Abstract
This paper provides a discussion to which extent the Mathematician David Hilbert could or should be considered as a Philosopher, too. In the first part, we discuss some aspects of the relation of Mathematicians and Philosophers. In the second part we give an analysis of David Hilbert as Philosopher.
Work partially supported by the Portuguese Science Foundation, FCT, through the projects The Notion of Mathematical Proof, PTDC/MHC-FIL/5363/2012, Hilbert’s 24th Problem, PTDC/MHC-FIL/2583/2014, and the Centro de Matemática e Aplicações, UID/MAT/00297/2013.
The title phrase was recorded at the entrance of the Platonic Academy. In a free English translation it reads: “Let none but geometers (i.e., mathematicians) enter here.” The earliest references to this phrase is from the sixth century and can be found, in slight variations, in works of John Philoponus (1897, p. 117.27) and Elias (1900, p. 118, 18–19). It is in line with Diogenes Laertius (1959, VI.10; p. 384) who reports that Xenocrate, the third leader of Platon’s Academy, classified Mathematics as part of the handles of philosophy.
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- 1.
- 2.
Hegel’s dissertation of 1801 finishes with a discussion—although not a “proof”—that there should be exactly eight planets in the solar system. His work was characterized as “Monumentum Insaniae Saec. XIX” by Ernest II, Duke of Saxe-Gotha-Altenburg and discussed as such by Gauß in the correspondence with his friend Schumacher in 1842 (Peters 1862, letters 763–765).
- 3.
Just look around at the modern philosophers, at Schelling, Hegel, Nees von Esenbeck and consorts—don’t their definitions make your hair stand on end? Read in the history of ancient philosophy what the men of the day, Plato and others (I except Aristotle), gave as explanations. And even in Kant matters are often not much better; his distinction between analytic and synthetic propositions seems to me to be either a triviality or false.
Gauß in a letter to Schumacher on 1 November 1844; translation from (Ewald 1959, p. 293); German original in (Peters 1862, letter 944).
- 4.
“Bolzano, der große Gegner Kants, ist seit Leibnitz[sic!] der erste philosophische Mathematiker und mathematische Philosoph.” (“Bolzano, the great opponent of Kant, is the first philosophical Mathematician and mathematical Philosopher since Leibniz.”) (Korselt 1903, p. 405).
- 5.
Von Plato (2016) pointed to such a rare case, when Kant triggered with his discussion of the equation \(7 + 5 = 12\) the development of modern recursive foundations of Arithmetic through the obscure figures of Johann Schultz and Michael Ohm; via Grassmann, Hankel, Schröder, Dedekind it finally reaches Peano, Skolem, and Bernays.
- 6.
The translation is ours; German original in (Peckhaus 1990, p. 166): “Von Hilberts philosophischen ‘Resultaten’ mach’ Dir nun lieber keine großen Hoffnungen. Ich bin bereits durch das, was er bisher davon produziert hat, recht enttäuscht. [...] Daß er das Bedürfnis nach mehr fühlt, ist ja auch sehr schön, aber so wie er das rein Mathematische verläßt, wird er einfach albern.”
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A detailed account to the development is given in (Toepell 1999).
- 9.
See (Brouwer 1927, Footnote 1).
- 10.
“Besonders interessiert hat mich der neue meta-mathematische Standpunkt, den Sie jetzt einnehmen und der durch die Gödelsche Arbeit veranlaßt worden ist.” (“I was particularly interested in the new meta-mathematical standpoint which you now adopt and which was provoked by Gödel’s work.”) Ackermann in a letter to Hilbert, August 23rd, 1933 (Ackermann 1933, p. 1f).
- 11.
Gentzen expressed it in these words: (Gentzen 1938, p. 237 in the english translation)
A foremost characteristic of Hilbert’s point of view seems to me to be the endeavour to withdraw the problem of the foundations of mathematics from philosophy and to tackle it as far as in any way possible with methods proper to mathematics.
- 12.
This was, in fact, one of the central points of origin of Brouwer’s criticism (see Brouwer 1927, Third insight).
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For the more positive evaluation of the influence of Kant on Hilbert, see Sinaceur (2018) in this volume.
- 14.
French original from Janet’s notebook: “[Hilbert] a émis l’idée qu’il serait heureux que toutes les bibliothéques du monde brũlassent, \(\ll \)les mathématiciens seuls pourraient reconstruire les mathématiques, les philosophes seraient bien embarrassés\(\gg \).” (Mazliak 2013, p. 55).
- 15.
This is explicit, for instance, in (Hilbert 1967, p. 479): “I would like to note further that P. Bernays has again been my faithful collaborator. He has not only constantly aided me by giving advice but also contributed ideas of his own and new points of view, so that I would like to call this our common work.” It is also known that the opus magnum of Hilbert and Bernays, the Grundlagen der Mathematik (Hilbert and Bernays 1934; 1939), was essentially entirely written by Bernays.
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“Um den Widerspruch in der Mengenlehre zu beseitigen, will er [Hilbert] (nicht etwa die Mengenlehre sondern) die Logik reformieren.” Nelson in a letter to Hessenberg, June 1905, cited in (Peckhaus 1990, p. 166).
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See the discussion of this point by Bernays (1935).
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According to Bernays (1935, p. 215) the first steps towards such a shift took already place before Gödel’s result became known.
- 19.
For the necessary “philosophical switch” see, for instance, [Bernays 1954, p. 4], and the letter of Ackermann cited in Footnote 10 above.
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Kahle, R. (2018). M
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In: Tahiri, H. (eds) The Philosophers and Mathematics. Logic, Epistemology, and the Unity of Science, vol 43. Springer, Cham. https://doi.org/10.1007/978-3-319-93733-5_6
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