FOM: Re: 87:Programs in Naturalism

[Penelope Maddy]

Dear Harvey,

I just have a couple of remarks on your 'Programs in Naturalism' (5/15/00).

First, speaking of your disagreements with my naturalism, you write

>In addition, I, personally, am still very interested in - even optimistic
>about - old fashioned intrinsic justifications for set theory, including
>large cardinal axioms, even though these have proved very hard to develop
>in any convincing way.

I'm sorry if something in my book suggests that I reject intrinsic 
justifications, because I don't.  My goal was to make room for extrinsic 
justifications as well, not to rule intrinsic justifications out.

Second, you write

>1. FRAGMENTATION.
>
>The idea here is that different branches of mathematics have such widely
>different goals that rational practitioners may come to very different
>preferences as to the choice of axioms. What may be a compelling reason for
>practitioners in one branch to conclude that they need new axioms - or even
>need to adopt specific new axioms - may not be a compelling reason for
>practitioners of another branch.

Yes, the particular goals of a particular branch of mathematics make it 
rational for mathematicians pursuing that branch to mold their definitions 
in certain ways, employ distinctive methods of proof, restrict their 
attention to structures of certain sorts, etc.  It might make good sense, 
for example, for a particular group of mathematicians to assume V=L, that 
is, to restrict their attention to constructible sets.  Given that 
contemporary set theorists have good reasons to adopt large cardinal axioms 
inconsistent with V=L, this might look like fragmentation.

But it isn't, because it remains the set theorist's job (that is, one of 
the set theorist's goals) to provide a unified arena for all of 
mathematics, including the practice of this particular group of 
V=L-assuming mathematicians.  And it's easy:  the set theorist interprets 
them as working within L.

Different branches of mathematics will develop in pursuit of their own 
goals, perhaps even to the extent of adopting different existential axioms, 
but the set theorist in turn will strive to develop an axiom system that 
permits him to encompass all these disparate practices.

Yours,

Pen