FOM: Re: Reply to Franzen
torkel at sm.luth.se
Tue Jan 20 02:23:27 EST 1998
John Steel comments (in a message originally intended for the list)
on an earlier posting of mine:
>If we agree that there are sets, then don't we agree that
>"there are sets" is true? That's one tiny truth of set theory;
>fortunately there are more interesting ones!
> Mathematics is about sets, numbers,.. , and not (except in the
>metamathematical fragment) about formal provability. The "if-thenist"
>understanding of what mathematicians do is woefully inadequate; it
>fails to deal with the fact that our choice of axioms matters.
I agree that "if-thenism" is inadequate. However, the view I had
in mind, that "there are not any questions of truth or falsity where
set theory is concerned, but only questions of what can or cannot
be proved from certain formal principles" is not incompatible with
the observation that there are sets, or that "there are sets" is
true. What it *is* incompatible with is the view that it is problematic
whether there are sets, that there may or may not be sets - we can't
be quite sure. That is, what is ruled out is the view that there are
questions of truth and falsity in set theory in the sense of
questions that have a correct answer that we may or may not find out.
Wittgenstein, for example, had no problem with the observation that
there are sets, or that "there are sets" is true. After all, we do say
these things in mathematics - "there is a set such that...", "... is a
set", and so on. Claiming that mathematicians are mistaken in this
regard would be pretty strange, and anyway not our business as
philosophers (from this broadly speaking Wittgensteinian viewpoint).
But nor do the sayings and doings of mathematicians establish the
existence of any realm of sets in a sense that would make it
meaningful to agonize over the true answer to questions left undecided
by our formalized mathematical principles.
To use your own earlier formulation, sets exist (from the point of
view I'm explaining) in the sense that there exists a continuing story
about sets. Denying the existence of sets is no more appropriate than
denying that Watson and Holmes shared rooms. But equally, our ignorance
in matters left undecided by the basic principles of our mathematical
story is not genuine ignorance, any more than our ignorance about any
habits of Watsons not described in the canonical corpus is genuine
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