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John von Neumann wrote the following in his essay The Mathematician:

As a mathematical discipline travels far from its empirical source, or still more, if it is a second and third generation only indirectly inspired by ideas coming from "reality" it is beset with very grave dangers. It becomes more and more purely aestheticizing, more and more purely I'art pour I'art. This need not be bad, if the field is surrounded by correlated subjects, which still have closer empirical connections, or if the discipline is under the influence of men with an exceptionally well-developed taste. But there is a grave danger that the subject will develop along the line of least resistance, that the stream, so far from its source, will separate into a multitude of insignificant branches, and that the discipline will become a disorganized mass of details and complexities. In other words, at a great distance from its empirical source, or after much "abstract" inbreeding, a mathematical subject is in danger of degeneration. At the inception the style is usually classical; when it shows signs of becoming baroque, then the danger signal is up. It would be easy to give examples, to trace specific evolutions into the baroque and the very high baroque, but this, again, would be too technical. In any event, whenever this stage is reached, the only remedy seems to me to be the rejuvenating return to the source: the re-injection of more or less directly empirical ideas. I am convinced that this was a necessary condition to conserve the freshness and the vitality of the subject and that this will remain equally true in the future.

So, did John von Neumann hate pure mathematics that became too abstract? In many ways, he seems to be the complete opposite of G.H Hardy.

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  • $\begingroup$ This and this are both examples of "very pure" mathematical topics. Von Neumann would almost certainly include the first example in the "This need not be bad" case and the second example in his "abstract inbreeding" case. $\endgroup$ Jun 17 '20 at 16:12
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    $\begingroup$ Does not he give an answer himself, in the passage that you cite? What other answer can you possibly hope for? von Neumann theories of unbounded operators, or of operator algebras are quite abstract. But they have a clear origin in physics. $\endgroup$ Jun 17 '20 at 17:01
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    $\begingroup$ There is nothing "opposite to Hardy". von Neumann's theories are much more abstract then those of Hardy, whose many works deal with intuitively clear notion of integer. von Neumann's inspiration came from physics and Hardy's from numbers, this is the only difference. Physics itself became much more abstract in 20th century, and von Neumann contributed to this. $\endgroup$ Jun 17 '20 at 17:08
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    $\begingroup$ @Alexandre Eremenko: von Neumann's theories are much more abstract --- Perhaps even more abstract than unbounded operators (which for me, at least, mostly means we're looking at differentiation operators, but I'm not all that knowledgeable about this) would be von Neumann's conception of and work in continuous geometry. $\endgroup$ Jun 17 '20 at 18:29
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    $\begingroup$ @GEP Halmos wrote about it here. $\endgroup$ Sep 26 '20 at 17:58

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