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Re: [HM] Sets & Categories
Posted:
Jul 19, 1999 1:24 PM
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Bill Barton wrote:
> Dear John,
> In your message to the HM list about Sets & Categories (3 July) you
> wrote that "Historically, there have been other proposals for the
> organisation of mathematics or parts of mathematics..."
> I would be interested in an expanded version of this comment - a
> couple of leads to follow through myself would be sufficient.
Dear Bill,
I have included the complete quote from page 407 of Mac Lane's book
below. Also, I have now had an opportunity to read Colin McLarty's, The
Uses and Abuses of the History of Topos Theory, Brit. J. Phil. Sci. 41
(1990), 351-375. This is a fascinating description of the development of
category theory, and in particular topos theory and its subsequent
misconstrual by some as having orginally developed as an alternative to
set theoretic foundations for mathematics. The paper gives one the
feeling of a view into live history as the original sources include
conversations with some of the developers of the subject (e.g. Mac Lane,
Freyd, Lawvere), and since it illustrates how the psychological
(intuitive) precedence of one concept, e.g. set, can tend to both
mediate and obscure another, e.g. category.
Best wishes,
John
Here is the paper's ABSTRACT:
"The view that toposes originated as generalized set theory is a figment
of set theoretically educated common sense. This false history obstructs
understanding of category theory and especially of categorical
foundations for mathematics. Problems in geometry, topology, and related
algebra led to categories and toposes. Elementary toposes arose when
Lawvere's interest in the foundations of physics and Tierney's in the
foundations of topology led both to study Grothendieck's foundations
for algebraic geometry. I end with remarks on a categorical view of the
history of set theory, including a false history plausible from that
point of view that would make it helpful to introduce toposes as a
generalization from set theory."
Here is the full paragraph from Mac Lane's book, p. 407:
"Historically, there have been other proposals for the organization of
Mathematics or of parts of Mathematics. For the Greeks, Mathematics was
geometry, and they formulated real numbers and algebraic operations only
in geometric terms. In the 18th century, Mathematics appeared largely in
the development of all aspects of the calculus; this was a natural
reflection of the wide opportunities this development offered for formal
manipulations and for extensive applications. Subsequently the
extraordinary fruitful properties of holomorphic functions made complex
variable theory a center about which (much of) Mathematics could
revolve. There were competing organizations. In analysis, rigor was
enshrined under epsilon and delta. In geometry, Felix Klein proposed
that the many varieties of space provided by non-Euclidean and other
geometries could be classified and hence organized in terms of their
groups of symmetries--the full linear group, the orthogonal group, the
projective group, and others. In a way this organization did include
complex analysis as the study of groups of conformal transformations;
this approach amounts to putting a heavy emphasis on geometric function
theory in contrast to analytic function theory and the overuse of
epsilon-delta methods. Currently the remarkable properties of group
representations and their use in arithmetic and physics again suggests
that group theory provides and effective organization."
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