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

Date: Fri Oct 10 2008 - 16:23:43 EDT

Date: Fri Oct 10 2008 - 16:23:43 EDT

On Fri, 10 Oct 2008 11:29:29 -0600 (MDT) gordon brown

<Gordon.Brown@Colorado.EDU> writes:

*> >>
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*> > I see a practical problem with this claim, for it means that all
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*> the
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*> > specific terms in the axioms are meaningless. As a consequence,
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*> any set
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*> > of consistent axioms, that is, empty terms with empty relations,
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*> would be
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*> > investigated. However, it seems that only a limited set of terms
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*> and
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*> > relations are worked with. Thus a plane is a two-dimensional
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*> structure,
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*> > with the earlier assumption that it is Euclidean restricted. What
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*> I can
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*> > draw on a sheet of paper fits Euclid's or Playfair's parallel
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*> axiom, with
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*> > the unexampled assumption that it be infinite. But it is equally
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*> possible
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*> > to deal with the surface of the earth as a Riemannian plane. Also,
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*> the
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*> > mathematical functions are essentially the same whether we deal
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*> with real
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*> > numbers, modular numbers or infinities, though there are
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*> 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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*> > ____________________________________________________________
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*> > Love Graphic Design? Find a school near you. Click Now.
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*> >
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*>
*

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*> >
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*>
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*> The preferred modern approach to geometry is to use some
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*> approximation to
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*> David Hilbert's axioms. The undefined terms are point, line, lie on,
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*>
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*> between, and congruent. These are the basis for defining all other
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*> geometric terms. Of course, the axioms also use nongeometric words
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*> such as
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*> if and then as well as terms from set theory and arithmetic, both of
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*> which
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*> have their own axioms. No matter what one thinks these undefined
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*> terms
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*> should mean, only those of their properties which are given by the
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*> axioms
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*> can be used in proofs. If one removes Hilbert's Parallel Postulate
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*> (Playfair's Postulate), both Euclidean and hyperbolic geometry
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*> satisfy the
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*> remaining postulates. Restore it and you have Euclidean geometry. If
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*> you
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*> replace it by the Hyperbolic Parallel Postulate, you have hyperbolic
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*>
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*> geometry.
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*>
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*> For a nongeometric example of different interpretations of terms
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*> producing
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*> valid models, we can have subsets of a given set together with
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*> unions,
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*> intersections, and complements, or we can have propositions with
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*> and, or,
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*> & negation, or we can have divisors of some given square-free
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*> integer n
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*> with least common multiples, greatest common divisors, and division
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*> into
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*> n.
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*>
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*> Gordon Brown (ASA member)
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*>
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So the current attitude is to deal with a purely formal system with terms

whose sole meaning is their abstract interrelationship. However, their

activity betrays something deeper, for I have not found them playing with

any random set of symbols in arbitrary combinations. Hilbert's axiom set

sprang from the desire to have proofs that do not depend on anything but

the logical relationships, whereas Euclid's version required deriving

some evidence from the diagrams. A similar restriction holds, I think,

with the modification of Peano's postulates so that they specifically

produce the sequence of integers, which was the original intent, rather

than multiple sequences.

Since mathematical calculi, like logical calculi, consist of tautologies,

any substitution instance that "begins" true will maintain truth. In the

case of logical calculi, a requirement is that a term and its complement

must cover the entire universe of discourse. This gives a problem with

terms which have no sharp line of distinction. While 'bald' is the usual

term cited, the problem applies broadly. We usually don't worry about

'red' and 'not-red', but there is also 'reddish' and 'partly red' among

empirical applications. It's easier to posit terms with no empirical

exemplification than to sweat applied mathematics.

Dave (ASA)

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Received on Fri Oct 10 16:28:56 2008

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