[FOM] When is it appropriate to treat isomorphism as identity?
Timothy Y. Chow
tchow at alum.mit.edu
Tue Apr 21 17:14:53 EDT 2009
Vaughan Pratt wrote:
>This raises the following question. It is clear that arguments
>purporting to show either the existence or uniqueness of any given
>entity beg the question of the necessity of the rules used in those
>arguments. That said, is the situation entirely symmetric between
>existence and uniqueness, or is there some reason to suppose that
>existence arguments might be supportable by rules having a more
>necessary character than those supporting uniqueness arguments?
If I understand what you mean by "necessary character," then I would say
that, from the point of view of a working mathematician, "necessity" just
means "necessary for studying the particular mathematical question I'm
interested in right now." The level of precision with which one treats
existence, uniqueness, or indeed any mathematical concept is always
dictated by the needs of the problem at hand. So I would say that there's
no particular reason to think that existence is any different from
uniqueness in this regard.
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.
In contrast, when the existence or uniqueness issue is something of
paramount mathematical interest---the existence and uniqueness of a
solution to a particular PDE, say---then proportionate care will be taken
to ensure that the appropriate level of precision is used to set up the
definitions.
In short, what drives "necessity" is not the *form* of the mathematical
statement, but the *interest* people have in the question.
Tim
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