FOM: Does Mathematics Need New Axioms?

[John Steel]

I would like to reply to some comments by Harvey. 

>View A. (Most closely associated with John Steel). For the purposes of
>the
>main question, we should ignore distinctions between different kinds of
>mathematics for various reasons.

       If any part of mathematics needs new axioms, then mathematics needs
new axioms. Of course, the importance of the need relates to the
importance of the areas in need.


>Firstly, it doesn't make sense to me to disucss the main question without
>considering the views and attitudes of the mathematics community. After
>all, who is going to adopt new axioms?

      Is the question what it would be rational to accept, or what will
sell? I don't think we need to be doing market research on the attitudes
of what Harvey calls "normal" or "core" mathematicians.
      In the law (and this panel process certainly resembles an exchange
of legal briefs), there's a rule against hearsay evidence. Witnesses are
supposed to speak for themselves. In this connection, I wonder what Harvey
himself believes concerning the rationality of accepting large cardinal
hypotheses. Was it rational to do this in, say, 1990?

> If one chooses to ignore
>mathematicians, then the question could have been put very differently:

>Does set theory need new axioms? Does mathematical logic need new axioms?

   If set theory needs new axioms, then mathematics needs new axioms.

>Secondly, the distinctions are not only reasonably well defined, but also
>are intrinsically important. There are apparently fixed fundamental views
>and attitudes towards mathematical abstraction and generality that have
>began in earnest in the 60's and have been strengthened since then.

   How can you know views which "began in earnest in the 1960's" are
fixed and fundamental? We need a much larger frame of reference if we
are to discuss the foundations of all mathematics for all time.

   Let me add that I think applications are very important. The more
concrete and pervasive, the better. Boolean relation theory may turn out
to be a great success story for large cardinals; I hope so. If it does,
then it may be the first application of large cardinals in obtaining some
systematic, interesting, non-metamathematical consequences which are
concrete enough to be "absolute", i.e. Sigma^1_2 or simpler. That would
be an important development. 
   
   Nevertheless, we had strong reasons to believe that mathematics needs
large cardinal axioms by 1970, if not before. By 1990, the evidence was
overwhelming.

   I will reply to Harvey's discussion of V=L vs. PD in a separate post.

John Steel