Foundations?

[F W Lawvere <wlawvere@buffalo.edu>]

	Down with "Foundations"! Up with algebra!

Dear Friends,

Presumably I am among those who are being "vilified" as "lovers of
categories as foundation". By avoiding any precise definition, such
a formulation might appeal to the widespread justified boredom induced by
the past hundred years of "foundations as justification".

	Whenever I used the word "foundation" in my writings over the past
forty years, I have explicitly rejected that reactionary use of the term
and instead used the definition implicit in the work of my teachers
Truesdell and Eilenberg. Namely, an important component of mathematical
practice is the careful study of historical and contemporary analysis,
geometry, etc. to extract the essential recurring concepts and
constructions; making those concepts and constructions (such as
homomorphism, functional, adjoint functor, etc.) explicit provides
powerful guidance for further unified development of all mathematical
subjects, old and new.

	What is the primary tool for such summing up of the essence of
ongoing mathematics? Algebra! Nodal points in the progress of this kind of
research occur when, as is the case with the finite number of axioms for
the metacategory of categories, all that we know so far can be expressed
in a single sort of algebra. I am proud to have participated with
Eilenberg, Mac Lane, Freyd, and many others, in bringing about the
contemporary awareness of

			Algebra as Category Theory

	Had it not been for the century of excessive attention
given to the alleged possibility that mathematics is inconsistent, with
the accompanying degradation of the F-word, we would still be using it in
the sense known to the general public: the search for what is "basic". We,
who supposedly know the explicit algebra of homomorphisms, functionals,
etc. are long remiss in our duty to find ways to utilize those concepts
also in guiding high school calculus.

	Best wishes, Bill


Bibliography:
- The Category of Categories as a Foundation for Mathematics,
La Jolla conference 1965, Springer (1966)

- Adjointness in Foundations, Dialectica, (1969),
to be reprinted in TAC

- Foundations and Applications: Axiomatization and Education,
Bulletin of Symbolic Logic, (2003) vol 9, pp 213-224

- Sets for Mathematics, w/ Bob Rosebrugh, Cambridge Univ. Press, (2003)


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F. William Lawvere
Mathematics Department, State University of New York
244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA
Tel. 716-645-6284
HOMEPAGE:  http://www.acsu.buffalo.edu/~wlawvere
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