Francesca Biagioli

Spatial perception occupied an important place in the early years of _Mind_. It not only hosted Hermann von Helmholtz’s controversy with Jan Pieter Nicolaas Land over the Kantian theory of space (1876–8), but the following decade also saw a resurgence of interest in the debate between empiricist and nativist approaches to different physiological spaces. This chapter will provide an overview of the controversy initiated by Helmholtz and of the most significant contributions to the subsequent debate as published in _Mind_ in the 1880s. It will be shown that, while the debate in the German-speaking world focused on novel attempts to conciliate a moderate version of empiricism with neo-Kantianism, in _Mind_ Helmholtz’s empirical approach received less known but interesting criticisms, both from philosophers trying to defend the Kantian theory, and from proponents of a nativist account of spatial perception, including Land and William James.

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.

The development of non-Euclidean geometry in the nineteenth century led mathematicians, scientists, and philosophers to reconsider the foundations of geometry. One of the issues at stake was to redefine the notion of geometrical axiom and to establish criteria of choice among different axiomatic systems in case of equivalent geometries. The possibility of considering a variety of hypotheses concerning physical space appeared to contradict Kant’s conception of geometrical axioms as a priori synthetic judgments. Therefore, Riemann called geometrical axioms hypotheses, and maintained that the geometry of physical space is a matter for empirical investigation. In order to support this view, Helmholtz pointed out the empirical origin of geometrical axioms. At the same time, he foreshadowed a conventionalist conception of geometrical axioms as definitions that can be abstracted from our experiences with solid bodies and their free mobility.This chapter is devoted to Helmholtz’s objections to Kant and to two different strategies for defending the aprioricity of geometrical axioms from a neo-Kantian perspective. Cohen was able to readapt his notion of the a priori to the case of geometrical axioms because he fundamentally relativized Kant’s notion. For the same reason, Cohen agreed with Helmholtz that the principles of geometry should be determined in connection with those of mechanics. Another line of argument goes back to Alois Riehl. He did not deny the empirical origin of three-dimensionality. However, he maintained that the remaining, formal properties of space suffice to assume that the geometry of space must be Euclidean. My suggestion is that Cohen developed a more plausible view, because he was not committed to any spatial structure independently of empirical science.

Implicit definitions have been much discussed in the history and philosophy of science in relation to logical positivism. Not only have the logical positivists been influential in establishing this notion, but they have addressed the main problems connected with the use of such definitions, in particular the question whether there can be such definitions, and the problem of delimiting their scope. This paper aims to draw further insights on implicit definitions from the development of this notion from its first occurrence in German language in Enriques’s “Principles of Geometry” (1907) to Schlick’s General Theory of Knowledge (1918). Enriques was one of the first to acknowledge that implicit definitions in mathematics are possible only for higher-order entities or structures, which can have infinitely many interpretations in terms of physical objects. While Schlick introduced coordinating principles to account for the scientific interpretations of implicit definitions, Enriques addressed the problem of bridging the gap between abstract and concrete terms in a different way: He identified, within mathematics, structural patterns that provide a clarification of conceptual relations, and so also serve (indirectly) the purposes of applied mathematics. My suggestion is that Enriques’s analysis of these patterns deserves deeper consideration also from a contemporary perspective on mathematical concept formation.

Several studies have emphasized the limits of invariance-based approaches such as Klein’s and Cassirer’s when it comes to account for the shift from the spacetimes of classical mechanics and of special relativity to those of general relativity. Not only is it much more complicated to find such invariants in the case of general relativity, but even if local invariants in Weyl’s fashion are admitted, Cassirer’s attempt at a further generalization of his approach to the spacetime structure of general relativity seems to obscure the fact that the determination of this structure requires a completely different approach that is derived from Riemann rather than Klein. This paper aims to reconsider these issues by drawing attention to the combination of subtractive and additive strategies (more in line with Riemann’s) in Klein’s own considerations about the determination of the structure of spacetime from 1897 to 1910. I will point out that Cassirer relied on Klein’s argument in some central passages from Substance and Function (1910), and elaborated further on his combined approach in Einstein’s Theory of Relativity (1921), also taking into account the application of Riemannian geometry in general relativity. My suggestion is that an appreciation of Cassirer’s continuing commitment to a variety of geometrical traditions may shed light on his particular understanding of a priori elements of knowledge and avoid some of the classical objections to the idea of a relativization of the a priori.

Hermann von Helmholtz has been widely acknowledged as one of the forerunners of contemporary theories of measurement. However, his conception of measurement differs from later, representational conceptions in two main respects. Firstly, Helmholtz advocated an empiricist philosophy of arithmetic as grounded in some psychological facts concerning quantification. Secondly, his theory implies that mathematical structures are common to both subjective experiences and objective ones. My suggestion is that both of these differences depend on a classical approach to measurement, according to which the arithmetic laws of addition define what is measurable as a particular domain for their application, and, at the same time, the extensibility of these laws to all known physical processes works as a heuristic principle for empirical research. Such an approach is worth reconsidering, not only because it lends plausibility to some of the controversial aspects of Helmholtz’s theory, but also because it offers a philosophical perspective on quantification problems that originated in the nineteenth-century.This paper draws insights on Helmholtz’s philosophical views from his engagement with the measurability of sensations via Fechner’s psychophysical law. This seems to be in contrast with the fact that the reception of Helmholtz’s theory culminated with the formulation of the theory of extensive measurement. I will contend that Helmholtz reached a no less important standpoint in the nineteenth-century debate on whether sensations are different (i.e., intensive) kinds of magnitudes and on how, if at all, they can be measured.

Helmholtz’s objections to Kant concerning the origin and meaning of geometrical axioms were influential in the later philosophical debate on the relationship between space and geometry. However, the discussion of Kant’s Transcendental Aesthetic in neo-Kantianism was rooted in earlier objections formulated by such philosophers as Kant’s successor at the University of Königsberg, Johann Friedrich Herbart, and the neo-Aristotelian Adolf Friedrich Trendelenburg. This chapter is devoted to the discussion of these objections in the early works of the founder of the Marburg School of neo-Kantianism, Hermann Cohen. Cohen’s defense of Kant was based on a sharp distinction between the psychological and the transcendental method: the aprioricity of some knowledge does not depend on our mental faculties, but rather on its role in the constitution of the objects of experience. In order to clarify this point, Cohen identified experience in Kant’s sense as scientific knowledge. His reading of Kant suggests that the search for the conditions of experience depends on the history of science. Since this consequence was made explicitly by Cohen’s student, Ernst Cassirer, the concluding section deals with Cassirer’s considerations in support of Cohen’s reading of Kant. These considerations were strictly related to Cassirer’s interpretation of the logic of knowledge in the history of science.

In 1870, Hermann von Helmholtz criticized the Kantian conception of geometrical axioms as a priori synthetic judgments grounded in spatial intuition. However, during his dispute with Albrecht Krause (Kant und Helmholtz über den Ursprung und die Bedeutung der Raumanschauung und der geometrischen Axiome. Lahr, Schauenburg, 1878), Helmholtz maintained that space can be transcendental without the axioms being so. In this paper, I will analyze Helmholtz’s claim in connection with his theory of measurement. Helmholtz uses a Kantian argument that can be summarized as follows: mathematical structures that can be defined independently of the objects we experience are necessary for judgments about magnitudes to be generally valid. I suggest that space is conceived by Helmholtz as one such structure. I will analyze his argument in its most detailed version, which is found in Helmholtz (Zählen und Messen, erkenntnistheoretisch betrachtet 1887. In: Schriften zur Erkenntnistheorie. Springer, Berlin, 1921, 70–97). In support of my view, I will consider alternative formulations of the same argument by Ernst Cassirer and Otto Hölder.

One of the issues at stake in the discussion about the origin and meaning of geometrical axioms was to establish the preconditions for the possibility of spatial measurement. A related issue was to analyze the concept of number to gain insights into its relation to that of magnitude. Despite the traditional definition of arithmetic as the theory of quantities, numbers cannot be identified as magnitudes. Numbers can only represent magnitudes in measurement situations. In order to justify the use of numbers in modeling measurement situations, some conditions are required. The study of these conditions is now known as measurement theory. Helmholtz has been acknowledged as one of the forefathers of measurement theory. However, the connection between Helmholtz’s analysis of measurement and his inquiry into the foundations of geometry has not received much attention.This chapter deals with the philosophical aspect of Helmholtz’s theory and with the psychological interpretation of Kant’s forms of intuition proposed by Helmholtz. The psychological part of Helmholtz’s theory of measurement may not overcome compelling objections. Nevertheless, I rely on the reception of Helmholtz’s views about measurement in neo-Kantianism to reconsider the transcendental structure of Helmholtz’s argument for the applicability of mathematics: additive principles can be established independently of the entities to be measured, although they are necessary for judgments about quantities to be valid.

Cassirer’s neo-Kantian epistemology has become a classical reference in contemporary history and philosophy of science. However, the historical aspects of his thought are sometimes seen to be in some tension with his defence of a priori elements of knowledge. This paper reconsiders Cassirer’s strategy to address this tension by positing functional dependencies at the core of the notion of objectivity. This requires the epistemologist to account for the determination of the objects of knowledge within given scientific theories, but also for conceptual changes over time.

This chapter gives a brief account of the debate about the foundations of geometry after general relativity, with a special focus on Cassirer’s view in 1921. Cassirer emphasized that the geometrical hypotheses of general relativity differed completely from those of Newtonian mechanics and of special relativity. Therefore, in 1921, he revised his argument for the aprioricity of geometry as stated in 1910. Nevertheless, Cassirer argued for continuity across theory change with regard to the symbolic function of geometry in its correlation with the empirical manifold of physical events. In this regard, he reaffirmed his conviction that neo-Kantian views were compatible with empiricist arguments such as Helmholtz’s. Not only did Helmholtz offer an argument for the generalization of the Kantian theory of space, but he influenced the view that the principles of measurement play some a priori role relative to physical theories and can be subject to revision for considerations of uniformity in the formulation of physical laws and empirical evidence.

Hermann von Helmholtz developed epistemological views in connection with his contributions to various branches of science, including physics, physiology, and the inquiry into the foundations of mathematics. Helmholtz’s reception of Kant goes back to his earliest epistemological considerations, and further developments are found in Helmholtz’s main epistemological writings. Helmholtz’s relationship to Kant was much discussed at the time and in more recent studies, arguably because Helmholtz formulated both compelling objections to Kant and rejoinders to those objections within the framework of Kant’s transcendental philosophy. It might seem that Helmholtz may help to address the question which aspects of Kant’s philosophy are in agreement with later scientific developments, including non-Euclidean geometry, and which ones ought to be refurbished or even rejected by someone who is willing to take such developments into consideration. In order to highlight this question, this chapter provides an overview of Helmholtz’s remarks on Kant regarding the law of causality, the theory of spatial perception, and the conception of space and time.

When non-Euclidean geometry was developed in the nineteenth century, both scientists and philosophers addressed the question as to whether the Kantian theory of space ought to be refurbished or even rejected. The possibility of considering a variety of hypotheses regarding physical space appeared to contradict Kant’s supposition of Euclid’s geometry as a priori knowledge and suggested the view that the geometry of space is a matter for empirical investigation. In this article, I discuss two different attempts to defend the Kantian theory of space against geometrical empiricism. Both Cohen and Riehl defended the apriority of geometrical axioms. Cohen pointed out that knowledge necessarily requires both sensibility and understanding. Therefore, he maintained that space can be proved to be a condition of experience without admitting that some statements about it are intrinsically necessary. By contrast, Riehl emphasized universality and necessity as intrinsic properties of a priori knowledge. He was one of the first philosophers to discuss the space problem in detail and to distinguish between a priori properties of space and empirical properties. Nevertheless, I suggest that Cohen developed a more plausible view because he was not committed to any spatial structure independently of empirical science.

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.

This book offers a reconstruction of the debate on non-Euclidean geometry in neo-Kantianism between the second half of the nineteenth century and the first decades of the twentieth century. Kant famously characterized space and time as a priori forms of intuitions, which lie at the foundation of mathematical knowledge. The success of his philosophical account of space was due not least to the fact that Euclidean geometry was widely considered to be a model of certainty at his time. However, such later scientific developments as non-Euclidean geometries and Einstein’s general theory of relativity called into question the certainty of Euclidean geometry and posed the problem of reconsidering space as an open question for empirical research. The transformation of the concept of space from a source of knowledge to an object of research can be traced back to a tradition, which includes such mathematicians as Carl Friedrich Gauss, Bernhard Riemann, Richard Dedekind, Felix Klein, and Henri Poincaré, and which finds one of its clearest expressions in Hermann von Helmholtz’s epistemological works. Although Helmholtz formulated compelling objections to Kant, the author reconsiders different strategies for a philosophical account of the same transformation from a neo-Kantian perspective, and especially Hermann Cohen’s account of the aprioricity of mathematics in terms of applicability and Ernst Cassirer’s reformulation of the a priori of space in terms of a system of hypotheses. This book is ideal for students, scholars and researchers who wish to broaden their knowledge of non-Euclidean geometry or neo-Kantianism.

Hermann Weyl adopted the Kantian definition of space as a form of intuition and referred to Edmund Husserl’s phenomenological approach for the philosophical characterization of space in the introduction to Raum-Zeit-Materie (1918) and other writings from the same period (1918–1923). At the same time, Weyl emphasized that subjective factors are completely excluded from the mathematical construction of physical reality in Albert Einstein’s general theory of relativity, with the sole exception of the setting of a coordinate system, which for Weyl is what remains of the original perspective of the self in becoming aware of one’s own intuitions. This paper reconsiders Weyl’s philosophical position as a possible response to the earlier debate on the relation between intuition and conceptual construction in the foundation of geometry, key figures of which, besides Husserl, included Hermann von Helmholtz, Felix Klein, and Moritz Schlick.

Hölder’s philosophy of geometry might appear to be the most problematic part of his epistemology. He maintains that geometry depends on experience also after Poincaré’s fundamental criticism of Helmholtz. Nevertheless, I think that Hölder’s view is worth discussing, for two reasons. Firstly, the related methodological considerations were crucial for the development of his epistemology. Secondly, Poincaré uses the opposition between Kantianism and empiricism to argue for his geometrical conventionalism. Nevertheless, Hölder shows that an alternative strategy is not excluded: he profits from Kantian objections in order to develop a consistent empiricism. At the same time, especially in Die mathematische Methode [Holder 1924], he vindicates the Kantian view that mathematics is synthetic. In this paper, I will consider Hölder’s defence of the deductive method in geometry in Anschauung und Denken in der Geometrie [Holder 1900] in connection with his approach to the theory of quantity [Holder 1901]. Moreover, I will discuss his connection with Kant. My suggestion is that Hölder’s methodological considerations enable him to foreshadow a relativized conception of the a priori.

It is well known that Hermann Cohen was one of the first philosophers who engaged in the debate about non-Euclidean geometries and the concept of space. His relation to Hermann von Helmholtz, who played a major role in the same debate, is an illuminating example of how some of the leading ideas of Marburg neo-Kantianism, although motivated independently of scientific debates, naturally led to the examination of scientific works and scientists’ epistemological views. This paper deals with Cohen’s view of magnitudes and measurement and with his – less known – review of Helmholtz’s paper “Zählen und Messen, erkenntnistheoretisch betrachtet”, which contains one of the first attempts to formulate a theory of measurement in the modern sense. The first part provides a brief introduction to this debate in its connection with the earlier discussion on geometrical axioms and the concept of space. The main sections deal with Helmholtz’s and Cohen’s approaches to the foundations of measurement. Cohen’s criticism of some of Helmholtz’s assumptions notwithstanding, my emphasis is on some unexpected affinities between these two approaches. In the concluding section, I rely on the constructive side of Cohen’s criticisms to reconsider the philosophical aspects of Helmholtz’s theory and draw a few comparisons with contemporary measurement theory.

Hermann von Helmholtz’s geometrical papers have been typically deemed to provide an implicitly group-theoretical analysis of space, as articulated later by Felix Klein, Sophus Lie, and Henri Poincaré. However, there is less agreement as to what properties exactly in such a view would pertain to space, as opposed to abstract mathematical structures, on the one hand, and empirical contents, on the other. According to Moritz Schlick, the puzzle can be resolved only by clearly distinguishing the empirical qualities of spatial perception from those describable in terms of axiomatic geometry. This paper offers a partial defense of the group-theoretical reading of Helmholtz along the lines of Ernst Cassirer in the fourth volume of The Problem of Knowledge of 1940. In order to avoid the problem raised by Schlick, Cassirer relied on a Kantian view of space not so much as an object of geometry, but as a precondition for the possibility of measurement. Although the concept of group does not provide a description of space, the modern way to articulate the concept of space in terms of transformation groups reveals something about the structure and the transformation of spatial concepts in mathematical and natural sciences.

Riehl was one of the leading figures in the neo-Kantian movement and the founder of critical realism. This view was characterised by a realist interpretation of Kant’s notion of a thing in itself based on a physiological account of sensation: although Riehl agreed with Kant that things in themselves are unknowable through reason alone, he maintained that such things do affect empirical intuitions and manifest themselves indirectly in empirical knowledge. This paper offers a discussion of Riehl’s argument by taking the example of his account of empirical factors in the representation of space. According to Riehl, Helmholtz’s empiricist approach to spatial intuition offered an argument for the accessibility of external reality. In order to clarify the position of the critical realist, I compare Riehl’s argument with Helmholtz’s own account of objectivity in terms of repeatability of measurement operations.