Professor of
Mathematics
University of Manchester, UK
4. What do you consider the
most neglected topics and/or contributions in late 20th century
philosophy?
Considering the number of papers, theses,
books, lectures in Philosophy over this period it would be hard to find
any topic or contribution with more than provincial aspirations which
had been truly neglected, even when that was what is thoroughly
deserved.
Down at the level my own, local, interests however, an area which seems
to me to have remained strangely quiescent for the past 40 years is
Inductive Logic in the sense of Carnap. Following its birth with the
papers of W.E. Johnson in the 1920's and 30's, its independent discovery
by Carnap some 20 years later and the subsequent flurry of activity with
important contributions from de Finetti, Jeffreys, Kemeny et al further
developments have indeed been few and far. Indeed that whole approach to
understanding induction now seems to be out of fashion. This comparative
lack of activity is especially surprising given that in Schilpp's well
known 1963 volume of collected papers on The Philosophy of Rudolf
Carnap Kemeny sets out a clear agenda for the future development of
the topic. In part this might be explained by the increasing technical
complexity of mathematics involved though there were certainly
philosophical logicians around for whom this would not have have been
overly daunting.
Failing to respond to this challenge seems to me to have been an
unfortunate missed opportunity. For one thing I believe that recent
research shows that there are philosophically, and mathematically
interesting new results in inductive logic so that without the
intervening (mainly) dark age this could by now have been a flourishing
and stimulating area. Apart from the obvious intellectual pleasure that
has been lost, or at least postponed, the period has seen the rapid
development of other disciplines, AI and Cognitive Science in
particular, who are covering much the same ground (basically predicate
probability logic). Though their emphasis and aims may be different to
have had in place a well thought through body of knowledge and
theory(ies) with which to inform would certainly not have done our
subject any harm.
A second philosophical problem, which is often remarked on by
Mathematicians though I am not aware of any convincing explanation,
concerns the way `natural mathematical examples' tend to occupy
`distinguished positions' in classifications. To give a specific, and
particularly vivid, example of this phenomenon, we can classify the
complexity of sets of natural numbers by their Turing degrees (for this
example it is not important to know what this actually means). In this
classification the bottom position is taken up by the class O of
recursive sets and above that we have it's so called jump, the class
O′ which contains, for example, the set of codes for Turing Machines
which halt (eventually) when run on their own code. There are known to
be many classes strictly between these two `distinguished positions'.
However it is striking that whenever one has a `natural', as opposed to
artificial, set of numbers which lies somewhere in this range then it
either lies in O or in O′, never in one of the many
intermediate classes between them.
