Abstract
In this article the problem of unification of mathematical theories is discussed. We argue, that specific problems arise here, which are quite different than the problems in the case of empirical sciences. In particular, the notion of unification depends on the philosophical standpoint. We give an analysis of the notion of unification from the point of view of formalism, Gödel's platonism and Quine's realism. In particular we show, that the concept of “having the same object of study” should be made precise in the case of mathematical theories. In the appendix we give a working proposal of a certain understanding of this notion.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
P. Bernays: 1964, On Platonism in Mathematics, reprinted in: P. Benacerraf and H. Putnam H. (eds): Philosophy of Mathematics, Prentice-Hall.
J. Brown: 1990, II in the Sky. In A.D. Irvine (ed): Physicalism in Mathematics. Kluwer Academic Publishers.
Cantor: 1932, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts. Hrsg. E. Zermelo, Verlag von Julius Springer, Berlin 1932; reprinted: Springer-Verlag, 1980.
P.J. Cohen: 1971, Comments on the Foundations of Set Theory. In D. Scott (ed): Axiomatic Set Theory, Proceedings in Symposia in Pure Mathematics, 13, part 1, AMS, Provindence, Rhode Island.
H.B. Curry: 1951, Formalist Philosophy ofMathematics. North-Holland, Amsterdam.
K. Gödel: 1947, What is Cantor's Continum Problem?, reprinted in: P. Benacerraf and H. Putnam (eds): Philosophy of Mathematics, Prentice-Hall.
G.H. Hardy: 1929, Mathematical Proof. Mind 38: 1–25.
K. Kunen: 1980, Set Theory. North-Holland.
I. Lakatos: 1976, Proofs and Refutations. Cambridge University Press, Cambridge.
S. Mac Lane: 1986, Mathematics: Form and Function. Springer-Verlag.
Y. Moschovakis: 1980, Descriptive Set Theory. North-Holland.
J. PARIS and L. Harrington: 1977, A Mathematical Incompleteness in Peano Arithmetic. In J. Barwise (ed): Handbook of Mathematical Logic. North-Holland, Amsterdam.
A. Robinson: 1964, Formalism 64. In Proceedings, International Congress for Logic, Methodology and Philosophy of Science.
W.V.O. Quine: 1964, Two Dogmas of Empiricsm, reprinted in: P. Benacerraf and H. Putnam (eds): Philosophy of Mathematics. Prentice-Hall.
W.V.O. Quine: 1969, Existence and Quantification. In Ontological Relativity and Other Essays. Columbia University Press, New York.
S.G. Simpson: 1987, Subsystems of Z2 and Reverse Mathematics. In G. Takuti (ed.): Proof Theory. North-Holland, Amsterdam.
S.G. Simpson: 1988, Partial Realisations of Hilbert's Program. Journal of Symbolic Logic 53: 349–363
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Wójtowicz, K. Unification of Mathematical Theories. Foundations of Science 3, 207–229 (1998). https://doi.org/10.1023/A:1009665705969
Issue Date:
DOI: https://doi.org/10.1023/A:1009665705969