Re: Epistemology 201: The Science of Science
From: Lester Zick (lesterDELzick_at_worldnet.att.net)
Date: 02/02/05
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Date: Wed, 02 Feb 2005 15:30:33 GMT
On Wed, 2 Feb 2005 03:21:04 +0000 (UTC), Neil W Rickert
<rickert+nn@cs.niu.edu> in comp.ai.philosophy wrote:
>"Jason" <jasonstevensNOSPAM@free.net.nz> writes:
>
>>> You are perhaps referring to First Order Predicate Calculus (FOPC).
>>> And indeed, mathematicians do use FOPC. However, mathematics is not
>>> FOPC, and FOPC is not sufficiently expressible to allow it to be used
>>> exclusively.
>
>>> Given a particular system of axioms, say PA (the Peano Axioms),
>>> mathematicians could in principle use FOPC applied to those axioms.
>>> But mathematics is not confined to working within a particular axiom
>>> system. Moreover, the discussion axiom system itself is part of
>>> mathematics.
>
>>Maths is an extension of FOPC, like PA.
>
>Not really. Mathematics is much older than FOPC, so it doesn't make
>sense to say it is an extension of FOPC.
Mathematics is a subset of differences and differences between
differences etc. regardless of age and as such is certainly a
derivative of predicates and predication in one form or another.
Age as a justification is just an historical anachronism.
>> The ZFC axioms are conventionally used
>>and assumed, as far as I am aware.
>
>Again, not really. Mathematicians often try to make do with minimal
>axioms.
>
>> If another system is used in maths then
>>people need to know about it. The ZF system without the axiom of Choice for
>>example, can lead to the creation of two spheres out of one in topology.
>
>I'm not sure of your point there.
>
>If you happen to be making a vague reference to the Banach-Tarski
>paradox, then you have it wrong. Banach-Tarski does depend on the
>axiom of choice.
>
>>The study of axioms don't take place in maths. It is meta-logic or meta-maths
>>that deals with this. Godels theorem for example is a meta-mathematical proof.
>
>While Goedel's theorem is meta-mathematics, nevertheless a lot of
>mathematics is effectively a study of axioms and their consequences.
>
>>> >Since mathematics has evolved along-side science and plays a large part in
>>> >describing and predicting how the world works, then as a formal system goes,
>>it
>>> >seems to be on the money as far as capturing something about the world.
>
>>> That's your opinion. As a mathematician, I have a different
>>> opinion. I consider it important that mathematics is not about the
>>> world. Roughly speaking, mathematics is about what would happen if
>>> reality did not intrude. We discover a lot about reality by seeing
>>> how it differs from the mathematical ideal.
>
>>Fair enough. The formal system of maths is ripe for exploration. People study
>>it divorsed from the world. But why spend so much time on maths and not some
>>other formal system? I think because of the close link maths has with the
>>world.
>
>There you go again. You talk about "the formal system of maths", but there
>is no such formal system. Then you suggest that we should instead
>study some other formal system. It is gibberish.
>
Regards - Lester
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