Path: utzoo!utgpu!news-server.csri.toronto.edu!cs.utexas.edu!asuvax!ncar!zaphod.mps.ohio-state.edu!rpi!uupsi!sunic!sics.se!sics.se!torkel From: torkel@sics.se (Torkel Franzen) Newsgroups: comp.ai.philosophy Subject: Re: Intuition and doubt (was Re: Minds, machines, and Godel) Message-ID: <1991Feb21.202449.22562@sics.se> Date: 21 Feb 91 20:24:49 GMT References: <25220.9102201346@s4.sys.uea.ac.uk> Sender: news@sics.se Organization: Swedish Institute of Computer Science, Kista Lines: 59 In-Reply-To: jrk@information-systems.east-anglia.ac.uk's message of 20 Feb 91 13:46:44 GMT In article <25220.9102201346@s4.sys.uea.ac.uk> jrk@information-systems.east- anglia.ac.uk (Richard Kennaway CMP RA) writes: >This is a second attempt to post this. Apologies if it did get out >the first time. This is getting ridiculous! I only saw this article the first time it appeared thanks to the generous nntp policy of the University of Minnesota, and I've failed since in trying to post both an article and a cancel message. There's been a lot of difficulties lately at my end with getting/sending articles. We're having serious trouble keeping this war up, and maybe it's a sign from heaven that it's time to give this a rest. So I'll just make one more attempt to get out with my second and more temperately worded response to your message. You write: >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. You seem to be saying that the fact that no inconsistency has yet been found in the axioms actually used by mathematicians is good evidence that no inconsistency exists in the full set of axioms. You don't claim to have any justification for this. You do have a couple of further comments. First, your remarks on generalizations and support. These are not helpful, since they make no distinction at all between theories. Take any theory T extending ZFC. Adapting your own words, we may argue that there is good empirical evidence that T is consistent, given that our mathematical theorems are provable in T. "You see the observations as supporting at most the consistency of PA; I take them as supporting the consistency of T." Your second observation consists in emphasizing that the induction axioms are not an arbitrary collection, but seem to form a natural whole. Here we must ask in what sense this impression is supposed to yield any "empirical evidence" for the consistency of the axioms? Also, for any set theory T there are people who regard its axioms as forming a natural whole. So, as far as the evidence for consistency is concerned, my conclusion is that your grounds for claiming that arithmetic is consistent are no less arbitrary, peculiar, or subjective, and no more empirical in any obvious sense, than just those appeals to intuition which you so sternly rejected in your original postings. You make much of my not presenting any theory of mathematical evidence, but this is surely a bit disingenuous, since you ignore the question of what, if anything, mathematical proofs achieve. As I have emphasized, there is nothing special about consistency theorems from the point of view of mathematical evidence. That 7+6=13, that addition is commutative, that every number is the sum of four squares, that a problem is NP-complete, that a problem is recursively undecidable, that a differential equation has a certain asymptotic behavior - in considering mathematical evidence we must consider what the evidence for such theorems of ordinary mathematics amounts to. All that emerges from your remarks is a determination to say that there is no evidence other than the (in some sense) empirical. There exists a considerable body of thought, not necessarily found in Usenet articles, that does a great deal better.