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?).