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Concluding Comments by Martin

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Martin Davis on Computability, Computational Logic, and Mathematical Foundations

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 10))

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Abstract

After a very brief comment on Yuri Matiyasevich’s contribution, I discuss at greater length proposals to use modal logic to clarify foundational issues in set theory. Finally, I very sadly bid farewell to my friend and collaborator Hilary Putnam.

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Notes

  1. 1.

    The cardinal in question is in fact quite large: a countable infinity of Woodin cardinals with a measurable cardinal above them.

  2. 2.

    Recent work by Joel Friedman on modalism [1] should also be mentioned.

  3. 3.

    “It is difficult to see that the word if acquires when written \(\supset \), a virtue it did not possess when written if.” [3], p. 156.

References

  1. Friedman, J. (2005). Modal platonism: An easy way to avoid ontological commitment to abstract entities. Journal of Philosophical Logic, 34, 227–273.

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  2. Hellman, G. (1989). Mathematics without numbers: Toward a modal-structural interpretation. Oxford.

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  3. Poincaré, H. (2012). Science and method, translated from French by Francis Maitland, Thomas Nelson and Sons, London 1914. Facsimile Reprint: Forgotten Books. http://www.forgottenbooks.org.

  4. Weyl, H. (1950). David Hilbert and his mathematical work. Bulletin of the American Mathematical Society, 50, 612–654.

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  5. Zermelo, E. (1996). Über Grenzzahlen und Mengenbereiche: neue Untersuchungen über die Grundlagen der Mengenlehre, Fundamenta Mathematicae, vol. 16, pp. 29–47. English Translation: On Boundary numbers and domains of sets: New investigations in the foundations of set theory. In W. B. Ewald (Ed.), From Kant to Hilbert: A source book in the foundations of mathematics (pp. 1219–1233). Oxford University Press.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, Berkeley, CA, USA

    Martin Davis

  2. Courant Institute of Mathematical Sciences, New York University, New York, NY, USA

    Martin Davis

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  1. Martin Davis
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Correspondence to Martin Davis .

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Editors and Affiliations

  1. University of Trieste, Trieste, Italy

    Eugenio G. Omodeo

  2. University of Udine, Udine, Italy

    Alberto Policriti

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Davis, M. (2016). Concluding Comments by Martin. In: Omodeo, E., Policriti, A. (eds) Martin Davis on Computability, Computational Logic, and Mathematical Foundations. Outstanding Contributions to Logic, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-319-41842-1_15

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