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I have tried vainly to understand the significance of the following statement attributed to David Hilbert:

The theory of complex multiplication is not only the most beautiful part of mathematics but also of the whole of science.

as quoted by J.S.Milne at the beginning of Sec.15 'Complex Multiplication for Elliptic Curves' in his book "Modular Functions and Modular Forms'. Can anyone throw some light on this comment, especially its second part?

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    $\begingroup$ I don't think he's comparing it to specific areas of physics or science, he's just emphasizing how strikingly beautiful it is (which it is :-) ). $\endgroup$
    – David Corwin
    Commented Nov 20, 2012 at 3:14
  • $\begingroup$ You could always ask Milne. My impression is that he's quite receptive such queries. $\endgroup$
    – Ramsey
    Commented Nov 20, 2012 at 3:46
  • $\begingroup$ @Ramsey Thank you for the suggestion. Yes Milne welcomes that. $\endgroup$
    – kks
    Commented Nov 20, 2012 at 4:10
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    $\begingroup$ I cast the final vote to close; in some sense the answer that he said it at a talk of somebody having just written on it sheds some light on the significance/context. But then as a general remark, well, people say subjective/emotional things, if they are famous some of them get preserved. And then it is not even so surprising. After all, it is a 'theorem' (well, a Satz) of Gauss that mathematics is the queen of science and nummber theory the queen of mathematics. So, this seems like some obvious corollary. :) $\endgroup$
    – user9072
    Commented Nov 20, 2012 at 11:04
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    $\begingroup$ Are you sure that Hilbert was talking specifically about complex multiplication for elliptic curves rather than the theory of complex numbers in general? I remember that Sir Michael Atiyah once gave a talk in which he started with the question "What was the greatest advance in mathematics in the last 1000 years?" and spent the rest of the talk defending his answer: The discovery of complex numbers. $\endgroup$
    – Robert Bryant
    Commented Nov 20, 2012 at 13:29

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On page 5 of his notes Complex Multiplication, J.S.Milne ( http://www.jmilne.org/math ) gives the following footnote:

As quoted by Olga Taussky in her obituary for Hilbert in Nature, 152 (1943), 182–183.

The following is from a letter she sent to me in October 1990:

"Yes it is true, Hilbert said this and I was in the audience when he said it and I was pleased he said it. It was at the Mathematiker Kongress Zürich 1932. Fueter ... had written an opus in 2 volumes: Vorlesungen über die singulären Moduln und die komplexe Multiplikation der elliptischen Funktionen, Teubner, 1924, 1927. Hilbert presided at Fueter’s lecture."

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  • $\begingroup$ If you want italics, wrap the text in * (asterisks). $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Nov 20, 2012 at 4:10

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