Abstract
Klein’s classification of geometries by the use of group theory inaugurated a new phase in the debate on the geometry of space. On the one hand, the conclusion of Riemann’s and Helmholtz’s inquiries into the foundations of geometry appeared to be confirmed: Euclidean geometry does not provide us with the necessary presuppositions for empirical measurement, because both Euclidean and non-Euclidean assumptions can be obtained as special cases of a more general system of hypotheses. On the other hand, Helmholtz had believed that he had shown that the free mobility of rigid bodies implied and was implied by a metric of constant curvature, which includes spherical and elliptic geometries. The group-theoretical approach enabled Sophus Lie to disprove Helmholtz’s argument and provide a mathematically sound solution to the same problem. The most challenging argument against Helmholtz’s empiricism, however, was formulated by Henri Poincaré: observation and experiment cannot contradict geometrical assumptions, because the application of geometrical concepts to empirical objects, including the characterization of solid bodies as “rigid,” already presupposes these kinds of assumptions. The present chapter is devoted to the reception of Poincaré’s argument in neo-Kantianism. In particular, I contrast Poincaré’s conclusion that geometrical axioms are conventions with Cassirer’s view that the interpretation of measurements depends on conceptual rules and ultimately on rational rather than conventional criteria. Cassirer relied on the group-theoretical analysis of space to infer such criteria from the relations of geometrical systems to one another.
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Notes
- 1.
The Erlangen Program is often mistaken for Klein’s inaugural address (see Rowe 1983). It is only after the second edition of Klein’s “Comparative Review of Recent Researches in Geometry” that this work, also called the Erlangen Program, became known as a retrospective guideline for Klein’s research (see Gray 2008, p.117).
- 2.
Klein specified the second condition in the 1893 version of the paper. In the first version of 1872, he adopted Jordan’s (1870, p.22) definition, which referred to finite groups of permutations. In that case, the closure of a set of elements relative to a fundamental operation (i.e., the first of the said conditions) is a sufficient condition for the set to form a group. Afterwards, Lie drew Klein’s attention to the fact that the existence of an inverse operation is required in the case of infinite groups (see Wussing 1969, p.139). Furthermore, it is noteworthy that both Galois and Jordan dealt with groups of permutations. A set G of permutations forms a group if: (i) G contains the (unique) identity permutation ε = \( \left(\begin{array}{cc}1& 2\dots n\\ {}1& 2\dots n\end{array}\right) \); (ii) together with every two permutations σ = \( \left(\begin{array}{ccc}1& 2\dots & n\\ {}{a}_1& {a}_2\dots & {a}_n\end{array}\right) \) and τ = \( \left(\begin{array}{ccc}1& 2\dots & n\\ {}{b}_1& {b}_2\dots .& {b}_n\end{array}\right) \) (where the equality τ = σ is not excluded), G contains the product τσ = \( \left(\begin{array}{ccc}1& 2\dots & n\\ {}{b}_1& {b}_2\dots & {b}_n\end{array}\right) \) · \( \left(\begin{array}{ccc}1& 2\dots & n\\ {}{a}_1& {a}_2\dots & {a}_n\end{array}\right) \) = \( \left(\begin{array}{ccc}1& 2\dots & n\\ {}{b}_{a_1}& {b}_{a_2}\dots & {b}_{a_n}\end{array}\right) \); (iii) together with every permutation σ, the set G also contains the inverse permutation σ – 1 = \( \left(\begin{array}{ccc}{a}_1& {a}_2\dots & {a}_n\\ {}1& 2\dots & n\end{array}\right) \) = \( \left(\begin{array}{ccc}1& 2\dots & n\\ {}{\alpha}_1& {\alpha}_2\dots & {\alpha}_n\end{array}\right) \), where α ai = i, i = 1, 2, …, n (see Yaglom 1988, pp.12–13) . The group-theoretical treatment of geometry showed that the same conditions generally apply to groups of operations. In this sense, Klein’s Erlangen Program can be considered a fundamental step in the development of the abstract concept of group (Wussing 1969 , pp.132–143).
- 3.
Leibniz’s goal was to develop a general science of situational relations in order to represent nonspatial relations (i.e., relations between monads). The reception of Leibniz’s ideas in the nineteenth century differed considerably from his original project and took place in very different contexts. Hermann Grassmann (1847) compared the analysis situs to his Theory of Extension (Grassmann 1844). Giuseppe Peano and his school contributed both to the rediscovery of Grassmann’s work in the second half of the nineteenth century and to the connection between Leibniz’s analysis situs and the vector calculus. On the other hand, “geometry of position ” was used to designate projective geometry as well. It was in that context that Johann Benedikt Listing , in a letter to his old school teacher dated 1836, introduced the term “topology,” which was substituted for “analysis situs” in the twentieth century. Listing introduced a new term because the phrase “geometry of position” had been used by Lazare Carnot (1803) . The fundamental ideas of algebraic topology go back to a fragment from Riemann’s discussions with Enrico Betti . The fragment appeared in Riemann (1876) with the title “Fragment Belonging to Analysis Situs.” The development of these ideas is due mainly to Poincaré and lies at the origin of the discipline now known as topology. Klein was arguably acquainted with Riemann’s fragment. However, in the Erlangen Program, Klein specifically referred to the continuous transformation group . On Leibniz’s project and its reception in the nineteenth century, see De Risi (2007, pp.XII, 111–114). On Poincaré’s work on topology, see Gray 2013, Ch.8.
- 4.
The question of the influence of Klein’s Erlangen Program is controversial. I rely on Hawkins (1984) for a historically well-documented reconstruction of the delayed reception of the Erlangen Program. Furthermore, Hawkins gives evidence of the role of other mathematicians, including Sophus Lie, Henri Poincaré and Gino Fano , in the implementation of a research program in line with Klein’s view. Cf. Birkhoff and Bennett (1988) for the view that Klein had a major influence on later mathematical researches, including his own. More recently, Gray points out the different backgrounds of these readings: whereas mathematicians, such as Birkhoff, often consider Klein’s Erlangen Program very influential, a number of historians, including Hawkins and Erhard Scholz , showed that the solid work establishing group theory between 1870 and 1890 was done by Camille Jordan , Sophus Lie and Henri Poincaré, among others. Klein himself did not implement or advance the view of the program (Gray 2008, p.117).
- 5.
- 6.
After a journey to Paris in 1882, Lie reported to Klein on his meeting with Poincaré and on the latter’s view of mathematics as a “tale about groups” (Gruppengeschichte). On that occasion, Lie informed Poincaré about Klein’s Erlangen Program. After that, Poincaré and Darboux remained in touch with Lie – who was based at the University of Leipzig – and promoted studies in the theory of continuous functions at the École normale supérieure in Paris (see Hawkins 1984, p.448).
- 7.
- 8.
For a reconstruction of how Russell’s and Poincaré’s different mathematical conceptions influenced their disagreement on philosophical matters, see Nabonnand (2000, p.259): “In presupposing the models of Euclidean and non-Euclidean geometries, Russell is led to restrict his consideration to the metrical concepts . Poincaré, who endorses the viewpoint of the transformation groups – which in the eyes of Russell is nothing more than a change in formulation without philosophical importance –, considers distance to be essentially an invariant of a group, whose content is equivalent to that of this group. The choice of the distance depends, therefore, on a convention no less than the choice of the group.”
- 9.
On the explanatory power of Poincaré’s talk of conventions when it comes to measurement, see Gray (2013). The contrast with Russell is illuminating: “Russell sought to define distance, or perhaps to elucidate our familiar concept of distance by saying what it is, while Poincaré could only do so by saying how the concept if distance is used, that is, measured” (Gray 2013, p.81).
- 10.
For a thorough comparison between Helmholtz and Lie, see Torretti (1987, pp.158–171).
- 11.
Klein’s account differs from Coffa’s, because Klein does not presuppose an extension of Beltrami’s two-dimensional model to the three-dimensional case. On the problems of interpreting Helmholtz’s thoughts experiments as a three-dimensional model of non-Euclidean geometry, see Sect. 3.2.3, and note 8.
- 12.
Color was one of Riemann’s examples of continuous manifolds in Section I.1. of “On the Hypotheses Which Lie at the Foundation of Geometry .” Another important source for Helmholtz’s considerations was Hermann Grassmann’s work on manifolds and its application to color mixtures (see Hatfield 1990 , p.218, note 106; Hyder 2009, Ch.4).
- 13.
Torretti (1978, pp.166–167) points out that the general properties of space may not have counted as axioms in the Euclidean tradition. This does not mean that they cannot be axiomatized at all. Following Helmholtz’s analogy with color mixtures, Torretti suggests that the characteristics required for the interpretation of the general properties of space may be specified in terms of of the theory of manifolds . He describes Helmholtz’s space as a differentiable , three-dimensional manifold. The axioms of quantity, on the other hand, are not determined by the form of space because their formulation presupposes the existence of the solid bodies we experience. Following a similar line of reasoning, Lenoir (2006, pp.201–202) describes Helmholtz’s form of spatiality as a differentiable , n-fold extended manifold.
- 14.
- 15.
However, there are important differences between Folina’s (1992) and Crocco’s (2004) readings. Whereas Folina emphasizes the parallel with Kant regarding for the idea that arithmetic and geometry have different kinds of synthetic judgments a priori (i.e., formulas and axioms , respectively), Crocco points out the deeper connection between space and time in Poincaré’s conception of synthesis a priori. The latter account has the advantage of offering a unitary perspective on Poincaré’s epistemology.
- 16.
Despite the fact there were developments in Poincaré’s thought throughout the 1890s, his choice to collect his writings on the foundations of mathematics in a comprehensive exposition suggests that his argument for the conventionality of geometry should be considered as a whole (see Ben-Menahem 2006 , p.40). The proposed reconstruction of the argument is mainly based on Poincaré (1902), as this appears to have been most popular in Germany and was translated into German by Ferdinand Lindemann in 1904. References to earlier writings are given in those cases in which there were significant changes.
- 17.
On the constitutive role of mathematics in Poincaré’s epistemology, cf. Crocco (2004) .
- 18.
See especially Folina (2006, p.288) .
- 19.
For a discussion of this aspect of Poincaré’s notion in contrast to Kant, see MacDougall (2010, p.140).
- 20.
Poincaré (1906, p.307) referred to Russell’s objection that the definition of irrational numbers as upper limits of sets of real numbers is non-predicative. Intuitively, the definition of two objects A and A′ is non-predicative, if A occurs in the definition of A′, and vice versa. In formal terms, this kind of definition can be derived from the schema of the axioms of comprehension : ∃Y∀X (X ∈ Y ↔ φ (X)). Russell and Poincaré believed that the use of non-predicative definitions caused the antinomies of set theory. The problem lies in the fact that these definitions seem to contradict some conditions for quantification. Poincaré indicated two possible ways to avoid antinomies. The domain under consideration can be restricted to those elements that could be specified independently of the quantification. Alternatively, in the case of indefinite domains, one can require that the classification of proper subsets of the set remain unvaried under quantification (see Heinzmann 1985 , pp.72–73).
- 21.
For an illustration of this example, see Heinzmann 1985, p.42.
- 22.
Regarding Poincaré’s objection, it must be noticed that the later axiomatization of set theory offered a possible solution to the problems concerning non-predicative definitions . In order to avoid antinomies, it suffices to integrate the schema of the axioms of comprehension with the schema of the axioms of separation . Given a set A, the elements that satisfy the property φ are separated and reunited in Y, which is a subset of A, according to the following schema: ∃Y∀X (X ∈ Y ↔ X ∈ A ∧φ (X)). See Heinzmann 1985, pp.10–11.
- 23.
It is revealing that Poincaré omitted his reference to Helmholtz in the revised version of the paper on the mathematical continuum as found in Science and Hypothesis.
- 24.
The reference to Helmholtz remains implicit, but it emerges clearly from Poincaré’s example. We turn back to Poincaré’s remark in the next chapter, as it appears to have influenced Schlick’s objection against Helmholtz’s definition of congruence: “This definition reduces congruence (the equality of two extents) to the coincidence of point pairs in rigid bodies ‘with the same point pair fixed in space’, and thus presupposes that ‘points in space’ can be distinguished and held fixed. This presupposition was explicitly made by Helmholtz […], but for this he had to presuppose in turn the existence of ‘certain spatial structures which are regarded as unchangeable and fixed’. Unchangeability and fixity (the term ‘rigidity’ is more usual nowadays) cannot for its own part again be specified with the help of that definition of congruence, for one would otherwise clearly go round in a circle. For this reason the definition seems not to be logically satisfactory” (Schlick in Helmholtz 1921, pp.192, 31, note 31). Schlick goes on to argue for Poincaré’s geometrical conventionalism as the only way to escape the circle.
- 25.
As pointed out by Gray (2013, pp.48–57) , the earlier formulation of the argument in Poncaré (1898b) sheds light on the priority of the concept of group over that of space in Poincaré’s philosophy of geometry. The same idea lies behind Poincaré’s definition of distance in terms of measurement. I argue in what follows that it was this aspect of Poincaré’s approach that especially influenced Cassirer in his reconstruction of the problem of space from Helmholtz to Poincaré. The leading idea of Cassirer’s account is that the construction of the concept of space required the mathematicians to take a step back and look at the more general concept of group.
- 26.
This again emerges most clearly in Poincaré’s 1898 formulation of the argument: “Have we the right to say that the choice between geometries is imposed by reason, and, for example, that the Euclidean geometry is alone true because the principle of the relativity of magnitudes is inevitably imposed upon our mind? […] Unquestionably reason has its preferences, but these preferences have not this imperative character. It has its preferences for the simplest because, all other things being equal, the simplest is the most convenient. Thus our experiences would be equally compatible with the geometry of Euclid and with a geometry of Lobachévski which supposed the curvature of space to be very small. We choose the geometry of Euclid because it is the simplest. If our experiences should be considerably different, the geometry of Euclid would no longer suffice to represent them conveniently, and we should choose a different geometry” (Poincaré 1898b, p.42). On Poincaré’s commitment to the problems concerning measurement, see also Hölder (1924, p.400) . On the physical equivalence of the geometries considered by Poincaré, see also Torretti (1978, pp.327, 336) and Ben-Menahem (2006, p.58) .
- 27.
In 1904, Hönigswald wrote his Dissertation On Hume’s Doctrine of the Reality of the External World at the University of Halle under the supervision of Riehl. Riehl’s influence is also apparent in Hönigswald’s Habilitation Thesis, Contributions to the Theory of Knowledge and Methodology (1906). In my opinion, both Hönigswald’s paper on geometry of 1908 and his essay On the Discussion about the Foundations of Mathematics (1912) show the importance of Riehl’s teaching for the development of Hönigswald’s philosophy of mathematics and of science, even though, after 1906, Hönigswald distanced himself from Riehl’s views about the possibility of knowledge of things in themselves . Arguably, Riehl influenced Bauch as well, when they met in Halle between 1903 and 1904. On that occasion, Bauch collaborated with both Riehl and Hönigswald. Bauch’s friendship with Hönigswald lasted until Bauch adhered to Nazism. In 1933, Hönigswald lost his professorship in Munich according to racist legislation. In 1939, he was deported to the Dachau concentration camp for 5 weeks and was only freed following international protests. He then emigrated to the United States (see Ollig 1979 , pp.73–81; 88–93).
- 28.
Cassirer (1910, p.186) referred in particular to Poincaré’s considerations regarding the measurement of time . As in the case of geometry, Poincaré maintained that one way to measure time cannot be “more true” than another; it can only be more convenient. He referred to the fact that the definition of time presupposes the laws of mechanics . Poincaré’s requirement was that time be defined in such a way that the equations of mechanics result as simply as possible (Poincaré 1898a, p.6). The difference with geometry lies in the fact that Poincaré did not consider geometry a part of mechanics. However, this difference does not affect Cassirer’s argument insofar as this regards the conditions of measurement in general.
- 29.
On the distinction between conventionality and arbitrariness, see Ben-Menahem (2006, p.54). Ben-Menahem (2006, Preface) interprets Poincaré’s conventionalism as an attempt to disentangle the concepts of convention and truth. The goal of such an approach is to avoid the tendency to consider some theoretical assumptions (e.g., the principles of geometry ) necessary when in fact they are free.
- 30.
Consider the following description of the idea of the semantic tradition: “Much of the most interesting philosophy of science developed in the past few decades has been inspired by the […] idea: Many fundamental scientific principles are not necessarily thought – indeed, it takes great effort to develop the systems of knowledge that embody them; but their denial also seems oddly impossible – they need not be thought, but if they are thought at all, they must be thought as necessary” (Coffa 1991 , p.55).
- 31.
The German term for “correlation ” is Zuordnung, and, in that context, indicates a mathematical function. “Univocal correlation” or “coordination” in Cassirer’s sense refers both to the one-to-one correspondence between series that occurs, for example, in Dedekind’s definition of irrational numbers and to the general idea of functional dependence between different elements of a theory. On mathematical and epistemological uses of “Zuordnung” between the nineteenth century and the first decades of the twentieth century, see Ryckman (1991, pp.58–60). As pointed out by Ryckman, the idea of a univocal correlation occupies a central place in such different epistemologies as Cassirer’s and Schlick’s.
- 32.
As pointed out in Chap. 4, the comprehensibility of nature is presupposed in Helmholtz’s physical interpretation of the homogeneity of the sum and the summands. The equivalent proposition for the composition of magnitudes to be considered in terms of arithmetical sum depends on the repeatability of measurements.
- 33.
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Biagioli, F. (2016). Euclidean and Non-Euclidean Geometries in the Interpretation of Physical Measurements. In: Space, Number, and Geometry from Helmholtz to Cassirer. Archimedes, vol 46. Springer, Cham. https://doi.org/10.1007/978-3-319-31779-3_6
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