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2013-09-01 01:17:18 UTC

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HORST STRUVE, UNIVERSITY OF COLOGNE

Geometry was considered until modern times to be a model science. To be developed more geometrico was a seal of quality for any endeavor, whether mathematical or not. In the 17th century, for example, Spinoza set up his Ethics in a more geometrico manner, to emphasize the perfection, certainty, and clarity of his pronouncements. Geometry achieved this status on the heels of Euclid's Elements, in which, for the first time, a theory was built up in an axiomatic-deductive manner. Euclid started with obvious axioms - he called them “common notions" and “postulates" -, statements whose validity raised no doubts in the reader's mind. His propositions followed deductively from those axioms, so that the truth of the axioms was passed on to the propositions by means of purely logical proofs. In this sense, Euclid's geometry consisted of “eternal truths". Given its prominence, Euclid's Elements was also used as a textbook until the 20th Century.

Today geometry has lost the central importance it had during earlier centuries, but it still is an important area of mathematics, and is truly fundamental for mathematics from a variety of points of view. The “Introduction to Geometry" by Ewald tries to address some of these points of view, whose significance will be examined in what follows from a historical perspective.

Looking back at the history of geometry, one finds that the first significant development going beyond Euclid's geometry was the discovery and later the systematic development of projective geometry at the beginning of the 19th century. It was motivated by problems of descriptive geometry, coming from the perspective representation of three-dimensional bodies, problems that had already been addressed by artists of the Renaissance. Monge, Poncelet, Steiner, Plücker, and their contemporaries realized that one can extend the Euclidean plane by adding elements “at infinity", points and a straight line, to get a consistent new structure, a projective plane. It turned out that geometry was more than Euclid's theory, with which one can eminently describe physical space. The space of visual perception itself can be described mathematically. Moreover, projective geometry displayed a characteristic property that Euclidean geometry lacked. It satisfies the so-called duality principle, which states that the dual of any valid statement of projective geometry is also valid. One obtains that dual statement (in the planar case) by interchanging the words “point" and “line" (while leaving the word “incidence" unchanged). The nature of the geometric objects, whether they be points or lines, seemed at once to matter less than the relations among them.

Another discovery undermined the special status of Euclidean geometry. Euclid's axioms came with a claim of being beyond doubt, as well as recognizable as true by anyone with common sense. For many mathematicians throughout history, Euclid's fifth (and last) postulate, which was later called the Parallel Postulate, did not fulfill this requirement. Even the syntactic formulation that Euclid had chosen to avoid the notion of a straight line extended indefinitely was very complicated. Reformulations of that axiom as more reasonably sounding statements, such as “Given a line a and a point P not on a, there exists a unique line through P that does not intersect a", were seen as unconvincing (arguing that such reformulations make statements about straight lines extended beyond known bounds, and as such are not verifiable). This led to attempts to prove the Parallel Postulate from the other axioms, to allow for its removal from the list of obvious axioms. Famous are the attempts by Lambert and Saccheri, to derive a contradiction from the assumption that the Fifth Postulate does not hold. In hindsight, one may say that they developed the beginnings of a non-Euclidean geometry. It was Lobachevsky and Bolyai who first saw the possibility of building a consistent theory with the negation of the Fifth Postulate in the first half of the 19th century. Its consistency was only postulated by them. It became accepted by the mathematical community only after the discovery of models within the well-known Euclidean geometry (Beltrami, Klein, Poincare). If Euclidean geometry is consistent, which no one doubted, then so must the new structures be. What was all the more remarkable was that the same structure had several models in which the "points" and "lines" were different objects (lines could be chords of a circle or circular arcs or plane sections of quadrics, i. e. substructures of the circle geometry developed by Möbius). This was a stark departure from Euclid's geometry, where points and lines were univocally determined through definitions. For the new structures - should they be still called geometry? - the relations among the objects were apparently more important than the nature of those objects.

In his Foundations of Geometry from 1899 Hilbert answered the question asked earlier by presenting - on the example of Euclidean geometry - a new conception of mathematics, the formalist one. For Hilbert, as for Euclid, a mathematical theory is built in the axiomatic-deductive deductive manner. The axioms themselves, however, are for Hilbert no longer self-evident statements which structure a given realm of phenomena, but rather they define a structure whose objects are left undefined. Since the axioms do not specify the nature of the geometrical objects, but only the relations in which they stand to each other, the fundamental notions are only "implicitly" defined. The realization that this is sufficient for the formulation of a mathematical theory had been prepared by the discovery of the projective and non-Euclidean geometries. For Hilbert, the theorems of a theory are not true or false in the sense of the correct description of a given domain of phenomena, but only in terms of the presence or the absence of a derivation from the axiom system. If mathematics is thus understood, then the above structures are indeed geometries, namely non-Euclidean geometries.

The historical development of geometry led to the modern conception of geometry, which in turn led to the modern conception of mathematics. This is without a doubt a major cultural achievement of geometry that is worth emphasizing. All modern mathematical theories are currently set up more geometrico. A side effect of its own pioneering role and of the success of the axiomatic method throughout mathematics was the fact that geometry lost its special position within mathematics.

Hilbert builds Euclidean geometry in a step-wise manner. There are five groups of axioms, regarding incidence, order, congruence, parallelism and continuity. Since mathematical structures do not provide descriptions of realms living outside of mathematics, but are rather defined by axioms, one may ask questions regarding the scope of the axioms as well as regarding the structures that do not satisfy all of the axioms but only subsets thereof. Connections between geometry and algebra were gradually discovered, e. g. the simple relationship between three-dimensional projective geometry and (skew) fields. This was the beginning of a new area of mathematics, the so-called foundations of geometry, to whose development contributed not only geometers but also logicians, such as Tarski. One of the insights that arose not long after Hilbert's work, was that it is possible to build geometry without notions of order or continuity. An essential tool in this direction was the calculus of reflections, an idea that owes much to Hjelmslev. Bachmann has later deepened the study of reflection geometry in a systematic way and coined the concept of a metric plane. Metric planes are a class of structures that capture the core of the orthogonality properties common to the Euclidean and the classical non-Euclidean planes, the hyperbolic and the elliptic planes.1 Bachmann provided axiomatizations both in terms of geometric objects, and in group theoretical terms. All Hilbert planes, i. e. all models of the plane axioms of Hilbert's axiom system, without the parallel axiom and the continuity axioms, turn out to be metric planes. Metric planes can be embedded in projective-metric planes, and thus can also be described analytically, i. e. in terms of coordinates. Reflection geometry emphasizes the interplay between geometry and group theory, the latter being a fundamental concept in many areas of mathematics.

The terms used by Hilbert for the axiomatization of Euclidean geometry, incidence, congruence, and order, indicate that even good old Euclidean geometry is a complex, multi-layered theory. Geometry is a discipline accurately described by Ewald as follows: "That man's mind is a creative mind is becoming beautifully apparent in geometry."

There are several books titled "Introduction to Geometry." The way in which the words "geometry" and "introduction" are interpreted depends on that book's author. Upon closer inspection, Günter Ewald's interpretation can be seen to be informed - both in substance and methodologically - by the major lines of development sketched above. English language books with the title "Introduction to Geometry" consist, following Coxeter's classic, of a wide variety of geometric problems, which offer a colorful, kaleidoscopic view of geometry.

These problems are dealt with on the basis of the school geometry knowledge of the reader. The "Introduction" by Ewald is of a different nature, and occupies a singular place in the English language literature. Ewald's book treats a central topic of geometry, the theory of metric planes in Bachmann's sense. It makes this theory accessible to readers of English, in a systematic manner, more geometrico, through an axiomatic-deductive approach.

Ewald defines metric planes in the sense of Bachmann both in geometric terms and in group theoretical language, and shows following Bachmann that metric planes can be embedded in projective-metric planes. For Euclidean and hyperbolic metric planes Ewald presents a stepwise construction (seen from the vantage point of the coordinate field, all the way to the field of real numbers) by adding order and continuity axioms. To make the book self-contained, the author first treats projective planes and models of non-Euclidean geometries. Models of hyperbolic and elliptic geometry are also treated as substructures of a circle geometry, the Möbius geometry. This geometry is also introduced axiomatically by using an axiom system of van der Waerden.

The publisher should be commended for the re-issue of this textbook, for the view of geometry presented therein cannot be found in any other English language textbook, and that view deserves to become better known to future generations of aspiring mathematicians.2

Translated from German by Victor Pambuccian.

1That metric planes can be axiomatized in terms of the notion of orthogonality alone was shown in V. Pambuccian, Orthogonality as single primitive notion for metric planes. With an appendix by Horst and Rolf Struve, Beiträge zur Algebra und Geometrie 48 (2007), 399-409.

2 For a survey of more recent developments of the foundations of geometry, see the review by V. Pambuccian of Hilbert's work on the foundations of geometry in Philosophia Mathematica (III), 21 (2013), 255-277.