In her 2013 study, Francesca Biagioli examines Helmholtz’s claim that space can be transcendental without the axioms being so. In 1870, Kant’s concept of geometrical axioms as a priori synthetic judgments based on spatial intuition was questioned by Hermann von Helmholtz, employing a Kantian argument that can be paraphrased as follows: for judgments about magnitudes to be generally valid, mathematical structures that can be described independently of the objects we experience are required.

Helmholtz, according to Biagioli, envisioned space as one of these structures, geometrical axioms only yielding objective knowledge when used in conjunction with mechanical principles. Geometry becomes scientifically testable in this way, and its certainty no longer relies on a priori intuition.

The interpretation of Helmholtz’s claim sparked an intensive debate. Alois Riehl, for example, utilized Helmholtz’s claim to argue that Kant’s theory might operate for intuitive space regardless of the results of physical space measurements. Moritz Schlick, following Riehl’s line of reasoning, connects Helmholtz’s “narrower specifications” to axioms of congruence about physical magnitudes. According to Schlick, these characteristics are distinct from topological aspects (three-dimensionality, continuity, etc.) which are thought to be based on spatial intuition. To retain Helmholtz’s distinction, Schlick considers space qualities and psychological processes in spatial perception to be indescribable.

Schlick’s interpretation is at odds with the Kantian theory: one needs not introduce a ‘‘pure’’ intuition besides the empirical one. Then why does Helmholtz deem space

F. Biagioli

‘‘transcendental’’? On the other hand, any rejection of Schlick’s interpretation is committed to another question: what are the general characteristics of space?

Because Kant never used the term “transcendental space”, Biagioli argues that Helmholtz is thinking in terms of “a priori”. Therefore the confusion about his conception of space in Schick’s interpretation, and the misleading claim that the qualities of sense perception are indescribable: “since the requirement of indescribability is questionable, some interpreters seek to make sense of the quality–quantity opposition by introducing the distinction between topological and metrical properties.”

Helmholtz deems sensations ‘‘signs’’ for their stimuli, therefor meaning is only an intuitive aquaintance. However, once acquired through the localization of them in space (and constructing the concept of space), the intuition of them can be transcendental. Although Helmholtz rejects Kant’s idea that the axioms of Euclid’s geometry must be valid for an empirical manifold, he uses a Kantian argument: the properties of space should supply us with general conditions of measurement, the results of those interpretation being “objective”, by depending on the system’s conditions.

Biagioli continues her analysis on Albrecht Krause, which rejects the attempt to draw spatiality out of sensations and questions the ways of expressing laws of spatial intuition with axioms, considering that measurements should not be trusted when they contracticted the axioms. While axioms are exact knowledge, their interpretations are an approximate character of natural laws. Helmholtz point’s out that that Krause’s assumptions, which are derived from a nativist theory of vision (the belief that humans are born with all the perceptual abilities needed), are not committed to the Kantian theory of knowledge, that can be summed up to his statement: “Thoughts without contents are empty; intuitions without concepts are blind.” So, from a philosophical standpoint, Krause’s argument can be refuted: once nativist assumptions are discarded, space can be transcendental even if the axioms are not.

While detailing the transcendental intuition in Holmheltz’s view, Biagioli summarizes that the distinction between space and axioms follow from his analysis of measurement, in which a transcendental argument of intuition is requiered in order to prove that quantitative relations are common to subjective and objective experiences: “Geometrical axioms cannot be derived from an innate intuition independently of experience.” A detailed analysis of his concept of number provide the arguments for a general theory of measurements and the general conditions of the numbering of external objects – of which distinction in attributes is defined as *magnitude*, that provide means for transcendence. Intuition depends on both inner and outer experience:

On the one hand, the structure of time is described as a fact, whose origin should be explained psychologically; on the other hand, this fact provides us with concepts, such as that of number and of sum, which can be proved to determine our conception of nature. Similarly, space entails the concept of fixed geometric structure. The axioms of arithmetic, as well as those of geometry, presuppose a transcendental intuition. The axioms are not transcendental because they provide us with definitions which, in order to be applied to empirical objects, require a physical interpretation.

Thus, mathematical structures can only be defined independently of the objects we experience, and generally valid judgments about magnitudes presuppose a physical interpretation of the same structures. Furthermore, geometrical axioms are related to space as arithmetical axioms, and some mathematical structures provide us with general conditions of experience. By providing her analysis of Helmholtz’s transcendence, Cassirer’s remarks on Helmholtz’s conception of number, and Hölder’s argument that magnitudes can be constructed as hypothetic-syntethic concepts, Biagioli considers that space is one such structure.

*What Does It Mean That ‘‘Space Can Be TranscendentalWithout the Axioms Being So’’?*, Francesca Biagioli

Published: August 2013

DOI: 10.1007/s10838-013-9223-7