Abstract
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.
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Notes
- 1.
Following the current English usage – which goes back to Russell (1903) – I call properties standing in the relation of being greater or less than something “magnitudes.” “Quantities” refers to objects possessing magnitudes. These terms are translated in German to “Mass” and “Quantität,” respectively. Since the meaning of these and of related concepts (e.g., of “Größe”) changed considerably between the second half of the nineteenth century and the beginning of the twentieth century, more details about the transformation of these concepts in the German-speaking world are given below.
- 2.
- 3.
Contextualizing Helmholtz’s approach to measurement, Michael Heidelberger (1993) suggests that Helmholtz must have had psychological measurement in the back of his mind when he formulated his theory of measurement.
- 4.
For a reformulation of Helmholtz’s condition s of measurement in terms of a more recent version of measurement theory, see Diez (1997, pp.171–175).
- 5.
On the distinction between the classical and the representational views, see Michell (1993, p.189).
- 6.
Since space is characterized as a threefold extended manifold of constant curvature, more specific properties of space include the three classical cases of such a manifold. For this interpretation of Helmholtz’s distinction between general and specific properties of space, s ee also Friedman (1997, p.33), Ryckman (2005, p.73), Pulte (2006, p.198), and Hyder (2009, pp.190–191). References to opposing interpretations, beginning with Schlick’s, are given in Chap. 6.
- 7.
As it will become clear after discussing Helmholtz’s comparison between space and time, my emphasis lies not so much in Helmholtz’s naturalized interpretation of the forms of intuition – which I consider problematic – as in the fact that his argument for the objectivity of measurement retains, nonetheless, the structure of a transcendental argument. Cf. DiSalle (2006, p.129) for a different account of Helmholtz’s relationship to Kant on this point: “Helmholtz’s derivation of the general form of the Pythagorean metric from the axiom of free mobility reaffirms an important part of Kant’s view, namely, that the visual perception of space and the geometry of space have a common basis. But if that basis is nothing more than an empirical fact that might have been otherwise, then the postulates of geometry have no claim to necessity.” It seems to me that DiSalle here fails to appreciate the significance of Helmholtz’s distinction between the general and specific properties of space: although acquired, the general notion of space provides us with necessary preconditions for the possibility of measurement, and, therefo re, plays some role in the constitution of the objects of experience. Only the specific properties might have been otherwise and have no claim to necessity.
- 8.
Krause’s description is an oversimplification of the theory of local signs, which would entail, for instance, that a child sees in smaller way than an adult, for his eyes are smaller. However, this assumption is contradicted by the most familiar experiences (Krause 1878, p.39). Not only did Helmholtz rule out such assumptions, but Krause overlooked that Helmholtz’s explanation of visual perception was psychological rather than physiologic al (see Hatfield 1990, p.182).
- 9.
Cf. Krause’s misunderstanding of the theory of local signs discussed above. Hatfield points out that Helmholtz considered spiritualist as well as materialist identifications of psychic activities with the material world to be metaphysical views, lacking explanatory power. By contrast, “[Helmholtz’s] explanation ascribed the origin of our spatial abilities to the acquisition of rules for generating spatial representations , the acquisition process being guided by causal commerce with external objects” (Hatfield 1990, p.191).
- 10.
For a comparison between Helmholtz and Kant on the concept of magnitude, see also Hyder (2006).
- 11.
For a reconstruction of Helmholtz’s argument in comparison with alternative formulations of the same argument by Hölder and Cassirer, see also Biagioli (2014).
- 12.
Recall that Helmholtz had already contrasted Kant’s theory of knowledge with the idealist philosophy of nature of Schelling and Hegel in Helmholtz (1855). Helmholtz’s conception of the interaction between subjective and objective factors of knowledge had its roots in his interpretation of Kant and in his reception of the philosophy of Fic hte (see Köhnke 1986, pp.151–153; Heidelbe rger 1994, pp.170–175).
- 13.
Darrigol suggests that Helmholtz was influenced by the Grassmann brothers, Hermann and Robert, who constructed numbers by iterated connection of a single unit or element. They defined operations and derived their properties by mathematical induction . Evidence for this suggestion is Helmholtz’s use of Grassmann’s axiom , along with the fact that he refers to the Grassmann brothers’ way of proceeding in the introductory section of “Counting and Measuring.”
- 14.
The German term Größe was used to translate both Latin terms magnitudo and quantitas. This translation overlooks that magnitudo is a more general concept, whereas quantitas entails both extension and the possibility of a numerical representation (see Du Bois-Reymond 1882, p.14, and no te). A similar consideration can be made with regard to the difference between continuous quantities and discrete ones, which in Latin were called quanta and quantitas, respectively. The German translation for both terms was Größe, and was introduced by Christian Wolff in his Mathematisches Lexicon (1716). Wolff did not explicitly distinguish between discrete and continuous quantities. Nevertheless, such a distinction is implicit in Wolff’s use of multiply/diminish (vermehren/vermindern) for those quantities whose comparison presupposes the laws of arithmetic, and make bigger or smaller (vergrößern/verkleinern) for geometric figures (see Cantù 2008). Notwithstanding the ambiguity of the term, Helmholtz addressed the problem of establishing the conditions for a numerical representation of magnitudes. In this sense, he contributed to clarifying a distinction which is explicit in the first axiomatic theory of measurement by the German mathematician Otto Hölder (1901). Hölder fo rmulated axioms of quantity (Quantität) and introduced the concept of measure (Maß) after his proof that the ratios of quantities of the same kind can be expressed by positive real numbers an d can be summed arithmetically (Hölder 1901, Sect. 10). Further references about Helmholtz’s influence on Hölder are given later on in this section.
- 15.
For Helmholtz’s distinction between the method of comparison and that of addition, see also Helmholtz’s introduction to his lectures on theoretical physics from 1893. After recalling that AI is a general definition of equality, he wrote: “The principle in its general formulation is clearly false. For example, an object can have the same weight as another, and the latter can have the same color as a third object. It does not follow that the first object equals the third one. But the principle is correct and it is of great importance, insofar as it applies to magnitudes that can be compared by using the same method of observation. We call such magnitudes homogenous relative to the method of observation” (Helmholtz 1903, p.27). Helmholtz clarified the distinction between the two methods as follows: “The method of comparison does not provide us with an answer to the question: Which of the unequal magnitudes is the greatest? […] Only the method of addition also determines the concepts of smaller and greater” (p.36).
- 16.
Darrigol (2003, p.553) pointed out that Helmholtz’s notion of divisibility apparently entails not only the possibility of regarding a given quantity as the sum of a number of equal quantities, but also the possibility of approximately expressing a given quantity as a multiple of a fixed unit, and of indefinitely improving the approximation by introducing a series of subunits. In other words, Helmholtz implicitly assumed the Archimedean property as well, as this is the property that, along with the existence of a difference, allows the arbitrarily precise approximation of ratios by ration al numbers.
- 17.
Hölder wrote: “The mathematical development of the theory of quantity presupposes that there subsist some relations between some objects, and specifically that there subsists a relation of composition between these objects, which satisfy the established laws formally. The empirical interpretation of the objects and of their relations per se does not affect the development of the theory. Therefore, the same theory can be applied to completely different objects. If, for example, we substitute degrees of sensation and their differences for the points and the segments of a line, and assume, firstly, that such differences can have equal or different magnitudes and, secondly, that the same laws apply as those which in the case of segments were formulated as axioms, we obtain the same theory of measurement. For the same reason, the usual distinction between ‘extensive’ and ‘intensive ’ magnitudes, as found in philosophical texts, is not essential to mathematics” (Hölder 1924, pp.78–79).
- 18.
On the relationship between Helmholtz and Hölder, see also Darrigol (2003, pp.563–565). Cf. Michell (1993, pp.195–196), who argues that Helmholtz and Hölder – insofar as he was influenced by Helmholtz – defended a classical conception of measurement as the discovery of a matter of fact. I think that reconsidering Helmholtz’s philosophical ideas might help us to see why it is hard to find a place for Helmholtz’s theory of measurement in the classical/representational dichotomy. On the one hand, Helmholtz removed numbers from external experiences, as in the representational approach: his attempt was to prove that the laws of addition are grounded in inner intuition and can, nevertheless, be extended to empirical manifolds. On the other hand, his view ent ailed something more than the representational view: in order to account for the use of numbers in measurement, Helmholtz formulated a transcendental argument, which finds an echo in Hölder’s argument that magnitudes that can be intuited can also be constructed. Regarding Hölder’s argument, see Biagi oli (2013). I am grateful to Joel Michell for letting me know that, after reading Darrigol 2003, he changed his view about his earlier classification of Helmholtz and Hölder as exponents of a classical view of measurement.
- 19.
For an illuminating comparison between Helmholtz and Marburg neo-Kantians on the theory of s igns and the a priori, see also Patton (2009).
- 20.
- 21.
See Potter (2000, pp.83–84): “Dedekind proved the validity of the method of defining functions by means of recursive equations and derived from this a demonstr ation that his characterization of the natural numbers is categorical; no trace of either result is to be found in Frege until his later Grundgesetze, and by the time he wrote that work he had access to Dedekind’s monograph. But it is worth noting that once these omissions from Frege’s account are made good, there is a clear sense in which the two treatments are equivalent.”
- 22.
See Gray (2008). Further points of disagreement emerged only after the discovery of the antinomies of set theory. “Frege’s and Dedekind’s views were rather similar, and were taken to be so until the whole question of the relation between a set and the extension of a concept went from being elementary to being very problematic indeed” (Gray 2008, p.168).
- 23.
- 24.
According to Dummett (1991, p.49), Dedekind’s idea, “widely shared by his contemporaries, was that abstract objects are actually created by operations of our mind. This would seem to lead to a solipsistic conception of mathematics; but it is implicit in this conception that each subject is entitled to feel assured that what he creates by means of his own mental operations will coincide, at least in its properties, with what others have created by means of analogous operations. For Frege, such an assurance would be without foundation: for him, the contents of our minds are wholly subjective; since there is no means of comparing them, I cannot know whether my idea is the same as yours.” More recent scholarship initiated by 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 m athematical domains (including spatial and temporal intuitions) in order to obtain a categorical characterization of mathematical structures. For further evidence that Dedekind distanced himself from a literal understanding of the notion of metal creation – and, therefore, anticipated the objection above – cf. Reck (2003, pp.385–394). In what follows, I argue that Dedekind’s response to Helmholtz sheds further light on his critique of psychologism. I rely on the aforementioned literature for the interpretation of his definition of number as an expression of a variant of “logicism” or “logical structuralism.”
- 25.
It is controversial whether Dedekind advocated a Kantian view or opposed Kant, also because in this and other quotes he did not explicitly refer to Kant in support of his own characterization of counting. Potter argues that Dedekind distanced himself from the philosophy of mathematics which emerged from the Critique of Pure Reason, since Dedekind’s view “abandons the passivity of intuition, and ascribes to our intellect the power to represent entities of our own creation directly. This interpretation is unKantian, not merely because it credits us with a capacity Kant took to belong only to God, but because, according to it, we are capable of intuiting objects not lying within any structure of which we have an a priori grasp” (Potter 2000, pp.102–103). However, it seems to me that Dedekind’s idea is reminiscent not so much of intellectual intuition, as of the idea of a purely conceptual or symbolic synthesis as advocated by Marburg neo-Kantians, among others. Regarding the dispensability of pure intuitions, Dedekind’s view was in line with many of his contemporaries’, including the neo-Kantians.
- 26.
This proof became one of the most controversial parts of Dedekind’s account of number after the discovery of Russell’s antimony. Even regardless of the antinomies of set theory, the same proof cannot provide a semantic consistency proof. Göde l later showed that neither Dedekind’s nor others’ proofs could overcome the problems concerning the provability of consistency for arithmetic. Furthermore, Dedekind’s notion of the totality of the objects of thought has been largely misunderstood as psychological (see, e.g., Potter 2000, p.100). I agree wi th Reck (2003) that the categorical characterization of numbers as specific kinds of simple infinities shows rather the characteristics of a logical proof in Dedekind’s sense. Regarding Dedekind’s definition of infinity, it is noteworthy that Dedekind took a decisive step towards a modern conception of mathematics as the study of abstract structures. As pointed out by Ferreirós, in contrast to the traditional definition, De dekind based the notion of natural number on a general theory of finite and infinite sets. “The familiar and concrete was thus explained through the un known, abstract, and disputable. […] H e was defining the infinite through a property that Galileo, and even Cauchy, regarded as paradoxical, for it contradicted the Euclidean axiom ‘the whole is greater than the part’” (Ferreirós 1999, p.233).
- 27.
For a similar account of the notion of abstraction in Dedekind’s w ork, see Tait (1996). The meaning of Dedekind’s “creation” in this connection has been clarified especially by Reck (2003, p.400): “[Simple infinity] is identified as a new system of mathematical objects, one that is neither located in the ph ysical, spatio-temporal world, nor coincides with the previously constructed set-theoretic infinities. […] what has been done is to determine uniquely a certain ‘conceptual possibility’, namely a particular simple infinity. Which one again, i.e., what is the system of natural numbers now? It is that simple infinity whose objects only have arithmetic properties, not any of the additional, ‘foreign’ properties objects in other simple infinities have.”
- 28.
Friedman refers in particular to Kant’s exchange with August Rehberg in 1790. For further details about Kant’s view of irrational magnitudes, see Friedman (1992, pp.110–112).
- 29.
The passage above clearly foreshadows the concept of symbol that lies at the center of Cassirer’s philosophy of symbolic forms . However, in 1910 and even in Cassirer (1929), Cassirer uses “sign” to refer to mathematical symbols in continuity with the mathematical tr adition of his time and with Helmholtz’s usage.
- 30.
The dual character of Helmholtz’s notions as both empirical and formal has been emphasized especially by DiSalle . However, I do not believe that Helmholtz’s account of mathematical notions can be compared to Kant’s notion of pure intuition in this respect (cf. DiSalle 2006). Not only did Helmholtz distance himself from Kant, but the main analogy from my point of view lies simply in the fact that the formal/empirical dichotomy was introduced only later. Such a dichotomy does not do justice to the dual direction of inquiry which is characteristic of Helmholtz’s approach: it is because of this characteristic of his approach that some of his notions appear to us to be both formal and empirical, depending on the direction of inquiry. In fact, Helmholtz’s definitions are neither formal nor empirical in the current understanding of these terms.
- 31.
For a clarification of the inductive aspect of Helmholtz’s approach, see Schiemann (2009). Pulte (2006, p.199) refers to Schiemann’s characterization of Helmholtz’s approach as an inductive or bottom-up conceptualization to point out that, according to such an approach, there can be no sharp separation of intuition and conceptual knowledge. In the following chapters, I rely on Pulte also for the observation that Schlick tacitly presupposed such a separation in his interpretation of Helmholtz’s epistemological writings on account of his own top-down approach to measurement. Cf. my former consideration about the formal/empirical dichotomy. For a thorough discussion of Schlick’s reading of Helmholtz, see also Friedman (1997).
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Biagioli, F. (2016). Number and Magnitude. In: Space, Number, and Geometry from Helmholtz to Cassirer. Archimedes, vol 46. Springer, Cham. https://doi.org/10.1007/978-3-319-31779-3_4
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