FOM: "categorical foundations" -- an oxymoron
Stephen G Simpson
simpson at math.psu.edu
Fri Nov 21 20:47:06 EST 1997
A comment on McLarty's posting entitled "categorical foundations for
linear transformations".
In the earlier discussion of Chow's lemma, McLarty has already shown
that his use of the term "foundations" is inconsistent with the way
the term has been generally used in the last 150-200 years in
academic/scientific circles. Now McLarty's discussion of
"categorical foundations for linear transformations" provides more
evidence of that inconsistency.
McLarty claims to be doing "categorical foundations". In actual
fact, all he gave is a set of axioms L1-L4 for a category of vector
spaces and linear transformations. The axioms aren't even even
specific to vector spaces. They aren't a foundation for linear
transformations or anything else. They don't do what foundations of
X are normally expected to do: explain X in terms of basic concepts
and indicate the essential relationships between X and the rest of
human knowledge. Imagine teaching undergraduate engineering
students the basics of linear transformations. Are you going to
begin by explaining that a linear transformation is any arrow in any
category satisfying axioms L1-L4? This is obviously nonsense. The
fact that, according to McLarty, some specialized graduate students
may find these categorical axioms useful is irrelevant to the
foundational issue. McLarty also adduces the fact that Grothendieck
won a Fields medal; this is also irrelevant.
Contrast McLarty's axioms L1-L4 with normal set-theoretic
foundations. The set-theoretic foundation of linear transformations
is to start with sets, define functions, algebraic structures of
various kinds, Abelian groups, fields, vector spaces, mappings, and
finally linear transformations, then to give examples, and then to
prove theorems. This places vector spaces and linear
transformations in a proper context, where they can be compared with
and merged with other types of mathematical structures (matrices,
vector bundles, etc), so that the connections with the rest of
mathematics (also explained set-theoretically) are apparent or can
straightforwardly be elucidated. I have no great love for
set-theoretic foundations, but in this and countless other
instances, it is very clear that set-theoretic foundations are far
superior to the structuralist or category-theoretic approach.
McLarty's discussion of "all" (as in the category of "all" R-linear
transformations) is also illuminating. The impetus of McLarty's
discussion of "all" is an obvious defect of McLarty's axioms L1-L4,
namely that they fail to capture the notion of linear
transformation, because they don't imply that ALL linear
transformations are obtained. In talking around this defect,
McLarty hedges, but he seems to be implying that the very concept of
"for all" is a set-theoretic concept, for which pure category-theory
has no use. I have to point out that, McLarty notwithstanding, "for
all" is a general logical concept, indispensable for all scientific
reasoning. Thus, on even this very general and basic level, McLarty
has thrown away a key foundational connection.
This entire posting by McLarty is an excellent example of how
structuralism is anti-foundational, because it disrupts virtually all
of the essential ties and links between various subjects or branches
of human knowledge.
Sincerely,
-- Steve
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