FOM: Does Mathematics Need New Axioms?

[Harvey Friedman]

Reply to Maddy 3:32PM 5/11/00.

>Putting these two together, my guess is that you think intrinsic
>justifications are of considerable interest, so the search for them should
>not be abandoned, but that extrinsic justifications are legitimate, too,
>and might even stand alone in some cases if they are strong
>enough.  Right?  (If so, we are in complete agreement here.)

This is an exactly correct statement of my position.

>Harvey's point is that mathematicians in other areas of mathematics do not
>now have identifiable goals that require them to adopt large cardinal
>axioms.  Quite right, but I agree with John that this fact doesn't
>undermine the set theorist's justification for adding them to his theory.

I agree that set theorists have enunciated a coherent system of goals that
makes it compelling for them to add large cardinals to their theories.

However, other set theorists - perhaps none of the ones in the profession
now - could have other goals that make it equally rational - perhaps
compelling - for them to add V = L to their theories.

The normal mathematician has some other goals that are quite different -
and perhaps incompatible - and for them, I claim that either adding no new
axioms or adding V = L are the most compelling moves (or nonmoves).

[See below - e.g., that the previous paragraph is about to change].

I will be saying that the adoption of V = L is not only rational, but
virtually imperative for normal mathematicians and mathematics as a whole -
except that it turns out - totally unexpectedly for the normal
matheamtician - to fail to meet his goals after all because of Boolean
relation theory. In the aftermath of this unexpected failure, acceptance of
large cardinals is perhaps inevitable by normal mathematicians. Certainly
reflection principles over large cardinals are inevitably going to be
accepted by normal mathematicians.

>This is the argument I try to spell out at the end of my book (where John's
>influence is much in evidence).  In fact, it seems to me that his outline
>contains an improvement of the crude idea of 'fair interpretation' proposed
>there:

These kinds of arguments make sense for the set theorist with their goals,
but not for the normal mathematician, with their goals. And not for the
mathematics community taken as a whole.

On this basis, it may well be the case that you and I do not have any
substantial disagreement. I would like to speculate that when you and your
collaborators do a detailed naturalistic investigation of the mathematics
community, you may well have a better understanding of the details of my
strong positions. The mathematics community thinks so differently from the
set theoretic community that you have studied so well, as to be shocking
beyond belief.