Re: Epistemology 201: The Science of Science

From: Lester Zick (lesterDELzick_at_worldnet.att.net)
Date: 02/07/05 

Date: Mon, 07 Feb 2005 15:28:49 GMT

On Sun, 06 Feb 2005 13:26:47 +1300, Jason
<jasonstevensNOSPAM@free.net.nz> in comp.ai.philosophy wrote:

>>>[snip]
>>>
>>>
>>>>>Yeah. We interpret "+" to mean addition that we are familiar with, but
>>>>>in the formal system(s) of maths it is just a mark on the page. The
>>>>>axioms show the legal moves that can be done with this mark in moving
>>>>
>>>>>from one sentence in the language of maths to another.
>>>>
>>>>
>>>>>Rules are defined axiomatically through great pains, but they are
>>>>>typically done in predicate logic and using the material implication to
>>>>>go from one string to another. Each step is a deduction. Even
>>>>>"mathematical induction" is deduction.
>>>>
>>>>
>>>>So I would have to assume that any mechanism used to define the rules
>>>>would be a step up the axiomatic ladder from math to logic?
>>>
>>>A step up the "meta" ladder. Meta-maths is where the rules of maths is
>>>looked at. This could be couched in another formal system, with or
>>>without logic. Or it might be quite an informal thing (although in the
>>>case of maths it isn't).
>>
>>
>> Well mathematikers seem especially anxious to keep anything with or
>> about rules with mathematical implications in the province of math. I
>> would say there is tautological logic and math derived from it rather
>> than trying to define metamath when we really aren't even sure what
>> math is. Of course we might say that math is whatever metamath says it
>> must be. But I don't see any recognition of tautological logic in such
>> an observation at least in mechanical terms.
>
>Well I guess there are any number of arithmetic axioms to choose from,
>studied in 'meta-maths'. The ones that survive are the ones that seem
>to work the best with how we understand arithmetic to work. So they
>start out to be descriptive of maths, and then become prescriptive at
>the limits of our mathematical intuition.

Yes, but can they describe the limits of our meta-intuition. Obviously
resort meta-science and meta-mathematics only regresses conceptual
problems never addressed or resolved by science or mathematics itself.

>I'm not sure there is such a thing as tautological logic. Tautologies
>only seem to arise when assumptions have already been laid down in
>advance. But Aristotelian logic, through to Frege's predicate logic
>certainly seems to have been a big influence in maths.

Well, I'm a lot surer there is tautological logic than mathematics or
science because tautologies are a lot easier to spell out and examine.
They're simple. Aristotelian logic of syllogistic inference is hugely
complex in comparison and wrong in many respects.

Regards - Lester

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