Re: Epistemology 201: The Science of Science

From: Jason (jasonstevensNOSPAM_at_free.net.nz)
Date: 01/25/05 

Date: Tue, 25 Jan 2005 23:22:43 +1300

> >> A proof is a proof (in mathematics) solely through the logical
> >> mechanisms that enables one step of the proof to be infered from an
> >> earlier step or steps. A rigorous proof is strictly formal, even if non
> >> formal means were used to discover it. Distinguish between discovery and
> >> justification. Discovery can be very empirical and heuristic, but
> >> justification (actual proof) is formal.
> >
> >Maths as a formal system is incomplete, so some statements cannot be proven
as
> >derivations from the axioms. Some of these statements are true or false
under
> >the standard interpretation of the language of mathematics. In these cases,
> >discovery IS justification. Proof is empirical:
> >
> >The four colour map problem was finally 'proved' by computer. That is, every
> >possible combination of neighbouring map shapes were tried and tested. This
is
> >empirical. There is (or at least was at the time) no know formal method to
> >prove it.
> >
> Oh, you miss the key thing here. A priori, the number of possible
> combinations is infinite. It was only *after* you had a proof that
> any of these combinations is equivalent to one from a smaller, finite
> set, that you could go and check the combinations in this finite set,
> one by one. Absent the proof, and yes, it was a formal proof, you
> could run all the computers in the world for eternity, getting
> nowhere.
>
> So, yes, there most certainly *was* a formal method to prove it,
> consisting of reducing the problem to a finite number of cases.

They key to the proof being empirical is that a computer is used. So if the
algorithm can be deductively proven to be correct, then I'll concede. Otherwise
all that can be done is to test it on different computers with different
programmes until we're satisfied, which is inductive.

My contention is not that the four colour problem is not formally provable. It
may well be by someone with a lot of time on their hands. But unless the
referees are prepared to spend the same amount of time, then they can only
appeal to a computer proof.

BTW, this is not to say that it is not admissible as a proof. I think it should
be. It is just another argument for quasi-empirical mathematics.

Relevant Pages

  • Re: Epistemology 201: The Science of Science
    ... > set, that you could go and check the combinations in this finite set, ... all that can be done is to test it on different computers with different ... My contention is not that the four colour problem is not formally provable. ... It is just another argument for quasi-empirical mathematics. ...
    (sci.physics)
  • Re: Epistemology 201: The Science of Science
    ... > set, that you could go and check the combinations in this finite set, ... all that can be done is to test it on different computers with different ... My contention is not that the four colour problem is not formally provable. ... It is just another argument for quasi-empirical mathematics. ...
    (sci.cognitive)