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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.Pledger@vuw.ac.nz Posts: 1,412 Registered: 12/3/04
Re: Euclid's Elements Book 1 Proposition 1 - something Euclid missed?
Posted: Feb 17, 2014 3:23 PM

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.

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2/14/14 ken quirici
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2/18/14 Ken.Pledger@vuw.ac.nz
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