csrrc@daisy.warwick.ac.uk (R M Howarth) (03/28/88)
In article <7521@boring.cwi.nl> jurjen@cwi.nl (Jurjen N.E. Bos) writes: >In article <1009@sdcc13.ucsd.EDU> ln63wgq@sdcc13.ucsd.edu.UUCP (Keith Messer) writes: >>You know, that's my logic too, Bob. A mathematical (even if we have to call >>it mathematical) property is useful whether or not it is proven. Supposing >>I find some regularity in a mathematical system by induction (but in a case >>where mathematical induction is not adequate proof) and decide to write a >>program to do analysis based on that regularity. Either the program will >>succeed and be useful to me or it will fail, disproving my hypothesis. The >>point is that I win either way. > >I think you're missing something here. Your program NEVER will succeed, >because there is an infinity of cases. It can only fail. I think you're missing something here. Keith is talking about a result being USEFUL not about PROVING it. > If you want to >prove something, you need a proof, not a 'convincing' number of tries. Sure, but we don't, do we? After all, we're computer scientists, not mathematicians :-) Bob and Jurjen are of course right that a proper mathmatical proof is intellectually satisfying and essential to make a hypothesis respectable no matter how many examples in favour of it you find. But that isn't what Steven Tate and Keith were talking about. If I am 99.99% confident in the validity of the Rieman hypothesis and write a program which depends on it to work, then I can be *reasonably* confident in my program even without a "proof". Hell, even if the hypothesis were proven there would still be a > 0.01% (to say the least!) chance of a bug in my program, so in a practical sense the proof of the hypothesis does nothing to increase confidence in my program. OK, so next I try to prove my program is correct, but then where's the proof that my "proof" is correct ... ? Humans are fallible beings, so we have to accept the fact that in principle any hypothesis/program/aeroplane engine etc may fail in some situation. While this may not be very satisfying, it doesn't stop us making use of the thing concerned. We learn to live with this possibity of failure (though of course this doesn't prevent us from trying to keep the probability of failure down as low as possible). -Rolf P.S. Apologies if this article is less than topical for most people. Our news connection is abysmally slow.
srt@duke.cs.duke.edu (Stephen R. Tate) (03/31/88)
In article <507@sol.warwick.ac.uk> rolf@flame.warwick.ac.uk writes: >Bob and Jurjen are of course right that a proper mathmatical proof is >intellectually satisfying and essential to make a hypothesis respectable >no matter how many examples in favour of it you find. But that isn't what >Steven Tate and Keith were talking about. If I am 99.99% confident in >the validity of the Rieman hypothesis and write a program which depends >on it to work, then I can be *reasonably* confident in my program even >without a "proof". Well.... that's *sort of* what I said, anyway. It's true that unproven hypotheses can provide useful results, just be careful how you use them. I seemed to be lumped in with people who underestimate the power of proofs above -- this just isn't true. Basically, in my work, if I can't prove something, it's just as bad as if it wasn't true. *But*, that doesn't contradict what I said about useful programs being possible -- it's just that I generally don't write programs. Lastly, I hope that if you're doing work using unproven hypotheses (even something as plausable as the RH) that you're not working on something critical... I can just see someone working on this SDI stuff saying "Well, it looks like it works", and then when the real test comes that one case they didn't consider fails. -- Steve Tate UUCP: ..!{ihnp4,decvax}!duke!srt CSNET: srt@duke ARPA: srt@cs.duke.edu