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Hilbert’s Axiomatisches Denken

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Abstract

We discuss some aspects of the impact of Hilbert’s talk on Axiomatic Thinking in the development of modern mathematical logic.

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Notes

  1. 1.

    See [7, p. 422]; Hilbert himself commented on this dead end in 1928 [35]: “Eine unglückliche Auffassung Poincarés, dieses Meisters mathematischer Erfindungskunst, betreffend den Schluss von n auf \(n+1\), eine Auffassung die überdies bereits zwanzig Jahre früher von Dedekind widerlegt war, verrammelte den Weg zum richtigen Vorwärtsschreiten.” English translation [39, p. 228]: “An unfortunate view of Poincaré concerning the inference from n to \(n+1\), which had already been refuted by Dedekind through a precise proof two decades earlier, barred the way to progress.”

  2. 2.

    See the footnote to the “first insight” in [9]:“Eine mündliche Erörterung der ersten Einsicht Herrn Hilbert gegenüber hat im Herbst 1909 in mehreren Unterhaltungen stattgefunden.” English translation [10, p. 491]: “An oral discussion of the first insight took place in several conversations I had with Hilbert in the autumn of 1909.”

  3. 3.

    Reid, mistakenly, dates this invitation to Hilbert’s visit in Zurich in spring.

  4. 4.

    See [34]. Hilbert’s enthusiasm for Cantor was already expressed in his obituary for his colleague and friend, Minkowski, with whom he had started his studies of set theory in Königsberg [32]: “[Minkowski] war der erste Mathematiker unserer Generation – und ich habe ihn darin nach Kräften unterstützt –, der die hohe Bedeutung der Cantorschen Theorie erkannte und zur Geltung zu bringen suchte.” English translation: “Minkowski was the first mathematician of our generation, who realized the high significance of Cantor’s theory and who sought to bring it to fruition—and I supported him diligently in this endeavor.”

  5. 5.

    Hilbert expressed this concern in a lecture of 1920 [33, p. 16]: “Insbesondere in der Mathematik, der wir doch die alte gerühmte absolute Sicherheit erhalten wollen, müssen Widersprüche ausgeschlossen werden.” English translation: “In particular in mathematics, for which we would like to retain the venerable vaunted absolute certainty, contradictions must be excluded.”

  6. 6.

    Hilbert commented in 1920 [33, p. 33] (published in [40, p. 363]): “Das Ziel, die Mengenlehre und damit die gebräuchlichen Methoden der Analysis auf die Logik zurückzuführen, ist heute nicht erreicht und ist vielleicht überhaupt nicht erreichbar.” English translation: “Today, the goal to reduce set theory and thereby the usual methods of analysis to logic is not attained, and it is possibly not attainable at all.” In 1928, he expressed it more pithily [36, p. 65]: “Die Mathematik wie jede andere Wissenschaft kann nie durch Logik allein begründet werden.” English translation: “Mathematics as any other science can never be founded by logic alone.”

  7. 7.

    See [50]. José Ferreirós, in his contribution to this volume, gives insight in the developments of axiomatics from Dedekind via Peano and Hilbert to Bourbaki [17, Chap. 6].

  8. 8.

    German original [35]: “Mit dieser Neubegründung der Mathematik, die man füglich als eine Beweistheorie bezeichnen kann, glaube ich die Grundlagenfragen in der Mathematik als solche endgültig aus der Welt zu schaffen, indem ich jede mathematische Aussage zu einer konkret aufweisbaren und streng ableitbaren Formel mache und dadurch den ganzen Fragenkomplex in die Domäne der reinen Mathematik versetze.”

  9. 9.

    English translation from [24, p. 237]; German original [23]: “Ein Hauptmerkmal des Hilbertschen Standpunkts scheint mir das Bestreben zu sein, das mathematische Grundlagenproblem der Philosophie zu entziehen und es soweit wie irgendmöglich mit den eigenen Hilfsmitteln der Mathematik zu behandeln.”

  10. 10.

    Nelson expressed his astonishment concerning Hilbert’s aim to reform logic in a letter to Hessenberg, June 1905, cited in [59, p. 166]: “Um den Widerspruch in der Mengenlehre zu beseitigen, will er [Hilbert] (nicht etwa die Mengenlehre sondern) die Logik reformieren.” English translation: “To eliminate the contradiction in set theory, [Hilbert] wants to reform (not set theory but) logic.”

  11. 11.

    See the textbook by Hilbert and Ackermann [42]; it is worth mentioning, that the technical elaboration is mainly due to Bernays.

  12. 12.

    This account profited a lot from previous work by Pasch.

  13. 13.

    In Hilbert’s own words [33, p. 16]: “Ein Widerspruch ist wie ein Bazillus, der alles vergiftet, wie ein Funke im Pulverfass, der alles vernichtet.” English translation: “A contradiction is like a bacillus, which poisons everything, as a spark in a powder barrel, which destroys everything.” For more on Hilbert’s discussion of the paradoxes, see [47].

  14. 14.

    The small roman numbers are added by us to simplify referencing.

  15. 15.

    50 years later, Bernays wrote an encyclopedia entry for “David Hilbert” [5, p. 500]. Obviously with the 1917 talk at hand, he recalled the given paragraph, however without mentioning (iii). We may take this as an indication that, by 1967, the question of formal proof-checking was considered solved.

  16. 16.

    See the discussion of inhaltliche and formale Axiomatik (contentual and formal axiomatics) at the beginning of [43], in particular [43, p. 2]: “Die formale Axiomatik bedarf der inhaltlichen notwendig als ihrer Ergänzung, weil durch diese überhaupt erst die Anleitung zur Auswahl der Formalismen und ferner für eine vorhandene formale Theorie auch erst die Anweisung zu ihrer Anwendung auf ein Gebiet der Tatsächlichkeit gegeben wird.” English translation [44, p. 2]: “Formal axiomatics requires contentual axiomatics as a necessary supplement. It is only the latter that provides us with some guidance for the choosing the right formalism, and with some instructions on how to apply a given formal theory to a domain of actuality.”

  17. 17.

    It is important to note that Hilbert was only looking for formalizability on principle; he never advocated doing mathematics in formal terms; to the contrary, the possibility of formalization, axiomatization together with consistency proofs for the formalized theories would justify the common informal reasoning in mathematics.

  18. 18.

    See the discussion in [49].

  19. 19.

    (vi) in our numbering of the quote above.

  20. 20.

    Arguably, the model of Turing machines has a distinguished status such that it appears reasonable to speak about the Church-Turing Thesis rather than just the Church Thesis.

  21. 21.

    See his Problem Lecture in Paris [29]: “Wenn uns die Beantwortung eines mathematischen Problems nicht gelingen will, so liegt häufig der Grund darin, daß wir noch nicht den allgemeineren Gesichtspunkt erkannt haben, von dem aus das vorgelegte Problem nur als einzelnes Glied einer Kette verwandter Probleme erscheint.” English translation [30, p. 443]: “If we do not succeed in solving a mathematical problem, the reason frequently consists in our failure to recognize the more general standpoint from which the problem before us appears only as a single link in a chain of related problems.”

  22. 22.

    We note in passing that he had also subsumed probability theory under the sixth problem, a theory which may well serve as prime example for the success of the axiomatic method, if one takes into account its axiomatization by Kolmogorov [52].

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  1. Carl Friedrich von Weizsäcker-Zentrum, Universität Tübingen, Keplerstr. 2, 72074, Tübingen, Germany

    Reinhard Kahle

  2. CMA, FCT, Universidade Nova de Lisboa, 2829-516, Caparica, Portugal

    Reinhard Kahle

  3. Eidgenössische Technische Hochschule, Zurich, Switzerland

    Giovanni Sommaruga

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  1. Departamento de Matemática, Faculdade de Ciências, Universidade de Lisboa, Lisboa, Portugal

    Fernando Ferreira

  2. Carl Friedrich von Weizsäcker-Zentrum, Universität Tübingen, Tübingen, Germany

    Reinhard Kahle

  3. Department of Mathematics, ETH Zurich, Zurich, Switzerland

    Giovanni Sommaruga

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Kahle, R., Sommaruga, G. (2022). Hilbert’s Axiomatisches Denken. In: Ferreira, F., Kahle, R., Sommaruga, G. (eds) Axiomatic Thinking I. Springer, Cham. https://doi.org/10.1007/978-3-030-77657-2_2

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