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I read somewhere a long time ago that Hilbert once said words (no doubt in German) to the effect that any mathematician worth his salt ought to be able to explain his results to any man in the street. Can anybody tell me where the primary source for this quote is?

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That would be this talk given by Hilbert in 1900 before the International Congress of Mathematicians. (You know, the one where he talked about those problems...) It is not due to him, though; he ascribes it to an old French mathematician he did not explicitly identify.

Here is the appropriate paragraph from that talk:

It is difficult and often impossible to judge the value of a problem correctly in advance ; for the final award depends upon the gain which science obtains from the problem. Nevertheless we can ask whether there are general criteria which mark a good mathematical problem. An old French mathematician said: "A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street." This clearness and ease of comprehension, here insisted on for a mathematical theory, I should still more demand for a mathematical problem if it is to be perfect ; for what is clear and easily comprehended attracts, the complicated repels us.


For reference, here's that part of the talk in the original German:

Es ist schwierig und oft unmöglich, den Wert eines Problems im Voraus richtig zu beurteilen; denn schließlich entscheidet der Gewinn, den die Wissenschaft dem Problem verdankt. Dennoch können wir fragen, ob es allgemeine Merkmale giebt, die ein gutes mathematisches Problem kennzeichnen. Ein alter französischer Mathematiker hat gesagt: Eine mathematische Theorie ist nicht eher als vollkommen anzusehen, als bis du sie so klar gemacht hast, daß du sie dem ersten Manne erklären könntest, den du auf der Straße triffst. Diese Klarheit und leichte Faßlichkeit, wie sie hier so drastisch für eine mathematische Theorie verlangt wird, möchte ich viel mehr von einem mathematischen Problem fordern, wenn dasselbe vollkommen sein soll; denn das Klare und leicht Faßliche zieht uns an, das Verwickelte schreckt uns ab.

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Many thanks. I've wondered about this for a looong time! –  Koilon Sep 2 '11 at 20:08
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Here's the archived version of Hilbert's Gesammelte Abhandlungen. Read 17. Mathematische Probleme, p.290ff. –  t.b. Sep 3 '11 at 6:07
    
Thanks @Theo! I was trying to comb through GDZ but it is somehow slow for me here. I've added a direct link to Mathematische Probleme. –  J. M. Sep 3 '11 at 6:18

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