GEOMETRY

Until 1800, mathematics was divided into two great branches: geometry and arithmetic. Both were commonly regarded as the more obviously secure repositories of human knowledge. At this stage, geometry could be suitably defined as "the science which investigates the properties and relations of magnitudes in space, as lines, surfaces, and solids" (Oxford English Dictionary ). However, with the enormous enrichment of mathematics in the nineteenth century, the scope of geometry was greatly expanded and diversified, its content disrupted, and its epistemic standing called into question.

The word "geometry" comes from a Greek word that literally means measurement of the earth and was originally applied to the art of land surveying. But around 500 BCE or even earlier, the spatial properties and relations that had been codified by land surveyors in Mesopotamia and Egypt became in Greece the starting-point of inquiries of a more abstract sort that soon took leave of their down-to-earth origins. In this guise, geometry appeared to Plato as a testimony of the other-worldly origin of the human soul (Meno ) and was included by him as a compulsory item in the curriculum for would-be philosopher-kings (Republic, VII, 526c–528d). For more than twenty centuries, philosophers regarded the geometry created by Greek mathematicians from Eudoxus, through Euclid, to Archimedes and Apollonius as the standard of indubitable truth and cogent reasoning. As a result of later developments, geometry, with the rest of mathematics, came to be seen as a capital example of the loss of certainty that currently pervades most areas of civilized life (Kline 1980). In more than one sense, this enhances, rather than diminishes its philosophical significance.

This entry is divided into three sections. The first section touches on some philosophically noteworthy aspects of ancient geometry. The second section deals briefly with geometry and philosophy from 1600 to 1800. And the third section describes those episodes in the history of geometry since 1800 that had the greatest impact on twentieth-century philosophy.

Topics in Ancient Geometry

geometry in the middle east

According to Herodotus (2.109), the Greeks learned land surveying (geōmētriē ) from the Egyptians, who used it to reassess taxes on properties partially washed away by the Nile. It appears that this art was first cultivated in the Middle East to cope with the consequences of floods in southern Mesopotamia. Archaeological evidence from both regions displays applications of the so-called theorem of Pythagoras, and a clay tablet now at Yale University (YBC 7289) gives the length of the diagonal of a unit square as 1.41421296, the same approximation to √2 that Ptolemy used some 2,000 years later. The Old Babylonian scribe who calculated it probably knew that he could improve on this figure, but it is highly unlikely that he suspected that no algorithm could ever yield a perfectly accurate one. No extant document from ancient Egypt or Mesopotamia contains the general statement of a geometric theorem or anything that even remotely resembles a geometric proof.

pythagoreans and irrationals

Thales of Miletus, "the first to philosophize," supposedly was also the first to prove a geometric theorem (namely, that a triangle with two equal sides also has two equal angles). The earliest proofs probably consisted of diagrams that plainly displayed the relations they were meant to prove (see Plato, Meno, 80d–86c). But Greek geometers soon produced purely discursive proofs (like the one given later in this paragraph). The Pythagoreans, intellectually and politically active in southern Italy throughout the fifth century BCE, worked intensely on mathematical problems, as they thought that numbers (i.e., the positive integers) are the principles of everything. This suggestive belief was supported by their discovery that musical chords are associated with simple numerical proportions. It broke down, however, when a member of the school, possibly Hippasus of Metapontum, showed that there are geometric magnitudes of the same kind whose relative sizes cannot be conveyed by numbers. Presumably, this was first demonstrated for the diagonal and the side of the regular pentagon; but it is proved more easily for the diagonal and the side of a square by the following argument transmitted in an appendix to Euclid's Elements.

Take the side of the square as the unit of length. Then, by Pythagoras's theorem, the length of the diagonal equals √2. But there are no two integers a and b such that (a /b )² = 2. For suppose there are. Then, by simplification of the fraction a /b we should find two integers p and q, with no common divisor, such that (p /q )² = 2. Then p ² = 2q ² and p is an even number, equal to 2n, say. (For the square of an odd number, say 2n + 1, is always odd, that is, 4n ² + 4n + 1). But then 2q ² = p ² = 4n ², and q ² = 2n ², so that q is also even. But this is impossible, for we assumed that p and q do not have a common divisor. Therefore, one cannot find two integers a and b, no matter how large, such that the diagonal of a square exactly equals a × 1/b of its side. Awareness of the existence of incommensurable lengths cut short dreams of grasping nature through numbers and opened a chasm between arithmetic and geometry.

eudoxus's theory of proportions

Eudoxus of Cnidos (c. 390–c. 337 BCE) invented a method for representing the visible motion of each planet in the sky (including Sun and Moon) as the resultant of the combined uniform rotations of several geocentric spheres. Eudoxus's planetary models are the earliest extant example of geometrical representation of natural processes for the sake of predicting their future evolution. Their moderate predictive success may have motivated Plato's change of mind from his early view that real planetary motions are essentially irregular and unpredictable (Republic VII, 529d7–530b4) to his later commendation of mathematical astronomy as an efficient servant of theology (Laws VII, 822a4–c5; X, 897c4–9; XII, 966d6–967d2) and his endorsement of Eudoxus's program as the proper way of "saving the phenomena" of the sky (Simplicius, 7.492.30–35). Eudoxus also originated the method of exhaustion employed by Archimedes for calculating volumes enclosed by curved surfaces, which was the first step toward the creation of the integral calculus. But Eudoxus's chief contribution to geometry was his theory of proportions, preserved in book 5 of Euclid's Elements. With it, geometry recovered the computational powers it had lost when separated from arithmetic, and the road was opened for rigorously conceiving and handling physical quantities of all sorts.

Two magnitudes a and b are said to have a ratio a :b to one another if there are integers m and n such that m × a > b and n × b > a. (The assumption that any two lengths have a ratio to one another is known as the Archimedean postulate.) Eudoxus produced definitions by virtue of which ratios can be added and multiplied, yielding new ratios, and any two ratios a :b and c :d satisfy trichotomy, that is, either a :b = c :d, or a :b > c :d, or c :d > a :b. In this last case, there will always be an integer n such that n (a :b ) > c :d. Thus, it is natural to regard all Eudoxean ratios as magnitudes that have ratios to one another. This paves the way for setting up equations that combine magnitudes of very different kinds, for example, masses, distances, and times, or volumes, temperatures, and pressures (as represented by their respective ratios with the appropriate units). However, it is not apparent that anyone saw this before the seventeenth century.

euclid's elements

In this, the most famous of mathematical textbooks, Euclid (c. 325–c. 265 BCE) organized the results and displayed the methods of fourth-century-BCE Greek geometry. It is usually taken for granted that the book is patterned after Aristotle's conception of a true science (epistēmē ). This must consist of a collection of universal statements (theorems) obtained by deductive inference from self-evident premises (axioms) and definitions using a few self-explanatory terms (primitives). However, Euclid's book, though prima facie it may seem to prove every theorem from five postulates and a short list of so-called common notions, often resorts to unspoken assumptions. Moreover, Euclid's deductions do not all fit into the narrow frame of Aristotle's logic, and use forms of inference first codified by George Boole (1815–1864), Augustus De Morgan (1806–1871), and Charles Sanders Peirce (1839–1914). Also, his primary definitions (e.g., "A straight is a line which lies evenly with the points in itself") would have to be further supplemented by axioms to be of use in deductions. It seems more likely, therefore, that Aristotle based his idea of a true science on his own grasp of what contemporary geometricians were doing (textbooks similar to Euclid's had been around since Aristotle was a student in Plato's Academy) but did not set a paradigm that they or their successors actually followed.

euclid's postulates

The first three postulates are not statements, but requests to allow certain constructions. The third—"to describe a circle with any center and any radius"—would require an infinite drawing board, which is not self-evidently available. The fifth is a conditional existential statement: "If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, intersect on that side on which are the angles less than the two right angles." Obviously, the condition here printed in italics can only be met on an endless plane. So in the finite world of Aristotelian and medieval cosmology, this postulate is vacuously true, and its existential consequent may be false (there may well not be any such intersection). Still, if Euclid's other postulates and the Archimedean postulate are true, denial of the consequent implies that a quadrangle with three right angles has an acute angle at the remaining corner, so there can be no rectangles. It also implies that polygons with the same shape also have the same size, in which case Aristotle's suggestion (Physics 207b29–34) that all geometrical theorems can be demonstrated in his bounded cosmos by suitably scaling down the diagrams employed would simply be wrong.

These seemingly counterintuitive implications kept geometers throughout the centuries trying to prove the fifth postulate from other principles until, shortly after 1820, Nikolay Lobachevsky and Janos Bolyai dared to deny it and independently published essentially the same system of non-Euclidean geometry. It might be a sign of Euclid's genius that he did not gloss over the fact that this assumption (without which the theorem of Pythagoras will not stand) is not self-evident.

Geometry and Philosophy at the Onset of Modernity

nature geometrized

Aristotle taught that natural science, to adequately grasp its proper subject, must employ terms that connote the peculiar matter of each thing, for example, "snub," which only applies to fleshy noses, rather than "concave," which connotes merely a geometric shape (Metaphysics Ε, 1, 1025b30–1026a7; Physics II, 2, 194a2–27). Still, he agreed (Metaphysics Λ, 8) with the purely geometric description of astronomical phenomena proposed by Eudoxus, presumably because he believed that ether, the stuff that the heavens are made of, can change only by rotation about the center of the Earth, and this is properly described in geometric terms. Anyway, Aristotle's strictures on science did not deter Archimedes (c. 287–212 BCE) from dealing mathematically with the equilibrium and the flotation of bodies. In the meantime, astronomers from Apollonius (third cent. BCE) and Hipparchus (second cent. BCE), through Ptolemy (second cent. CE), to Copernicus (1473–1543) developed ever more complex geometric models of planetary motion, involving diverse circular motions about different centers (none of which coincides with that of the Earth).

After Galileo Galilei's telescope showed that there are mountains on the Moon and fleeting spots on the Sun, the distinction between celestial and terrestrial physics became pointless, and each took cues from the other. Thus Johannes Kepler (1571–1630) sought to explain the motion of planets (including the Earth) by forces exerted on them from the Sun, while Galileo (1564–1642) proposed a chronogeometrical model of free fall on the surface of the Earth, which he conceived as uniformly accelerated rectilinear motion. Lasting success was finally achieved by Isaac Newton (1642–1727), by dint of his mathematical genius and his consummate command of geometry. In the course of these efforts, Kepler (1609) had the words "God is always doing geometry" printed on the front page of his masterpiece, and Galileo wrote that the book of nature "is written in mathematical language, and its characters are triangles, circles and other geometrical figures, without which we cannot understand a word of it" (1623, sec. 6). René Descartes's contention that extension is the one and only clearly and distinctly conceivable attribute of bodies surely called for a comprehensive and thoroughgoing geometrization of physics and might have led to it had geometry been ripe enough to deal with its strenuous demands.

descartes's revolution in geometry

Except for his first law of motion (the principle of inertia) and his work on the refraction of light, Descartes's direct contribution to physics, subjected to unsparing criticism by Christian Huygens, Gottfried Wilhelm Leibniz, and Isaac Newton, failed to gain admission into the classical canon. But modern mathematical physics would not have been possible without Descartes's indirect contribution to it, through his two great inventions in geometry: coordinates (independently introduced also by Pierre Fermat) and the algebra of lengths.

Coordinates are quantitative labels employed for identifying points in space. By means of them the relations among the points can be quantitatively represented and investigated. Nowadays geometric coordinates are drawn from the field of real numbers ℝ, which we regard as a natural extension of ℚ, the field of rationals, which, in turn, is constructed from the familiar integers. But this understanding of these matters was still far off in Descartes's time (although his geometric algebra was a decisive step toward it). To avoid anachronism, one must regard Descartes's original coordinates as oriented lengths or, more exactly, as Eudoxean ratios between such lengths and a conventionally chosen unit length. To assign so-called Cartesian coordinates to a point P in space, one takes the three distances x, y, and z from P to three mutually perpendicular planes (listed in a conventional order, the same for all points), and prefixes to each a plus sign or a minus sign, according to the side of the respective plane that faces P (again by convention). The Cartesian coordinates of P then form an ordered triple of oriented lengths, say 〈+x, −y, −z 〉. (There are other ways of defining coordinates: Oblique coordinates depend on three planes not at right angles to each other. Polar coordinates label a point P by its absolute distance from a fixed point O, the angle made by OP with a fixed plane Γ through O, and the angle made by the perpendicular projection of OP on Γ with a fixed line through O on that same plane.)

In Euclid's Elements (1956), segments are added to segments in an obvious way to obtain new segments; multiplying a segment s by an integer n amounts to adding n copies of s end to end; a straight segment or a length is never multiplied by another one. Until not too long ago, it was usually understood that such multiplications do occur in Euclid's book, but then the product of such a multiplication had to be an area. Descartes followed Euclid on the addition of lengths and defined the multiplication of a length a by a length b so as to yield still another length ab. Here is how. Draw two straight lines from a point O. Mark points F and H on one line so that OF has unit length and OH has length a. Mark point G on the second line so that OG has length b. Draw the straight line segment FG. Let the parallel to FG through H cut the second line at K. Clearly, then, OH /OF = OK /OG. Therefore, OG × OH = OK × OF ; in other words, OK has length ab (Figure 1).

By this procedure, entirely based on elementary geometrical knowledge available to Euclid, Descartes and his successors were able to represent all geometrical relations by equations or inequalities between given and unknown

quantities, and to solve geometrical problems algebraically. In the algebra all such quantities were handled in the same way as the positive integers and they were therefore called numbers. Newton explains: "By a number we do not understand a multitude of units, but rather the abstract ratio of any quantity to another quantity of the same kind, which is taken as a unit. There are three varieties of number: integers, rationals and irrationals" (1707, p. 2). Eventually, they were called real numbers, to distinguish them from the imaginary ones, that is, the multiples of √−1, which also turned up as solutions of algebraic equations.

The method of coordinates soon suggested the idea of a space with n dimensions, whose points would be labeled by n quantities. In particular, if 〈x, y 〉 denotes an arbitrary point on a plane, a straight line on that plane can now be defined as the set of points satisfying the linear equation y = ax + b, and a circle with radius r and center at 〈0, 0〉 as the set of points satisfying the quadratic equation x ² + y ² = r ². These two equations take care of all points on the plane that can be constructed with a ruler and a compass, which were the only points contemplated by Euclid. But after Descartes, mathematicians felt free to consider any curve defined by an algebraic equation or indeed by a convergent series, such as y = sin x, or y = e^x (where e is the base of the natural logarithms). Even though Euclid never countenanced the plethora of points obtainable in such ways and it does not follow from his postulates, what we normally call "Euclidean space" comprises them all.

kant's philosophy of geometry

The overwhelming success of geometry in physics and astronomy induced some seventeenth-century philosophers to follow its example in ethics and metaphysics. The foremost instance of this is Benedict de Spinoza's Ethica ordine geometrico demonstrata (Ethics demonstrated in geometric order; 1677), but John Locke too believed that "if men would in the same method, and with the same indifferency, search after moral as they do mathematical truths," then "a great part of morality might be made out with that clearness, that could leave, to a considering man, no more reason to doubt, than he could have to doubt of the truth of propositions in mathematics, which have been demonstrated to him" (1690, IV.iii.20, xii.8).

Immanuel Kant, however, thought otherwise. Invidiously comparing geometry, as a science that "excels all others in certainty and distinctness," with metaphysics, which "has only just started out on the path to these goals" (1902–, 2: 168), he recommended, in 1763, that the latter stop imitating the former, in order to progress along that path. He soon went further. In his Latin dissertation of 1770, Kant taught that confusion and stagnation in metaphysics were due to the contamination of the human intellect with the sensuous notions of time and space. By thoroughly avoiding them, metaphysics will escape the temptations of materialism and determinism and become a secure science of God, freedom, and immortality. Yet in the Critique of Pure Reason (1781), Kant likened the purified understanding he advocated in 1770 to a bird that, tired by the resistance of the air, sets out to fly in a vacuum. In his mature view, the basic concepts of human thought—one and many, reality and negation, substance and cause—are not obtained from sensuous experience, but they can refer to objects only when applied to it, under the conditions of human sensibility, namely, space and time.

This decisive turnabout in the history of philosophy is closely related to Kant's reflections on geometry and its use in physics. In 1746 Kant spoke of a general or "supreme" geometry, adapted to a space with any number of dimensions. That the space we live in has only three dimensions is due to the empirical fact that all material particles are linked by forces governed by Newton's inverse-square law (1902–, 1: 34). But in 1768 he made a discovery that, he thought, put an end to all such explanations of space and spatial structure from the physical relations between bodies in space. No description of a shoe in terms of its different parts and the relations between them will allow us to tell a left shoe from the matching right shoe; the difference between the two shoes can be grasped only by considering their respective orientation in the space that embraces them. Kant understood this to imply that the bodily structure of bodies depends on that of space as a whole, which therefore is presupposed by them, rather than being only an expression of their interactions.

Kant was then faced with the following dilemma: Either (a) space itself is a substance, of which bodies are modes (a position that, according to Kant, results in Spinoza's unchristian and immoral deification of space), or (b) space must be thought of as possessing a novel, hitherto unheard of manner of existence (which implies a corresponding adjustment of the ontological standing of bodies as such). About 1769 Kant lighted on alternative (b), which he described around 1791 as one of the two hinges on which metaphysics must turn (the other one being the reality of freedom). He claims that "space is not something objective and real, neither a substance, nor an accident, nor a relation; it is rather a subjective and ideal scheme, so to speak—which issues from the nature of the mind according to a stable law—for coordinating everything that is sensed externally" (1902–, 2: 403). Or, in mantra form, space is one of the forms of human sensibility (time is the other one). As a consequence of this, things are bodies only insofar as they are actual or potential objects of our sense perception, but not as they are in themselves. (Indeed, Kant figured out in the early 1770s that the standard assumption that things in themselves are spatial would make it impossible to solve the contradictions regarding the limit of the physical world and the divisibility of its content, which he later set forth in the first two items of the Antinomy of pure reason.)

Kant's conception of space is the key to his philosophy of geometry (and is in turn reinforced by it). The epistemological problem of geometry lies in explaining how it can furnish us with precise quantitative information about things we have never met in real life and which anyway we could not measure accurately, for example, the exact size of the angles of a trillion-sided regular polygon. Plato proposed that this knowledge is remembered from another life in which we had direct access to the intelligible "form" of things. The fact that geometry contains such knowledge nourished similar hopes for metaphysics and ethics, which, however, were crushed by Kant's approach. In his view, geometry rests on our natural awareness of the conditions under which alone the manifold appearances displayed through our external senses "can be ordered into certain relations" (1787, B 34) and thus shaped into corporeal phenomena. Such awareness is not intellectual but intuitive, as we may gather from the example of the pair of shoes, described above, and also from the fact that geometrical proofs proceed by the "construction of concepts." Kant explains this expression somewhat intriguingly as follows: "To construct a concept means to exhibit a priori the intuition that corresponds to it: the construction of a concept therefore requires a non-empirical intuition which … as intuition is a particular object, but nevertheless, as the construction of a concept (a general idea), must convey universal validity for all possible intuitions that belong under the same concept" (1781/1787, A 713/B 742).

Anyone not put off by these opaque notions could well regard them as a proper explanation of the amazing success of geometry in physics. For, if geometry spells out the ordering that is required for us to grasp external phenomena, then it is no wonder that all external phenomena comply in every detail with the teachings of geometry. Soon, however, innovations in geometry moved the ground from under Kant's position and made it untenable. Before we turn to them, it should be emphasized that among these innovations, the best known one—the derivation of a consistent geometry from the denial of Euclid's fifth postulate—does not challenge Kant's view but somehow corroborates it. For Kant, geometry provides information, conveyed by what he called synthetic propositions, and this implies that any of its unproven principles can be denied without selfcontradiction.

From Gauss to Hilbert and beyond

non-euclidean geometry

The fact that Euclid's fifth postulate is not self-evident prompted several mathematicians to try to prove it. John Wallis (1616–1703) succeeded in inferring it from the assumption that for any given figure there is another one, similar to it, of any arbitrary size. This assumption is neither necessarily true nor empirically obvious, but it does provide a perspicuous characterization of Euclidean space.

Girolamo Saccheri (1733[1986]) sought to prove the postulate indirectly. He devised a quadrilateral thus constructed on a plane: Draw straight lines m and n through points P and Q, making right angles with the segment PQ. Mark points A and B on m so that AP = PB. Mark points C and D on n so that CA and DB are both perpendicular to m (See Figure 2).

If one assumes the Archimedean postulate (which Saccheri tacitly does), the fifth postulate will hold if and only if ∠ACQ and ∠BDQ are right angles. Saccheri assumed that these angles are obtuse and easily proved that, if so, two points can be joined by more than one line, which he considered absurd. He then assumed that both

angles are acute and derived from this hypothesis many surprising propositions that did not appear to be contradictory, until at last he reached one he pronounced "repugnant to the nature of the straight line."

The consequences that Saccheri drew from the acute-angle hypothesis reappeared in the nineteenth century in the private papers of Carl F. Gauss (1777–1855) and in independent publications by Nikolay I. Lobachevsky (1793–1856) and Janos Bolyai (1802–1860). These authors treated these consequences as theorems of an alternative system of geometry, based on the straightforward denial of Euclid's fifth postulate (with the others retained). This system has received various names, but by priority of publication, it should be called Lobachevskian geometry. In this geometry, the three interior angles of a triangle add up to less than two right angles, the difference being proportional to the area of the triangle. Therefore, similar triangles are congruent. Consider again the segment PQ perpendicular to straight line m at P. By the denial of the fifth postulate, there is a set S of straight lines through Q that form an acute angle with PQ on one or the other side of it and yet do not meet m on that side (let alone on the side where they form an obtuse angle with PQ ). Let α be the smallest of these angles. By symmetry, there are two lines in S that form angle α on either side of PQ. In Lobachevsky's terminology (independently adopted also by both Gauss and Bolyai), these two lines are called the parallels of m through Q, and α is the angle of parallelism for PQ. The size of α decreases as PQ grows. On any Lobachevskian plane, there is a unique length h such that the angle of parallelism for any segment of length h equals 45°. The length h provides an absolute standard of length for that plane. Kant's friend Johann Heinrich Lambert, who around 1766 worked on this subject along lines similar to Saccheri's, said there was "something alluring about this consequence which readily arouses the desire that the [acute angle] hypothesis be true!" (1786/1895, p. 162). Note that if, in the case discussed, PQ = h, the two parallels to m through Q are mutually perpendicular. Lambert thought this was an intolerable paradox.

The absence of contradiction in a long series of theorems inferred from the denial of the fifth postulate does not, of course, imply that Lobachevskian geometry is consistent. Lobachevsky proposed an argument for proving that his geometry is at least as tenable as Euclidean geometry. He showed that there is a logically formal correspondence between the equations of Lobachevskian trigonometry and the familiar equations of spherical trigonometry. By virtue of it, any contradiction derived from the former will be matched by one flowing from the latter. Such a contradiction would entail that the said standard trigonometric equations are false, and this in turn would entail the falsehood of the Euclidean principles from which these equations follow.

Lobachevsky also tried to ascertain whether his own geometry or Euclid's is true of physical space. He used astronomical data to calculate the sum of the internal angles of the triangle formed by three stars and concluded that the difference between the result obtained in a Lobachevskian space and the Euclidean value was well within the margin of observational error. Decades would pass before Hermann Lotze (1879, p. 774) pointed out that all such attempts are vain, for if astronomical measurements do not agree with Euclidean geometry, the disagreement can still be accounted for by a deviation of stellar light from its supposedly rectilinear trajectory.

groups and invariants

Shortly before Lobachevsky's earliest publication on his geometry, Jean-Victor Poncelet's Treatise on the Projective Properties of Figures (1822) started a way of doing geometry that seemed more intuitive and tame but which ultimately was much more radical and would have deeper consequences than the denial of Euclid's fifth postulate. It is based on adding to each straight line m a "point at infinity" that m shares with every straight line parallel to it and treating all such "points" as belonging to a single "plane." This assumption enormously simplified the statement and the proof of geometric theorems concerning relations of incidence, collinearity, and coplanarity among points, lines, and planes. Metric features like distance and metric relations like congruence were totally ignored. Natural (initially tacit) assumptions regarding the neighborhood relations between the points at infinity and the standard points implied that ordinary space differed drastically from the new projective space in which it was now embedded. For example, a left shoe traveling indefinitely in a fixed direction would, after crossing the plane at infinity, return to its original location from the opposite side, in the guise of a right shoe. In this way, projective geometry disposed of Kant's claim about the irreducible difference between the two kinds of shoes, its ontological implications, and its intuitive roots.

Projective geometry grew in scope and sophistication at the hands of August Moebius (1790–1868), Julius Plücker (1801–1868), Karl Georg Christian von Staudt (1798–1867), Arthur Cayley (1821–1895), and others. Different sorts of numerical coordinates were introduced as sheer labeling devices, for in this metric-free context they plainly did not represent distances. The use of coordinates consisting of complex numbers made it possible to introduce more points, in addition to the familiar real points (those labeled by real numbers). These "complex points" are linked to real points and among themselves by relations of collinearity (if their coordinates satisfy the same linear equations) and vicinity (by dint of the neighborhood relations between real and nonreal numbers on the complex plane). The beautiful vistas opened by such developments inspired further flights of mathematical freedom leading to the creation of still other branches of geometry.

Moved by the confusing variety of geometrical methods and approaches, Felix Klein formulated his celebrated Erlangen Program, in which he seeks to unify all forms of geometry under a single overarching point of view. This is provided by the notion of transformation group and the related notion of invariant.

Let S be a set of points. A transformation (or permutation ) T of S assigns to each point p of S one and only one point T (p ) of S, in such a way that every point of S equals T (p ) for some p. In other words, a transformation of S is a one-to-one mapping of S onto itself. We say that T (p ) is the value of T at p. T is said to send p to T (p ). If M is a subset of S, T is said to send M to the set T (M ) = {T (p ): p ∈ M }. For every transformation T, there is an inverse transformation T ⁻¹ that, for each p in S, sends T (p ) back to p. The identity transformation Id_S sends every p in S to itself. Given two transformations T ₁ and T ₂, their product T ₂T ₁ is the transformation that sends each p in S to the value of T ₂ at T ₁(p ). The product of transformations is clearly associative, that is, (T ₃T ₂)T ₁ = T ₃(T ₂T ₁) for any three transformations T ₁, T ₂, and T ₃. A set G is a group of transformations of S if every element of G is a transformation of S and G contains (1) the product of any two of its elements, (2) the inverse of every one of its elements, and (3) the identity transformation Id_S. Any subset of G that meets conditions (1) through (3) is said to be a subgroup of G.

Given a group G of transformations of a set S, let R be an n -adic predicate (n ≥ 1) such that, for any points p ₁, …, p_n in S and any transformation T in G, R (p ₁, …, p_n ) implies that R (T (p ₁), …, T (p_n )). We say then that R is an invariant of group G or that R is G-invariant. Likewise, a function f on Sⁿ is said to be G-invariant if, for every n -tuple 〈p ₁, …, p_n 〉 of elements of S, f (p ₁, …, p_n ) = f (T (p ₁), …, T (p_n )). G is said to preserve its invariants.

Klein's Erlangen Program for systematically ordering geometries is based on the following simple idea: Each geometry is the study of the invariants of a group, and the relations of inclusion between groups and their subgroups determine a hierarchy of geometries. Starting from the group of all possible transformations of an arbitrary set, whose sole invariant is the cardinality of the set, one descends, through multiple branches, right down to the trivial group, which is a subgroup of every group and preserves every property and relation, for it only comprises the identity transformation. In particular, projective geometry studies the invariants of the group of collineations, that is, the set of transformations that send straight lines to straight lines. This is a subgroup of the group of continuous transformations, whose invariants are the topological properties of projective space. Drawing on work by Arthur Cayley (1859), Klein (1871, 1873) found a way of defining different real-valued functions on point pairs that behaved, on well-defined regions of projective space, precisely like the distance functions of, respectively, Lobachevskian geometry (which he called hyperbolic ), Euclidean geometry (which he called parabolic ), and a third geometry (which he called elliptic ). Each of these functions was an invariant of a certain subgroup of the said group, comprising the collineations that map a specific quadric surface onto itself.

Klein's result led Russell (1897) to assert that the general "form of externality" is disclosed to us a priori in projective geometry, but its metric structure—which Russell wrongheadedly claimed can only be Lobachevskian, Euclidean, or elliptic—must be determined a posteriori by experiment. Poincaré took another view of this matter: If geometry is nothing but the study of a group, "one may say that the truth of the geometry of Euclid is not incompatible with the truth of the geometry of Lobachevsky, for the existence of a group is not incompatible with that of another group" (1887, p. 290). Euclidean geometry has seemed preferable only because the rotations and translations of the Euclidean group reflect, to a comfortable approximation, the motions of ordinary hard bodies in our environment.

riemannian manifolds

In his lecture "On the Hypotheses Which Lie at the Foundation of Geometry" (1867), Bernhard Riemann took an approach to geometry that did not fit into Klein's Erlangen Program. He noted that traditional geometry rested on assumptions summed up in Pythagoras's theorem (by which the distance between a point with Cartesian coordinates 〈x, y, z 〉 and the origin 〈0, 0, 0〉 equals the positive square root of x ² + y ² + z ²). These assumptions had been corroborated by using light rays to line up things and rigid bodies to measure the length of lines, and therefore were bound to break down on very small scales, where such physical objects are not available, and perhaps also on very large scales, where the errors of observation generated by using such instruments might become intolerable.

Riemann therefore proposed to proceed from more general assumptions toward a more flexible geometry that physicists could later resort to when they needed it. He took his cue from Carl Friedrich Gauss's work on the intrinsic geometry of surfaces (1828), which he extended to general spaces of n -dimensions (n-manifolds, for short). Though Riemann supposedly addressed his lecture to humanists (hence its meager use of mathematical symbolism), the meaning and reach of the lecture first became clear through its further elaboration by other outstanding mathematicians (e.g., Elwin Bruno Christoffel, Friedrich Schur, Wilhelm Killing, Élie Cartan, Hermann Weyl), and a whole century would pass before it was satisfactorily explained to undergraduate students (Spivak 1979). The following rough sketch of Riemann's breakthrough owes much to the light shed on it by such later developments.

An n -manifold M is furnished with coordinate systems or charts, by which different regions or patches of M are mapped continuously and one-to-one onto subsets of ℝⁿ (the set of all n -tuples of real numbers, endowed with the neighborhood relations it inherits from the real-number field ℝ). Two charts defined on overlapping patches are said to be compatible if the coordinate transformation between them is a smooth function from one open subset of ℝⁿ onto another (possibly the same) one. An atlas of M is a collection of compatible charts for M such that every point of M lies on the patch of at least one chart. Any atlas A of M determines a corresponding maximal atlas A _max comprising every conceivable chart of M compatible with those in A. The way the charts of A _max combine with each other in coordinate transformations reflects the overlapping and intertwining of the patches on which they are defined and thus specifies the global topology, the shape, of M. (In the realm of 2-manifolds, or surfaces, the atlas of a pretzel differs from that of a donut or a bun.) The lengths of curves drawn in M (which are best thought of as continuous mappings of an open interval of ℝ into M ) can then be defined in an endless variety of ways by rules that assign to each point p of M an appropriate function on the coordinate differentials at p, in a manner that varies smoothly from point to point.

Riemann was aware that this approach gave the mathematician enormous freedom, and he proposed restricting the admissible functions, for the time being, to quadratic functions on coordinate differentials, which, on a small neighborhood of each point of M, would yield a definition of length in optimal agreement with the n -dimensional version of the Pythagorean theorem. To conform to the standard concept of length, he also required his quadratic functions to be positive-definite, that is, to take their values only among the nonnegative real numbers. This requirement was subsequently relaxed in the spacetime geometries of the theory of relativity, which, for this reason, are often called semi-Riemannian.

From the standpoint of twentieth-century mathematics, the characterization of an n -manifold M by means of an atlas of the kind described makes it possible to assign an n -dimensional real vector space to each point p of M, the tangent space at p (thus called by analogy with the plane tangent to a smooth surface at any point of it). A Riemannian metric on M assigns to each p of M a tensor of rank 2, that is, a bilinear function on its tangent space, that varies smoothly from point to point. Riemann's quadratic functions on the coordinate differentials at each point can be naturally obtained as appropriate representations, relative to one or another locally defined coordinate system, of such coordinate independent objects.

A given Riemannian metric μ on M determines a smoothly varying assignment, to each p of M, of a tensor of rank 4, that is, a quadrilinear function on its tangent space. Such an assignment is in effect a field of tensors (one at each point), but, for short, this assignment is called the Riemann tensor of the Riemannian manifold 〈M, μ〉. This is a natural generalization to n -manifolds of the analytically defined yet fairly intuitive concept of the Gaussian curvature of a surface (a real-valued function that is positive and constant on a sphere, variable but always positive on an ovoid, alternatively positive and negative on the surface of a saddle, and equal to 0 on a plane or a cylinder), and is therefore also called the curvature tensor, a term that mathematicians wield nonchalantly but that among philosophers has been a source of endless worries.

On this same analogy, n -manifolds of constant 0 curvature are said to be flat. In particular, Euclidean space is a flat 3-manifold, while Lobachevskian space is a 3-manifold of constant negative curvature, and Klein's elliptic space is a 3-manifold of constant positive curvature. But a generic Riemannian manifold 〈M, μ〉 has variable curvature, and therefore the only group of transformations of M that will preserve the metric μ is the trivial group consisting only of the identity transformation. Already for this reason, Riemannian geometry obviously cannot fall under the Erlangen Program. Another excluding reason is the fact that a geometric inquiry that considers general spaces of endlessly different shapes cannot be characterized by a group of transformations of one of these spaces onto itself. Still, there are Riemannian manifolds endowed with interesting symmetries, and group theory has been the tool of choice for studying them.

hilbert's foundations

Euclid's putative program for logically inferring the truths of geometry from a sufficient list of unproven premises was fondly imitated by scientists and philosophers in the seventeenth century, but it was first properly carried out by Moritz Pasch (1882). He gathered what he regarded as the empirical foundation of geometry into a few undefined concepts concerning the shape, size, and reciprocal position of bodies and a few axioms that linked these concepts among themselves and with other concepts defined in terms of them. Pasch's axioms "state what has been observed in certain very simple diagrams" (p. 43). All other geometric statements should be proved from the axioms by the strictest deductive methods.

Pasch dealt with projective geometry. The first rigorous axiomatization of Euclidean geometry was given in David Hilbert's Foundations of Geometry (1899), a book that had a major influence on twentieth-century mathematics and philosophy. Hilbert invited the reader to consider three arbitrary sets of objects, which he called points, straights, and planes; three undefined relations of incidence between a point and a straight line, between a straight line and a plane, and among three points; and two undefined relations of congruence between two pairs of points (segments,) and between two equivalence classes of point triples (angles ). Hilbert linked these objects and relations through nineteen axioms, which—when supplemented with the "axiom of completeness" added in the second edition—are sufficient for characterizing the said objects and relations up to isomorphism. This means that if we have two threefold collections of points, straights, and planes having the prescribed relations of incidence and congruence in agreement with Hilbert's twenty axioms, there will always be a one-one mapping of the points, straights, and planes of one collection respectively onto the points, straights, and planes of the second that preserves all five sorts of relations. Such a structure-preserving mapping between structured sets is called an isomorphism. Evidently it can hold between two systems of intuitively very dissimilar objects.

Hilbert availed himself of this feature of axiomatic theories for studying the independence of some axioms from the rest. To prove such independence, he proposed actual instances (models ) of the structure determined by all the axioms but one, plus the negation of the omitted one. Gottlob Frege complained that the geometric axioms retained in these exercises could be applied to Hilbert's far-fetched models only by tampering with the natural meaning of words. Hilbert replied, on December 29, 1899, "Every theory is only a scaffolding or schema of concepts together with their necessary mutual relations, and the basic elements can be conceived in any way you wish. If I take for my points any system of things, for example, the system love, law, chimney-sweep, … and I just assume all my axioms as relations between these things, my theorems, for example, the theorem of Pythagoras, also hold of these things.… This feature of theories can never be a shortcoming and is in any case inevitable" (Frege 1967, p. 412). Hilbert's declaration of independence from sense experience and ordinary usage concisely expresses the modern view of mathematics as a universal "science of patterns" (Resnik 1997), in which geometry is barely distinguishable from its other branches, except on historical grounds.

the geometry of the universe

This approach to pure geometry and mathematics gives physicists enormous freedom to choose the abstract structures they judge most suitable for representing (modeling) the phenomena under inquiry. Yet, as Albert Einstein (1921) pointed out, so long as physics remains unable to provide, from microphysical principles, an exact theoretical construction of the instruments it uses for measuring distances and times, it will continue to need a practical geometry, which must be suggested and corroborated by experience.

According to Einstein, the stability of sharp spectral lines justifies the postulate that any two ideal clocks, once running beside each other at the same rate, will always do so, no matter where and when they are brought together again for comparison. Under this postulate, and for spatiotemporal regions sufficiently small that gravity is practically homogeneous in them, experience has amply vindicated the validity of the flat semi-Riemannian geometry that Hermann Minkowski (1909) initially proposed for the whole world. For broader regions, Einstein's theory of general relativity assumes a semi-Riemannian spacetime geometry whose variable curvature reflects the variations in gravity.

By solving Einstein's equations of the gravitational field under cosmologically plausible special symmetry requirements, Alexander Friedmann (1922, 1924) produced big-bang models of the universe. These big-bang models were ready at hand to explain the systematic recession of the galaxies away from us when discovered by Vesto Slipher and Edwin Hubble before 1930 and the current low temperature of the background thermal radiation when discovered by Arno Penzias and Robert Wilson in 1964. It is worth emphasizing that the explanation of these phenomena from the Einstein field equations is purely geometrical—a consequence of the shape of the universe under the postulated symmetry requirements. Purely geometric arguments also support the proof by Stephen Hawking and Roger Penrose (1970) that, under physically very plausible assumptions, a generic relativistic space-time most likely contains black holes.

See also Aristotle; Black Holes; Boole, George; Copernicus, Nicolas; De Morgan, Augustus; Descartes, René; Einstein, Albert; Frege, Gottlob; Galileo Galilei; Hilbert, David; Kant, Immanuel; Kepler, Johannes; Leibniz, Gottfried Wilhelm; Locke, John; Lotze, Rudolf Hermann; Mathematics, Foundations of; Newton, Isaac; Peirce, Charles Sanders; Plato; Poincaré, Jules Henri; Pythagoras and Pythagoreanism; Russell, Bertrand Arthur William; Space; Spinoza, Benedict (Baruch) de; Thales of Miletus; Weyl, (Claus Hugo) Hermann.

Bibliography

references

Aristotle. The Complete Works. Revised Oxford translation. 2 vols., edited by Jonathan Barnes. Princeton: Princeton University Press, 1984.

Cayley, Arthur. "A Sixth Memoir upon Quantics." Philosophical Papers of the Royal Society of London 149 (1859): 61–90.

Einstein, Albert. Geometrie und Erfahrung. Berlin: Julius Springer. Reprinted in The Collected Papers of Albert Einstein. Vol. 7, 383–402. Princeton, NJ: Princeton University Press, 2002. Translated as "Geometry and Experience" in the matching supplementary volume.

Frege, Gottlob. Kleine Schriften, herausgegeben von I. Angelelli. Darmstadt, Germany: Wissenschaftliche Buchgesellschaft, 1967.

Friedmann, Alexander. "Über die Krümmung des Raumes." Zeitschrift für Physik 10 (1922): 377–386.

Friedmann, Alexander. "Über die Möglichkeit einer Welt mit konstanter negativer Krümmung des Raumes." Zeitschrift für Physik 21 (1924): 326–332.

Galileo Galilei. Il saggiatore nel quale con bilancia esquisita e giusta si ponderano le cose contenute nella Libra astronomica e filosofica di Lotario Sarsi Sigensano. Roma: Giacomo Mascardi, 1623. Partial English translation in Stillman Drake, Discoveries and Opinions of Galileo. New York: Doubleday, 1957 (The Assayer ).

Gauss, Carl Friedrich. Disquisitiones generales circa superficies curvas. Göttingen, Germany: Dieterich, 1828. English translation by A. Hiltebeitel and J. Morehead: General Investigations of Curved Surfaces. Hewlett, NY: Raven Press, 1965.

Hawking, Stephen W., and Roger Penrose. "The Singularities of Gravitational Collapse and Cosmology." Royal Society of London Proceedings A 314 (1970): 529–548.

Herodotus, with an English translation by A. D. Godley. 4 vols. Cambridge, MA: Harvard University Press, 1946.

Hilbert, David. "Die Grundlagen der Geometrie." In Festschrift zur Feier der Enthüllung des Gauss-Weber Denkmals. Leipzig, Germany: B.G. Teubner, 1899, pp. 3–92.

Kant, Immanuel. (A), Critik der reinen Vernunft. Riga: J.F. Hartknoch, 1781. Repeatedly translated into English as Critique of Pure Reason.

Kant, Immanuel. (B), Critik der reinen Vernunft. Zweyte hin und wieder verbesserte Auflage. Riga: J.F. Hartknoch, 1787. Second edition of Kant (A); variants normally included in English translations.

Kant, Immanuel. Kant's gesammelte Schriften, herausgegeben von der Preussischen Akademie der Wissenschaften (Bde. 1–22), der Deutschen Akademie der Wissenschaften zu Berlin (Bd. 23), und der Akademie der Wissenschaften zu Göttingen (Bde. 24, 25, 27–29). Berlin, 1902–. Page numbers of this edition are printed on the margin of The Cambridge Edition of Kant's Works in Translation.

Kepler, Johannes. Astronomia nova aitiologetos seu physica coelestis, tradita commentariis de motibus stellæ Martis. Pragæ, 1609. English translation by William H. Donahue: New Astronomy. Cambridge, U.K.: Cambridge University Press, 1992.

Klein, Felix. "Über die sogenannte Nicht-Euklidische Geometrie." Mathematische Annalen 4 (1871): 573–625.

Klein, Felix. "Über die sogenannte Nicht-Euklidische Geometrie (Zweiter Aufsatz)." Mathematische Annalen 6 (1873): 112–145.

Klein, Felix. Vergleichende Betrachtungen über neuere geometrische Forschungen. Erlangen, Germany: A. Duchert, 1872.

Kline, Morris. Mathematics: The Loss of Certainty. New York: Oxford University Press, 1980.

Lambert, Johann Heinrich. "Theorie der Parallellinien." Magazin für die reine und angewandte Mathematik 2 (1786): 137–164; 3 (1786): 325–358. Reprinted in P. Stäckel and F. Engel, Die Theorie der Parallellinien von Euklid bis Gauss. Leipzig, Germany: B.G. Teubner, 1895, pp. 152–207; our quotation refers to this edition.

Locke, John. An Essay concerning Humane Understanding. London: Thomas Basset, 1690.

Lotze, Hermann. Metaphysik: Drei Bücher der Ontologie, Kosmologie und Psychologie. Leipzig, Germany: S. Hirzel, 1879.

Minkowski, Hermann. "Raum und Zeit." Physikalische Zeitschrift 10 (1909): 104–111.

Newton, Isaac. Arithmetica universalis, sive De compositione et resolutione arithmetica liber. Cambridge, England: Typis academicis, 1707.

Pasch, Moritz. Vorlesungen über neueren Geometrie. Leipzig, Germany: Teubner, 1882.

Plato. Laches - Protagoras - Meno - Euthydemus, with an English translation by W. R. M. Lamb. Cambridge MA: Harvard University Press, 1990.

Plato. Laws, with an English translation by R. G. Bury. 2 vols. Cambridge MA: Harvard University Press, 1984.

Plato. Republic, with an English translation by Paul Shorey. Cambridge MA: Harvard University Press, 1982.

Poincaré, Henri. "Sur les hipothèses fondamentales de la géométrie." Bulletin de la Société mathématique de France 15 (1887): 203–216.

Poncelet, Jean-Victor. Traité des propriétés projectives des figures. Paris: Bachelier, 1822.

Resnik, Michael D. Mathematics as a Science of Patterns. Oxford, U.K.: Clarendon Press, 1997.

Riemann, Bernhard. "Über die Hypothesen, welche der Geometrie zugrunde liegen." Abhandlungen der Kgl. Gesellschaft der Wissenschaften zu Göttingen 13 (1867): 133–152. For English translation, see under Spivak.

Russell, Bertrand. An Essay on the Foundations of Geometry. Cambridge, U.K.: Cambridge University Press, 1897. Unaltered reprint: New York, Dover, 1956.

Saccheri, Girolamo. Euclides ab omni nævo vindicatus. Mediolani: e typographia Pauli Antonii Montani, 1733. Reprinted as Euclides vindicatus, with facing English translation by G. B. Halsted, New York: Chelsea, 1986.

Simplicius. In Aristotelis de Caelo Commentaria, edidit I. L. Heiberg. Berlin: G. Reimer, 1894.

Spinoza, Benedictus. Ethica ordine geometrico demonstrata, in Opera posthuma. Amstelodami: J. Rieuwertsz, 1677.

Spivak, Michael. A Comprehensive Introduction to Differential Geometry. 2nd ed., Berkeley: Publish or Perish. 5 vols, 1979. Includes an excellent English translation, with mathematical commentary, of Riemann's lecture "On the hypotheses that lie at the foundation of geometry"; see Vol. 2, pp. 135ff.

suggestions for further reading

Aaboe, Asger. Episodes from the Early History of Mathematics. New York: Random House, 1964.

Blumenthal, Leonard M. A Modern View of Geometry. San Francisco: W. H. Freeman, 1961.

Boi, Luciano. Le problème mathématique de l'espace: Une quête de l'intelligible. Berlin: Springer, 1995.

Bonola, Roberto. Non-Euclidean Geometry: A Critical and Historical Study of Its Development. English translation with additional appendices by H.S. Carslaw. New York: Dover, 1955. Includes English translations of works by Bolyai and Lobachevsky.

Buroker, Jill Vance. Space and Incongruence: The Origin of Kant's Idealism. Dordrecht, Netherlands: D. Reidel, 1981.

Coxeter, Harold S. M. Introduction to Geometry. 2nd ed. New York: John Wiley and Sons, 1969.

Descartes, René. The Geometry. Translated from the French and Latin by D. E. Smith and M. L. Latham, with a facsimile of the first edition of 1637. New York: Dover, 1956.

Ellis, George F. R., and Ruth M. Williams. Flat and Curved Space-Times. Oxford, U.K.: Clarendon Press, 1988.

Heath, T. L. The Thirteen Books of Euclid's Elements. Translated from the text of Heiberg with introduction and commentary. New York: Dover. 3 vols.

Heilbron, J. L. Geometry Civilized: History, Culture, and Technique. Oxford: Clarendon Press, 1998.

Hilbert, David and S. Cohn-Vossen. Geometry and the Imagination. Translated from the German book Anschauliche Geometrie by P. Nemenyi. New York: Chelsea, 1952.

Modenov, P. S. and Parkhomenko, A. S. Geometric Transformations. Translated and adapted from the first Russian edition by M. B. P. Slater. 2 vols. New York: Academic Press, 1965.

Nagel, Ernest. "The Formation of Modern Conceptions of Formal Logic in the Development of Geometry." Osiris 7 (1939): 142–224.

Neugebauer, Otto. The Exact Sciences in Antiquity. 2nd ed. New York: Dover, 1969.

Rosenfeld, B. A. A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space. Translated from the Russian by A. Shenitzer. New York: Springer, 1988.

Torretti, Roberto. Philosophy of Geometry from Riemann to Poincaré. Dordrecht, Netherlands: D. Reidel, 1978.

Roberto Torretti (2005)