[FOM] When is it appropriate to treat isomorphism as identity?
meskew at math.uci.edu
Wed Apr 22 23:45:16 EDT 2009
On Tue, Apr 21, 2009 at 2:14 PM, Timothy Y. Chow <tchow at alum.mit.edu> wrote:
> Why is ZFC so much more powerful than what we need for most mathematics?
> I would say that part of the reason is that many times, to carry out some
> mathematical argument, one needs the existence of something (an algebraic
> closure or a set of subsets, say) whose existence is really "not very
> interesting"---i.e., it's just a technicality that is tangential to the
> real problem of interest. Thus mathematicians simply brush away such
> technicalities by saying, in effect, "Whenever we need something silly
> like that, we'll just assume that it exists." That way they don't have to
> think too hard about something that is not what they're really interested
> in. Existences of this sort are often not "necessary" in your sense.
This characterization of mathematical existence seems wrong. The
mathematicians I know all seem very careful with their existence
claims. For instance it's not completely trivial to show that every
field has a (unique up to isomorphism) algebraic closure, and I
haven't seen an introduction to field theory that skips over the
argument for it. The rule, "whenever we need something silly like
that, it exists," is not only imprecise but too liberal. Such a
stance seems to license assuming the existence of a countably
additive, translation invariant measure on the real line, which is an
intuitive principle with some interesting implications, but is of
course not possible in the presence of the axiom of choice. Also,
some interesting statements about real numbers can be proved from
large cardinals, but most mathematicians don't seem that eager to make
large cardinal assumptions. I think existence is not so silly.
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