Abstract
In this paper, I will discuss a conceptual distinction, or a contrast, between two opposing ideas, that has played an extremely important role in the foundations of mathematics. The distinction was first formulated explicitly, though not quite generally, by Leon Henkin in 1950.1 He called it a distinction between the standard and the nonstandard interpretation of higher-order logic. I will follow his terminology, even though it may not be the most fortunate one. One reason for saying that this nomenclature is not entirely happy is that it is not clear which interpretation is the “standard” one in the sense of being a more common one historically. Another reason is that one can easily characterize more than one nonstandard interpretation of higher-order logic, even though Henkin considered only one.2 Furthermore, it turns out that the distinction (rightly understood) is not restricted to higher-order logics.3 Last but not least, it is far from clear how Henkin’s notion of standard interpretation or standard model is related to logicians’ idea of the standard model of such first-order theories as elementary arithmetic, in which usage “standard model” means simply “intended model”.
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Notes
Leon Henkin: 1950, ‘Completeness in the Theory of Types’, Journal of Symbolic Logic 15, 81–91. For a correction to Henkin’s paper, see Peter B. Andrews: 1972, ’General Models and Extensionality’, Journal of Symbolic Logic 37, 395–397.
See section 2.
Cf. here Jaakko Hintikka: 1980, ‘Standard vs. Nonstandard Logic: Higher Order, Modal and First-Order Logics’, in E. Agazzi (ed.), Modern Logic: A Survey, D. Reidel, Dordrecht, 283–296.
The following paragraphs as well as section 3 below follow closely the exposition in Jaakko Hintikka and Gabriel Sandu: 1992, ‘The Skeleton in Frege’s Cupboard: The Standard versus Nonstandard Distinction’, Journal of Philosophy 89, 290–315.
See ‘Ober eine bisher noch nicht benützte Erweiterung des finiten Standpunktes’, in Solomon Feferman et al. (eds.), Kurt Gödel: Collected Works, Vol. 2: Publications 1938–1974,Oxford University Press, New York, 1990, pp. 240–251. (Cf. also pp. 217–241.)
Cf., e.g., Jaakko Hintikka: 1955, ‘Reductions in the Theory of Types’, Acta Philosophica Fennica 5, 59–115.
Cf. here Hintikka and Sandu, op. cit. note 4.
Cf. the end of section 4 below.
This question is tantamount to the question of the validity of Leibniz’s Law.
Quoted in Gregory H. Moore: 1982, Zermelo’s Axiom of Choice, Springer-Verlag, Berlin-Heidelberg-New York, p. 314.
Quoted in op. cit., p. 318.
For instance, it appears that some commentators have misinterpreted Frege because he does not assume the definability interpretation. (For Frege, functions exist objectively independently of their representability in language.) From this they have in effect mistakenly inferred that Frege accepted the standard interpretation. See here Hintikka and Sandu, op. cit.
Jon Barwise and Solomon Feferman (eds.): 1986, Model-theoretical Logics, Springer-Verlag, Berlin-Heidelberg-New York.
Quoted in Umberto Bottazzini, The “Higher Calculus”: A History of Real and Complex Analysis from Euler to Weierstrass, Springer-Verlag, Berlin-Heidelberg-New York, 1986, p. 33.
I. H. Anellis, A History of Mathematical Logic in Russia and the Soviet Union,unpublished.
See L. Euler: 1990, Introduction to Analysis of the Infinite, Book 11,translated by John D. Blanton, Springer-Verlag, Berlin-Heidelberg-New York, p. 6, section 9.
See his paper, `Mathematical Ideas, Ideas, and Ideology’, The Mathematical Intelligencer 14(2) (Spring 1992), 6–19 (here p. 7b).
The first quotation is from Judith V. Grabiner: 1981, The Origins of Cauchy’s Rigorous Calculus, The MIT Press, Cambridge MA, pp. 89–90. The second is from Thomas Hawkins: 1970, Lebesgue’s Theory of Integration, University of Wisconsin Press, Madison, p. 4.
P. Dugac: 1973, “Eléments d’analyse de Karl Weierstrass”, Archive of the History of Exact Sciences 10, 41–176. (See p. 71; quoted in Bottazzini, op. cit., p. 199.)
In `Kronecker’s View of the Foundations of Mathematics’, in David E. Rowe and John McCleary (eds.), The History of Modern Mathematics,vol. I, Academic Press, San Diego, pp. 67–77. (See here p. 74.)
For Weierstrass’s work, see Felix Klein: 1927, Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert, vol. 1, Springer-Verlag, Berlin-Heidelberg, pp. 276–295.
Leopold Kronecker: 1886, `Ober einige Anwendungen der Modulsysteme auf elementare algebraische Fragen’, Journal far reine und angewandte Mathematik, vol. 99, pp. 329–371, especially p. 336. Quoted in Joseph W. Dauben: 1979, Georg Cantor: His Mathematics and Philosophy of the Infinite, Harvard U.P., Cambridge MA, p. 68.
Michael Hallett: 1984, Cantorian Set Theory and Limitation of Size, Clarendon Press, Oxford.
See here Jaakko Hintikka: 1994, `What is Elementary Logic? Independence-friendly Logic as the True Core Area of Logic’, in K. Gavroglu et al. (eds.), Physics, Philosophy and Scientific Community: Essays in Honor of Robert S. Cohen, Kluwer Academic, Dordrecht, pp. 301–326.
See Stewart Shapiro: 1991, Foundations without Foundationalism, Clarendon Press, Oxford.
Bertrand Russell: 1908, `Mathematical Logic as Based on the Theory of Types’, American Journal of Mathematics,vol. 30, pp. 222–262, reprinted in Bertrand Russell: 1956, Logic and Knowledge: Essays 1901–1950,ed. by Robert C. Marsh, Allen Unwin, London, pp. 59–102.
Bertrand Russell and Alfred North Whitehead: 1910–1913, Principia Mathematica I-1I1, Cambridge University Press, Cambridge; second ed., 1927.
Op. cit., note 26, second edition, vol. 1, Appendix B.
Frank P. Ramsey: 1925, `The Foundations of Mathematics’, Proceedings of the London Mathematical Society, Ser. 2, vol. 25, part 5, pp. 338–384. Reprinted (among other places) in F. P. Ramsey: 1978, Foundations, ed. by D. H. Mellor, Routledge and Kegan Paul, London, pp. 152–212.
Op. cit., p. 173 of the reprint.
Op. cit., p. 165 of the reprint.
See Maria Carla Galavotti (ed.): 1991, Frank Plumpton Ramsey, Notes on Philosophy, Probability and Mathematics, Bibliopolis, Napoli, Appendix, and Mathieu Marion’s contribution to the present volume.
For a discussion of the history of this contrast, see Gregory H. Moore: 1988. Moore: 1988, “The Emergence of First-Order Logic”, in William Aspray and Philip Kitcher (eds.), History and Philosophy of Modern Mathematics (Minnesota Studies in the Philosophy of Science, vol. II ), University of Minnesota Press, Minneapolis, 1988, pp. 95–135.
See here Merrill B. Hintikka and Jaakko Hintikka: 1986, Investigating Wittgenstein,Basil Blackwell, Oxford, chapters 2 and 4.
Cf. Gregory H. Moore, op. cit., note 10 above.
Bertrand Russell: 1919, Introduction to Mathematical Philosophy,Allen Unwin, London, chapter 12, especially p. 126.
Op. cit., p. 309.
See op. cit., note 5 above, and cf. Jaakko Hintikka: 1993, ’’Gödel’s Functional Interpretation in Perspective’, in M. D. Schwabl (ed.), Yearbook of the Kurt Gödel Society, Vienna, pp. 5–43.
See Hintikka, op. cit., note 3 above.
Michael Dummett, Elements of Intuitionism,Clarendon Press, Oxford, 1977, pp. 52–53 and 314.
Op. cit., note 10 above, especially pp. 64–76.
Op. cit., note 20 above, p. 71.
See here Jaakko Hintikka: 1988, `What Is the Logic of Experimental Inquiry?’, Synthese 74, 173–190.
For an early discussion of the nature and role of conclusiveness conditions, see Jaakko Hintikka, The Semantics of Questions and the Questions of Semantics (Acta Philosophica Fennica vol. 28, no. 4) Societas Philosophica Fennica, Helsinki, 1976, especially ch. 3. The analysis presented there is now being generalized, especially to questions whose answers are functions.
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Hintikka, J. (1998). Standard vs. Nonstandard Distinction: A Watershed in the Foundations of Mathematics. In: Language, Truth and Logic in Mathematics. Jaakko Hintikka Selected Papers, vol 3. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2045-8_6
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