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Foundations and Applications: Axiomatization and Education

Published online by Cambridge University Press:  15 January 2014

F. William Lawvere
F. William Lawvere*
Affiliation:
Mathematics Department, Suny at Buffalo, 244 Mathematics Building, Buffalo, Ny 14260-2900, USA.E-mail:[email protected]
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Abstract

Foundations and Applications depend ultimately for their existence on each other. The main links between them are education and the axiomatic method. Those links can be strengthened with the help of a categorical method which was concentrated forty years ago by Cartier, Grothendieck, Isbell, Kan, and Yoneda. I extended that method to extract some essential features of the category of categories in 1965, and I apply it here in section 3 to sketch a similar foundation within the smooth categories which provide the setting for the mathematics of change. The possibility that other methods may be needed to clarify a contradiction introduced by Cantor, now embedded in mathematical practice, is discussed in section 5.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 9 , Issue 2 , June 2003 , pp. 213 - 224
Copyright
Copyright © Association for Symbolic Logic 2003

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References

REFERENCES

[1] Cartier, P., Groupes algébriques et groupes formels, Colloq. théorie des groupes algébriques, (Bruxelles 1962) (Paris), Gauthier-Villars, pp. 86–111.Google Scholar
[2] Eilenberg, S. and Steenrod, N., Foundations of algebraic topology, Princeton University Press, 1952.Google Scholar
[3] Grothendieck, A., Techniques of construction in analytic geometry, Cartan seminar, 1960.Google Scholar
[4] Hurewicz, W., k-spaces, unpublished lecture referred to by Gale, David, Compact Sets of Functions and Function Rings, Proceedings of the American Mathematical Society, vol. 1, 1950, pp. 303–308.Google Scholar
[5] Isbell, J., Adequate subcategories, Illinois Journal of Mathematics, vol. 4 (1960), pp. 541–552.Google Scholar
[6] Lawvere, F.W., Thesis: Functorial semantics of algebraic theories, Columbia University, 1963, (reprinted Buffalo Workshop Press, POB 171, Buffalo, N.Y. 14226).Google Scholar
[7] Lawvere, F.W., The category of categories as a foundation for mathematics, La Jolla conference on categorical algebra, Springer-Verlag, 1966, pp. 1–20.Google Scholar
[8] Lawvere, F.W., Categories of space and of quantity, The space of mathematics: Philosophical, epistemological and historical explorations, De Gruyter, Berlin, 1992, pp. 14–30.CrossRefGoogle Scholar
[9] Lawvere, F.W., Cohesive toposes and Cantor's “lauter Einsen”, Philosophia Mathematica, vol. 2 (1994), pp. 5–15.Google Scholar
[10] Lawvere, F.W., Toposes of laws of motion, 1997, video transcript: http://www.acsu.buffalo.edu/~wlawvere/downloadlist.html.Google Scholar
[11] Lawvere, F.W., Volterra's functionals and the covariant cohesion of space, Categorical studies in Italy, Rendiconti del Circolo Matematico di Palermo. Serie II. Supplemento, vol. 64 (2000), pp. 201–214.Google Scholar
[12] Truesdell, C., Mechanical foundations of elasticity and fluid mechanics, Journal of Rational Mechanics and Analysis, vol. 1 (1952), pp. 125–171 & 173–300.Google Scholar
[13] Volterra, V., Sur une généaeralisation de la théorie des fonctions d'une variable imaginaire, Acta Mathematica, vol. 12 (1889), pp. 233–286.Google Scholar