FOM: reply to the "list 2" crowd
Charles Silver
csilver at sophia.smith.edu
Fri Jan 23 07:47:22 EST 1998
I appreciated Vaughan Pratt's explanation of the differences
between a set-theoretic and a category-theoretic approach to foundations,
and I'm trying to digest his points. It would help me to understand the
notion of "continuum" that he explains is basic to categorical
foundations. I'll pluck out a quote from what he said, to provide
motivation. Also, I want to register a mild disagreement below.
Vaughan Pratt:
> Whereas set theory starts from the notion of a discrete collection
> and aims for the notion of continuum without ever quite reaching it
> satisfactorily, category theory does the exact opposite. It starts
> from the notion of "continuum," conceived just as fundamentally as
> "collection", and aims for the notion of collection without ever quite
> reaching it satisfactorily.
This may just be obtuseness on my part, but I don't understand how
the notion of a "continuum" is used here as a foundational concept. I
admit that I've been "corrupted" by set theory, and it may be that I have
a slightly different notion of "continuum" in my head.
Here's the part I mildly disagree with:
Vaughan Pratt:
> To avoid misunderstanding it should be said that ZF itself does not start
> with a preconceived notion of collection and go from there. ZF assumes
> only the language of first order logic with equality and an uninterpreted
> binary relation (membership). In fact there is no a priori assumption
> at all about the universe being described by the axioms, other than the
> traditional (but unnecessary) assumption that it is nonempty. ZF weaves
> the entire notion of set from the whole cloth of first order logic.
>
> How one *reads* ZF is another matter.
I think the picture you present above is of ZF tumbling out of
first-order logic on its own and later being interpreted (or '*read*') by
us in a certain way. I don't think this is quite correct. It seems to me
that the axioms are constructed by us in the first place in order to
capture the conception of interest. For example, Hilbert's axiomatization
of geometry is an axiomatization *of* the facts of geometry. One wouldn't
say of a first-order axiomatization of geometry that it "weaves the entire
notion of geometry from the whole cloth of first order logic" and that
"how one *reads* that axiomatization is another matter". To me, that
image suggests that axiomatizations just pop out of logic, like a
jack-in-the-box, and that later we examine them to see what these
axiomatizations are axiomatizations *of*. The reason for my disagreeing
with this at some length is to suggest that there's an underlying
*conception* that stimulates the search for axioms that will capture that
conception. I know what I'm saying is vastly oversimplified. For example,
the position can be taken that the axiomatic versions of set theory were
developed merely to salvage Frege's theory and to avoid paradox--and that
there was *no* real conception during that period. This last point could
buttress your view that the "conception" popped up later (which I think is
not quite correct). I think--but I cannot argue persuasively for this
point--that even during the period of uncertainty after Frege's theory
there was still something like an "underlying conception" of "set" even if
it was in a relatively nebulous state at this time.
Charlie Silver
Smith College
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