From: D. F. Siemens, Jr. <dfsiemensjr@juno.com>

Date: Thu Oct 09 2008 - 17:28:52 EDT

Date: Thu Oct 09 2008 - 17:28:52 EDT

On Thu, 9 Oct 2008 13:31:18 -0600 (MDT) gordon brown

<Gordon.Brown@Colorado.EDU> writes:

*> On Wed, 8 Oct 2008, D. F. Siemens, Jr. wrote:
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*>
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*> > As to proof in mathematics, note that it depends absolutely on the
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*> axioms
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*> > assumed. Some of these are so commonsensical that we do not
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*> usually
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*> > recognize that they cannot be proved except by reiteration.
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*> > Dave (ASA)
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*> >
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*>
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*> The modern attitude toward axioms in mathematics is different from
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*> the
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*> classical one. The key development in this change was the discovery
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*> that
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*> by changing the definition of certain terms in Euclidean geometry
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*> one
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*> could achieve a geometry in which Euclid's parallel postulate did
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*> not hold
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*> but would not lead to a contradiction unless Euclidean geometry did.
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*> Now
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*> rather than viewing axioms as being true commonsensical statements,
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*> the
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*> basic terms are taken as being undefined, and the question is not
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*> whether
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*> the axioms are true but rather whether they are consistent.
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*>
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*> Gordon Brown (ASA member)
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*>
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I see a practical problem with this claim, for it means that all the

specific terms in the axioms are meaningless. As a consequence, any set

of consistent axioms, that is, empty terms with empty relations, would be

investigated. However, it seems that only a limited set of terms and

relations are worked with. Thus a plane is a two-dimensional structure,

with the earlier assumption that it is Euclidean restricted. What I can

draw on a sheet of paper fits Euclid's or Playfair's parallel axiom, with

the unexampled assumption that it be infinite. But it is equally possible

to deal with the surface of the earth as a Riemannian plane. Also, the

mathematical functions are essentially the same whether we deal with real

numbers, modular numbers or infinities, though there are different

consequences. So I hold that there are, despite claims to avoid

explanations, tacit assumptions about the underlying meanings.

Dave (ASA)

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Received on Thu Oct 9 17:31:50 2008

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