II'i=~.~--w_,~[,[,~ David E. Rowe, Editor 1 t takes a good deal of historical imag- a system of ideas ultimately grounded Euclidean ~ ination to picture the kinds of de- in undefined terms and arbitrary ax- bates that accompanied the slow ioms. A modern-day Platonist would ve- process which ultimately led to the ac- hemently object to this characterization, Geometry and ceptance of non-Euclidean geometry a which puts too much emphasis on little more than a century ago. The dif- purely arbitrary constructions rather ficulty stems mainly from our tendency than conceiving of geometrical figures Physical to think of geometry as a branch of pure as idealized instantiations of perfect mathematics rather than as a science forms. Space with deep empirical roots, the oldest Those who might like to see what natural science so to speak. For many such a debate looked like around 1900 DAVID E. ROWE of us, there is a natural tendency to need only read the correspondence be- think of geometry in idealized, Platonic tween the philosopher Gottlob Frege terras. So to gain a sense of how late- and the mathematician David Hilbert nineteenth-century authorities debated [Gabriel 1980]. Their dispute began over the true geometry of physical when Frege wrote Hilbert after reading space, it may help to remember the the opening pages of Hilbert's Foun- etymological roots of the word geome- dations of Geometry [Hilbert 1899], the try: geo plus metria literally meant to work that did so much to make the measure the earth. In fact, Herodotus modern axiomatic approach fashion- reported that this was originally an able. Although one of the founding fa- Egyptian science; each spring the Egyp- thers of modern logic, Frege simply tians were forced to remeasure the land could not accept Hilbert's contention after the Nile River flooded its banks, that the fundamental concepts of geom- altering property lines. Among those etry had no intrinsic meaning when engaged in this land survey were the seen from a purely logical point of view. legendary Egyptian rope-stretchers, the For Frege, points, lines, and planes harpedonaptai, who were occasionally were not simply empty words. They depicted in artwork relating to Egyp- were in some deep sense real; geome- tian ceremonials. try was the science that studied the We are apt to smile when reading properties of real figures composed of Herodotus's remarks, dismissing these them. Hilbert, to be sure, was by no as just another example of the Greek means adw)cating a modern formalist tendency to think of ancient Egypt as approach to geometry that broke with the fount of all wisdom. Herodotus was the classical tradition. In essence, his ax- famous for repeating such lore, and ioms for Euclidean geometry were here he was apparently confusing merely a refinement of those presented geometry with the science of geodesy, in Euclid's Elements. In 1905 he em- and the latter has little to do with the phasized that "the aim of every science former; at least not anymore. We do not is... to set up a network of concepts customarily think of circles, triangles, or based on axioms to which we are led the five Platonic solids as real figures: naturally by intuition and experiencg' they are far too perfect, the products of [Corry 2004, 124]. Hilbert thus recog- the mind's eye. Of course, there is still nized the empirical roots of geometri- plenty of room for disagreement. A for- cal knowledge, but he also emphasized Send submissions to David E. Rowe, malist will stress that geometrical figures that the question as to how and why Fachbereich 17--Mathematik, are mere conventions or, at best, im- Euclidean geometry conformed to our Johannes Gutenberg University, ages we attach to fictive objects that spatial perceptions lay outside the reahn D55099 Mainz, Germany. have no purpose other than to illustrate of mathematical and logical investiga- 2006 Springer Science+Business Media, Inc., Volume 28, Number 2, 2006 51 tions. For these, all that mattered at bot- Herodotus wrote about the Egyptian before the time of Pythagoras. It tells us tom was proving the consistency of a roots of the science of geometry. something fundamental about planar certain set of axioms. This reductionist measurements in right triangles: the viewpoint was sheer anathema in the Gauss, Measurement, and square on the hypotenuse equals the eyes of Gottlob Frege. the Pythagorean Theorem sum of the squares on the two other Most mathematicians, to the extent Gauss, after all, was not only a mathe- sides. Some have conjectured that the that they grasped what was at stake, matician and astronomer, he was also Egyptian harpedonaptai, whom Dem- sided with Hilbert in this debate. Over a professional surveyor who at least oc- ocritus once praised, used the converse the course of the next three decades the casionally waded through the marshy of the Pythagorean Theorem to lay out status of the continuum nevertheless hinterlands of Hanover taking sightings right angles at the corners of temples played a major role in the larger foun- in order to construct a net of triangles and pyramids. The claimants suggest dations debates between formalists and that would span this largely uncharted that these professional surveyors used intuitionists. Those rather esoteric dis- region [BOhler 1981, 95-103]. This work a rope tied with 12 knots at equal dis- cussions, however, left traditional real- helped inspire a profound contribution tance from each other; by pulling the ist assumptions about the nature of geo- to pure geometry: Gauss's study of the rope taut, they could form a 3-4-5 right metrical knowledge behind. For despite intrinsic geometry of surfaces, which triangle. It's a nice idea, but nothing their strong differences, the proponents helped launch a theory of measurement more. Archaeologists can still measure of formalism and intuitionism were both in geometry that opened the way to the angles of Egyptian buildings, of guided by their respective visions of probing the geometry of space itself. course, but our access to the mathe- pure mathematics, independent of its Only about a half century earlier, two matical knowledge that lay behind the relevance to other disciplines, like as- leading French mathematicians, Clairaut architectural splendours of ancient tronomy and physics. and Maupertuis, had studied the shape Egypt is highly limited. Papyri can eas- This suggests that a fundamental shift of the earth's surface, showing that it ily disintegrate with time, and only two took place around 1900 regarding the formed an oblate spheroid. As one have been found that provide much in- status of geometrical knowledge. This moved northward, they discovered, the sight into the mathematical methods of reorientation was certainly profound, curvature of the earth flattened, just as the time: the Rhind and Moscow papyri, but it seems to have been rather quickly Newtonian theory predicted. Mauper- which were presumably used as train- forgotten in the wake of other, even tuis's celebrated expedition to Lapland ing manuals for Egyptian scribes. Nei- more dramatic developments. Soon af- brought him fame and the nickname of ther contains anything close to the terward Einstein's general theory of rel- "the earth flattener." It also provided the Pythagorean Theorem. ativity would lead to a flurry of new dis- French Academy with stunning proof Historians of Mesopotamian mathe- cussions about the interplay between that Descartes's theory of gravity could matics have been luckier; they have had space, time, and matter. Leading math- not be right, thereby overcoming the plenty of source material available ever ematicians like Hilbert and Hermann last major bastion of resistance against since it became possible to decipher the Weyl became strong proponents of Ein- Newtonianism in France. Thus precise clay cuneiform tablets archaeologists stein's ideas, even as they sharply dis- measurements of the earth's curvature began turning up a little more than a agreed about epistemological issues re- had already exerted a deep impact on century ago. Some of the Babylonian lating to the mathematical continuum, a modern science. mathematical texts reveal not just a concept of central importance for the In the 1820s Gauss took the mea- passing familiarity with the Pythagorean geometer. surement of the earth as his point of Theorem but even a masterful use of it Looking backward from the 1920s, it departure for an abstract theory of sur- for numerical computations, like ap- would seem that the opposing views of faces, asking whether and how a sci- proximating the value of ~ or calcu- formalists and intuitionists actually re- entist could determine the curvature of lating Pythagorean triples. Herodotus flect distinctly modern attitudes about an arbitrary surface through measure- claimed that "the Greeks learnt the the nature of geometrical knowledge ments made only along the surface it- rroAog, the gnomon, and the twelve that would have been scarcely think- self, without knowing anything at all parts of the day from the Babylonians" able prior to 1900. Up until then, geom- about the way in which the surface [Heath 1956, vol. 1, 370]. etry was always conceived as somehow might be embedded in space. To talk Still, justly or not, we tend to credit wedded to a physical world that dis- about curvature as an intrinsic property the Greeks with being the first to give played discernible geometrical features. of a surface requires a careful recon- a proof of this ancient theorem. But Take the developments that led to the sideration of concepts like the measure since nearly all information about early birth of modern science in the seven- of distances between points and angles Greek mathematical texts is lost, we can teenth century: anyone who studies the between curves.