From: gordon brown <Gordon.Brown@Colorado.EDU>

Date: Fri Oct 10 2008 - 13:29:29 EDT

Date: Fri Oct 10 2008 - 13:29:29 EDT

On Thu, 9 Oct 2008, D. F. Siemens, Jr. wrote:

*>
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*> On Thu, 9 Oct 2008 13:31:18 -0600 (MDT) gordon brown
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*> <Gordon.Brown@Colorado.EDU> writes:
*

*>> On Wed, 8 Oct 2008, D. F. Siemens, Jr. wrote:
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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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*>>
*

*>> 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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*>>
*

*>> Gordon Brown (ASA member)
*

*>>
*

*> I see a practical problem with this claim, for it means that all the
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*> specific terms in the axioms are meaningless. As a consequence, any set
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*> of consistent axioms, that is, empty terms with empty relations, would be
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*> investigated. However, it seems that only a limited set of terms and
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*> relations are worked with. Thus a plane is a two-dimensional structure,
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*> with the earlier assumption that it is Euclidean restricted. What I can
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*> draw on a sheet of paper fits Euclid's or Playfair's parallel axiom, with
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*> the unexampled assumption that it be infinite. But it is equally possible
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*> to deal with the surface of the earth as a Riemannian plane. Also, the
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*> mathematical functions are essentially the same whether we deal with real
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*> numbers, modular numbers or infinities, though there are different
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*> consequences. So I hold that there are, despite claims to avoid
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*> explanations, tacit assumptions about the underlying meanings.
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*> Dave (ASA)
*

*> ____________________________________________________________
*

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*

*>
*

The preferred modern approach to geometry is to use some approximation to

David Hilbert's axioms. The undefined terms are point, line, lie on,

between, and congruent. These are the basis for defining all other

geometric terms. Of course, the axioms also use nongeometric words such as

if and then as well as terms from set theory and arithmetic, both of which

have their own axioms. No matter what one thinks these undefined terms

should mean, only those of their properties which are given by the axioms

can be used in proofs. If one removes Hilbert's Parallel Postulate

(Playfair's Postulate), both Euclidean and hyperbolic geometry satisfy the

remaining postulates. Restore it and you have Euclidean geometry. If you

replace it by the Hyperbolic Parallel Postulate, you have hyperbolic

geometry.

For a nongeometric example of different interpretations of terms producing

valid models, we can have subsets of a given set together with unions,

intersections, and complements, or we can have propositions with and, or,

& negation, or we can have divisors of some given square-free integer n

with least common multiples, greatest common divisors, and division into

n.

Gordon Brown (ASA member)

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Received on Fri Oct 10 13:32:12 2008

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