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16 votes

Why did Hilbert believe consistency implies existence?

Hilbert wrote this observation some decades before the formal theories of completeness (and incompleteness) took shape, so you can't expect him to be overly precise with regard to such notions. The ...
Mikhail Katz's user avatar
  • 6,131
11 votes

Source for Hilbert's famous quote "Mathematics in Göttingen? There really is none anymore"

In my grandfather's book, lately translated from German: Recollections of a Jewish Mathematician in Germany, by Abraham A. Fraenkel, edited by Jiska Cohen-Mansfield, translated by Allison Brown. ...
Avraham Fraenkel's user avatar
11 votes
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What were Hilbert's weekly 1933 lectures on "matters of general intellectual interest" about?

Here is MacLane's sentence about the lectures in its entirety: "Although the leading mathematician, David Hilbert, had retired and came only once a week to lecture on matters of general ...
Conifold's user avatar
  • 77.6k
9 votes
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Is there any reason why Hilbert's problems are ordered in the way they are?

Yes, they are ordered by the broad areas of mathematics. If you read his paper, you will see that he hints this. 1,2 - foundations, set theory. 3,4, and 5 to some extent - foundations of geometry. ...
Alexandre Eremenko's user avatar
9 votes

Hilbert's problem list did not include Fermat's last theorem. Why?

Hilbert included in his list a much more general Problem 10: Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process ...
Alexandre Eremenko's user avatar
9 votes

Was Hilbert ambivalent about set theory?

No, he was not, as one can see from the full passage from Hilbert's lecture On the Infinite delivered June 4, 1925, before a congress of the Westphalian Mathematical Society in Munster, and published ...
Conifold's user avatar
  • 77.6k
9 votes

Are there any records that show how Hilbert came to "invent" or "discover" Hilbert spaces?

The notion of Hilbert space comes from Hilbert's theory of integral equations. Of course, it was partially motivated by physics, by the theory of oscillations in classical mechanics, but this theory ...
Alexandre Eremenko's user avatar
8 votes

Why did Hilbert believe consistency implies existence?

I just want to add some context to Katz's very nice answer. Hilbert's work on foundations occurred in the aftermath of the intuitionistic criticisms by Brouwer and Weyl. (Weyl in particular must have ...
Michael Weiss's user avatar
7 votes
Accepted

Are there any records that show how Hilbert came to "invent" or "discover" Hilbert spaces?

From Steen’s paper mentioned in a very similar MO question just yesterday: Hilbert himself was astonished that the spectra of his quadratic forms should come to be interpreted as atomic spectra. “...
Francois Ziegler's user avatar
4 votes
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Why did Einstein get a better hearing in Göttingen than Berlin?

There were a number of chemists in the Prussian Academy of Sciences, most notably Nernst and Fischer. They had hoped that Einstein would engage in atomic theory and bring about a better understanding ...
Jan Peter Schäfermeyer's user avatar
4 votes
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Hilbert's problems

Contrary to what you say, Hilbert very rarely says anything which can be interpreted as a "rating". There is no objective criterion to "rate" the problems except by the time it took to solve them. ...
Alexandre Eremenko's user avatar
4 votes

Why did Hilbert believe consistency implies existence?

Think back to when you learned linear algebra: \begin{align} x + y + z & = 2 \\ 6x - 4y + 5z & = 31 \\ 5x + 2y + 2z & = 13 \end{align} Does a solution to this system exist? (In this case, ...
Michael Hardy's user avatar
4 votes
Accepted

David Hilbert's paper: Substitution of the group of cyclotomic field

I think in modern notation, the element $\Theta$ (which is literally defined as $\displaystyle e^{\frac{2\pi i}{l^h}}$ i.e. a primitive $l^h$-th root of unity; $l$ is a prime here) would often be ...
Torsten Schoeneberg's user avatar
2 votes

Why did Hilbert believe consistency implies existence?

There is a historical context involved arising from the influence of Kant. There had been a number of reasons for anti-Kantian backlash in the nineteenth century. However, his wider influence led to ...
mls's user avatar
  • 29
1 vote

Historical background with Weber and Kronecker's Jugendtraum

The history is described in the book of Serge Vladut, Kronecker's Jugendtraum and modular functions, Overseas Publishers Assoc., 1991.
Alexandre Eremenko's user avatar
1 vote

Why did Hilbert believe consistency implies existence?

@Dr. Katz The original poster concludes with the question: "Why does Hilbert believe that consistency of axioms for real numbers would provide him with existence of the reals?" So, the issue ...
mls's user avatar
  • 21
1 vote

What was the connection between David Hilbert and Stefan Banach?

There is no known record of any personal encounter between Banach and Hilbert. But the not-so-random connection between the two was Hugo Steinhaus (Banach's discoverer, and later collaborator and ...
Margaret Friedland's user avatar

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