How effective indeed is present-day mathematics?

Athanassios Tzouvaras

DOI: http://dx.doi.org/10.12775/LLP.2006.008

Abstract


We argue that E. Wigner’s well-known claim that mathematics is unreasonably effective in physics (and not in the natural sciences in general, as the title of his article suggests) is only one side of the hill. The other side is the surprising insufficiency of present-day mathematics to capture the uniformities that arise in science outside physics. We describe roughly what the situation is in the areas of (a) everyday reasoning, (b) theory of meaning and (c) vagueness. We make also the point that mathematics, as we know it today, founded on the concept of set, need not be a conceptually final and closed system, but only a stage in a developing subject.

Keywords


effectiveness of mathematics; everyday reasoning; vagueness; meaning

Full Text:

PDF

References


Aczel, P., Non-well-founded sets, CSLI Lecture Notes, No 14, 1988.

Ajdukiewicz, K., Pragmatic logic, D. Reidel P.C. 1974.

Barwise, J., The situation in logic, CSLI Lecture Notes, No 17, CSLI Publications, Stanford 1989.

Barwise, J., and J. Etchemedy, The Liar. An essay on truth and circularity, Oxford University Press 1989.

Barwise, J., and L. Moss, Vicious circles: on the mathematics of circular phenomena, CSLI Lecture Notes 1996.

Barwise, J., and J. Perry, Situations and attitudes, Cambridge, MA, MIT Press, 1983.

Carnap, R., Meaning and necessity, Univ. of Chicago Press, 1956.

Chomsky, N., Syntactic structures, Mouton, Hague 1957.

Castoriadis, C., L’ institution imaginaire de la société, Editions du Seuil, Paris 1975.

Castoriadis, C., Domaines de l’ homme, Editions du Seuil, Paris 1986.

Frege, G., “On sense and reference”, in: D. Davidson and G. Harman (eds.), The logic of grammar, Dickenson, Encino, California 1975, pp. 116–128.

Heyting, A., Intuitionism: An introduction, North Holland, Amsterdam 1956.

Marciszewski, W., (ed.), Dictionary of logic as applied in the study of language. Concepts, methods, theories, Martinus Nijhoff Publishers 1981.

Montague, R., Formal Philosophy: selected papers of Richard Montague, R.H. Thomason (ed.), Yale U.P., 1974.

Rota, G.-C., “The barrier of meaning”, Letters in Math. Physics 10 (1985), 97–99.

Smith, N., Noam Chomsky – Ideas and ideals, Cambridge Univ. Press, 1999. [17] Tzouvaras, A., “Modeling vagueness by nonstandardness”, Intern. J. of Fuzzy Sets and Systems 94 (1998), 385–396.

Tzouvaras, A., “An axiomatization of “very” within systems of set theory”, Studia Logica 73 (2003), 413–430.

Wigner, E., “The unreasonable effectiveness of mathematics in the natural sciences”, Communications in Pure and Applied Mathematics 13 (1960). Available also in URL: http://nedwww.ipac.caltech.edu/level5/March02/Wigner/Wigner.html

Zimmermann, H.-J., Fuzzy set theory and its applications, 2nd Edition, Kluwer Academic Publishers, 1991.




Bookmark and Share




Print ISSN: 1425-3305
Online ISSN: 2300-9802

Partnerzy platformy czasopism