Hilbert and Hindsight

I just ran across a quote by Hilbert from 1930:

The real reason for Comte’s failure to find an unsolvable problem is, in my opinion, that an unsolvable problem does not, altogether, exist.

I assume that Comte is Auguste Comte, the sociologist, but I don’t know what remark Hilbert is alluding to. Gödel published his incompleteness theorem in 1931.

9 thoughts on “Hilbert and Hindsight

  1. Maybe Hilbert was talking about Comte’s 1835 statement that humans would never know the composition of the stars? A couple decades later, astronomers used spectrography to figure out that they were mostly hydrogen; this is still cited as one of the most embarrassing moments in the history of philosophy.

  2. and of course, for a platonist, godel’s thm just implies that no set of rules will be sufficient to determine truth. but this does not say anything about human ability to find it, unless you assume humans are algorithms. so it is still possible that all problems can be solved, only not mechanically.

  3. What about this one from
    Notices of the AMS, Vol. 50, #3; The Riemann Hypothesis, J. Brian Conrey, Page 344

    “On one occasion [Hilbert] compared three unsolved problems:
    the transcendence of 2^sqrt(2), Fermat’s Last Theorem,
    and the Riemann Hypothesis. In his view, RH would
    likely be solved in a few years, Fermat’s Last Theorem
    possibly in his lifetime, and the transcendence
    question possibly never. Amazingly, the transcendence
    question was resolved a few years later by
    Gelfond and Schneider, and, of course, Andrew
    Wiles recently proved Fermat’s Last Theorem.”

  4. I believe Comte made his statement, “We can never by any means investigate their [the stars’] chemical composition,” in Cours de Philosophie Positive (1842). Comte died in 1857; Gustav Kirchhoff announced that each chemical element had a different spectrum two years later.

  5. You don’t need a specific quote for this – showing that every problem in mathematics is decidable was a large part of Hilbert’s program in the 1920’s (though perhaps the formal goal was just to show that established mathematics was in fact consistent – completeness just would have been a nice touch).

  6. I found the quote in a book review of L. C. Young’s book on the calculus of variations: not the most authoritative source. That source attributed it to Hilbert’s 1930 speech Naturerkennen und Logik. This article attributes similar sentiments to that same speech. It looks like Scott’s guess is right.

  7. As a treat for mathematical history lovers—who are likely pretty much the only people following this thread—I have uploaded an audio recording of Hilbert’s 1930 lecture, as it was broadcast on the radio. Also uploaded is Hilbert’s famous concluding phrase “We must know, we will know” as a separate (much shorter) audio clip.

    With no preview, I must apologize in advance if the above HTML doesn’t work. And I am not absolutely certain it is the same Hilbert lecture that Scott refers too, but I think so.

  8. Additional information on Hilbert’s speech, including a translation of the above recording, can be found on Prof. James T. Smith’s web site, under “document’s available for browsing.”

    The following translation of Hilbert’s lecture is Prof. Smith’s:

    We must not believe those, who today with philosophical bearing and deliberative tone prophesy the fall of culture and accept the ignorabimus. For us there is no ignorabimus, and in my opinion none whatever in natural science. In opposition to the foolish ignorabimus I offer our slogan: We must know, We will know.

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