Re: Epistemology 201: The Science of Science

mmeron_at_cars3.uchicago.edu
Date: 01/25/05 

Date: Tue, 25 Jan 2005 04:17:39 GMT

In article <IliJd.11286$mo2.865083@news.xtra.co.nz>, "Jason" <jasonstevensNOSPAM@free.net.nz> writes:
>> > I repeat, some mathematical proofs are by empirical means...
>>
>> 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.

Mati Meron | "When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same"

Relevant Pages

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  • Re: Epistemology 201: The Science of Science
    ... Discovery can be very empirical and heuristic, ... > justification is formal. ... the standard interpretation of the language of mathematics. ... The four colour map problem was finally 'proved' by computer. ...
    (sci.math)
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