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Mathematics, the empirical facts, and logical necessity


We have argued that mathematical statements are a posteriori synthetic statements because they are ultimately based on empirical facts or on empirical hypotheses. In some cases, mathematical statements can be verified by direct inspection of an empirical model of the relevant mathematical structure. In other cases, they must be inferred from the axioms of the relevant axiomatic theories, on the assumption that these axioms are consistent. Yet, the consistency of these axioms is always an empirical assumption: it may admit of empirical verification (by exhibiting an empirical model for these axioms), or at least it will admit of potential empirical falsification (by deriving contradictory implications from them).

On the other hand, mathematical statements are a posteriori synthetic statements of a very special sort, because they do not depend on the contingent features of the empirical world, but rather are logically necessary properties of the mathematical structures realized or potentially realized in the empirical world (as shown by the invariance of these properties under isomorphism). Moreover, even though they are synthetic statements, they resemble analytic statements in being logical consequences of the axiomatic definition of the mathematical structure they are dealing with. For this reason, we have proposed the term structure-analytic statements to describe them.

Their logical status as structure-analytic statements gives them a logical position intermediate between truly analytic statements and ordinary empirical statements. This explains the nontrivial and nontautological character of many important mathematical theorems, which often gives them the quality of a priori quite unexpected “brute facts.” This is an aspect of mathematics very hard to explain on the logical positivist assumption that mathematical statements are truly analytic.

We have also discussed some of the philosophic and mathematical problems posed by various limitational theorems. We have argued that for Peano arithmetic the danger of inconsistency can be minimized (though it cannot be fully eliminated), and the problem of noncategoricity can be fully overcome, by stating it in the form of a quantifier-free recursive theory.

On the other hand, in the case of set theory, we have argued that the Skolem paradox shows that we are logically free to reject the existence of absolutely nondenumerable sets, yet that, both on intuitive and on pragmatic grounds, it is preferable to admit their existence, as most set theorists do.

Finally, we have found that we are logically free to opt either for dualism (or for pluralism) or for monism in set theory. If we opted for the former position, this would mean only that the theory of finite sets can be extended to infinite sets in two (or more) different but equally admissible ways-just as other mathematical theories can often be generalized in more ways than one. (Moreover, if infinite sets have the nature of ideal elements, as Hilbert has suggested, it cannot really surprise us if we find ourselves to be logically free to invent two or more different but equally consistent stories about them.) Indeed, we have argued, again both on intuitive and on pragmatic grounds, that it seems preferable to admit a need for both Cantorian and non-Cantorian set theories.

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  • BenacerrafP.: 1965, ‘What numbers could not be’, Philosophical Review 74, 47–73.

    Google Scholar 

  • BenacerrafP.: 1973, ‘Mathematical truth’, Journal of Philosophy 19, 661–679.

    Google Scholar 

  • BenacerrafP., and PutnamH. (eds.): 1964, Philosophy of Mathematics, Prentice-Hall, Englewood Cliffs.

    Google Scholar 

  • BendaM.: 1980, ‘On strong axioms of induction in set theory and arithmetic’, in A. I.Arruda et al. (eds.), Mathematical Logic in Latin America, North-Holland, Amsterdam, pp. 43–56.

    Google Scholar 

  • Vernays, P.: 1964, ‘Comments on Ludwig Wittgenstein's Remarks on the Foundations of Mathematics’ in Benacerraf and Putnam (1964), pp. 510–528.

  • CohenP.: 1963, ‘A minimal model for set theory’, Bulletin of the American Mathematical Society 69, 537–540.

    Google Scholar 

  • CohenP.: 1966, Set Theory and the Continuum Hypothesis, Addison-Wesley, Menlo Park.

    Google Scholar 

  • CohenP.: 1971, ‘Comments on the foundations of set theory’, in D.S.Scott (ed.), Axiomatic Set Theory, Am. Math. Soc., Providence, pp. 9–15.

    Google Scholar 

  • GentzenG.: 1936, ‘Die Wiederspruchsfreiheit der reinen Zahlentheorie’, Mathematische Annalen 112, 493–565.

    CAS  PubMed  Google Scholar 

  • Gödel, K.: 1933, ‘Eine Interpretation des intuitionistischen Aussagekalküls’, Ergebnisse eines mathematischen Kolloquiums, 4.

  • Gödel, K.: 1940, The Consistency of the Continuum Hypothesis.

  • GödelK.: 1962, On Formally Undecidable Propostions in Principia Mathematica and Similar Systems, Basic Books, N.Y.

    Google Scholar 

  • Gödel, K.: 1964, ‘What is Cantor's continuum hypothesis?’, in Benacerraf and Putnam (1964), pp. 258–273.

  • Gödel, K.: 1967, ‘The completeness of the axioms of the functional calculus of logic’, in van Heijenoort (1967); pp. 582–591.

  • LakatosI.: 1978, ‘A renaissance of empirism in the recent philosophy of mathematics’, in his Mathematics, Science and Epistemology, Cambridge U.P., Cambridge, pp. 24–42.

    Google Scholar 

  • MostowskiA.: 1957, ‘On recursive models of formalized arithmetic’, Bulletin de l'Académie Polonaise des Sciences, cl. III, Vol. V, 705–710.

    Google Scholar 

  • ParisJ., and HarringtonL.: 1977, ‘A mathematical incompleteness in Peano arithmetic’, in J.Barwise (ed.), Handbook of Mathematical Logic, North-Holland, Amsterdam.

    Google Scholar 

  • ParsonsC.: 1964, ‘Infinity and Kant's conception of the possibility of experience’, Philosophical Review 73, 182–197.

    Google Scholar 

  • ParsonsC.: 1977, ‘What is the iterative conception of set?’ in R.E.Butts et al. (eds.), Logic, Foundations of Mathematics, and Computability Theory, Reidel, Dordrecht, pp. 336–367.

    Google Scholar 

  • PutnamH.: 1975, Mathematics, Matter, and Method, Cambridge U.P., Cambridge.

    Google Scholar 

  • Putnam, H.: 1979, ‘Analyticity and apriority’, in Midwest Studies in Philosophy, IV, 1979, pp. 423–441.

  • Skolem, T.: 1967, ‘Some remarks on axiomatized set theory’, in van Heijenoort (1967), pp. 290–301.

  • vanHeijenoortJ. (ed.): 1967, From Frege to Gödel, Harvard U.P., Cambridge, MA.

    Google Scholar 

  • WestonT.: 1976, ‘Kreisel, the continuum hypothesis and second order set theory’, Journal of Philosophical Logic 5, 281–298.

    Google Scholar 

  • WittgensteinL.: 1974, Bemerkungen über die Grundlagen der Mathematik, Suhrkamp, Frankfurt am Main.

    Google Scholar 

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Harsanyi, J.C. Mathematics, the empirical facts, and logical necessity. Erkenntnis 19, 167–192 (1983).

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  • Mathematical Statement
  • Mathematical Structure
  • Empirical Fact
  • Axiomatic Theory
  • Peano Arithmetic