Path: utzoo!utgpu!news-server.csri.toronto.edu!cs.utexas.edu!wuarchive!sdd.hp.com!spool.mu.edu!uunet!shelby!agate!ucbvax!information-systems.east-anglia.ac.uk!jrk From: jrk@information-systems.east-anglia.ac.uk (Richard Kennaway CMP RA) Newsgroups: comp.ai.philosophy Subject: Re: Intuition and doubt (was Re: Minds, machines, and Godel) Message-ID: <25220.9102201346@s4.sys.uea.ac.uk> Date: 20 Feb 91 13:46:44 GMT Sender: daemon@ucbvax.BERKELEY.EDU Lines: 122 This is a second attempt to post this. Apologies if it did get out the first time. I (Richard Kennaway) asked: >So I ask you once more: are you claiming that there is better evidence than >the empirical for the consistency of arithmetic? And you (Torkel Franzen) cagily replied: > Certainly; of the same kind as the evidence for theorems of >ordinary mathematics in general - whatever may be the nature of that >evidence. I also asked what such evidence might be, concerning which you said: > Now it is of course perfectly legitimate to ask: if the evidence for >the consistency of arithmetic or analysis is not empirical, what is >it? And does it exist? These are classical questions in the philosophy >of mathematics. I don't believe any very good answers have been >arrived at So you believe that (1) There is good evidence but (2) You don't know what it is. So much for the intuitive evidence of consistency. I turn to the empirical evidence, i.e. the fact that no inconsistency has yet turned up. Your rebuttal of the empirical argument, by means of the hierarchy of induction axioms, appears to be this: (1) The induction axiom can be split into a strict hierarchy of stronger and stronger induction axioms. (2) All existing mathematics that can be carried out in arithmetic, can be carried out using only the first few members of this hierarchy (plus the other axioms). (3) Therefore, the absence of contradictions in such mathematics is at best only evidence in favour of the consistency of those first few induction axioms. It is quite possible (leaving aside the mysterious other evidence, such as it might be) that those first few could be consistent, yet later ones inconsistent. There is a problem with this argument, a well-known and fundamental one: what is empirical evidence evidence of? Suppose that, by definition, all A's are B's, and I hypothesise that all B's are C's. If I test this hypothesis by observing some B's, and find that they are all C's, what grounds do I have for believing my hypothesis? If it happens that all the B's I looked at were A's, should I only consider the hypothesis that all A's are C's to be supported, the stronger one being in as much doubt as before? Suppose I only define A after having made the observations, and define it to be exactly the set of observed instances of B? Were one to argue so, one would be denying the possibility of ever supporting a generalisation by particular evidence. Yet that is the situation in your argument. A = the first few induction axioms in the hierarchy, B = the whole lot, C = does not lead to a contradiction. (Or alternatively: A = mathematical arguments which have been performed in arithmetic, B = all possible mathematical arguments which could be performed in arithmetic, C as before.) You see the observations as supporting at most the consistency of A; I take them as supporting the consistency of B. I can't give any satisfactory justification for my choice, any more than you can for the non-empirical evidence. I don't believe anyone else can. This is just the problem of induction (the comp.ai.philosophy relevance), and we are as unlikely to get an answer to it in this discussion as we are to get an answer concerning what other evidence there is for the consistency of arithmetic. [Footnote: not that anyone reading this should need to be told, but just in case: the "induction" referred to in "the problem of induction" has only coincidentally the same name as the "induction axioms" under discussion.] >Since you make no attempt to >suggest the existence of anything corresponding to physical theory in >the case of the consistency statement... Let me rectify this omission. What corresponds in mathematics, and specifically the hierarchy of induction axioms, to that background of physical theory that makes prediction of the sunrise more than just "what has been, will be"? The fact that those induction axioms are not an arbitrary collection, but seem to form a natural whole. Now, I am being very vague (because I cannot be more precise), and as a result this may seem an unutterably feeble argument. But no more feeble than the corresponding claim in the physical sciences. Successive sunrises are as logically independent of each other as are the higher induction axioms from the lower. The connection between sunrises is not in the world but in our theory. What grounds do we have for believing any of the predictions of that theory at all? Those grounds, whatever they might be (and I agree that a mere "what has been, will be" is no good), do we have for believing in the consistency of arithmetic. >...it is very easy to delude oneself into regarding >evidence that is in fact based on our abstract understanding of the >axioms as somehow empirical... If you like, you can call the reason for choosing the conjecture one takes the empirical evidence to support, "intuition" or "abstract understanding", but this doesn't solve the problem, it just gives it a name. (BTW, I suspect delusions in the reverse direction are just as easy.) We seem to have reached philosophical brick walls in each direction. Or perhaps we are on different sides of the same one. I believe (in the sense of today's .signature quote) that the empirical evidence supports the consistency of arithmetic, but cannot say why I conjecture that consistency, rather than that of some weaker theory. You believe there is non-empirical evidence, but cannot say what it is. Can we leave it there? -- Richard Kennaway SYS, University of East Anglia, Norwich, U.K. Internet: jrk@sys.uea.ac.uk uucp: ...mcsun!ukc!uea-sys!jrk "Believe", vb.: Doubt, as in "I believe it's going to rain".