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Syntax, Semantics, and the Problem of the Identity of Mathematical Objects

Published online by Cambridge University Press:  01 April 2022

Gian-Carlo Rota ,
David H. Sharp  and
Robert Sokolowski
Gian-Carlo Rota
Affiliation:
Department of Mathematics Massachusetts Institute of Technology
David H. Sharp
Affiliation:
Theoretical Division Los Alamos National Laboratory
Robert Sokolowski
Affiliation:
School of Philosophy The Catholic University of America
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Abstract

A plurality of axiomatic systems can be interpreted as referring to one and the same mathematical object. In this paper we examine the relationship between axiomatic systems and their models, the relationships among the various axiomatic systems that refer to the same model, and the role of an intelligent user of an axiomatic system. We ask whether these relationships and this role can themselves be formalized.

Type
Research Article
Information
Philosophy of Science , Volume 55 , Issue 3 , September 1988 , pp. 376 - 386
Copyright
Copyright © 1988 by the Philosophy of Science Association

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Footnotes

†

The authors wish to thank Professors Hilary Putnam and John A. Wheeler for their valued comments on this paper.

‡

Work supported by the U.S. Department of Energy.

References

REFERENCES

Birkhoff, G. (1967), Lattice Theory, 3d ed. Providence: American Mathematical Society.Google Scholar
Birkhoff, G., and von Neumann, J. (1936), “The Logic of Quantum Mechanics”, Annals of Mathematics 37: 823–43.CrossRefGoogle Scholar
Fitting, M. (1970), “Intuitionistic Model Theory and the Cohen Independence Proofs”, in A. Kino, J. Myhill, and R. E. Vesley (eds.), Intuitionism and Proof Theory. Proceedings of the Summer Conference at Buffalo, New York, 1968. Amsterdam: North Holland, pp. 219–226.Google Scholar
Hilbert, D., and Bernays, P. (1970), Grundlagen der Mathematik II. Berlin: Springer.CrossRefGoogle Scholar
Husserl, E. (1969), Formal and Transcendental Logic. Cairns, D. (trans.). Hague: Nijhoff.CrossRefGoogle Scholar
Husserl, E. (1971), Logical Investigations, 2 Vol. J. Findlay, N., (trans.). London: Routledge and Kegan Paul.Google Scholar
Kitcher, P. (1984), The Nature of Mathematical Knowledge. New York: Oxford University Press.Google Scholar
Lakatos, I. (1976), Proofs and Refutations. The Logic of Mathematical Discovery. Cambridge: Cambridge University Press.10.1017/CBO9781139171472CrossRefGoogle Scholar
Lakatos, I. (1978), “A Renaissance of Empiricism in the Recent Philosophy of Mathematics?”, in Mathematics, Science, and Epistemology. Cambridge: Cambridge University Press, pp. 24–42.10.1017/CBO9780511624926.003CrossRefGoogle Scholar
Loomis, L. (1955), “The Lattice-Theoretic Background of the Dimension Theory”, Memoirs of the American Mathematical Society, No. 18A.CrossRefGoogle Scholar
Mac Lane, S. (1973), Categories for the Working Mathematician. New York: Springer.Google Scholar
Putnam, H. (1979a), “Mathematics without Foundations”, in Mathematics, Matter and Method, 2d ed. Cambridge: Cambridge University Press, pp. 43–59.CrossRefGoogle Scholar
Putnam, H. (1979b), “What is Mathematical Truth?”, in Mathematics, Matter and Method. 2d ed. Cambridge: Cambridge University Press, pp. 60–78.CrossRefGoogle Scholar
Smullyan, R. M. (1968), First-Order Logic. Berlin: Springer.CrossRefGoogle Scholar
Stone, M. H. (1937), “Topological Representation of Distributive Lattices and Brouwerian Logics”, Casopis Pro Pestovani Matematiky A Fysiky 67: 1–25.Google Scholar