Abstract
In 1871, the German mathematician Felix Klein used the concept of a projective metric to classify geometries into elliptic, hyperbolic, and parabolic. This chapter deals with the question whether metrical projective geometry can provide a classification of hypotheses concerning physical space. Such philosophers as Bertrand Russell argued that projective geometry provides us with a priori knowledge in Kant’s sense, insofar as projective properties are common to all concepts of spaces. However, Russell did not attribute the same status to metrical properties or metrical projective geometry: the former depend on empirical factors; the latter rests upon a definition of distance that must be stipulated arbitrarily. Therefore, he considered Klein’s classification a merely technical result. By contrast, Ernst Cassirer attached great philosophical importance to this result for the clarification of the distinction between the general properties of space and the specific axiomatic structures. Following a line of argument that goes back to Helmholtz, Cassirer used Klein’s classification to generalize the Kantian notion of space to a system of hypotheses, including both Euclidean and non-Euclidean geometries. This generalization offered one of the clearest examples of Cassirer’s interpretation of the notion of the a priori in terms of a range of hypotheses for the use of mathematics in physics.
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The construction described above goes back to Philippe de la Hire and is also found in Poncelet (1822, p.82) . Von Staudt was the first to use it in the definition of the harmonic relation between two pairs of points on a projective line. In order to prove that there is one and only one point that is in such a relation to three given points, von Staudt repeated the construction of the quadrangle on another plane through the line. He proved that the second diagonals of the two quadrangles intersect with the line in the same point. The proof follows from a generalization of Desargues’s theorem about perspective triangles to perspective quadrangles (i.e., distinct triangles or quadrangles, which are projections of the same figure). The theorem states that if the lines through the corresponding vertexes of the figures intersect with the same point, the points of intersection of the corresponding sides lie in the same line, and reciprocally (Staudt 1847, pp.40–43). For a modern presentation of the theorems of Desargues and of von Staudt, see Efimov (1970) .
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This aspect of von Staudt’s way of proceeding is apparent if one considers later axiomatizations of projective geometry. In this sense, Otto Hölder (1911, p.67, and note) maintained that in order to develop projective geometry in the manner of von Staudt (i.e., without metric foundations), one had to presuppose all axioms of plane linear geometry (connection, order and the axiom of parallel lines ), except the axioms of congruence and the Archimedean axiom . Notice, however, that the first axiomatic treatment of (elementary) geometry goes back to Moritz Pasch (1882) , and he did not provide an axiomatization in the modern sense. It was only Hilbert (1903) who formulated a set of axioms that are sufficient to characterize geometrical objects and relations up to isomorphism. Furthermore, von Staudt presupposed continuity as well. Later presentations of the proof of the fundamental theorem – beginning with Klein’s (1874) – usually adopted an equivalent formulation of Dedekind’s Archimedean continuity (see Darboux 1880; Pasch 1882, pp.125–127; Enriques 1898) . Alternatively, Friedrich Schur (1881, p.253) used Thomae’s (1873, p.11) definition of projectivity in terms of prospectivity to prove the fundamental theorem in a manner which is independent of the Archimedean axiom. On von Staudt’s proof of the fundamental theorem of projective geometry and its development from 1847 to 1900, see Voelke (2008) .
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It can be hypothesized that Klein elaborated on a remark made by Weierstrass during Klein’s stay in Berlin. Federigo Enriques (1907) reported that Weierstrass discussed the same subject in one of his lectures at the University of Berlin. Even though Enriques did not mention the date, Voelke (2008, p.288) supposes that the discussion might have taken place during the seminar attended by Klein in 1870. To support his conjecture, Voelke (2008, p.258) points out that it was Weierstrass who introduced the notion of a limit point in his proof that every bounded infinite set of real numbers have at least one limit point.
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For a detailed presentation of this way of proving the fundamental theorem, see also Darboux (1880). This way of proceeding differs from Schur’s (1881), because it presupposes Archimedean continuity: every point of the projective lines under consideration is thought of as a limit point of an infinite series of harmonic elements.
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Arguably, Klein’s terminology relates to the fact that every linear transformation that maps a line onto itself can be associated with a characteristic quadratic equation. The transformation is elliptic, parabolic or hyperbolic, if the discriminant of this equation is less than, equal to or greater than 0 – namely, if the conic represented by this equation is an ellipse, a parabola or a hyperbola (see Torretti 1978, p.131).
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The Erlangen Program is often mistaken for Klein’s inaugural address as a newly appointed Professor at the University of Erlangen (see Rowe 1983). Klein’s comparative review of the existing directions of geometrical research circulated as a pamphlet when he gave his inaugural address and became known as Erlangen Program, arguably because, after the second edition of 1893, Klein himself (e.g. in Klein 1921, pp.411–114) presented it as a retrospective guideline for his research (see also Gray 2008, pp.114–117).
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On the delayed reception of Klein’s Erlangen Program, see Hawkins (1984, pp.451–463).
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The debate concerned the compatibility of the cosmological model developed by the Dutch astronomer Willem de Sitter with the principles of general relativity. Einstein argued for his own cosmological model by appealing to a principle borrowed from Mach . In a letter to de Sitter dated 24 March 1917, Einstein formulated the principle as follows: “In my opinion, it would be unsatisfactory, if a world without matter were possible. Rather, the g μν -field should be determined by matter and not be able to exist without the latter.” Klein and Hermann Weyl showed that De Sitter’s model provided a counterexample to this principle. See the editorial note on “The Einstein-De Sitter-Weyl-Klein Debate,” in Einstein (1998, pp.351–357).
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See Corry (1996) ; Avigad (2006). I refer to Avigad and Corry in particular, for a thorough account of how Dedekind’s successive revisions of his theory of ideals shed light on his structuralist approach. According to Avigad (2006, p.168), the progression from Dedekind’s first version of the theory of ideals to his last version represents a steady transition from Kummer’s algorithmic style of reasoning to a style that is markedly more abstract and set-theoretic.
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As referred to in Chap. 4, such scholars as Dummett (1991) and, more recently, Benis-Sinaceur (2015) draw logicism properly speaking back to Frege , and sharply distinguish the latter’s logicism from Dedekind’s view that “abstract objects are actually created by operations of our mind” (Dummett 1991, p.49). By contrast, Tait (1996) and Ferreirós (1999) reconsidered the logical aspect of Dedekind’s abstraction from all the properties that have to do with a particular representation of mathematical domains (including spatial and temporal intuitions) in order to obtain a categorical characterization of mathematical structures. Cf. also Reck (2003) for a structuralist rather than psychological account of Dedekind’s notions of abstraction and of creation. In this regard, Dedekind’s view has been called logical structuralism. I especially rely on Ferreirós’s broadening of logicism to include parallel versions of it, such as Dedekind’s and Frege’s.
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Dedekind’s definition of irrational numbers can be considered a decisive step in the clarification of the mathematical notion of continuity. Notice, however, that especially in the introductory part of “Continuity and Irrational Numbers” he used “continuity” in a broader and more intuitive sense. The property of the line he was dealing with is not continuity but connectedness, which intuitively corresponds with the idea of having no breaks. A set is disconnected if it can be divided into two parts such that a point of one part is never a limit point of the other part; it is connected if it cannot be so divided. I am thankful to Jeremy Gray for pointing out to me that Dedekind wished to explain the connected character of the line.
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As already mentioned, Dedekind’s use of “abstraction” and “creation” has sometimes been misunderstood as psychological (Cf. Dummett 1991, 49). In Chap. 4 and in the present chapter, I refer to more recent interpretations of the same operations as logical ones by Tait (1996) , Ferreirós (1999) , and Reck (2003).
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Although Dedekind’s conception appears to be naïve when compared to modern set theory, a set-theoretical approach is largely implicit in his logical foundation of arithmetic and became influential especially after Hilbert’s reception of Dedekind (see Gray 2008, pp.148–151).
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Further developments in Klein’s discussion with Pasch are found in their correspondence, which has been recently made available by Schlimm (2013). In particular, in a letter from October 19, 1891, Pasch fundamentally agreed with Klein’s remarks about the notion of axiom in the concluding part of Klein (1890). However, Pasch distanced himself from Klein’s defense of the role of intuition in mathematics. Although Pasch admitted that figures are commonly used in working on the axioms, he maintained that “the use of figures is merely a facilitation of the work; otherwise, the work would exceed our powers, or at least would progress much too slowly, or would not progress far enough. The consideration must be possible even without the figures, in other words: that which is derived from the figures must already be contained in the axioms, for otherwise the axioms are not complete” (Pasch in Schlimm 2013, p.193) .
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This classification goes back to Paul du Bois-Reymond’s General Theory of Function (1882). Regarding the fundamental notions of the calculus, du Bois-Reymond distinguished between idealism and empiricism as follows. Idealism is the view that limits exist as a logical presupposition of the calculus, although neither infinite nor infinitesimal quantities are imaginable in the sense of concrete intuition. By contrast, empiricism is the view that knowledge is grounded in immediate perception. Therefore, in the empiricist view, every representation in science must be referred to the objects of perception. In the case of such abstract concepts as the concept of limit, the representation can be obtained indirectly by the use of geometric constructions (see du Bois-Reymond 1882, pp.58–87). Du Bois-Reymond’s approach differed from Klein’s because the former did not propose a synthesis between two opposing views. The aforementioned sections of his work provided clarification on the assumptions of two equally possible views about the foundations of the calculus.
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Ihmig (1997, pp.306–326) refers to Cassirer’s comparison between the transcendental and the mathematical method to indicate a series of analogies between critical idealism and Klein’s Erlangen Program. Notwithstanding the significance of this comparison for reconsidering Cassirer’s relationship to Klein, it seems to me to be reductive to restrict the consideration to Klein’s general idea of a group-theoretical treatment. My suggestion is to reconsidered the importance of Klein’s projective model throughout his writings on non-Euclidean geometry. Not only did metrical projective geometry offer the first example of a classification of geometries by the use of the theory of invariants, but Klein used this example to support his epistemological views about the relationship between pure and applied mathematics. In this regard, I believe that there are more substantial points of agreement between Cassirer and Klein than the analogies between the Erlangen Program and critical idealism.
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Biagioli, F. (2016). Metrical Projective Geometry and the Concept of Space. In: Space, Number, and Geometry from Helmholtz to Cassirer. Archimedes, vol 46. Springer, Cham. https://doi.org/10.1007/978-3-319-31779-3_5
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