when classical analysis ceased to be a comfortable unifying and central area of graduate mathematical education (so that reading a good Cours d'Analyse would set you up for research), and became a bunch of technically-refined areas for specialists. This is also the period when the Lebesgue integral became a necessity, and when Brouwer had first provided needed foundations for topology (e.g. simplicial approximation). The generation of universalist wannabes that followed Poincaré and Hilbert would include names such as Weyl, von Neumann, Weil and Kolmogorov. But you can see from that list that such great talents have already "spread out", not trying to comment on everything.

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The question doesn't ask that one be able to understand mathematics at a level high enough to engage in research. It asks at what point it became impossible to understand the majority of the mathematical enterprise at a high enough level to understand a paper published in (almost) any subject. What you've said doesn't really answer the question, since much of the algebra developed in the late 19th century, as well as things like Galois theory and number theory are not really covered at all by an understanding of classical analysis. –  Harry Gindi Jun 13 '10 at 10:08
    
So, wouldn't a better way to think about this would be to say, identify the time where the average thing you learn in current grad school was no longer cutting-edge research? Then, that would roughly indicate where a generic education would be insufficient and require significant specific study in a particular area, to the exclusion of the others? –  jeremy Jun 13 '10 at 10:16
    
If you say so! –  Harry Gindi Jun 13 '10 at 10:19
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Well, I answered the question as I understood it, with the word "most" of mathematics interpreted as it would have seemed to a mathematician in the year X. In those days all the important guys could meet at the ICM. I think it is incorrect to "read back" our understanding to the 1910s, and I came to this view by reading a solid history of mathematics through, hitting Harald Bohr as a "pain barrier". Algebra wasn't the major issue right then: arguing that in our terms they should have felt it was risks anachronism. Of the Hilbert problems only 14 and 17 are close to our abstract algebra. –  Charles Matthews Jun 13 '10 at 10:29
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I see. Yes, I just realized that the way it was written before could be interpreted in two ways depending on whether you applied the modifier "at the level of contemporary research" to the word "understanding" or to the word "mathematics". –  Harry Gindi Jun 13 '10 at 11:54
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