Friday, November 26, 2004

No Royal Road to Geometry?




All those who have written histories bring to this point their account of the development of this science. Not long after these men came Euclid, who brought together the Elements, systematizing many of the theorems of Eudoxus, perfecting many of those of Theatetus, and putting in irrefutable demonstrable form propositions that had been rather loosely established by his predecessors. He lived in the time of Ptolemy the First, for Archimedes, who lived after the time of the first Ptolemy, mentions Euclid. It is also reported that Ptolemy once asked Euclid if there was not a shorter road to geometry that through the Elements, and Euclid replied that there was no royal road to geometry. He was therefore later than Plato's group but earlier than Eratosthenes and Archimedes, for these two men were contemporaries, as Eratosthenes somewhere says. Euclid belonged to the persuasion of Plato and was at home in this philosophy; and this is why he thought the goal of the Elements as a whole to be the construction of the so-called Platonic figures. (Proclus, ed. Friedlein, p. 68, tr. Morrow)


It was interesting to me that I find some thread that has survived through the many centuries , that moves through the hands of individuals, to bring us to a interesting abstract world that few would recognize.



While Euclid is not known to have made any original discoveries, and the Elements is based on the work of his predecessors, it is assumed that some of the proofs are his own and that he is responsible for the excellent arrangement. Over a thousand editions of the work have been published since the first printed version of 1482. Euclid's other works include Data, On Divisions of Figures, Phaenomena, Optics, Surface Loci, Porisms, Conics, Book of Fallacies, and Elements of Music. Only the first four of these survive.

Of interest, is that some line of departure from the classical defintions, would have followed some road of developement, that I needed to understand how this progression became apparent. For now such links helped to stabilize this process and the essence of the departure form this classical defintion needed a culmination reached in Einstein's General Relativity. But long before this road was capture in it's essence, the predecessors in this projective road, develope conceptual realizations and moved from some point. To me, this is the fifth postulate. But before I draw attention there I wanted to show the index of this same projective geometry.

A theorem which is valid for a geometry in this sequence is automatically valid for the ones that follow. The theorems of projective geometry are automatically valid theorems of Euclidean geometry. We say that topological geometry is more abstract than projective geometry which is turn is more abstract than Euclidean geometry.

The move from the fifth postulate had Girolamo Saccheri, S.J. (1667 - 1733) ask the question?

What if the sum of the angles of a triangle were not equal to 180 degrees (or p radians)?" Suppose the sum of these angles was greater than or less than p. What would happen to the geometry we have come to depend on for so many things? What would happen to our buildings? to our technology? to our countries' boundaries?




The progression through these geometries leads to global perspectives that are not limited to the thread that moves through these cultures and civilizations. The evolution dictates that having reached Einstein GR that we understand that the world we meet is a dynamical one and with Reason, we come t recognize the Self Evident Truths.

At this point, having moved through the geometrical phases and recognitions, the physics of understanding have intertwined mathematical realms associated with Strings and loop and other means, in which to interpret that dynamical world called the Planck Length(Quantum Gravity).

Reichenbach on Helmholtz