[FOM] Why not this theory be the foundational theory of mathematics?

Colin McLarty colin.mclarty at case.edu
Sun Mar 31 08:18:37 EDT 2019

> > Thomas Klimpel wrote (with much else)

> My remarks about category theory were triggered by Muller's paper
> (http://philsci-archive.pitt.edu/1372/1/SetClassCat.PDF), especially
> by the following passage in the middle of page 11:
> > When a category-theoretician talks about **Set**, *he means all sets,
> i.e. all sets available in the domain of discourse of mathematics*, which
> has now become V+, not V  \subset V+, because old V comprises less than a
> thee spoon of the sets in the indefinitely expanding cosmos V+. When a
> category-theoretician hears there are sets available of cardinality i1,
> 2^i1, i1^\omega, \alpha_i1, etc., he means to include *these too* in
> **Set**, because these are also sets according to these stronger
> set-theories, not mathematical objects distinct from sets.

Of course Muller quotes no category theorists saying such things, because
none does.   To the contrary, in that same paragraph, Muller quotes MacLane
saying his (MacLane's) foundation does not give a category of absolutely
all groups, for example.  Muller could have concluded that he misunderstood
what MacLane meant by *all sets* or *all groups* in the first place.  But
instead Muller says MacLane's idea is "is artificial and barks at his
explicit intentions."  Notice when Muller says "explicit intentions" he
means "implicit intentions," because MacLane never did utter the naive
claim Muller attributes to him.

This passage gave me the impression that foundations of category
> theory could turn into a fight over words.

Indeed the discussion often has become just such a fight.   Personally I
believe the best way to avoid fighting over words is to actually quote the
words of people you disgree with.   If you believe category theorists make
some claim, find a quote of one making it.

When I talked about category theory as too vague for a foundational
> theory, I thought about the passage at the top of page 2:
> > Lawvere [1966] proposed **CAT**, the category of all categories (save
> itself), as the domain of discourse of mathematics, and embarked on the
> endeavour to axiomatise **CAT** directly
> Independent of whether Lawvere came up with a nice set of axioms for
> **CAT** or not, this just feels too vague to me.

Here you and I agree.  Muller's sentence is entirely too vague to count as
an argument for a foundation for math.  But then, it was not supposed to be
such an argument.  To learn about axiomatizations of CAT you should read
papers that offer them.

best, Colin

> It is one thing to
> formalize category theory like group theory or lattice theory. If ten
> people tried this independently, at least eight would probably come up
> with equivalent formalizations. But if the same ten people tried to
> formalize **CAT**, they would end up with ten different non-equivalent
> formalizations.
> (If ten people tried to formalize set theory, none would come up with
> ZFC. Some would come up with inconsistent systems like Frege, some
> with second order formalizations like Zermelo, some with systems
> equiconsistent with finite order arithmetic like Russell and
> Whitehead, some with even weaker systems. Yes, set theory is also
> vague, but ZFC is not.)
> When I say "last word quantification", I try to capture an interesting
> property of the full semantics of second order quantification (which
> is absent from Henkin semantics). The second order induction axiom
> characterizes the natural numbers up to isomorphism. Initially,
> neither addition, nor multiplication or exponentiation is defined,
> only successor. But as soon as a symbol and axioms for an operation
> like exponentiation is added, the second order induction axiom
> includes it too (as a way to define properties). Since we know pretty
> well what we mean by the natural numbers, having the last word might
> not be bad in this case. However, a similar game could also be played
> for the real numbers (defined as a complete ordered field), but that
> does not mean that we know similarly well what we mean by the real
> numbers.
> Thomas
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