FOM: Toposy-turvey
Colin Mclarty
cxm7 at po.cwru.edu
Sun Jan 18 13:30:16 EST 1998
Reply to message from friedman at math.ohio-state.edu of Sat, 17 Jan
Friedman ends his post by saying meaningful discussion will
be possible only if I "start by admitting" that I am "grossly
mistaken" (the quote is included at the end). I have always
suspected this is the best possible argument against categorical
foundations: simply refuse to consider them.
But he may have overstated his demand, and anyway others
may want to know how I reply to his more particular arguments. So
I will reply.
>I can begin to see how one can, quite falsely, begin to think that the
>notion of category helps in understanding the notion of function. But I
>have been completely unable to see how one can begin to think that the
>notion of category helps in understanding the notion of collection.
And I do not see why it is a problem. Do you see how the
notion of continuous map helps in understanding the idea of space?
And it did so before anyone had a set theoretic analysis of space,
indeed when lots of mathematicians believed space could not be made
up of points let alone be a set of points.
Certainly this does not PROVE you can understand sets by way of
functions. But it is an analogy and I want to see where our ideas begin
to differ.
>Obviously, the way that the concept of collection is assimilated in little
>children is by placing two or three actual physical objects into a bunch or
>group and talking about the bunch or group instead of just talking about
>the actual physical objects. Of course, later there is the leap that the
>items in a collection can themselves be of a rather arbitrary character -
>e.g., a collection itself. Of course, this leads to all kinds of
>interesting things including the paradoxes, but that is another story for
>another time.
Yes, precisely. Nothing you say here contradicts categorical
foundations. The differences only come up when you get to that "other
story", the cumulative heirarchy or something like it.
>Probably the first thing you need to know about collections is that any two
>collections are identical if and only if they have the same members. At
>least, that's the first thing that's normally taught after the concept is
>"explained."
Notice it is also the first thing the student is told to ignore
when math textbooks use sets. The book defines or posits the integers,
defines rationals as equivalence classes of pairs, defines real numbers
as classes of Cauchy sequences--and then carries on as if all integers
are rational numbers and real numbers. Some books comment on the abuse
of notation, others seem not to notice. Either way the sets Z, Q, R
are carefully defined, and then the student is asked to ignore the
identity of the elements.
>And probably the first thing you need to know about functions is whether or
>not one is talking about functions as rules, or functions as black boxes.
>The latter obeys extensionality, whereas the former does not. In other
>words, do you only care about the input/output relationship, or do you care
>about the process? The history of mathematics shows that it is more
>productive to concentrate on the extensional concept, and analyze the
>intensional concepts in terms of extensional concepts. E.g., analyze the
>notion of algorithm in terms of ordinary extensional concepts in the theory
>of partial recursive functions, programming languages, and models of
>computation, etcetera.
And yet millenia of brilliant mathematicians managed to create
virtually all the mathematics that anyone but professional mathematicians
ever learns today, without knowing this distinction. Even grossly
abusing it. And university mathematics majors today routinely get
degrees without understanding it. You may deplore the fact but it is
true.
It is an important distinction, and must be made. It should
be more widely understood than it is. But you do not need to start
with it.
>>And now I have completely stated the category axioms.
>
>You haven't stated the axioms completely since you have hidden some
>complications. E.g., that it is a many sorted theory, with partially
>defined operations.
I haven't formalized the statement. We are talking about
motivation here, and I do not believe the motivation for ZF or topos
foundations or anything else lies in the formalized version. I believe
formalization is a tool for reflective study of motivations. As
to details of formalization, I prefer a multisorted partial operator
language for category theory, but you can as well use a single
sorted language with predicates. These are routine details.
> Also, you only have an axiomatic system, and that
>doesn't explain the concepts of collection and functions at all.
Yes, formal axiomatics is not explanation. That is why, in
the part just quoted above, I concentrated on understanding and no
formalization.
> You see,
>an axiomatic theory of something doesn't do much to explain the underlying
>concepts unless a little bit goes a long way; i.e., a few simple basic
>axioms allow one to derive an unexpectedly massive amount of diverse
>important facts. Without that, it goes almost nowhere in terms of an
>explanation.
This is the deep point and I hope we can discuss it in depth.
It is certainly vast, and I'm sure you have a sense of it that I do
not and I'd like to learn more about that. But obviously we cannot
go anywhere serious with it if you are convinced that categorical
foundations are in fact logically circular, ill-posed, hostile to
logic, and can only be discussed after an admission of being
"grossly mistaken". We'll have to clear up those issues first.
>One major reason for this gross difference is the following. The usual set
>theoretic foundations leverages over the universal language for all of
>logical thought - for all, there exists, and, or, not, if then, iff,
>equals. This counts for zero complexity since it underlies all systematic
>thought. (Of course, there is much much much more to thought than just
>this!). Categorical foundations seeks to replace or explain these notions
>in terms of other "notions." This may be a nice idea for certain technical
>constructions, which may be of some use in limited contexts in mathematics
Certainly not replace, but explain yes. And give new technical
tools for formal treatment. This began with Grothendieck and is one
aspect of SGA 4 that has become standard in advanced geometry and
number theory (while the idea of topos itself is still marginal to a
lot of people). It works to good ends.
My favorite use in logic is Makkai's Stone theorem. By the
original Stone theorem, given a preorder you can tell whether it is
(order isomorphic to) the preorder of models of a sentential theory,
and if it is you can find a theory whose models give it. It hs long
been known that the category of models of a first order theory plus the
elementary embeddings is not enough to determine the theory (not even
up to definitional extensions et c). No analogue to Stone's theorem
was known for the first order case. By Makkai's theorem here is the
information you ned: You need a category plus a kind of infinitary
"product" structure on it. Then you can tell whether it is the
category of models of a first order theory, with the "product" being
ultraproduct, and if it is you can find the theory (up to definitional
extension).
>
>Some category theorists have suggested that category theoretic foundations
>may become competitive because categorical foundations doesn't need: for
>all, there exists, and, or, not, if then, iff, equals, whereas set
>theoretic foundations does. But the actual development of actual
>mathematics done this way would be incomprehensible.
These people are talking about what you need to get started.
Of course you'll need all that to get all of math. But some of us,
including Peter Aczel I now learn, like the way categorical foundations
can get started with very simple logic. Zermelo set theory can hardly
do anything without its axiom scheme of separation--thus explicit
reference to the whole apparatus of formal first order logic. Topos
theory is finitely axiomatized.
>>...I even grant that you need informal
>>understanding of a lot of things: arithmetic, functions, explicit
>>deductive reasoning, and more, before any kind of foundational axioms
>>will make sense to you.
>
>Explicit deductive reasoning is a universal given, across the intellectual
>landscape. You should make the most effective use of it, since it is
>inevitable. Don't hide it inside some jargon. Of course, I admit that
>hiding it in some jargon creates a technical generalization - which may
>well have some technical uses in limited contexts.
You cannot mean that undergraduates universally come equipped
with a grasp of explicit deductive reasoning. Of course it is vital
to teach the idea.
Several times you have suggested that category theorists are
somehow against logic. I know people with various ideas about
formalizing it, which you might disagree with. But I don't know anyone
who wants to hide it or abandon it. Can you be more specific?
>
>>I do not believe that discrete structureless
>>collections stand out among mathematical objects as THE ONES you have
>>to know about to understand a foundation.
>
>The finite structureless collections seem to play a special role in thought
>and intellectual development. Bow to the inevitable!
They "seem to" to you. Not to me.
Bertrand Russell had a nice definition of naturalist ethics:
An ethical naturalist gives a theory of how people act, and then
declares 'immoral' any action which tends to show his theory is false.
Let's not just insist on a certain theory of how people understand
math, and then declare "incomprehensible" any approach to fom that
tends to show this theory is false.
>
>> I wonder if that much disagreement really does itself preclude
>>meaningful discussion. It might but I hope not.
>
>You can always admit that you are grossly mistaken. Then we can have a
>meaningful discussion on how you became grossly mistaken. How about it?
>
Colin McLarty
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