FOM: iterative conception of set: are sets natural for everything?
Charles Silver
csilver at sophia.smith.edu
Sun Mar 1 08:56:16 EST 1998
On Sat, 28 Feb 1998, Vaughan Pratt wrote:
> From: Charles Silver <csilver at sophia.smith.edu>
> >I think the fact that this question can be raised shows that not
> >any set theory can be claimed to be foundationally superior to, say, CAT,
> >in being firmly based on an underlying conception. One can *say* that set
> >theory is based on "collection", but not have it turn out that the
> >resultant set theory is a satisfactory explication of this concept. To my
> >mind, this raises further questions about how to judge whether an
> >intuitive conception is successfully embodied within a technical
> >formalism.
>
> I interpret Charlie to be questioning how far one can push the notion
> of set as a universal representation while retaining the naturality that
> motivated it in the first place. This is a very good question.
Thank you. What you say below, though interesting and
informative, is not exactly what I had in mind. Unfortunately, what I had
in mind is quite foggy and pretty simple-minded. Harvey has said, for
example, that kids learn about collections quite early. I agree with
this, sort of. I think kids learn to *count* *things*. I think kids
learn to put things (say blocks) in rows, in piles, and in more complex
configurations. To my way of thinking, these items are akin to primitive
data structures (which relates to some things you said). I see children
learning to recognize some "number" of things this way (essentially by a
kind of intuitive matching; something like a 1-1 correspondence), and I
also think they firm-up concepts of identity in this way. For example, a
child learns that one block with the letter 'A' on it is "the same as"
another one that also has the letter 'A' on it. So, he or she may prefer
to use the newer of the two blocks in building up a structure, where
either block could be used. Well, I don't wish to go on with this. I'm
wondering where "collection" is actually needed here. I see numbered and
identified "things" (like blocks), but I don't see "collections." This
corresponds somewhat with your point about programming languages (though I
don't agree entirely with what you say about them).
At any rate, the above attempts a crude connection between
mathematical concepts and early block-stacking of children. Connections
can also be made to the counting done in primitive societies. But, I want
to switch to a different point. In beginning set theory books, terms like
'collection', 'assemblage', and even 'group' are used to get the basic
idea off the ground. Let's call the *fundamental* idea that of
"collecting things". As I understand it, set theory is the technical
elucidation of the informal concept of "collecting things". How faithful
does 'set' represent a "collection"? Let me give an example that may
help. I remember once hearing from a well-known set theorist that Cantor
provided a philosophical explication of the concept of "same number of
elements". That is, before Cantor, we pretty much knew what it meant for
two finite sets to have the same number of elements, but what wasn't known
was what it could mean for two infinite sets to have the same number of
elements. According to this set theorist, Cantor, in providing his
definition of cardinality, actually "solved" a philosophical problem. In
the field of ethics, this would be somewhat like giving a successful
philosophical explication of the concept of "good".
Anyway, when the idea of Cantor having solved a philosophical
problem with his definition of cardinality was presented to me, I thought
about it and tried to assess it. When I thought about it, it seemed
flat-out right! To me at the time, this felt great! A successful
solution to a philosophical problem! Wow! (Though I must confess that
the to-my-mind strange proofs of the Schroder-Bernstein Theorem diminished
my enthusiasm somewhat. I didn't know then and don't know now why there
shouldn't be a more straightforward proof of this intuitively correct
result [which seems closely related to Cantor's definition of same Card].)
Getting back to the informal notion of "collection" and the formal
concept of "set": I wonder how faithful the notion of "set" is to the
informal concept it is supposed to represent. Does "set" capture
"collection" in a philosophically satisfying way? Right away, I realize
that "set" can't cover *all* of what we mean by "collection". Some
features of our thinking about "collection" probably should be thrown out
immediately. On the other hand, "set" *acquires* new meanings that don't
seem to really be part of the notion of "collection". For example, take
the collection of the number 3, a singleton. Then, take the collection of
two things, one being the number 3 and the other being
the-collection-of-the-number-3. Is there an informal "collection" for
this at all? Of course, we *do have* the set of this, namely {3,{3}}.
But, in this case, it looks as though "set" has acquired a meaning that is
not rooted in the informal concept of a "collection". I don't know how to
judge these seeming differences between the informal concept and its
technical "representation". So, I wondered whether anyone had some views
on how to judge whether an intuitive conception is successfully embodied
within a technical formalism. Is "set" a satisfying philosophical
embodiment of "collection"? This seems to me to be a hard but interesting
question, though I know that it is also a fuzzy one.
Then there is the further point about the cumulative hierarchy. I
am supposing that any set theory worth its salt is supposed to embody this
in some direct way. It seems as though it is accepted these days that a
"real" set theory must in some sense be *about* the cumulative hierarchy,
and that set theories that seem to diverge from this goal (of capturing
the cumulative hierarchy in a straightforward way) are *not* "real" set
theories at all. This assumption on my part provoked my asking Randall
Holmes (and Thomas Forster, by the way) whether Quine's NF was a "real"
set theory in this sense (given my impression that the set theory in _Set
Theory and its Logic_ is *not* a real set theory in this sense).
I also have some quibbles about what you say about data structures
in programming languages, but I think I've said enough. I am assuming
that a lot of what I've said above represents some confusions and
misunderstandings about set theory and its relationship to the fundamental
notion of a "collection". So, I would appreciate any corrections and
explanations that anyone wishes to provide.
Charlie Silver
Smith College
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