Intensionality in Mathematics
Do mathematical expressions have intensions, or merely extensions? If we accept the standard account of intensions based on possible worlds, it would seem that the latter is the case – there is no room for nontrivial intensions in the case of non-empirical expressions. However, some vexing mathematical problems, notably Gödel’s Second Incompleteness Theorem, seem to presuppose an intensional construal of some mathematical expressions. Hence, can we make room for intensions in mathematics? In this paper we argue that this is possible, provided we give up the standard approach to intensionality based on possible worlds.
Work on this paper was supported by Grant No. 17-15645S of the Czech Science Foundation.
- Auerbach, D. 1992. How to say things with formalisms. In Proof, logic and formalization, ed. M. Detlefsen, 77–93. London: Routledge.Google Scholar
- Boolos, G. 1995. The logic of provability. Cambridge: Cambridge University Press.Google Scholar
- Carnap, R. 1947. Meaning and necessity. Chicago: University of Chicago Press.Google Scholar
- Frege, G. 1892a. Über Sinn und bedeutung. Zeitschrift für Philosophie und philosophische Kritik 100: 25–50.Google Scholar
- Frege, G. 1892b. Über Begriff und Gegenstand. Vierteljahrschrift für wissentschaftliche Philosophie 16: 192–205.Google Scholar
- Kripke, S. 1963a. Semantical considerations on modal logic. Acta Philosophica Fennica 16: 83–94.Google Scholar
- Kripke, S. 1965. Semantical analysis of modal logic II (non-normal modal propositional calculi). In The theory of models, eds. L. Henkin, J.W. Addison, and A. Tarski, 206–220. Amsterdam: North-Holland.Google Scholar
- Montague, R. 1974. Formal philosophy: Selected papers of R. Montague. New Haven: Yale University Press.Google Scholar
- Peregrin, J. 2006. Extensional vs. intensional logic. In Philosophy of logic, Handbook of the philosophy of science, vol. 5, ed. D. Jacquette, 831–860. Amsterdam: Elsevier.Google Scholar