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.