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FOUNDATIONS AS TRUTHS WHICH ORGANIZE MATHEMATICS

Published online by Cambridge University Press:  09 May 2012

COLIN MCLARTY
COLIN MCLARTY*
Affiliation:
Departments of Philosophy and Mathematics, Case Western Reserve University
*
*DEPARTMENT OF PHILOSOPHY, CASE WESTERN RESERVE UNIVERSITY, CLEVELAND OH USA 44106 E-mail: [email protected]
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Abstract

The article looks briefly at Feferman’s most sweeping claims about categorical foundations, focuses on narrower points raised in Berkeley, and asks some questions about Feferman’s own foundations. Among many different senses of foundations, the one that mathematics needs in practice is a recognized body of truths adequate to organize definitions and proofs. Finding concise principles of this kind has been a huge achievement by mathematicians and logicians. We put ZFC and categorical foundations both into this context.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 6 , Issue 1 , March 2013 , pp. 76 - 86
Copyright
Copyright © Association for Symbolic Logic 2012

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References

BIBLIOGRAPHY

Cantor, G. (1897). Beiträge zur Begründung der transfiniten Mengenlehre. Mathematische Annalen, 46, 481–512and 49, 207–246. Reprinted by Ernst Zermelo in Cantor 1932, 282–356.CrossRefGoogle Scholar
Cantor, G. (1932). Gesammelte Abhandlungen mathematischen und philosophischen Inhalts. Berlin, Germany: Julius Springer. Edited with notes by Ernst Zermelo.Google Scholar
Dedekind, R. (1930–1932). Gesammelte mathematische Werke. Friedr. Braunschweig, Germany: Vieweg & Sohn. Three volumes.Google Scholar
Eisenbud, D. (1995). Commutative Algebra. New York: Springer-Verlag.Google Scholar
Feferman, S. (1977). Categorical foundations and foundations of category theory. In Butts, R. and Hintikka, J., editors. Logic, Foundations of Mathematics, and Comput- ability Theory. Dordrecht, The Netherlands: D. Reidel, pp. 149–169.CrossRefGoogle Scholar
Feferman, S. (2011). Foundations of category theory: What remains to be done. Slides of the Berkeley talk, on Feferman’s website at math.stanford.edu/ feferman/papers.html.Google Scholar
Feferman, S., & Kreisel, G. (1969). Set-theoretical foundations of category theory. with an appendix by G. Kreisel. In MacLane, S., editor. Reports of the Midwest Category Seminar III, Lecture Notes in Mathematics 106. New York: Springer-Verlag, pp. 201–247.CrossRefGoogle Scholar
Freyd, P. (1964). Abelian Categories: An Introduction to the Theory of Functors. Harper and Row. Reprinted with author commentary in: Reprints in Theory and Applications of Categories, No. 3 (2003) pp. 23–164, available on-line at /www.emis.de/journals/TAC/reprints/articles/3/tr3abs.html.Google Scholar
Godement, R. (1958). Topologie Algébrique et Théorie des Faisceaux. Paris, France: Hermann.Google Scholar
Gottschalk, W., & Hedlund, G. (1955). Topological Dynamics. Providence, RI: American mathematical Society.Google Scholar
Jech, T. (2006). Set Theory. New York: Academic Press.Google Scholar
Kan, D. (1958). Adjoint functors. Transactions of the American Mathematical Society, 87, 294–329.Google Scholar
Kanamori, A. (1994). The Higher Infinite. New York: Springer-Verlag.Google Scholar
Kelley, J. (1955). General Topology. New York: Van Nostrand.Google Scholar
Kreisel, G. (1952). Some elementary inequalities. Indagationes Mathematicae, 14, 334–338.Google Scholar
Kreisel, G. (1971). Observations on popular discussion of foundations. In Scott, D., editor. Axiomatic Set Theory. Providence, RI: American Mathematical Society, pp. 189–198CrossRefGoogle Scholar
Krömer, R. (2007). Tool and Object: A History and Philosophy of Category Theory. Basel, Switzerland: Birkhäuser.Google Scholar
Kunen, K. (1983). Set Theory: An Introduction to Independence Proofs. Amsterdam, Netherlands: North-Holland.Google Scholar
Lang, S. (2005). Algebra. Reading, MA: Addison-Wesley.Google Scholar
Lawvere, F. W. (1964). An elementary theory of the category of sets. Proceedings of the National Academy of Science of the United States of the America, 52, 1506–1511.Google Scholar
Lawvere, F. W. (1965). An Elementary Theory of the Category of Sets. Lecture notes of the Department of Mathematics, University of Chicago. Reprint with commentary by the author and Colin McLarty in: Reprints in Theory and Applications of Categories, No. 11 (2005) pp. 1–35, on-line athttp://138.73.27.39/tac/reprints/articles/11/tr11abs.html.Google Scholar
Lawvere, F. W. (1969). Adjointness in foundations. Dialectica, 23, 281–296.Google Scholar
Lee, J. (2009). Manifolds and Differential Geometry. Providence, RI: American Mathematical Society.CrossRefGoogle Scholar
Lie, S. (1888–1893). Theorie der Transformationsgruppen. Leipzig, Germany: Teubner. Three volumes, edited by Engel, Friedrich.Google Scholar
Mac Lane, S. (1964). Axiomatic Set Theory. Notes prepared by Murray Schacher and Richard Shaker, University of Chicago.Google Scholar
Mac Lane, S. (1969a). Foundations for categories and sets. In Category Theory, Homology Theory and their Applications, II (Battelle Institute Conference, Seattle, Wash., 1968, Vol. Two). New York: Springer-Verlag, pp. 146–164.Google Scholar
Mac Lane, S. (1969b). One universe as a foundation for category theory. In Reports of the Midwest Category Seminar. III. New York: Springer-Verlag, pp. 192–200.Google Scholar
Mac Lane, S. (1971). Categories for the Working Mathematician (first edition). New York: Springer-Verlag.Google Scholar
Mac Lane, S. (1986). Mathematics: Form and Function. New York: Springer-Verlag.Google Scholar
Mac Lane, S. (1998). Categories for the Working Mathematician (second edition). New York: Springer-Verlag.Google Scholar
Marquis, J.-P. (2009). From a Geometrical Point of View: The Categorical Perspective on Mathematics and its Foundations. New York: Kluwer.Google Scholar
McLarty, C. (1988). Defining sets as sets of points of spaces. Journal of Philosophical Logic, 17, 75–90.Google Scholar
McLarty, C. (1990). The uses and abuses of the history of topos theory. British Journal for the Philosophy of Science, 41, 351–375.CrossRefGoogle Scholar
McLarty, C. (1992). Failure of cartesian closedness in NF. Journal of Symbolic Logic, 57, 555–556.Google Scholar
McLarty, C. (1993a). Anti-foundation and self-reference. Journal of Philosophical Logic, 22, 19–28.CrossRefGoogle Scholar
McLarty, C. (1993b). Numbers can be just what they have to. Noûs, 27, 487–498.CrossRefGoogle Scholar
McLarty, C. (1994). Category theory in real time. Philosophia Mathematica, 2, 36–44.Google Scholar
McLarty, C. (2007). The last mathematician from Hilbert’s Göttingen: Saunders Mac Lane as a philosopher of mathematics. British Journal for the Philosophy of Science, 58, 77–112.Google Scholar
McLarty, C. (2008a). “There is no ontology here”: Visual and structural geometry in arithmetic. In Mancosu, P., editor. The Philosophy of Mathematical Practice. Oxford, UK: Oxford University Press, pp. 370–406.Google Scholar
McLarty, C. (2008b). What structuralism achieves. In Mancosu, P., editor. The Philosophy of Mathematical Practice. Oxford, UK: Oxford University Press, pp. 354–369.CrossRefGoogle Scholar
McLarty, C. (2011). A finite order arithmetic foundation for cohomology. Preprint on the mathematics arXiv, arXiv:1102.1773v3.Google Scholar
Mendelson, E. (1964). Introduction to Mathematical Logic. Princeton, NJ: Van Nostrand Co.Google Scholar
Munkres, J. (2000). Topology (second edition). Upper Saddle River, NJ: Prentice Hall.Google Scholar
Spivak, M. (1999). A Comprehensive Introduction to Differential Geometry (third edition). Wilmington, DE: Publish or Perish.Google Scholar
Weber, H. (1895). Lehrbuch der Algebra (first edition), Vol. 1. Braunschweig, Germany: F. Vieweg und Sohn.Google Scholar
Weber, H. (1896). Lehrbuch der Algebra (first edition), Vol. 2. Braunschweig, Germany: F. Vieweg und Sohn.Google Scholar
Whitehead, A., & Russell, B. (1910). Principia Mathematica, Vol. 1. Cambridge, UK: Cambridge University Press.Google Scholar
Ye, F. (2010). The applicability of mathematics as a scientific and a logical problem. Philosophia Mathematica, 18, 144–165.CrossRefGoogle Scholar
Ye, F. (2011). Strict Finitism and the Logic of Mathematical Applications. Number 355 in Synthese Library. New York: Springer.Google Scholar