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Philosophical Uses of Categoricity Arguments
Published online by Cambridge University Press: 02 December 2023
Summary
This Element addresses the viability of categoricity arguments in philosophy by focusing with some care on the specific conclusions that a sampling of prominent figures have attempted to draw – the same theorem might successfully support one such conclusion while failing to support another. It begins with Dedekind, Zermelo, and Kreisel, casting doubt on received readings of the latter two and highlighting the success of all three in achieving what are argued to be their actual goals. These earlier uses of categoricity arguments are then compared and contrasted with more recent work of Parsons and the co-authors Button and Walsh. Highlighting the roles of first- and second-order theorems, of external and internal theorems, the Element concludes that categoricity arguments have been more effective in historical cases that reflect philosophically on internal mathematical matters than in recent questions of pre-theoretic metaphysics.
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- Online ISBN: 9781009432894Publisher: Cambridge University PressPrint publication: 21 December 2023
References
Barwise, K. J. (1972). The Hanf number of second order logic. Journal of Symbolic Logic, 37, 588–594. https://doi.org/10.2307/2272748.Google Scholar
Benacerraf, P., & Putnam, H. (Eds. 1983). Philosophy of Mathematics: Selected readings (2nd ed.). Cambridge University Press, Cambridge.Google Scholar
Bernays, P. (1935). On platonism in mathematics, (Translated from the French by C. Parsons, Benacerraf and Putnam (1983), pp. 258–271)Google Scholar
Boolos, G. (1971). The iterative conception of set. Reprinted in Benacerraf and Putnam (1983), pp. 496–502. https://doi.org/10.2307/2025204.CrossRefGoogle Scholar
Bowler, N. (2019). Foundations for the working mathematician, and for their computer. In Centrone, S., Kant, D., and Sarikaya, D. (Eds.), Reflections on the Foundations of Mathematics: Univalent Foundations, Set Theory and General Thoughts (pp. 399–416). Springer, Cham. https://doi.org/10.1007/978-3-030-15655-8_18.Google Scholar
Button, T., & Walsh, S. (2018). Philosophy and Model Theory. Oxford University Press, Oxford. https://doi.org/10.1093/oso/9780198790396.001.0001 (With a historical appendix by Wilfrid Hodges).Google Scholar
Chu, J., Cheung, P., Schneider, R., Sullivan, J., & Barner, D. (2020). Counting to infinity: Does learning the syntax of the count list predict knowledge that numbers are infinite? Cognitive Science, 44, 1–30.CrossRefGoogle ScholarPubMed
Corcoran, J. (1980). Categoricity. History and Philosophy of Logic, 1, 187–207. https://doi.org/10.1080/01445348008837010.Google Scholar
Dean, W. (2021). Computational complexity theory. In Zalta, E. N. (Ed.), The Stanford Encyclopedia of Philosophy (Fall 2021 ed.). https://plato.stanford.edu/archives/fall2021/entries/computationalcomplexity.Google Scholar
Dean, W. (2018). Strict finitism, feasibility, and the sorites. Review of Symbolic Logic, 11(2), 295–346. https://doi.org/10.1017/S1755020318000163.Google Scholar
Dedekind, R. (1890). Letter to Keferstein. (Translated from German by H. Wang and Bauer-Mengelberg, S.. van Heijenoort, 1967, pp. 99–103)Google Scholar
Dedekind, R. (1872). Continuity and irrational Numbers. (Translated from German by W. Beeman and Ewald, W. Ewald. (2005), pp. 766–779.)Google Scholar
Dedekind, R. (1888). Was sind und was sollen die zahlen?. (Translated from German by W. Beeman and Ewald, W. Ewald. (2005), pp. 790–833.)Google Scholar
Dummett, M. (1963). The philosophical significance of Gödel’s theorem Ratio, 5, 140–155. Reprinted in Dummett (1978), pp.186–201.Google Scholar
Dummett, M. (1975). Wang’s paradox. Synthese, 30(3–4), 201–32. Reprinted in Dummett (1978), pp. 248–268. https://doi.org/10.1007/BF00485048.CrossRefGoogle Scholar
Ebbinghaus, H.- D. (2007). Ernst Zermelo: An Approach to his Life and Work. (In cooperation with V. Peckhaus) Springer, Berlin.Google Scholar
Ewald, W. B. (Ed. 2005). From Kant to Hilbert: A Source Book in the Foundations of Mathematics, volume 2. Oxford University Press, Oxford.Google Scholar
Feferman, S. (1972). Infinitary properties, local functors, and systems of ordinal functions. In Hodges, W. (Ed.) Conference in Mathematical Logic: London ’70. Lecture notes in mathematics (Vol. 255, pp. 63–97).Google Scholar
Field, H. (2001). Postscript to “Which undecidable mathematical sentences have determinate truth values?.” In Field, H., Truth and the Absence of Fact (pp. 351–360). Oxford University Press, Oxford.Google Scholar
Hellman, G. (1989). Mathematics without Numbers: Towards a Modal-structural Interpretation. Oxford University Press, Oxford.Google Scholar
Kanamori, A. (2010a). Introduction to Zermelo (1930a). In Zermelo (2010), pp. 432–433.Google Scholar
Kanamori, A. (2010b). Introduction to Zermelo (1930c). In Zermelo (2010), pp. 390–399.Google Scholar
Kreisel, G. (1967). Informal rigour and completeness proofs. In Lakatos, I. (Ed.), Problems in the Philosophy of Mathematics (pp. 138–157). North-Holland, Amsterdam.CrossRefGoogle Scholar
Kreisel, G. (1969). Two notes on the foundations of set-theory. Dialectica, 23(2), 93–114. https://doi.org/10.1111/j.1746-8361.1969.tb01184.x.CrossRefGoogle Scholar
Lavine, S. (1994). Understanding the Infinite. Harvard University Press, Cambridge, MA.Google Scholar
Maddy, P. (2011). Defending the Axioms: On the Philosophical Foundations of Set Theory, Oxford University Press, Oxford. https://doi.org/10.1093/acprof:oso/9780199596188.001.0001.Google Scholar
Maddy, P. (2018). Psychology and the a priori sciences. In Bangu, S. (Ed.), Naturalizing Mathematical Knowledge: Approaches From Philosophy, Psychology and Cognitive Science, pp. 15–29, Routledge, New York. Reprinted in P. Maddy, A Plea for Natural Philosophy and Other Essays (pp. 262–293). Oxford University Press, New York.CrossRefGoogle Scholar
Maddy, P. (2022). Enhanced if-thenism. In Maddy, P., A Plea for Natural Philosophy and Other Essays (pp. 262–293). Oxford University Press, New York.Google Scholar
Martin, D. (2018). Completeness or incompleteness of basic mathematical concepts (draft). www.math.ucla.edu/dam/booketc/efi.pdfGoogle Scholar
McGee, V. (1997). How we learn mathematical language. Philosophical Review, 106(1), 35–68. https://doi.org/10.2307/2998341.CrossRefGoogle Scholar
Moore, G. H. (1982). Zermelo’s Axiom of Choice: Its Origins, Development, and Influence. Springer, New York. https://doi.org/10.1007/978-1-4613-9478-5.Google Scholar
Parsons, C. (1990). The uniqueness of the natural numbers. Iyyun: The Jerusalem Philosophical Quarterly, 39, 13–44. http://www.jstor.org/stable/23350653.Google Scholar
Parsons, C. (2008). Mathematical Thought and its Objects. Cambridge University Press, Cambridge.Google Scholar
Quine, W. V. (1970). Philosophy of Logic. (Sixth printing) Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Reck, E. (2003). Dedekind’s structuralism: An interpretation and partial defense. Synthese, 137, 369–419.CrossRefGoogle Scholar
Relaford-Doyle, J., & Núñez, R. (2017). When does a “visual proof by induction” serve a proof-like function in mathematics. In Howes, A. and Tenbrink, T. (Eds.), Proceedings of the 39th Annual Meeting of the Cognitive Science Society (pp. 1004–1009). Cognitive Science Society, London.Google Scholar
Relaford-Doyle, J., & Núñez, R. (2018). Beyond Peano: Looking into the unnaturalness of natural numbers. In Bangu, S. (Ed.), Naturalizing Logicomathematical Knowledge (pp. 234–251). Routledge, New York.CrossRefGoogle Scholar
Relaford-Doyle, J., & Núñez, R. (2021). Characterizing students’ conceptual difficulties with mathematical induction using visual proofs. International Journal of Research in Undergraduate Mathematics Education, 7, 1–20.CrossRefGoogle Scholar
Shapiro, S. (1991). Foundations without Foundationalism: a Case for Second-order Logic. Oxford University Press, New York.Google Scholar
Shapiro, S. (2012). Higher-order logic or set theory: A false dilemma. Philosophia Mathematica, 20(3), 305–323. https://doi.org/10.1093/philmat/nks002.Google Scholar
Sieg, W., & Morris, R. (2018). Dedekind’s structuralism: Creating concepts and deriving theorems. In Reck, E. (Ed.) Logic, Philosophy of Mathematics and their History (pp. 251–301). College, London.Google Scholar
Simpson, S. G., & Yokoyama, K. (2013). Reverse mathematics and Peano categoricity. Annals of Pure and Applied Logic, 164(3), 284–293. https://doi.org/10.1016/j.apal.2012.10.014.Google Scholar
Spelke, E. (2000). Core knowledge. American Psychologist, 55, 1233–1243.CrossRefGoogle ScholarPubMed
Väänänen, J. (2019). An extension of a theorem of Zermelo. Bulletin of Symbolic Logic, 25(2), 208–212. https://doi.org/10.1017/bsl.2019.15.Google Scholar
Väänänen, J., & Wang, T. (2015). Internal categoricity in arithmetic and set theory. Notre Dame Journal Formal Logic, 56(1), 121–134. http://dx.doi.org/10.1215/00294527-2835038.CrossRefGoogle Scholar
van Heijenoort, J. (Ed. 1967). From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Harvard University Press, Cambridge, MA.Google Scholar
Walmsley, J. (2002). Categoricity and indefinite extensibility. Proceedings of the Aristotelian Society, 102(3), 217–235.CrossRefGoogle Scholar
Wang, H. (1974). The concept of set. (Reprinted in Benacerraf & Putnam, 1983, pp. 530–570).Google Scholar
Weston, T. (1976). Kreisel, the continuum hypothesis and second order set theory. Journal of Philosophical Logic, 5(2), 281–298. https://doi.org/10.1007/BF00248732.CrossRefGoogle Scholar
Wittgenstein, L. (1978). Remarks on the Foundations of Mathematics (Revised ed.) (Edited by G. H. vonWright, Rhees, R. and G. E. M. Anscombe, Translated from the German by G. E. M. Anscombe). MIT Press, Cambridge, MA.Google Scholar
Yessenin-Volpin, A. S. (1970). The ultra-intuitionistic criticism and the antitraditional program for foundations of mathematics. In Vesley, R. E., Kino, A., and Myhill, J. (Eds.) Intuitionism and Proof Theory (Proceedings of the Summer Conference at Buffalo, NY., 1968). North Holland, Amsterdam (pp. 3–45).CrossRefGoogle Scholar
Zermelo, E. (1908a). A new proof of the possibility of a well-ordering. (Translated from German by S. Bauer-Mengelberg. Reprinted in van Heijenoort (1967), pp. 183–198, and in Zermelo (2010), pp. 120–159). https://doi.org/10.1007/BF01450054.Google Scholar
Zermelo, E. (1908b). Investigations in the foundations of set theory I. (Translated from German by S. Bauer-Mengelberg. Reprinted in van Heijenoort (1967), pp. 200–215, and in Zermelo (2010), pp. 188–229). https://doi.org/10.1007/BF01449999.Google Scholar
Zermelo, E. (1930a). Report to the emergency association of German science. (Translated from German by E. de Pellegrin. In Zermelo (2010), pp. 434–443).Google Scholar
Zermelo, E. (1930b). On the set-theoretic model. (Translated from German by E. de Pellegrin. In Zermelo (2010), pp. 446–453).Google Scholar
Zermelo, E. (1930c). On boundary numbers and domains of sets. (Translated from German by E. de Pellegrin. Reprinted in Zermelo (2010), pp. 400–430).Google Scholar
Zermelo, E. (2010). Collected Works, volume I. (Ebbinghaus, H.-D and Kanamori, A., Eds. Springer, Berlin). https://doi.org/10.1007/978-3-540-79384-7.Google Scholar
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