The Foundations of Geometry

Date 1899

David Hilbert’s The Foundations of Geometry established the basic axiomatic-formalist approach to systematizing mathematics, initiated by compactly deriving a formal axiomatic model for Euclid’s geometry.

Locale Göttingen, Germany

Key Figures

• David Hilbert (1862-1943), German mathematician and logician
• Felix Klein (1849-1925), German mathematician and educator
• Moritz Pasch (1843-1930), German mathematician
• Giuseppe Peano (1858-1932), Italian mathematician and logician

Summary of Event

The modern hypothetico-deductive method in mathematics and the concurrent drive toward abstraction, formalization, and establishing universally applicable foundations may be traced to two principal and near-contemporary sources. One was the development of diverse non-Euclidean geometries and the “Erlanger Programme” efforts at their reconciliation. The other was the group of paradoxes in formal logic and set theory that followed from the invention of quantification theory by Gottlob Frege and the axiomatization of arithmetic by Richard Dedekind and Giuseppe Peano. Carl Friedrich Gauss, Nikolay Ivanovich Lobatchevsky, and János Bolyai in the first three decades of the nineteenth century developed alternate non-Euclidean geometries following the realization that logical negation of Euclid’s parallel postulate need not lead to contradiction. Subsequently, they and others found that although the theorems resulting from the new geometric axioms were at odds with observational results of everyday experience, or Immanuel Kant’s intuitions, none of the expected logical contradictions appeared in these new geometries.

During the late 1860’s, new attention was drawn to non-Euclidean geometry by publications of Hermann von Helmholtz and Ernesto Beltrami. In 1870, Felix Klein discovered a more general model or interpretation of non-Euclidean geometries in the 1859 work of Alexander Cayley, by means of which Klein was able to identify systematically all the primitive objects and relations of the new geometries with corresponding primitives in Euclidean geometry. In 1872, Klein published an important paper, Vergleichende Betrachtungen über neuere geometrische Forschungen (equivalency considerations of recent geometric research), which expounded his so-called Erlanger Programme, strongly influencing several generations of mathematicians. This program for the “algorithmic systematization” of geometry and other areas of mathematics expounded the thesis that diverse non-Euclidean geometries could be unified and classifiably related by reconsidering geometry as the more general study of the particular “forms,” or formal algebraic properties of spatial configurations of a “manifold” that are left unchanged (invariant) by an underlying group of transformations (such as rotation, translative motion, and the like).

The new group-theoretical viewpoint of Klein, varyingly adopted by Sophus Lie, Henri Poincaré , Friedrich Schur, Eli Cartan, and others, was subsequently applied to their work on the theory of equations, automorphic functions, and complex function theory, all considered by Klein as “higher geometry.” This liberal and novel use of the term “geometry” reflected not only disdain for ancient Euclidean axiomatics employing verbal definitions but also broader visions of the group-theoretic concept as a unifying principle for all mathematics. This view found even greater cultivation in Klein’s later work at the University of Göttingen during the 1890’s.

Concurrent and partially linked to the Erlanger Programme was reorganization of Euclidean projective geometry by the German mathematician Moritz Pasch in 1882. In the light of the variety of non-Euclidean geometries and the new symbolic logic, concurrent with but independent of Giuseppe Peano, Pasch underscored the distinctions between “explicit” and “implicit” definitions in geometry. An explicit definition expresses a new term by means of terms already accepted in the technical vocabulary at hand. By contrast, implicit definitions broadly define a new term from the total context in which it occurs, recognizing that it is logically impossible to define all terms explicitly without infinite regress or vicious circularity.

Whereas Euclid attempted explicit definitions for all his basic terms such as “point,” “line,” and “plane,” Pasch accepted these terms as primitive or “nuclearly irreducible” implicit definitions. Although the origin of these nuclear propositions in geometry might be based directly on assertions of physical or psychological origins, Pasch emphasized that these propositions, insofar as they are stated and used in mathematics, are stated totally without any regard to extra-geometrical aspects. Following Pasch, Italian mathematician Peano in 1889 gave a new and rigorous reinterpretation of Euclidean geometry using his new symbolic logic. Like Pasch, Peano based his treatment on specific primitive terms and their relations to the intuitive notion of “betweenness.” Effectively translating Pasch’s treatise into more compact symbolic notation, Peano’s geometry remained a purely formal calculus of relations between variables, which remained without any ready applications or continuations by other mathematicians.

As a student at Königsberg between 1884 and 1890, and later as a professor at Halle University in 1891, David Hilbert was exposed to the theory of invariants and notably a wide variety of abstract formalistic approaches to the new axiomatizations of (non-) Euclidean geometry. Some of Hilbert’s earliest university lectures, between 1889 and 1891, concerned algebraic and projective geometry. In the fall of 1891, Hilbert attended a lecture by German mathematician Hermann Wiener, where he first learned about the more general validity and scope of the axiomatic method. According to Constance Reid’s biography of Hilbert, as early as 1894 in his summer lectures on the foundations of geometry, Hilbert intended to produce the purest possible algebraic system of exact axiomatic non-Euclidean geometry, with Euclid’s geometry as a special case.

After extensive studies at the University of Göttingen of number theory and algebra, Hilbert was reinspired to continue his axiomatic foundations of geometry in early 1898, as noted through communications with Schur and Klein. In the summer of 1898, Hilbert gave a lecture series on elements of Euclidean geometry. Hilbert tried not only to synthesize more effectively other efforts to reorganize geometry axiomatically but also to reverse partially the trends toward purely abstract symbolization in geometry, by returning to Euclid’s points, lines, and planes and the basic relations of incidence, order, and congruence. Yet, instead of considering only Euclidean geometry, Hilbert began his lectures by explaining that the specific content of Euclid’s definitions, interpretation of which had proven so difficult in the case of non-Euclidean geometries, was irrelevant for mathematics and not for the purposes of philosophy, psychology, or physics.

For Hilbert, proper definitional meanings or interpretations of geometrical entities emerge only via their interconnections with whatever basic axioms are selected to define a given system of geometry. As Hilbert and others emphasized, all meanings in geometry are implicit and context-dependent. The “objects” are not to be determined a priori by an individual’s psychological or historical perceptions of geometrical shapes in the real world. Hilbert explicitly states that the intuitive basis of fundamental geometrical concepts is mathematically insignificant and that their interconnections come into consideration only through the axioms. In his lectures, Hilbert subsequently proposed to set up on this foundation a simple and (unlike Euclid) complete set of independent axioms, by means of which it would be deductively possible to prove systematically all the long-held theorems of Euclid’s geometry, and by implication to do the same for any other non-Euclidean geometry. By employing an algebra utilizing a minimum of new abstract symbols and by keeping his examples in Euclid’s axioms, Hilbert was able to formulate and present his (non-Archimidean) conception of the new axiomatic method more clearly and convincingly to a wider audience than did his predecessors.

In-depth analyses of Hilbert’s published lecture notes, in Grundlagen der Geometrie (1899; The Foundations of Geometry, 1902), have been given by many authors. A number of key themes and methodological conclusions common throughout Hilbert’s later efforts can be identified readily from Hilbert’s work. For the axiomatic reconstruction of any mathematical theory, Hilbert asserted three main requirements to be met by the system of axioms: algebraic independence, set-theoretic completeness, and logical consistency. The requirement of independence asserts that it must be possible to prove any one of the axioms from any others alone or in combination. Hilbert called a geometrical system of axioms complete if it suffices for the verification of all geometric theorems therein. The specific problematics surrounding the completeness requirement are related directly to the subsequent work by Kurt Gödel and Alan Mathison Turing in foundational studies, as well as A. N. Kolmorgorov’s axiomatization of Andrey Andreyevich Markov’s probability theory.

For Hilbert, the consistency of an axiomatic system like geometry is directly derivable from that of naïve arithmetic. As well known since René Descartes, analytical geometry simply assigns pairs of real number spatial coordinates to points in plane geometry, with two- and three-variable linear equations defining lines and planes. In 1900, Hilbert stated, as a more general methodological conclusion of his axiomatization of geometry, that it would be possible ultimately to prove the consistency of, for example, Georg Cantor’s continuum hypothesis, subsequent arithmetic axioms, and those of mathematical physics, by establishing the correctness of the solutions through a finite number of steps, based on a finite number of hypotheses that must be exactly formulated. A formal axiomatic system is construed not only as a system of specific statements about a given subject matter but also as a system of general conditions for what has been called a general “relational structure,” such that the infinite number of formulas in mathematics can be defined completely and consistently by a finite number of formal axioms. Further application of such relational structures to a specific domain of natural science is thus taken to be made by means of a further intuitive or other interpretation of the formal objects and relations of the axioms.

Significance

Within months of its original German publication, The Foundations of Geometry was translated into French, Italian, and English, with a number of important shorter and longer effects on the reformulation and pedagogy of geometry and number theory, which Hilbert next sought to axiomatize. Hilbert and others later showed that other geometries of higher than three dimensions, as well as several other areas of mathematics, were both consistent and complete in Hilbert’s sense. Notwithstanding the ingenuity, clarity, and brevity of his system, Hilbert received a number of criticisms, for example, that his axioms were semantically empty denatured symbols dealing with totally abstract things in a rarefied formal system to which no known kind of certain reality or truth could apparently be attached.

A related question was in what sense the logical (nongeometric-specific) meanings of key terms such as “and,” “is,” “not,” and “when” are to be defined. As a response, after a decade of mainly applications-oriented developments in mathematical physics, over the next two decades, Hilbert and his students redoubled further development of his Formalist program by inquiring directly into the logical and philosophical structure of the new foundations. Hilbert subdivided mathematics into three levels: Level 1 is ordinary operational mathematics considered as mathematics; level 2 is a system of formal symbols employed by level 1, and defined by level 3, which is an informal meta-mathematical theory about level 2.

An equally serious and longer-lasting criticism of Hilbert’s program, by L. E. J. Brouwer and others, was that the question of the absolute consistency of arithmetical axioms employed in Hilbert’s relatively consistent foundations of geometry was left unanswered. Under the impact of the antinomies in Cantor’s, Bertrand Russell’s, and Frege’s systems, several critics demonstrated how far, despite all claims, mathematics actually went beyond levels 1 and 2 meaning, expunged formal axioms purportedly divorced from linguistic and psychologistic considerations of evidence or truth. Although the results of Gödel and others suggest that Hilbert’s program may never be carried out fully, the practical advantages of Hilbert’s abstract approach to mathematics were adopted by numerous physicists as well as mathematicians through his publication Methoden der mathematischen Physik (1931-1937; Methods of Mathematical Physics, 1953, with Richard Courant).

Bibliography

Blanché, Robert. Axiomatics. Translated by G. B. Kleene. London: Routledge & Kegan Paul, 1962. One of the most cited texts embodying, as well as describing, Hilbert’s axiomatic method as comparatively embodied by Euclid, Hilbert, and others.

Hilbert, David, and S. Cohn-Vossen. Geometry and the Imagination. Translated by P. Nemenyi. New York: Chelsea House, 1952. Discusses methodological ideas in a popular presentation.

Hilbert, David, and Leo Unger. The Foundations of Geometry. 2d ed. La Salle, Ill.: Open Court, 1971. The high technical level and brevity of Hilbert’s book makes most of its discussion inaccessible to all but those with graduate-level background in abstract algebra and geometry. Primarily of historical interest.

Mlodinow, Leonard. Euclid’s Window: The Story of Geometry from Parallel Lines to Hyperspace. New York: Free Press, 2001. Details the development of geometry and Hilbert’s contribution.

Poincaré, Henri. The Foundations of Science. Translated by George B. Halsted. Lancaster, Pa.: Science Press, 1946. Although written somewhat at odds with Hilbert’s Foundations (cited above), Poincaré gives an intuition-based introduction to non-Euclidean geometries as well as some of the background to the debates between the formalist (Hilbert), logicist (Russell), and intuitionist (Brouwer) camps.

Reid, Constance. Hilbert. New York: Springer-Verlag, 1970. The most informative and readily obtainable English-language account of Hilbert’s education, developments, debates, and publications. Discusses Hilbert’s controversies with Frege on Hilbert’s semantic relativism, as well as an expert summary of Hilbert’s main contributions by his onetime student, Hermann Weyl.

Wilder, Raymond L. The Foundations of Mathematics. 2d ed. New York: J. Wiley, 1965. An extensive and accessible treatment. Contains a comprehensive bibliography.