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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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 David Bernier Posts: 3,884 Registered: 12/13/04
Re: Euclid's Elements Book 1 Proposition 1 - something Euclid missed?
Posted: Feb 18, 2014 12:10 PM

On 02/17/2014 03:23 PM, Ken Pledger wrote:
> It's hard to know what to quote from this discussion, so I shall
> chicken out of doing so.
>
> Historically, there was a profound change in understanding of the
> foundations of mathematics around 1900, especially with Hilbert and the
> rise of modern logic. We now set up theories which are purely sets of
> formulae and rules for manipulating them. Then we relate those to
> models, and consider various metalogical matters (consistency,
> completeness, etc.). But we can't understand how earlier
> mathematicians thought if we try to force their work into this modern
> framework.
>
> The Greeks regarded geometrical objects as idealized physical
> objects. A line drawn with a pencil can be imagined getting thinner
> and thinner until it is an ideal line of zero thickness (Euclid I Def.
> 2). Geometrical objects can even be imagined as moved around and
> fitted on top of one another (Euclid I.4 proof), in which case they are
> equal (Euclid I Common Notion 4). It's easy to see why Plato liked
> geometry, because it gave such good examples of his theory of ideals.
>
> But thinking that way carries the danger of relying too much on
> physical intuition when looking at actual diagrams. That seems to have
> been the trap which Euclid fell into in I.1, when he assumed that the
> two circles would intersect. A clear view of such things was achieved
> by Pasch in the 19th century. I think Euclid I.16 is an even more
> significant example. Near the end of the proof he claims that one
> angle is greater than another (Common Notion 5: "The whole is greater
> than the part"). This is just an intuition based on the diagram, and
> in elliptic geometry it is actually false.
>
> However, remember that logically structured mathematics began in the
> 5th century B.C. (probably with Hippocrates), and was pursued by quite a
> small number of people in each generation. Just over a century later,
> those few dozen people had reached the level of Euclid's "Elements".
> Then in the 3rd century B.C. came the formidable achievements of
> Archimedes, not to mention Apollonius. Rather than emphasize the
> faults which we can see by modern hindsight, I prefer to admire the
> enormous achievement of the Greeks who reached such a high level in such
> a short time.
>
> Ken Pledger.
>

It's a blessing to have contributors such as you in sci.math .

With respect to Archimedes of Syracuse,
since he discovered the law of floating, partially submerged
and completely submerged uniform-density "bodies" that
goes by the name of Archimedes's Principle,
I've been wondering why he doesn't seem to be called
mathematical physicist or just plain theoretical physicist.

Or I guess I'm intrigued by why they say Galileo started physics, (the
science);
and Archimedes, well , I'm not exactly sure but obviously he's
famous with mathematicians (if he was a scientist, doesn't
that make him a physicist?).

David Bernier

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Date Subject Author
2/14/14 ken quirici
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2/17/14 YBM
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2/17/14 ken quirici
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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
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2/18/14 Wizard-Of-Oz
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2/19/14 Wizard-Of-Oz
2/19/14 Brian Q. Hutchings
2/18/14 thenewcalculus@gmail.com
2/18/14 thenewcalculus@gmail.com
2/18/14 ken quirici
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