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*To*: fom@math.psu.edu*Subject*: FOM: natural examples*From*: S B Cooper <pmt6sbc@amsta.leeds.ac.uk>*Date*: Thu, 29 Jul 1999 16:59:47 +0100 (BST)*Reply-To*: s.b.cooper@leeds.ac.uk*Sender*: owner-fom@math.psu.edu

Just a short remark in regard to part 2 of Steve's summary of Harvey's comments (circulated in his Wed 28 July FOM message): One very often sees something along the lines "The only natural examples [of r.e. sets and degrees] are the original ones, i.e. the halting problem and the complete r.e. degree", and a qualification (even if providing nothing new to most of us) is appropriate. There are of course a number of different definitions of 'natural' in this context, depending on different notions of 'natural' information content or of degree theoretic context. It is true that all the known canonical c.e. sets (e.g. those associated with standard first-order axiomatic theories, the halting problem, etc) turn out to be computable or of complete c.e. degree, and that for a pure mathematician that is very significant. It may also turn out that 0 and 0' are the only Turing definable c.e. degrees. However, there are mathematical criteria according to which *all* c.e. sets and Turing degrees potentially contain 'natural' information content which may be encountered in specific contexts - just to mention two well-known examples: 1) (Feferman, Hanf) All c.e. degrees contain (finitely) axiomatisable theories, and 2) (Matiasevich) All c.e. sets are diophantine. In recognising such mathematically immediate levers to incomputable information content, one may then (and this leads on to a purely personal response along the lines of h-i of Steve's/Harvey's part 1 - Defending Specialised Subjects) allow the probability of the existence of material counterparts. And then one would have to expect consequences. There are a number of results (many already mentioned) pointing in this direction, but the persisting speculative nature of the discussion that makes one ask how such 'real world' incomputabilities are to be recognised. If one looks to the history of mathematics and the way it gives often unexpected explanations of physical phenomena, it is the development of theory which opens the perceptual doors. (For instance, the acceptance of gravitational 'action at a distance' implicit in Newtonian dynamics forced itself on people mainly by the explanatory power of the theory in relation to the observed motion of the moon and planets.) In the case of computability theory, if the apparent permeation of the material and epistemological universe by nascent incomputabilities is confirmed, then the Turing model for computationally complex environments has immediate relevance. And that relevance may only become widely recognised through the development of the theory of that model to the point where it is capable of explaining something people actually care about. Of course, there is no guarantee that every arcane body of theory, even if built on fundamental concepts, will eventually explain anything. But those of us who are led by intellectual curiosity (and that is 'the *real* reason', maybe incapable of the kind of reduction Steve is looking for) into (what may appear to be) more abstruse research topics, can look to one of Gian-Carlo Rota's characteristic quotes (taken from an interview with MIT Tech Talk) for encouragement: ________________________________________________________________________ Applications are found after the theory is developed, not before. A math problem gets solved, then by accident some engineer gets hold of it and says, 'Hey, isn't this similar to...? Let's try it.' For instance, the laws of aerodynamics are basic math. They were not discovered by an engineer studying the flight of birds, but by dreamers -- real mathematicians -- who just thought about the basic laws of nature. If you tried to do it by studying birds' flight, you'd never get it. You don't examine data first. You first have an idea, then you get the data to prove your idea. ____________________________________________________________________ Of course, Turing's 1936 paper and the development of the stored-memory computer is a famous example of how ostensibly esoteric 'pure' research can lead to important applications. In fact, it is hard to think of a publication in any other area of mathematical logic that has had so celebrated an influence outside of pure mathematics. Steve may propose perfectly valid criteria of relevance for assessing the long-term value of a particular field of research. Unfortunately (as computability theory tells us) those criteria may not be effectively implementable - and while one may on occasion be forced to attempt such implementation, in so doing one must be aware that more damage than benefit may result. And one such damaging effect can be demoralisation among people producing some very nice mathematics - not necessarily a good basis for a 'productive' change of direction, just for diminished research activity. -- Barry ____________________________________________________________________________ S Barry Cooper Tel: UK: (0113) 233 5165, Int: +44 113 233 5165 School of Mathematics Fax: UK: (0113) 233 5145, Int: +44 113 233 5145 University of Leeds Email: s.b.cooper@leeds.ac.uk Leeds LS2 9JT Home tel: (0113) 278 2586, Int: +44 113 278 2586 U.K. WWW: http://www.amsta.leeds.ac.uk/~pmt6sbc ____________________________________________________________________________

**Follow-Ups**:**Steve Stevenson**- FOM: natural examples

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