Re: Epistemology 201: The Science of Science
From: Jason (jasonstevensNOSPAM_at_free.net.nz)
Date: 01/25/05
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Date: Wed, 26 Jan 2005 11:04:13 +1300
> [context is discussion of 4 color problem]
>
> >Well, they verified the cases by computer.
>
> Lots of mathematics is verified with pencil and paper. Does the use
> of pencil and paper make it empirical.
Yes, the idea can cover pencil and paper as well. It doesn't make their use
empirical, but it does show how fragile the concept can become.
> > But how do you prove an algorithm
is
> >correct?
>
> The same way that you demonstrate that a mathematical proof is
> correct. Oh, by the way, there are many incorrect proofs in the
> published literature.
Yes, I think you're right. There is no algorithm that can check algorithms, but
we could do it on a case-by-case basis I suppose. I didn't do this when they
accepted the proof though, by the sounds.
> > The 'proof' of the
four
> >colour problem is partially inductive.
>
> If that is correct, then all logic is partially inductive.
I'm not quite sure I follow your reasons here, but I don't think there is
disagreement for the most part.
> >The computer literally did the colouring of maps and counting of the colours.
>
> The computer was used as a book keeping tool, to keep track of
> details to numerous for ordinary human attention. This is not an
> empirical investigation, except in the strange meanings you seem to
> be giving to "empirical".
While I'd like to take credit for the strangeness, but it was Chaitin (and
Tymoczko for the four colour problem). The strange meaning of "empirical" here
is the lack of correctness proof in algorithms, trust in their implementation,
trust in computers. You wouldn't run the proof on one computer once, it would
be run on different computers using different algorithms several times before
accepting the proof.
If it turns out that something is not provable due to incompleteness, then why
shouldn't it be proved empirically and a new axiom added?
> >It was accepted after they tried it on other computers with different
> >programmes.
>
> Traditional proofs are accepted only after people have worked through
> the details of the proof. Recoding the program and running on
> different computers is just part of the normal working through a
> proof.
>
> > But again, this is inductive evidence. That it is a legitimate
> >proof is controversial.
>
> You are misusing "inductive", much as you have been misusing "empirical".
I don't think so. These concepts are not so easily nailed down and defined.
What inductive reasoning doesn't use deductive reasoning and vice-versa?
>
> Although there was some initial controversy over the idea of using a
> computer in the proof, I have not seen much evidence that it is
> currently considered controversial.
Compsci and maths are merging in discrete maths. I'm sure mathematicians are
more willing to use computers as they learn more about algorithms and
vice-versa.
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