FOM: "categorical foundations" -- an oxymoron
Colin Mclarty
cxm7 at po.cwru.edu
Mon Nov 24 10:37:31 EST 1997
Reply to message from simpson at math.psu.edu of Fri, 21 Nov
>
>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.
I have said from the start of this thread I am not trying to
establish the whole case for categorical foundations. I am trying
to establish that major parts of mathematics can be axiomatized
without presupposing classes or sets. And, by the way, that you can
define linear transformations before linear spaces (not only in
principle, but very usefully).
>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 are if you specify that each non-zero scalar has an
inverse. But when I talk about linearity I include modules over rings
(and indeed over sheaves of rings and more) among the examples.
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?
In fact, engineers almost always learn what a linear
transformation is long before they ever learn what a linear
space is. But I think this pedagogical fact is irrelevant to
Feferman's concerns--as he has always stressed, foundations as a
subject in its own right need not focus too much on practice let alone
pedagogy. We will not start the engineers on his theory of operations
and collections either--which I suppose you agree is foundational?
>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.
All of this is just as available on categorical foundations
as set theoretic. The only difference is that categorical foundations
put the transformations before the spaces, and put more useful math
into the characterizaton of linear transforms per se--you can connect
with whatever other math you want, but you can get genuinely useful
theorems without connecting.
That is, we often remark that mathematicians spend little
time deriving theorems from the first order group axioms, e.g.
Rather they spend most of their time one representation theorems
and various higher order definable classes of models. But they do
in fact spend time on proofs in the first order theory of an
Abelian category.
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.
This is the kind of sweeping claim that I will not argue
with. It is no use arguing about "love", the "very clear" and the
"far superior".
>
>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.
As Bill Tait has said, there is no use worrying about "all"
sets (and a fortiori about "all vectors spaces over IR") in any
absolute sense. However many ordinals you think there are, someone
can come along and posit more. I concede there is a ZF set theoretic
notion of "all" (conveyed by the universal quantifier in ZF) and
many people have it in mind when they say "all sets". I note that
it is equally axiomatizable in categorical set theory, and that
there are other notions as well (notably "all" in an Abelian category
with some internal completeness, or in a given topos).
But the "general logical" notion of "all" is just the one
I get easily for these axioms: They talk about "all" transformations
right from the start.
>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.
This is the kind of broader point I have argued elsewhere,
and might here sometime, but I hope to focus this thread on two
of Feferman's specific claims.
best, Colin McLarty
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