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Topic: Euclid's Elements Book 1 Proposition 1 - something Euclid missed?
Replies: 62   Last Post: Feb 22, 2014 12:11 AM

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 ken quirici Posts: 336 Registered: 1/29/05
Re: Euclid's Elements Book 1 Proposition 1 - something Euclid missed?
Posted: Feb 17, 2014 4:09 PM

On Monday, February 17, 2014 9:42:41 AM UTC-5, David Hartley wrote:
>
> Quirici <kquirici@yahoo.com> writes
>

> >It would be an interesting exercise (at least it seems to me) to
>
> >imagine this Euclidean axiomatic system INDEPENDENT of ANY pictures.
>
> >How many 'realities' would this system model? I suspect there are many
>
> >that have absolutely nothing to do with geometry. That is, take the
>
> >axioms as simply words that relate to each other in logical ways and
>
> >see what they can represent.
>
>
>
> I was going to suggest this to you, but you've got there on your own.
>
> Imagine trying to explain Euclid's proofs to a blind man. If it can't be
>
> done solely from the axioms and postulates, with no diagrams, then it's
>
> not a valid proof.
>
>
>
> It is indeed true that the explicit axioms and postulates in the
>
> Elements are not sufficient for even the simplest of his proofs.
>
>
>
> Most trivially, there's nothing to ensure that there even exist any
>
> points or lines.
>
>
>
> More importantly, as was raised at the start of this thread, without an
>
> axiom that ensures that certain circles intersect, most of the
>
> constructions fail.
>
>
>
> Consider a system consisting of a single straight line and the points
>
> thereon. It satisfies all Euclid's explicit postulates (and Pasch's
>
> axiom) but circles have only two points in their circumferences and
>
> circles with a common radius do not intersect. There are no equilateral
>
> triangles.
>
>

there's problem in that his definitions include (at least in Heath's translation) surface - that which has length and breadth only.

>
> So even if you take Euclid's postulates and add the existence of point
>
> and line, Pasch's axiom and any necessary order axioms, you still don't
>
> have enough to prove Proposition 1.
>
>
>
> Add an axiom that ensures there is a point not on a given line and it
>
> may be possible, I don't know.
>
>
>
> --
>
> David Hartley

Thanks. I'm hoping Moise's book on Elementary Geometry from an Advanced Standpoint will provide a sufficiently rigorous approach. Have you read it? If so, is it sufficiently rigorous?

Regards,

Ken

Date Subject Author
2/14/14 ken quirici
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2/15/14 Brian Q. Hutchings
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2/17/14 ken quirici
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2/17/14 ken quirici
2/17/14 thenewcalculus@gmail.com
2/17/14 ken quirici
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2/17/14 ken quirici
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2/17/14 YBM
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2/17/14 ken quirici
2/17/14 David Hartley
2/17/14 ken quirici
2/19/14 David Hartley
2/20/14 thenewcalculus@gmail.com
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2/18/14 thenewcalculus@gmail.com
2/17/14 ken quirici
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2/17/14 Ken.Pledger@vuw.ac.nz
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2/18/14 David Bernier
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2/18/14 Ken.Pledger@vuw.ac.nz
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2/18/14 ken quirici
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2/18/14 Wizard-Of-Oz
2/18/14 thenewcalculus@gmail.com
2/18/14 Wizard-Of-Oz
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2/19/14 Wizard-Of-Oz
2/19/14 Brian Q. Hutchings
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2/18/14 thenewcalculus@gmail.com
2/18/14 ken quirici
2/18/14 thenewcalculus@gmail.com
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