geometry

Galileo's aphorism that "the Book of Nature is written in the characters of Geometry" is as old as Plato, as old as Pythagoras, and as old perhaps as the wisdom of the Egyptians.

But is geometry, like the idea of nature as a book, a human invention? Interestingly enough, Euclid's Elements was one of the first books printed. After being lost to the West when the Northern tribes overran the Roman Empire, the original Greek texts were preserved in Byzantium and were translated into Latin during the Renaissance. Not only was Euclid's Elements one of the first published books, but after the bible, it is one of the most often republished.

"Geometry lies at the crossroads of a physics problem and an affair of the State." (Thousand Plateaus, p.489)

"We owe geometry to the tax collector." (J.L. Heilbron, Geometry Civilized, p.1) According to the Greek historian Herodotus, the Egyptian king Sesostris divided all the land in Egypt equally among its inhabitants in return for an annual rent. But every year the flood of the Nile washed away parts of the plots. Those whose lands had disappeared naturally objected to paying the rent on what they had lost. 'Upon which, the king sent persons to examine and determine by measurement the exact extent of the loss; and thenceforth only such a rent was demanded of him as was proportionate to the reduced size of his land. From this practice, I think, geometry first came to be known in Egypt, whence it passed into Greece.' "Geometry expresses in Greek what the Greeks received from the Egyptians, retaining its root meaning of land measurement." (Heilbron)

The Greeks, or rather Euclid, gave geometry a second meaning. By treating geometry as a deductive science, the Greeks transformed it into a mental discipline, an abstract discourse, and the exemplar of rigorous reasoning. In his commentary on Euclid, Proclus preserved a statement of Eudemos, according to which Pythagoras was the first to raise mathematics to the dignity of a liberal education by returning to its general principles and treating its problems as purely intellectual, with no restrictions to partiucular matters. This was the real stroke by which an empirical art of mensuration was made all at once at the very center of a true, philosophical knowledge. According to Proclus' Commentary on the first book of Euclid's elements, Ptolemy I, king of Egypt once asked Euclid for a shorter way to mastering geometry than working through the Elements. Euclid answered, "There is no royal road to geometry."

Plato and Euclid developed an indissoluble partnership between geometrical and philosophical ideas of truth. The Platonic concept of the theory of ideas was possible only because Plato had continually in mind the static shapes discovered by Greek mathematics. On the other hand, Greek gemetry did not achieve completion as a real system until it adopted Plato's manner of thinking. (see Ernst Cassirer, The Problem of Knowledge.) The concepts and propositions that Euclid placed at the apex of his system were a prototype and pattern for what Plato called the process of synopsis in idea. What is grasped in such synopsis is not the peculiar, fortuitous, or unstable; it possesses universal necessary and eternal truth. (see transcendence / immanence)

Critics of Euclid, such as the Greek philosopher Epicurus, used the pack donkey's geometric intuitions to ridicule Euclid's insistence on "proving things that have no need of proof." Euclid's science is ridiculous, Epicurus claimed, pointing to a proposition half way through the first book of the Elements, in which Euclid labors to show that no side of a triangle can be longer than the sum of the other two sides. "It is evident even to an ass." For a hungry ass will go directly to a bale of hay at B, without passing through any point C outside the straight line.

The exemplary role of geometry after Euclid enjoyed uncontested supremacy for centuries, until the discovery of non-Euclidean geometry introduced entirely new questions for mathematical thought and forced it to a new interpretation of its own logical structure. (see scientific space)

In the preface to book vi of De Architectura, Vitruvius recounts the story of Aristuppus, the Socratic philosopher, who was shipwrecked but reached the shore of Rhodes. There he noticed geometric figures drawn in the sand, and is said to have shouted to his companions, "there is hope, for I see traces of men." Vestigium hominis video, quoted by Kant, Critique of Judgement, sect. 64. If a footprint would have signified a man, a circle signified a mind. (see W. Oechslin in Daidalos 1) Yet geometric forms are not a monopoly of men. (see natural form ) Still, for Kant, the Aristuppus story means that here we must assume an actual causality according to a concept and look for the agent.
Moonprint


For Kant, there is one and only one internally consistent set of a priori and synthetic judgments. It consists of the axioms of Euclidean geometry, all true arithmetical propositions, and certain assumptions of Newtonian physics. (See Critique of Pure Reason) For Kant mathematics (and geometry) "carries with it apodictic certainty; that is to say, absolute necessity, not based on experience, and consequently a pure product of reason." (Prologemena § 6.) Here Kant adheres to the mechanical postulates of classical physics, in which geometry stood outside nature, claiming to offer an a priori analysis of Euclidean space, which was regarded as the scene of all natural phenomena but not thought to be involved in them.

Viollet-le-Duc simultaneously acknowledges the aspects of geometry that are of human creation and its submission to natural laws. "when the first man traced in the sand with a stick pivoting upon its axis, he in no way invented the circle; he merely discovered a figure already existing. All discoveries in geometry have resulted from observations, not creations." (from " Style" in Dictionnaire Raisoné )

The "Origin of Geometry" has been a subject of inquiry by, among others, Edmund Husserl. In The Crisis in European Sciences and Transcendental Phenomenology. Husserl sought to develop a theory of rational progress through isolating the ways in which eidetic sciences-- of which mathematics and geometry were the prime examples-- could arise from previously more material or concrete contexts and could become idealized or autonomous from the material conditions. Thus geometry (geo-metria ) arises as natural boundaries give way to the "limit shapes" of builder and surveyer. In the famous appendix to the Crisis he describes this origin of geometry: "...in the life of practical needs, certain particularizations of shape stood out and (...) a technical praxis always aimed at the production of preferred particular shapes..." (p.375) quoted in Don Ihde, Technology and the Lifeworld. p.28)(see also technique in technology) For Cassirer, the most radical removal of geometry from experience had already occurred with Euclid, which was already based on figures that are removed from all possibility of experiment. Not only the idealizations of point, line, and plane, but the idea of similar triangles, whose differences are considered inconsequential or fortuitous, and that become identified as "the same" mark an immense step away form ordinary perception. Perspective and Cartesian rationality provided the classical regime of visuality, which was meant to be founded on the geometric certainties of optics.

Geometry has been called the science of space. Does geometry stabilize architecture? Geometry is thought to be a constitutive part of architecture, but not dependent on it. (see Robin Evans, The Projective Cast) The objects of classical geometry are for the most part not self-similar like the forms of life. in fact for most architects, the best geometry for firmness and stability is surely a dead geometry. Evans calls "Dead geometry is an inoculation against uncertainty." (pxxvii) The thesis of Evans' book, however, is that the question of architecture's relation to geometry is best asked of projection, of zones of instability, rather than in compostion, in the shape of buildings. The axioms of geometrical similarity allow a general movement of abstraction away from the immediate objects of sensuous experience. For architecture, the universalizing quality of geometry allows individual works to be part of a general art.