Re: Epistemology 201: The Science of Science

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

Date: Tue, 25 Jan 2005 20:24:40 GMT

In article <U4pJd.11424$mo2.873618@news.xtra.co.nz>, "Jason" <jasonstevensNOSPAM@free.net.nz> writes:
>> >> 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.

Nope. The key to the proof above is the step where you reduce the
apriori infinite set of possibilities to a finite one, at which point
the individual cases can be enumerated and checked one by one. This is a
standard technique used in math for centuries. Now, would the finite
set consist of just few or, at most, few dozen cases, you could've
checked them by hand. Since the number, though finite, is still
rather large, you use a computer. The computer here is not a "key",
just a time saving device. That's all.
>
>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.
>
And if you take a proof aof a problem which reduces to few hundred
cases, with somebody checking them manually one by one, then you can
either believe that he made no error in checking or spend the same
amount of time repeating the verification. And even if, after
spending all the time needed, you got the same result, there is still
a non-zero (though very small) probability that the original checker
made an error in some place and you just happened to repeat same
error. This doesn't make it "empirical".

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
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  • Re: Epistemology 201: The Science of Science
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  • Re: Epistemology 201: The Science of Science
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