Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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

 Messages: [ Previous | Next ]
 thenewcalculus@gmail.com Posts: 1,361 Registered: 11/1/13
Re: Euclid's Elements Book 1 Proposition 1 - something Euclid missed?
Posted: Feb 17, 2014 8:32 PM

On Monday, February 17, 2014 10:23:14 PM UTC+2, Ken Pledger wrote:

KP: Historically, there was a profound change in understanding of the
foundations of mathematics around 1900, especially with Hilbert and the
rise of modern logic.

JG: A very unfortunate turn of events.

KP: We now set up theories which are purely sets of formulae and rules for manipulating them.

JG: That's just a trashy platitude.

KP: Then we relate those to models, and consider various metalogical matters (consistency, completeness, etc.).

JG: Irrelevant rot of course.

KP: But we can't understand how earlier mathematicians thought if we try to force their work into this modern framework.

JG: Agreed. You can't force sound mathematics on illogical ideas.

KP: The Greeks regarded geometrical objects as idealized physical objects.

JG: You obviously don't know anything about Ancient Greek thought.

KP: 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).

JG: Getting thinner? Are you sure that you read that in the Elements? Because it's not there.

KP: 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).

JG: And so?

KP: It's easy to see why Plato liked geometry, because it gave such good examples of his theory of ideals.

JG: Nonsense. Geometry was not yet realised in Plato's time.

KP: But thinking that way carries the danger of relying too much on physical intuition when looking at actual diagrams.

JG: I am glad the Greeks did not rely on any intuition.

KP: That seems to have been the trap which Euclid fell into in I.1, when he assumed that the two circles would intersect.

JG: There is no trap and Euclid made no such assumptions. That's just your irrational brain at work. If I were you, I would worry more about the traps you fall into with Cantorian rot.

KP: 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").

JG: I don't know much about Pasch, neither do I care. The whole is greater than the part. But of course if you can't properly distinguish between the part and the whole, well then naturally you will reach incorrect conclusions.

KP: This is just an intuition based on the diagram, and in elliptic geometry it is actually false.

JG: Were you trying to say something with that statement? It's completely irrelevant. The Elements does not deal with elliptic geometry.

KP: 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.

JG: Wrong! It began with Euclid.

KP: Just over a century later, those few dozen people had reached the level of Euclid's "Elements".

JG: Bollocks! I have yet to meet a professor of mathematics who knows what is a number, and last time I checked my calendar, we are in 2014 already.

KP: Then in the 3rd century B.C. came the formidable achievements of Archimedes, not to mention Apollonius.

JG: They are still formidable.

KP: 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.

JG: There is NO FAULT in the Works of Archimedes or Apollonius. Well, none that I know of.

Date Subject Author
2/14/14 ken quirici
2/14/14 ross.finlayson@gmail.com
2/15/14 thenewcalculus@gmail.com
2/15/14 bert
2/15/14 thenewcalculus@gmail.com
2/16/14 ross.finlayson@gmail.com
2/16/14 thenewcalculus@gmail.com
2/15/14 thenewcalculus@gmail.com
2/15/14 Brian Q. Hutchings
2/15/14 ken quirici
2/15/14 FredJeffries@gmail.com
2/15/14 ken quirici
2/15/14 thenewcalculus@gmail.com
2/16/14 ken quirici
2/16/14 ken quirici
2/16/14 thenewcalculus@gmail.com
2/17/14 ken quirici
2/17/14 thenewcalculus@gmail.com
2/17/14 ken quirici
2/17/14 thenewcalculus@gmail.com
2/17/14 ken quirici
2/17/14 thenewcalculus@gmail.com
2/17/14 ken quirici
2/17/14 thenewcalculus@gmail.com
2/17/14 YBM
2/18/14 thenewcalculus@gmail.com
2/17/14 David Hartley
2/17/14 thenewcalculus@gmail.com
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
2/17/14 ross.finlayson@gmail.com
2/18/14 thenewcalculus@gmail.com
2/17/14 ken quirici
2/17/14 thenewcalculus@gmail.com
2/17/14 Brian Q. Hutchings
2/17/14 thenewcalculus@gmail.com
2/17/14 Ken.Pledger@vuw.ac.nz
2/17/14 thenewcalculus@gmail.com
2/18/14 David Bernier
2/18/14 thenewcalculus@gmail.com
2/18/14 Ken.Pledger@vuw.ac.nz
2/18/14 thenewcalculus@gmail.com
2/18/14 thenewcalculus@gmail.com
2/18/14 ken quirici
2/18/14 thenewcalculus@gmail.com
2/18/14 Wizard-Of-Oz
2/18/14 thenewcalculus@gmail.com
2/18/14 Wizard-Of-Oz
2/19/14 thenewcalculus@gmail.com
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
2/18/14 thenewcalculus@gmail.com
2/18/14 Brian Q. Hutchings
2/18/14 thenewcalculus@gmail.com
2/18/14 Brian Q. Hutchings
2/20/14 Brian Q. Hutchings
2/22/14 thenewcalculus@gmail.com