FOM: Response to Soare's long posting

[Joe Shipman]

Dear Professor Soare,

Thanks very much for posting your response to Simpson on the FOM.  The
whole point of the FOM list, which people seem to forget at "slow" times

when only a few people post regularly, is to stimulate discussion and
interchange on the foundations of mathematics rather than promote the
moderator's particular views.

I think Simpson (among others) was disturbed by the apparently insular
trend of recursion theory over the last three decades, and became
suspicious that the recent terminological changes (computability theory
rather than recursion theory) were an attempt to artificially hide this
by linking classical recursion theory with a popular subject (theory of
computation, in particular computational complexity) which was really
distinct from it.  The other way of looking at it, which you may yet
persuade Simpson of (anyway you can persuade the rest of the FOM list),
is that recursion theory is, as all healthy mathematical subjects
periodically do, returning from a phase of "internal" development to a
fruitful "external" phase where connections to other subjects become
stronger, and that the connections to computation theory in particular
have become significantly stronger recently, justifying a formal
remarriage of the two subjects which had developed relatively
independently for a quarter century or so.

It is of considerable interest that the connections to and applications
of classical recursion theory involve its *methods* rather than its
*results* (which is probably related to the absence of natural examples
of degrees between 0 and 0', or indeed of any natural degrees not in
{0,0',0'',...,0^(omega),0^(omega+1)...}.  This is not a major criticism
of classical recursion theory; but if this absence of natural examples
continues for much longer, it will be reasonable to contend that
classical recursion theory is only important for the methods it
inspired, and that it is not "about" a subject matter of intrinsic
importance.  What is your opinion on the existence of "natural" examples

of intermediate degrees?

I look forward to further posts on the FOM list from you.

-- Joe Shipman