Jeff Paris


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.

Read the remaining part of Jeff Paris' interview in the book Formal Philosophy

ISBN-10    87-991013-1-9    hardcopy
ISBN-10    87-991013-0-0    paperback
Published by Automatic Press ● VIP, 2005

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