ARTICLE

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

Helmholtz’s Claim in Context

Francesca Biagioli

� Springer Science+Business Media Dordrecht 2013

Abstract 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 uber 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 mag-

nitudes 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

(Zahlen und Messen, erkenntnistheoretisch betrachtet 1887. In: Schriften zur Erkenntnis-

theorie. Springer, Berlin, 1921, 70–97). In support of my view, I will consider alternative

formulations of the same argument by Ernst Cassirer and Otto Holder.

Keywords Geometrical empiricism � Hermann von Helmholtz � Measurement �Transcendental intuition

1 Introduction

The well-known lecture Hermann von Helmholtz gave in Heidelberg in 1870, ‘‘Uber den

Ursprung und die Bedeutung der geometrischen Axiome’’, includes a presentation of his

inquiry on the foundations of geometry for a wider audience and Helmholtz’s objection to

Kant. Helmholtz rejects Kant’s conception of the principles of geometry as a priori syn-

thetic judgments. On Helmholtz’s view, geometrical axioms provide us with objective

knowledge only when taken in connection with the principles of mechanics. In this regard,

F. Biagioli (&)Institut fur Humanwissenschaften: Philosophie, Universitat Paderborn, 33098 Paderborn, Germanye-mail: francescabiagioli1983@gmail.com

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J Gen Philos SciDOI 10.1007/s10838-013-9223-7

geometry becomes empirically testable and its certainty does not depend on a priori

intuition.

In the following, I will discuss attempts to defend the Kantian theory of space, such as

Jan Pieter Nicolaas Land’s attempt of 1877 and Albrecht Krause’s attempt of 1878. They

are worth discussing because in the debate Helmholtz makes it clear that he is not willing

to reject the Kantian theory of space altogether. He now maintains that ‘‘space can be

transcendental without the axioms being so’’ (Helmholtz 1977, 149). Helmholtz’s claim

appears in the title of his reply to Krause, which is published as the second Appendix to

Helmholtz’s paper ‘‘Die Tatsachen in der Wahrnehmung’’ (1878b). There Helmholtz

distinguishes between the general form of spatial intuition and its narrower specifications,

which are the axioms of geometry.

The interpretation of Helmholtz’s claim immediately became a matter of controversy.

Philosophers such as Alois Riehl used Helmholtz’s claim to argue that Kant’s theory may

work for intuitive space independently of the results of measurements in physical space

(Riehl 1904, 263). Following Riehl’s line of argument, Moritz Schlick goes a step further.

In his comments to the celebrated centenary edition of Helmholtz’s Schriften zur Er-

kenntnistheorie (1921), Schlick identifies Helmholtz’s ‘‘narrower specifications’’ with

axioms of congruence about physical magnitudes. Schlick sharply distinguishes these

properties from topological properties, such as three-dimensionality, continuity etc., which

are supposed to be grounded in spatial intuition. In order to uphold Helmholtz’s distinction,

Schlick deems the general characteristics of space indescribable, psychological factors in

spatial perception (see Schlick’s comment in Helmholtz 1977, 128, note 33).

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

‘‘transcendental’’? On the other hand, any rejection of Schlick’s interpretation is committed

to another question: what are the general characteristics of space? Furthermore, a Kantian

interpretation is supposed to give reasons for the transcendental role of space. It is difficult

to see what Helmholtz means because Kant himself never used the expression ‘‘tran-

scendental space’’: arguably, Helmholtz is thinking in terms of ‘‘a priori’’ here. His choice

of ‘‘transcendental’’ might be explained by the fact that it ought to be proved that the

general characteristics of space play a role in the constitution of the object of experience.

Indeed, such a proof would require a transcendental argument in Kant’s sense.

Let us begin with the second question, i.e. the question what the general characteristics

of space are. Helmholtz’s conception of space can be specified in different ways. Roberto

Torretti 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. Fol-

lowing Helmholtz’s analogy with color-mixtures, Torretti suggests that the required

characteristics may be specified by use of the theory of manifolds.1 He describes Helm-

holtz’s space as a differentiable three-dimensional manifold. The axioms of quantity are

not determined by the form of space because their formulation presupposes the existence of

the solid bodies we experience (Torretti 1978, 166–167). This interpretation can be called

1 Helmholtz’s conception of the relationship between space form and reality does not coincide with that ofmany followers of Kant. But Helmholtz does not exclude that space may be ‘‘a form of intuition in theKantian sense, and yet not necessarily involve the axioms’’ […]. ‘‘To cite a parallel instance, it undoubtedlylies in the organisation of our optical apparatus that everything we see can be seen only as a spatialdistribution of colours. This is the innate form of our visual perceptions. But it is not in the least therebypredetermined how the colours we see shall co-exist in space and follow each other in time’’ (Helmholtz1878a, 213). Helmholtz’s source is Hermann Grassmann’s work on manifolds and its application to color-mixtures (see Hatfield 1990, 218, note 106; Hyder 2009, chap. 4).

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into question for several reasons. On the one hand, a priori properties cannot be identified

with topological properties. One of such properties, i.e. three-dimensionality, follows from

Helmholtz’s analysis of sense perception, and requires intentional motion. So the form of

spatiality may be described more generally as a differentiable n-fold extended manifold

(Lenoir 2006, 201–202). On the other hand, there is evidence that Helmholtz ascribes to

space both three-dimensionality and constant curvature, which, as a result of his inquiries

on the foundations of geometry of 1868 and 1870, is supposed to follow from the free

mobility of rigid bodies. He distinguishes between space and axioms because, if one

defines space as a differentiable three-dimensional manifold of constant curvature, a choice

has to be made between the three classical cases of such manifolds (see Friedman 1997, 33;

Ryckman 2005, 73).

I believe that the confusion about Helmholtz’s conception of space shows the influence

of Schlick’s interpretation. Schlick’s claim that the qualities of sense perception are

indescribable is misleading: 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. However, Helmholtz does not take

into account the topological–metrical distinction. And his conception of the qualities of

sense perception is quite different from Schlick’s. Helmholtz deems sensations ‘‘signs’’ for

their stimuli. So it might seem that we can only achieve intuitive acquaintance of their

meaning. However, Helmholtz requires that the meaning of sensations be grasped by the

understanding (1867, 797). This point has been clarified by Michael Friedman, who

interprets Helmholtz’s conception of space as follows: the localization of the objects in

space provides us also with a construction of the concept of space (Friedman 1997, 33).

Helmholtz’s argument would be that the forms of intuition, once acquired, can be tran-

scendental. This leads us to our first question, i.e. the question why Helmholtz deems space

‘‘transcendental’’. I do not believe that a psychological analysis of the concept of space

might give reasons for the claim that space is transcendental. Helmholtz rejects Kant’s

view that the axioms of (Euclid’s) geometry are necessarily valid for the empirical man-

ifold. Nevertheless, Helmholtz’s replies to Land and Krause show a Kantian line of

argument: the characteristics of space should provide us with general conditions of mea-

surement, and Helmholtz deems ‘‘objective’’ those results of measurement whose inter-

pretation depends on such conditions. Schlick’s quality–quantity opposition does not play

any role here: the metrical aspect is crucial to Helmholtz’s analysis of the concept of space,

and cannot be overlooked without trivializing his argument. As pointed out by Helmut

Pulte, Helmholtz’s spatial intuition is richer than Schlick’s, because it operates with the

free mobility of rigid bodies and therefore entails that space is a manifold of constant

curvature (Pulte 2006, 198).

In the first part of the paper, I will present Helmholtz’s discussion with Land and

Krause. The defect of their views, according to Helmholtz, is that they are not able to

account for measurement. In the second part of the paper, I will reconstruct the above-

mentioned Kantian line of argument by reading Helmholtz’s defense of scientific mea-

surements in connection with his theory of measurement of 1887. I agree with neo-

Kantians, such as Ernst Cassirer, that Helmholtz’s psychological interpretation of Kant’s

forms of intuition should be rejected for philosophical reasons. Firstly, a psychological

origin of such forms would exclude their general validity. Secondly, the use of psychology

can be made superfluous, if Helmholtz’s issue is reformulated in mathematical terms. I will

argue that, nevertheless, Helmholtz’s conditions of measurement may play a constitutive

function in Kant’s sense, insofar as Helmholtz foreshadows a formal analysis of mea-

surement, such as Otto Holder’s theory of quantity of 1901. I am aware that I reject an

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important part of Helmholtz’s view, and yet I do not see my criticism as an external one.

My claim is that the line of argument suggested by Holder and Cassirer clarifies the

metrical aspect overlooked by Schlick: Helmholtz does not admit incommensurable

magnitudes, because measurability requires that general principles be formulated and

coherently extended to empirical domains independently of the specific nature of the

objects to be measured.

2 Helmholtz’s Reply to Jan Pieter Nicolaas Land

In 1876, 6 years after his public lecture in Heidelberg, Helmholtz revises his paper on the

origin and meaning of geometrical axioms and translates it into English for the British

Journal Mind. In the English version of his paper, Helmholtz summarizes the outcome of

his inquiry as follows:

1. The axioms of geometry, taken by themselves out of all connection with mechanical

propositions, represent no relations of real things. When thus isolated, if we regard

them with Kant as forms of intuition transcendentally given, they constitute a form

into which any empirical content whatever will fit and which therefore does not in any

way limit or determine beforehand the nature of the content. This is true, however, not

only of Euclid’s axioms, but also of the axioms of spherical and pseudospherical

geometry.

2. As soon as certain principles of mechanics are conjoined with the axioms of geometry

we obtain a system of propositions which has real import, and which can be verified or

overturned by empirical observations, as from experience it can be inferred. If such a

system were to be taken as a transcendental form of intuition and thought, there must be

assumed a pre-established harmony between form and reality. (Helmholtz 1876, 321)

This passage foreshadows Helmholtz’s distinction between space and geometrical axioms:

in the first case, space is deemed transcendental, and, in the second case, our system of

propositions is said to have real import. Notice that these cases exclude each other: in the

first case, the axioms of geometry are not synthetic for Helmholtz, and, in the second case,

the assumption of a transcendental form of intuition presupposes an idealistic argument he

usually rejects (see, e.g., Helmholtz 1903, 164). So the claim that space can be

transcendental without the axioms being so requires an explanation. The first step in this

direction is Helmholtz’s reply to Land: Helmholtz argues for the view that geometrical

axioms are synthetic from the standpoint of both realism and idealism.

Let us present Land’s remarks. In his paper ‘‘Kant’s Space and Modern Mathematics’’

(1877), Land maintains that Helmholtz overlooks the distinction between objectivity and

reality. Whereas common sense regards the phenomena as real things, science regards

them as signs for real things, because objective knowledge presupposes some interpretation

of the data of sense perception. Physics agrees with common sense as far as metrical

properties are concerned and we are counting and measuring. But we cannot attach real

import to analytic geometry, which ‘‘has but a conventional connection with the data of

intuition, and merges into pure arithmetic’’ (Land 1877, 41).

Land admits that the axioms of geometry, taken by themselves out of all connection

with mechanical propositions, represent no relation between physical objects. Axioms

which concern parts of space do not determine the deportment of bodies which fill such

parts at a given moments. In this regard, Helmholtz is right: Euclid’s axioms do not differ

from those of spherical or pseudospherical geometry. Nevertheless, Land maintains that the

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form of spatial intuition which is actually given is that analyzed in Euclid’s axioms (Land

1877, 46).

Helmholtz’s reply appears in Mind as a second part of his paper on the origin and

meaning of geometrical axioms (1878a). A revised version of this part is published in

German in the same year as the third Appendix to the paper on the facts in perception. In

his reply, Helmholtz clarifies his use of expressions such as ‘‘relations between real things’’

and ‘‘real import’’. He calls physically equivalent those spatial magnitudes in which under

like conditions, and in like periods of time, like physical processes can exist and run their

course. In order to prove that two such magnitudes are equal, we usually superpose them

and verify whether they can be brought into congruent coincidence. As a general outcome

of our experiences, two such magnitudes show their equivalence also with respect to all

other known physical processes. Therefore Helmholtz maintains that physical equivalence

is ‘‘a perfectly definite’’ and ‘‘objective’’ attribute of spatial magnitudes (1878a, 218).

These considerations would give us a kind of geometry Helmholtz calls physical. Were

our spatial intuition endowed with an exact geometrical structure, there would also be a

pure geometry in which congruence can be defined independently of any motion. Land’s

argument can now be formulated as follows: these two kinds of geometries cannot con-

tradict each other. Whereas congruence, according to our inner intuition, depends on the

makeup of our mind, objective congruence depends on the physical properties of the bodies

we experience. But it might be asked whether the two different kinds of equality neces-

sarily coincide. Helmholtz’s answer is ‘‘no’’: such a coincidence would be only hypo-

thetical. Physical space and spatial intuition could also be supposed to be related to each

other as is actual (Euclidean) space to its (non-Euclidean) image in a convex mirror.

Helmholtz’s conclusion is the following:

If we really had an innate and indestructible form of space-intuition involving the

axioms within it, their objective scientific application to the phenomenal world

would be justified only in so far as observation and experiment made it manifest that

physical geometry, grounded in experience, could establish universal propositions

agreeing with the axioms. (Helmholtz 1878a, 221)

This is a realist description of the situation Helmholtz had already discussed in 1876. He

now maintains that his argument holds true also from an idealist point of view. He

distinguishes between the topogenous factors of localization and the hylogenous ones: the

former ones specify at what place in space an object appears us; the latter ones cause our

belief that at the same place we perceive at different times different material things having

different properties. Helmholtz formulates his argument as follows:

When we observe that the most diverse physical processes may go on during equal

periods of time in similar fashion at different, but congruent, parts of space, the real

meaning of such perception is, that there may be in the sphere of reality equal

sequences and aggregates of hylogenous moments combining with certain distinct

groups of topogenous moments, which letter we then call physically-equivalent. We

may thus discover by observation what special figures appearing in our perception

correspond with physically-equivalent topogenous moments; and experience tells us

that they are equivalent for all physical processes. (1878a, 224)

The German version of the paper is more explicit: the proposition that any combination or

sequence of hylogenous factors which can exist and run its course in combination with one

group of topogenous factors, is also possible with any other physically equivalent group of

topogenous factors is said to have ‘‘real content’’, and provides us with a proof that

Space Can Be Transcendental Without the Axioms

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topogenous factors influence the course of real processes (Helmholtz 1977, 161).

Helmholtz deems his argument idealistic, because the objectivity of scientific measure-

ments does not presuppose that any specific structure is found in actual space. All that is

presupposed is that some regularity is found in the phenomena. If space is identified with

the most general structure or group of topogenous factors underlying such regularities,

Helmholtz’s argument shows the role of this structure in the constitution of the object of

experience, and may be regarded as a transcendental argument in Kant’s sense.

3 Helmholtz’s Reply to Albrecht Krause

These considerations may help us to explain why Helmholtz sees his claim that space is

transcendental without the axioms being so as a reply to Krause in particular: Helmholtz

uses the notion of ‘‘transcendental’’ in connection with his attempt to justify the objectivity

of measurements. First of all, let us present Krause’s comparison between Kant and

Helmholtz. Krause’s essay Kant und Helmholtz uber den Ursprung und die Bedeutung der

Raumanschauung und der geometrischen Axiome (1878) begins with the following ques-

tion: can one state different properties of space and, consequently, different geometrical

axioms? In order to answer this question, Krause considers the relationship between the

sense organs and the brain. He maintains that the Kantian theory of space is compatible

with the requirement that spatial relations be univocally determined through their con-

nection with the brain, whatever form or size the sense organs may have. Krause’s view is

that any variation or hypothesis of different spaces is based on one and the same space,

whose properties depend immediately on higher cognitive functions. Otherwise the form of

our intuition would vary according to our sense organs, whose spatial features are con-

tingent. Therefore Krause rejects Helmholtz’s attempt to draw spatiality out of sensations.

Krause criticizes, in particular, Helmholtz’s argument that a comparison between our space

and its image in a convex mirror should provide us with intuitions we never had. According

to Krause, such intuitions are impossible. He considers plain surfaces, as well as curved

ones, as the boundaries of a three-dimensional body. Straight lines would not correspond to

the geodesics of spherical or pseudospherical geometry. And ‘‘straightest’’ lines between

pairs of points would mean straight lines in the proper sense of the term. For the same

reason, Krause calls into question the extension of the concept of curvature to more than

two-dimensional manifolds: the curvature of space cannot be measured because anything

endowed with direction already lies in space (Krause 1878, 84).

Krause’s further question is: to what extent are the laws of spatial intuition expressed by

the axioms certain? His answer is that, since spatial intuition is necessary for the construction

of every geometric object, the certainty of geometrical axioms cannot be called into question.

The skeptical outcome of the so-called ‘‘Riemann-Helmholtz’s theory’’ is that there would be

no geometrical axiom properly speaking. Krause’s view entails that we should not trust our

measurements, when they contradicted the axioms, because measurements share the

approximate character of natural laws. By contrast, axioms are exact knowledge.

In his reply, Helmholtz makes it clear that the empiricist theory of vision does not entail

that the spatial features of our sense organs determine form and size of the objects.2 More

2 Krause’s description is an oversimplification of the theory of local signs, which would entail, for instance,that a child sees smaller that an adult, for his eyes are smaller. But this assumption is contradicted byempirical facts we are most familiar with (Krause 1878, 39). Krause overlooks that Helmholtz’s explanationof visual perception is psychological rather than physiological (see Hatfield 1990, 182).

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notably, he points out that the Kantian theory of knowledge is not committed to Krause’s

assumptions, which are derived from a nativist theory of vision. So Krause’s argument can

be falsified from a philosophical point of view: once nativist assumptions are rejected,

space can be transcendental without the axioms being so.

Regarding Krause’s objections to nineteenth-century inquiries on the foundations of

geometry, Helmholtz replies that the measure of curvature is a well-defined magnitude and

generally applies to n-dimensional manifolds. This consideration nullifies Krause’s attempt

to show that Euclidean three-dimensional space is a necessary assumption. Helmholtz’s

point is that we must give reasons for our assumptions. Curiously enough, Krause does not

take into account the results of scientific measurements because of their limited accuracy.

But he does not need measurements to be convinced of the correctness of those axioms

which are supposed to be grounded in transcendental intuition. In this case, Krause reas-

sures himself with appraisals by ‘‘visual estimation’’. That is, for Helmholtz, ‘‘measuring

friend and foe by different standards!’’ (1977, 151).

Helmholtz does not say much about the convenience of regarding space as a tran-

scendental concept. Nevertheless, his considerations suffice to exclude Schlick’s later

interpretation. Helmholtz makes it clear that, if the form of intuition is transcendental, it

cannot be given ‘‘by visual estimation’’. Otherwise it could not provide us with conditions

of experience. Furthermore, it should be noticed that Helmholtz does not use Riemann’s

distinction between metrical and extensive properties. In fact Krause overlooks this dis-

tinction. But Helmholtz’s objection to Krause goes deeper: by dismissing well-defined

magnitudes, such as the measure of curvature, and by mistrusting scientific procedures,

Krause fails to explain measurement.

We shall return to this point later on. It may be helpful to notice first that Helmholtz’s

reply was anticipated in many ways by Benno Erdmann in his essay Die Axiome der

Geometrie: Eine philosophische Untersuchung der Riemann-Helmholtz’schen Raumtheo-

rie (1877). Erdmann calls both Helmholtz’s and Riemann’s epistemology a kind of formal

empiricism, according to which our representations are only partial images of the things,

which coincide with them in every quantitative relation (space, time, and natural laws)

while differing from them in every qualitative one. The assumption of a pre-established

harmony between sensations and their causes is called into question because our mental

activities are supposed to originate from our interaction with the world.3 The empirical

occasion does not provide us with spatial determinations; rather, we form spatial concepts

in order to organize our sensations. The form of space must be distinguished from its

empirical content.

Regarding the philosophical meaning of the inquiries on the foundations of geometry,

Erdmann argues that both Riemann’s inquiry on the hypotheses underlying geometry

(1854) and Helmholtz’s mental experiments of 1870 contradict the rationalist opinion that

our spatial intuition is independent of experience. If rationalists were right, space could not

undergo any changes. By contrast, Riemann and Helmholtz show that space admits dif-

ferent geometries. However, they neither answer the question of whether our inference

from our representations to the existence of the things is correct (which is a matter of

controversy between idealism and realism), nor do they exclude other kinds of empiricism.

3 Cf. Krause’s above mentioned misunderstanding of the theory of local signs. Hatfield points out thatHelmholtz considered spiritualist as well as materialist identifications of psychic activities with the materialworld as metaphysical views, lacking explanatory power. By contrast, ‘‘[Helmholtz’s] explanation ascribedthe origin of our spatial abilities to the acquisition of rules for generating spatial representations, theacquisition process being guided by causal commerce with external objects’’ (Hatfield 1990, 191).

Space Can Be Transcendental Without the Axioms

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Besides formal empiricism, a choice is given between sensism, according to which our

representations are images of the things, and a refined kind of apriorism, which assumes

that our representations, even though they are completely different from the things, may

correspond to them in each and every single part. Erdmann argues for apriorism as follows.

He maintains that the concept of space can be specified both geometrically and analyti-

cally. On the one hand, a system of metric relations can be derived from our spatial

intuition, which is supposed to be singular and directly given, and yet capable of infinitely

many variations. On the other hand, Riemann shows that a generalized metric form can be

developed also analytically, so that our original system becomes a special case. Now, this

prompts the question of how the geometrical and analytical interpretations of geometric

concepts are related. In order to answer this question, Erdmann uses the whole-part

opposition, which is characteristic of his apriorism. He writes:

The fact that our spatial intuition is single is not contradicted: we can only con-

ceptualize the general intuition of a pseudospherical or spherical space of a certain

measure of curvature. Such uniqueness, however, is not absolute anymore, because

we can fix homogeneous parts of those spaces intuitively and compare them with the

metrical relations between partial representations of our space. But the concepts of

such spaces show in their development all the clearness and distinction enabled by

the discursive nature of conceptual knowledge. Therefore we may also speak about a

concept of space. At the same time, however, we clearly cannot form it directly

without a diversion into the concept of quantity. (Erdmann 1877, 135)

This passage refers to Helmholtz’s mental experiments. Helmholtz’s world in the convex

mirror shows that an intuitive comparison between different metrics is possible, though

only locally: in order to make such a comparison, one should not start from space itself but

from its parts. Helmholtz adopts Riemann’s approach: the concept of space presupposes

that of quantity, not vice versa. At the same time, Erdmann does not reject the Kantian

hypothesis that space as a whole is a singular intuition, not a concept.

Helmholtz rejects Kant’s supposition. As Helmholtz puts it, the resolution of the con-

cept of intuition into the elementary processes of thought is the most essential advance in

the recent period (1977, 143). Nevertheless, he appreciates Erdmann’s work on the axioms

of geometry as a reliable discussion of that subject-matter in philosophical terms (Helm-

holtz 1977, 149). In my opinion, Helmholtz’s appreciation is due to the fact that Erdmann,

unlike Land and Krause, seeks to explain how space and quantity are related to each other.

Similarly, in order to construct the concept of space, Helmholtz begins with the most basic

relationship between spatial magnitudes—namely, their congruence. The broader specifi-

cations of space, especially constant curvature, are supposed to be determined by the free

mobility of rigid bodies, which is required for spatial magnitudes to be congruent. Since

manifolds of constant curvature admit different geometries, narrower specifications, such

as the axioms of congruence, must be distinguished from the general principles of

measurement.

To summarize, Helmholtz’s distinction between space and axioms follows from his

analysis of measurement. This distinction does not provide an argument for Helmholtz’s

connection with Kant. Helmholtz’s argument goes from the parts to the whole, and is not

compatible with Kant’s conclusion that space is an intuition, not a concept. Nevertheless,

Helmholtz’s formal-empiricist attempt to prove that quantitative relations are common to

subjective and objective experience might require a transcendental argument. As we shall

see, such argument is explicit in Helmholtz’s paper ‘‘Zahlen und Messen, er-

kenntnistheoretisch betrachtet’’ (1887). I suggest that a reconstruction of his argument may

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help us to explain his use of the notion of ‘‘transcendental’’. Firstly, the same expression is

used here with regard to time. Secondly, and more notably, in 1887 Helmholtz develops a

general theory of measurement or composition of physical magnitudes, so that congruence

can be treated as a special case. In the dispute with Krause, the issue at stake is the same. I

shall then add some remarks on Helmholtz’s conception of space and its reception after

presenting his full analysis of measurement.

4 Transcendental Intuition in Counting and Measuring

At the beginning of ‘‘Zahlen und Messen’’, Helmholtz summarizes his remarks on tran-

scendental intuition as follows. Geometrical axioms cannot be derived from an innate

intuition independently of experience. Helmholtz does not reject Kant’s view of space as a

transcendental form of intuition, but rather that particular specification of his view which

was influenced by the metaphysical endeavors of his successors.4 We already noticed that a

legitimate idealistic interpretation of the transcendental role of space in Helmholtz’s sense

should take into account an inner-outer opposition, which is reflected in the opposition

between physical and pure geometry. Helmholtz’s point in 1887 is that such interpretation

must be completed by an inquiry on the foundations of arithmetic, for several reasons.

Firstly, physical equivalence between spatial magnitudes is supposed to admit a numerical

expression. Secondly, Helmholtz’s goal is to use his theory of knowledge to account for the

origin and meaning of the axioms of arithmetic. He writes:

[…] if the empiricist theory—which I besides others advocate—regards the axioms

of geometry no longer as propositions unprovable and without need of proof, it must

also justify itself regarding the origin of the axioms of arithmetic, which are cor-

respondingly related to the form of intuition of time. (Helmholtz 1977, 72)

The parallelism with space suggests that time can be deemed transcendental in the same

sense: a transcendental argument is necessary for the axioms of arithmetic to be valid for

the empirical manifold. They are the following propositions:

AI. If two magnitudes are both alike with a third, they are alike amongst themselves.

AII. The associative law of addition: (a ? b) ? c = a ? (b ? c).

AIII. The commutative law of addition: a ? b = b ? a.

AIV. Like added to like gives like.

AV. Like added to unlike gives unlike.

Helmholtz’s argument proceeds as follows. First of all, he notices that arithmetic tea-

ches the logical application of a system of signs (i.e., numbers) and explores which dif-

ferent combinations (i.e., calculative operations) lead to the same results. Such a system,

however, neither warrants the inner coherence of our thought nor sets out rules of play for

fictitious objects. The axioms of arithmetic are laws of addition. And additive principles of

the same kind are required for physical magnitudes to be compared. Helmholtz’s purpose is

to provide us with a natural basis for our use of symbols and a proof of their applicability.

Therefore he deems arithmetic ‘‘a method constructed upon purely psychological facts’’

4 Recall that already in 1855, in a celebrated lecture entitled ‘‘Uber das Sehen des Menschen’’, Helmholtzcontrasts Kant with later attempts to solve nature into a system of subjective forms, as pursued by nine-teenth-century philosophers of identity such as Hegel. Helmholtz’s conception of the interaction betweensubjective and objective factors of knowledge has its roots in his interpretation of Kant as well as in hisreception of the philosophy of Fichte (see Kohnke 1986, 151–153; Heidelberger 1994, 170–175).

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123

(Helmholtz 1977, 75). For Helmholtz the clarification of this point requires a complete

analysis of the concept of number. In a certain sense, it is clear that the ‘‘naturalness’’ of

the number series is merely an appearance: the choice of number signs is a matter of

stipulation and the so-called ‘‘natural’’ numbers are but arbitrarily chosen signs. All the

same, their series is impressed on our memory much more firmly than any other series of

objects as a consequence of its frequent repetition. Ordinal numbers acquire a paradigmatic

role in the recollection of all other sequences in our memory, and their series shows the

characteristics of inner intuition: each representation entering our consciousness is nec-

essarily subject to an irreversible relationship with the preceding and the following rep-

resentations in the series and it is uniquely singled out by its position in time. This requires

us to designate each step in the series by means of a system of signs that allows neither

interruptions nor repetitions, as in the decimal system. Helmholtz maintains that the

complete disjunction thereby obtained is ‘‘founded in the essence of the time sequence’’

(1977, 77). He expresses this fact as follows:

AVI. If two numbers are different, one of them must be higher than another.

AVI entails that ordinal relations are asymmetric and transitive. From AI it follows that

equality is transitive and symmetric instead. From transitivity (i.e., if a = b and b = c,

then a = c) the validity of AI for the series of integer numbers follows. And a generalized

form of the remaining axioms can be derived from:

aþ bð Þ þ 1 ¼ aþ bþ 1ð Þ:

This is Grassmann’s axiom, which provides us with a definition of addition. For example,

the associative law of addition is generalized as follows:

Rþ aþ bþ S ¼ Rþ aþ bð Þ þ S;

where capital letters denote the sum of arbitrarily many numbers.

Once addition has been defined, Helmholtz introduces the following axiom:

AVII. If a number c is higher than another one a, then I can portray c as the sum of a and

a positive integer number b to be found.

In the following, Helmholtz extends the laws of addition, especially AVII, to cardinal

numbers. He describes the method of numbering off for the purpose of addition as coor-

dinating an ordered sequence (n ? 1), (n ? 2)… to the series of integer numbers. He then

combines a first series preserving a certain sequence with a second series having variable

sequences. Given two numbers n and (n ? 1), on the one side, and two symbols e and f, on

the other, there are two possible manners of coordination:

aÞ n! e; nþ 1ð Þ ! f

or bÞ n! f; nþ 1ð Þ ! e:

If a) is substituted for b), the second series a, b, c, etc. can be put into one-to-one

correspondence with the series (n ? 1), (n ? 2), etc. By continued exchange of neigh-

boring members of a group, we bring about any possible sequence of its members without

gaps or repetitions. According to Helmholtz’s theorem:

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Attributes of a series of elements which do not alter when arbitrarily neighbouring

elements are exchanged in order with each other, are not altered by any possible

alteration of the order of the elements. (1977, 85)

As a consequence of this theorem, cardinal numbers can be defined as follows: if the

complete number series from 1 to n is needed in order to coordinate a number with each

element of the group, then n is called the cardinal number of the members of the group.

The corresponding proposition for AVII is that the total number of the members of two

groups that have no member in common equals the sum of the cardinal numbers of the

members of the two single groups. Another consequence is that the commutative law of

addition can be generalized as follows. Given the associative law of addition, by AIII, and

by his theorem, Helmholtz infers that:

Rþ aþ bþ S ¼ Rþ aþ bð Þ þ S ¼ Rþ bþ aþ S:

Summing up, Helmholtz develops a theory of ordinal number: the laws of addition

apply, first of all, to ordinal numbers, but he needs a theorem in order to prove that the

same laws apply to cardinal numbers as well. It might seem that this technicality does not

provide us with a justification of Helmholtz’s empiricist theory. Note, however, that

Helmholtz’s conception of addition is not arithmetical strictly speaking. Arithmetical

addition is but a (possibly paradigmatic) case. But there are different kinds of numbering

also within the theory of numbers, and we do not know from the start whether the laws of

addition apply to the specific case of numbering that ascertains the cardinality of a set of

things. If Helmholtz’s theory of knowledge is to be justified, the same principles must be

extended to empirical domains. So it is not surprising that Helmholtz resumes his con-

siderations on transcendental intuition after presenting his theory of number. He writes:

The concept of addition described above […] coincides with the concept of it which

proceeds from determining the total cardinal number of several groups of numerable

objects, but has the advantage of being obtainable without reference to external

experience. One has thereby proved, for the concepts of number and of a sum—taken

only from inner intuition—from which we started out, the series of axioms of

addition which are necessary for the foundation of arithmetic; and also proved, at the

same time, that the outcome of this kind of addition coincides with the kind which

can be derived from the numbering of external numerable objects. (Helmholtz 1977,

87)

The concepts of number and of a sum, which are fist taken from inner intuition, show a

constitutive function, insofar as they provide us with the general conditions of the

numbering of external objects. In this sense, Helmholtz’s argument can be taken as a

transcendental one.

Let us see how this argument leads to a general theory of measurement. Helmholtz

defines magnitudes as those objects, or attributes of objects, which allow a distinction into

greater, equal or smaller when compared with similar ones. This requires us to give a

physical interpretation of AI and AVII. Note that Helmholtz already analyzed physical

equality between spatial magnitudes in his reply to Land. This analysis is completed in

1887: if two such magnitudes are to be measured, it must be possible to assign a cardinal

number to them. Helmholtz makes it clear that AI provides us with a definition of equality:

as a definition, it does not have objective meaning itself. Measurement presupposes, in

addition, that two like objects, when interacting under suitable circumstances, allow the

observation of a particular outcome which does not occur as a rule between other pairs of

Space Can Be Transcendental Without the Axioms

123

similar objects. Helmholtz calls a procedure which enables us to accomplish such obser-

vation method of comparison. For example, the simplest geometrical structure for which a

magnitude is specifiable is the distance between a pair of points. The condition for

assigning a numerical value is that the points remain fixedly linked for at least the time of

our measurement. The method of comparison, in that case, is to verify whether pairs of

points can be brought into congruent coincidence. It may seem that metrical properties are

thereby induced from our experiences with solid bodies. Indeed, the ‘‘well-known’’ method

of coincidence admittedly requires time (Helmholtz 1977, 92). This reading would be

correct with regard to Helmholtz’s theory of local signs. However, his argument in the

context of the theory of measurement is a different one: the dynamic factor is mentioned

because measurement requires us to assign numerical coordinates to the points within a

system of reference which needs to remain identifiable in motion.

Note also that Helmholtz’s argument would be incomplete without a second step. Once

we have found a suitable method of comparison, the particular outcome of the interaction

between two like objects is supposed to remain unaltered if the two objects are interchanged.

Furthermore, if two such objects a and b are proved to be equal, and we know by previous

observation that a equals c, then b must also equal c. This procedure must be generalized, so

that objects which have proved to be equal are also mutually substitutable in any further

cases. To clarify this point, let us return to Helmholtz’s example. The concept of length

gives something more than that of distance. Whereas distance allows only a distinction of

equal and unequal, length also entails an opposition of greater and smaller. If two pairs of

points a, b and a, c, of differing distance, coincide at a and are placed in a straight line, so

that a portion of this line is common to both, then either b falls upon the line ac or c upon the

line ab. This fact leads to a more general consideration: once we know whether two

magnitudes are equal or unequal, in order to measure them in the second case, the greater

must be calculable as the sum of the smaller and their difference. That is to say, if AI is to

acquire objective meaning, it must be also given a physical interpretation of AVII or the

principle of homogeneity of the sum and the summands. Equality (Gleichheit) can be

attributed to two or more objects only if they are compared from some point of view. Their

comparison with regard to magnitude requires that equal or possibly unequal magnitudes be

homogeneous. This consideration is purely logical and shows that homogeneity (Gle-

ichartigkeit) is a more fundamental property than equality. In addition, Helmholtz gives the

following physical interpretation: it does not suffice to have a method of comparison

whatsoever; the issue of whether the result of connection remains the same, when parts are

exchanged, must be decided by the same method of comparison with which we ascertained

the equality of the parts to be exchanged. Given the homogeneity of the sum and the

summands, the remaining laws of addition can be applied to the composition of physical

magnitudes. Helmholtz’s theorem corresponds to the following proposition:

A physical method of connecting magnitudes alike in kind can be regarded as

addition, if the result of the connection—when likened as a magnitude of the same

kind—is not altered either by exchanging individual elements with each other, or by

exchanging terms of the connection with alike magnitudes of like kind. (Helmholtz

1977, 96)

In the following, Helmholtz shows that magnitudes which can be added are also

divisible. Every occurring magnitude can be regarded as a sum of a cardinal number of

equal parts to be chosen as units. This choice is conventional, and it might happen that the

magnitudes under consideration are not expressible without remainder. Nevertheless, also

in that case, the unit can be divided again in the usual manner, so that any degree of

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123

precision can be attained. Complete precision is attainable only for rational proportions,

and the value of irrational ones can be enclosed between arbitrarily reducible limits. So we

can calculate all continuous differentiable functions of irrational magnitudes occurring in

geometry and physics, where Weierstrass’ everywhere continuous but nowhere differen-

tiable functions have not yet been encountered.

We already mentioned that Helmholtz overturns Kant’s part-whole opposition. Now his

analysis of measurement gives a specific reason to begin with the parts and their rela-

tionships to each other. In a well-known passage of the Transcendental Aesthetic, Kant

maintains that parts of space always belong to one and the same space (1787, 39–40). Their

homogeneity seems to be derived a priori from the infinite divisibility of space: there

would be no need to prove that the composition of spatial magnitudes follows the laws or

arithmetic. If Helmholtz’s problem is a real one, the Kantian theory provides no solution.

Helmholtz does not only require divisibility in its metrical aspect or into equal parts, but,

contrary to Kant’s argument, homogeneity implies infinite divisibility, and not vice versa

(see Darrigol 2003, 548–549; Hyder 2006, 33–36).

To conclude, the parallelism between space and time may help us to clarify Helmholtz’s

claim that space can be transcendental. In both cases, it is clear that the form of intuition is

not directly given. Such a form depends on an interaction between 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.

There is evidence that Helmholtz might have understood his parallelism in this way.

Indeed, in a celebrated passage of his lecture of 1870 he already noticed that the concept of

fixed geometric structure might be regarded as a transcendental one and the axioms of

geometry as propositions given a priori in transcendental intuition. He also noticed that, in

that case, the axioms of geometry would not be synthetic propositions in Kant’s sense: they

would only assert something which followed analytically from the definition of fixed

geometrical structures (Helmholtz 1977, 24–25). However, there is an important differ-

ence: in the case of time, Helmholtz does not distinguish between broader and narrower

specifications. Nevertheless, he distinguishes between time and arithmetical axioms

because, if the constitutive function of fundamental structures is to be proved, their laws

must be generalized progressively. Even though Helmholtz admits that irrational propor-

tions lack a numerical expression, he rejects the view that continuity is an intrinsic property

of some magnitudes. His point is that the extension of additive principles proceeds inde-

pendently of the supposition that there might be a difference in nature between extensive

and intensive magnitudes (Helmholtz 1977, 99). For the same reason, he does not treat

from the outset physical magnitudes as ones only composed out of units. Firstly, quantities

other than measurable magnitudes (e.g., coefficients) are used in physics. Secondly, and

more notably, the domain of validity of the laws of arithmetic should not be unnecessarily

restricted (Helmholtz 1977, 73).

These considerations may help us to explain Helmholtz’s distinction between tran-

scendental intuition and axioms also in the case of space. More precisely, since the

composition of spatial magnitudes is treated as a special case of measurement, Helmholtz’s

parallelism between space and time is better understood as an attempt to show how these

two concepts are related to each other. The claim that space can be transcendental without

Space Can Be Transcendental Without the Axioms

123

the axioms being so follows from a more general argument: Helmholtz deems transcen-

dental those structures which make measurement possible independently of the specific

nature of the objects to be measured. On the other hand, measurement requires principles

which are formulated for each measuring situation presently under consideration, and

which are coherently generalized to all known physical processes. These are not a priori

given because such extension depends on the advancement of empirical science.

My emphasis lies on the formal aspect of Helmholtz’s analysis of measurement, rather

than on the psychological one. I do not deny that there is such an aspect, but it seems to me

to be more problematic. In order to support my view, I shall benefit from the discussion of

Helmholtz’s theory of ordinal number in Ernst Cassirer’s Substanzbegriff und Funk-

tionsbegriff: Untersuchungen uber die Grundfragen der Erkenntniskritik (1910). For

Cassirer’s interpretation of the claim that space is transcendental, I shall refer to the fourth

volume of his Das Erkenntnisproblem in der Philosophie und Wissenschaft der neueren

Zeit (1957).

5 Cassirer’s Remarks on Helmholtz’s Transcendental Forms of Intuition

Let us first consider Cassirer’s remarks of 1910 on Helmholtz’s conception of number.

According to Cassirer, Helmholtz supports a kind of refined ‘‘copy’’ theory of knowledge

(Abbildtheorie): the numerical relations that we do not find directly in comparing external

objects acquire an objective meaning because we have previously developed a system of

signs which works as a substitute for them. Cassirer’s objection is that Helmholtz is not

able to explain how different signs could represent different things. From a psychological

point of view, signs would differ from each other because of their size and arrangement.

They were judged merely according to what they sensuously are, and not according to

what they intellectually mean. Judgments about numbers would be derived from judgments

about numerals. Cassirer writes:

It is only the ambiguity in the concept of symbol, only the circumstance that under it

can be understood, now the bare existence of a sensuous content, and now the ideal

object symbolized by the latter, which makes possible this reduction to the nomi-

nalistic schema. Leibniz, whose entire thought was concentrated upon the idea of a

‘‘universal characteristic,’’ clearly pointed out in opposition to the formalistic the-

ories of his time, the fact that is essential here. The ‘‘basis’’ of the truth lies, as he

says, never in the symbols but in the objective relations between ideas. If it were

otherwise, we would have to distinguish as many forms of truth as there are ways of

symbolizing. (Cassirer 1923, 43)

The nominalistic schema is untenable, because a meaningful use of symbolism

presupposes a logical basis or, as Leibniz puts it, objective relations between ideas. In

deeming such relations ‘‘objective’’, Cassirer overcomes Helmholtz’s opposition between

inner and outer experience. Cassirer reformulates Helmholtz’s Kant-oriented argument as

follows:

The consideration of the ‘‘cardinal numbers’’ […] occasions the discovery of no new

property and no new relation, which could not have been previously deduced from

the bare element of order. The only advantage is that the formulae developed by the

ordinal theory gain a wider application, since they can henceforth be read in two

different languages. (Cassirer 1923, 42)

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123

What is right in Helmholtz’s argument is that the theory of cardinal numbers does not

produce new objects, but rather realizes a new logical function: a finite sequence of unities

earlier regarded as a series can be regarded also as a whole—namely, a system, along with

the operations acting on it. This consideration must be completed by a logical analysis of

the system of natural numbers. Therefore, Cassirer uses Dedekind’s definition of number of

1888. Dedekind requires that the system of natural numbers be an infinite one, namely a set

which can be put into one-to-one correspondence with a proper subset of itself (e.g., the

even numbers). Given a fundamental element 1, a relation \, and an injective function ufrom any element to its successor, the series of natural numbers N can be generated by uN,

and their system can be defined as: {1 [ uN}.

Note that there is a specific assumption in Helmholtz’s axiomatization which is

unnecessary, namely AVI. And it is precisely the one which is supposed to be grounded in

the essence of the time sequence. The system of natural numbers can be treated inde-

pendently of the time sequence, because the structure analyzed by Dedekind is the more

general concept of a progression whatsoever. Dedekind shows that it suffices to introduce

‘‘\’’ as a transitive and asymmetric relation. Numbers are generated as those elements

which are in such a relation to each other. But Helmholtz’s connection with Kant is

misleading for another reason as well. Kant does not provide the foundations of arithmetic

in the Transcendental Aesthetic, and in the chapter on Schematism he writes:

The pure image of all magnitudes (quantorum) for outer sense is space; for all

objects of the senses in general, it is time. The pure schema of magnitude (quanti-

tatis), however, as a concept of the understanding, is number, which is a represen-

tation that summarizes the successive addition of one (homogeneous) unit to another.

Thus number is nothing other than the unity of the synthesis of the manifold of a

homogeneous intuition in general, because I generate time itself in the apprehension

of the intuition. (Kant 1787, 182)

Cassirer maintains that Dedekind’s conception of number is Kantian in spirit, because

every singular quantity is produced by addition of the preceding one in the series with the

unit. Helmholtz is right to point out that the cardinal aspect presupposes the ordinal one.

However, Cassirer prefers Dedekind’s definition of number because it makes it clear that

order is not intuitively given, but follows from the use of ‘‘u’’, which is an operation of the

thought. This way of thinking enables us to study the structural properties of numbers

independently of their specific nature. The set of numbers is determined and arguably

extended by the specification of the operations acting on it.

Cassirer maintains that the objects of geometry can be classified in a similar way. In

order to prove, in Euclidean geometry, that two figures are equal, we presuppose that they

can be brought into congruent coincidence without changes in shape and size during their

displacement. But this kind of congruence can also be defined as that relation between

spatial magnitudes that is invariant under a specific group of transformations, which is

called ‘‘the Euclidean group’’. This idea goes back to Sophus Lie and Felix Klein. The

general issue at stake in Klein’s Erlanger Programm of 1872 is, given a manifold and a

transformation group acting on it, to investigate those properties of figures on that manifold

that are invariant under all transformations of that group. The broader the group is, the

fewer properties are left unvaried. For example, the affine group is broader than the

Euclidean one, and does not make a difference between a circle and an ellipse, but only

between finite and infinite conic sections. From the point of view of the projective group,

all conic sections are classified as the same figure. And the group of continuous trans-

formations makes no difference between a cone-shaped figure and a cube.

Space Can Be Transcendental Without the Axioms

123

In the fourth volume of the Erkenntnisproblem, Cassirer maintains that Helmholtz

presupposes such a classification. The claim that space can be transcendental without the

axioms being so is interpreted by Cassirer as follows. On the one hand, the general

characteristics of space provide us with conditions of experience. In Cassirer’s interpre-

tation, they are three-dimensionality and constant curvature, and correspond to the pro-

jective group. Since we know by the above-mentioned classification that such group is

broader than the Euclidean group, Helmholtz is right to point out that the specific axi-

omatic structure of our space is not a priori determined and might be Euclidean or non-

Euclidean. On the other hand, he maintains that measurement also presupposes our

experiences with rigid bodies and the supposition of their free mobility, which is necessary

for spatial magnitudes to be compared by superposition. The problem with this specific

method of comparison is that it cannot prove that two such magnitudes are equal, unless

the Euclidean group is presupposed from the outset. As pointed out by the French math-

ematician Henri Poincare in La Science et l’Hypothese (1902), Helmholtz deems ‘‘rigid’’

those bodies whose properties are invariant under the transformations of the said group.

Then how are these transformations to be defined? If their definition presupposes the

existence of rigid bodies, Helmholtz’s reasoning is circular (Poincare 1902, 60).

Cassirer agrees with Poincare. Nevertheless, Cassirer emphasizes that Helmholtz, as in

the case of arithmetic, foreshadows a formal treatment of that subject-matter: the concept

he implicitly uses to define geometric objects and their properties is not that of space, but

that of group (Cassirer 1957, 50). This treatment shows that geometries are hypothetic-

deductive systems, whose application is a matter of empirical science. The issue of a

complete classification of hypotheses can be solved a priori, and must be distinguished

from that of a search for methodological criteria for choosing among them. Cassirer’s point

here is important. However, it might be objected that his characterization of Helmholtz’s

issue is wide enough to admit solutions which might be incompatible, such as Kantianism,

empiricism or conventionalism. In this regard, the parallelism with the philosophy of

arithmetic may help us because, in that case, Cassirer has a more specific argument for a

Kantian interpretation of Helmholtz’s theory of ordinal number: if singular quantities are to

be constructed, the structural properties of a numerical domain must be proved to have a

constitutive function. A similar consideration can be made with regard to Helmholtz’s

theory of measurement: the aspect in which his analysis admits a Kantian interpretation is

the formal one. In this regard, it is worth noting Helmholtz’s influence on the German

mathematician Otto Holder (1859–1937), who developed a formal analysis of measure-

ment and, at the same time, a synthetic conception of mathematics and geometry.

6 Helmholtz’s Influence on Holder

Let us first notice that Holder’s classification of the sciences differs from Helmholtz’s

classification because Holder maintains that arithmetic concepts admit a purely logical

construction. Arithmetic for Holder begins with definitions such as Grassmann’s formula,

which means that the sum of a number a and the successor b ? 1 of a number b is the

successor of the number a ? b in the series of integer numbers. By contrast, some geo-

metric objects must be given in intuition or experience. Their relations with each other can

be extended to all similar cases according to general principles to be postulated (i.e., the

axioms). Whereas geometric inferences are hypothetic-deductive, arithmetic inferences are

purely deductive and need no axioms.

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123

Helmholtz’s parallelism between space and time, and their relationship to the axioms of

geometry and arithmetic, respectively, does not play any role here. Nevertheless, in a paper

of 1901, ‘‘Die Axiome der Quantitat und die Lehre vom Mass’’, Holder faces the same

issue as Helmholtz (1887)—namely, how to formulate general conditions for a numerical

representation of magnitudes. This inquiry does need axioms. And, at the beginning of the

paper, Holder appeals to Helmholtz on this point, in support of his own conception of the

axioms of quantity (Holder 1901, 1, note). These are the following propositions:

QI. Given any two quantities a and b, one and only one of the following is true: (1) a is

identical with b (a = b and b = a); (2) a is greater than b and b is smaller than

a (a [ b and b \ a); (3) b is greater than a and a is smaller than b (b [ a and a \ b).

QII. For every quantity there exist one that is smaller.

QIII. Any two quantities a and b, also in case that a = b, when added in a definite order,

give a univocally determined sum a ? b.

QIV. a ? b is greater than a and greater than b.

QV. If a [ b, then there exist two quantities, x and y, such that a ? x = b and

y ? a = b.

QVI. (a ? b) ? c = a ? (b ? c).

QVII. If all quantities are divided into two classes such that (1) each quantity belongs to

exactly one class, and (2) each quantity of the first class is smaller than any quantity of

the second class, then there exist a quantity n, such that every quantity n0\ n belongs to

the first class and every quantity n00[ n belongs to the second class (n may belong to

either, depending upon the case).

QVII is an equivalent formulation of Dedekind’s axiom of continuity, which is not

deducible from Helmholtz’s axioms. Dedekind calls cuts all partitions of rational numbers

in two such classes, and proves that each rational number corresponds to one and only one

cut. He introduces irrational numbers by requiring that each cut correspond to one and only

one number (Dedekind 1872, 13). It is noteworthy that, except for QVI (i.e., the associative

law of addition), Holder’s axiomatization is completely different from Helmholtz’s. So it is

not easy to see the connection with Helmholtz. It may be helpful to illustrate Holder’s

theorem first. Holder uses QVII to prove a theorem which is equivalent to the Archimedean

axiom:

Given two quantities a and b, and a \ b, then there exist an integer number n, such that

na [ b.

This theorem, along with the theory of irrational numbers, enables Holder to develop a

complete theory of proportions. Suppose that a and b are magnitudes of the same kind and

that lm is called a lower fraction in relation to their ratio a:b, if ma [lb; and an upper

fraction, if ma B lb. By the Archimedean principle, there exist both a positive integer m,

such that ma [ b, and a positive integer l, such that a \lb. Thus, in relation to the ratio

a:b, there exist both lower and upper fractions, and in every case the lower fractions are

less than the upper ones. It follows from Dedekind’s continuity that:

For every ratio of quantities a:b, i.e., for each two quantities which are given in a

determined order, there exist a well-determined cut, i.e., a determined number in the

general sense of the word. (Holder 1901, 23)

Holder’s theorem justifies Newton’s requirement that the ratio of quantities of the same

kind be expressed by positive real numbers: since order and operations with cuts can be

developed arithmetically, the laws of addition apply to all quantities that satisfy the axiom

of continuity and the remaining axioms of quantity.

Space Can Be Transcendental Without the Axioms

123

In the second part of the paper, Holder presents a model of a non-Archimedean con-

tinuum. This is the most original part of the paper. So it is surprising that Holder

emphasizes the importance of the said theorem above all. Nevertheless, his emphasis is

interesting in the present context because it shows Helmholtz’s influence (see also Michell

1993). Holder notices that Newton’s requirement entails a more specific one, which cor-

responds to Helmholtz’s requirement that the composition of physical magnitudes be

expressed by the arithmetical sum of equal parts of the same. Therefore, Holder uses

Dedekind’s (Archimedean) continuity, which is necessary for physical measurements.

Holder’s proof that divisibility can be derived as a consequence of continuity entails

Helmholtz’s proof that magnitudes that can be added can also be divided into equal parts.

But there is also a methodological aspect of Holder’s approach that shows Helmholtz’s

influence. Though Holder’s theory of quantity provides us only with formal conditions of

measurement, he agrees with Helmholtz that arithmetic should be related to the theory of

quantity. Therefore, Holder adopts a non-formalistic approach (1901, 2, note). This point is

strictly related to Holder’s considerations on geometry in the inaugural lecture he gave in

Leipzig in 1899, ‘‘Anschauung und Denken in der Geometrie’’. The text of the lecture was

published in 1900 with Holder’s additional notes. His proof that both divisibility and the

Archimedean property can be derived from the axiom of continuity is first presented in this

essay, for the following reason. We already mentioned that geometry, according to Holder,

presupposes some given concepts. One of such concepts is that of a spatial magnitude.

Holder’s proof shows that given concepts can be constructed so that arithmetical reasoning

plays a role in geometry and arguably in physics. His approach corresponds to Helmholtz’s

Kant-oriented argument that the laws of arithmetic can be progressively extended to non-

numerical domains. The structure of inner experience is thereby proved to have a con-

stitutive function.

There is evidence that Holder might have seen his argument as a development con-

sistent with the Kantian philosophy of mathematics. This point is made explicit in his book

on mathematical method (1924). There Holder adopts Kant’s terminology and calls those

concepts whose development does not require assumptions other than the operations of our

thought, such as the concept of coordination, series, number, and group, purely synthetic.

Those concepts of geometry and mechanics which require special assumptions in order to

be constructed are called hypothetically synthetic. Holder rejects formalism because, if

those thoughts which may lead to the introduction of new symbols were represented

through another symbolic computation, there would be an infinite regress. Holder’s view is

that symbols represent concepts, and these can be constructed (1924, 5–6). Holder’s

argument is drawn from Paul Natorp, Kantian philosopher and master of Cassirer in

Marburg (see Natorp 1910, 5). This common reference to Natorp might explain why

Cassirer developed an argument similar to Holder’s in 1910 already. However, Holder does

not mention Cassirer, and, arguably, was not aware of his argument.

As in Helmholtz, the use of Kant’s terminology and Kantian arguments does not prevent

Holder from supporting geometrical empiricism. In 1899 he summarizes the different

views on geometrical intuition as follows. Kant’s view is that a pure intuition endowed

with subjective laws makes experience possible. By contrast, according to nineteenth-

century views, such as those developed by Julius Baumann and Wilhelm Wundt, geo-

metrical intuition can be induced by experience. Both views look at intuition as a source of

geometry and as an indispensable tool for proof. Since an indispensable role of intuition in

geometrical proof is excluded by the deductive character of geometrical inferences, it

might seem that intuition is necessary at least to provide us with specific objects. However,

it can be replaced by purely deductive inferences. We have already seen Holder’s example:

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the concept of magnitude can be constructed. Holder uses his methodological analysis to

defend Helmholtz’s philosophy of geometry. Helmholtz is said to avoid any commitment

to a supposedly intuitive character of geometry: he explains basic geometric concepts ‘‘in a

more physical way’’ (Holder 2013, 16). In order to avoid a vicious circle, such explanation

should be based on observations that do not presuppose geometry. For example, approx-

imately rigid bodies should be distinguished from non-rigid ones because only the former

can be easily brought back to their initial position after displacements.5 Helmholtz’s free

mobility can be derived from the fact that parts of two such bodies can be brought to

coincidence and that such experience can be repeated at any time. So Helmholtz’s ‘‘facts’’

underlying geometry are better understood as ‘‘rules’’ for inferring facts in any further

cases. For Holder the formulation of such rules requires a kind of ‘‘complicated induction’’

preceding both deduction and induction properly speaking (2013, 31). In this regard there

is no disagreement with Kant. Holder writes: ‘‘any single fact of experience, if expressed

by means of concepts—and how could one want to express it otherwise—is the result of a

mental elaboration of experience’’ (2013, 28). He argues against Kant’s philosophy of

geometry as follows:

It will no longer appear contradictory that, though we use [the concept of space] in

some cases in order to interpret experience, we nevertheless consider it possible to

check this concept—whose adequateness is hypothetical—for correspondence with

experience in order to reshape the concept, if necessary, as we do with physical

concepts. (Holder 2013, 46)

This argument entails that the use of the concept of space presupposes a physical theory.

The empiricist theory requires that assumptions about the geometrical part of the theory

have the same degree of probability as other (optical or mechanical) parts of it.

Holder does not discuss Helmholtz’s claim that space can be transcendental without the

axioms being so. Nevertheless, his methodological considerations suggest the following

interpretation. In the context of pre-relativistic physics, space can be supposed to have a

curvature that approximately equals zero. At the same time, the conjecture that space is

generally flat must be rejected: since geometrical inferences are deductive, it would nec-

essarily follow from this conjecture that Euclidean axioms are true. But these axioms are

only hypothetically true, and non-Euclidean hypotheses might also be considered. So there

must be a broader characterization of space that includes Euclidean geometry as a special

case. Helmholtz’s inquiry on the foundations of geometry provides us with such a char-

acteristic—namely, constant curvature. This follows from free mobility, which is Helm-

holtz’s rule for the interpretation of spatial measurements.

7 Concluding Remarks

Helmholtz argues that space is a three-fold extended manifold of constant curvature in his

inquiries on the foundations of geometry of 1868 and 1870, and there is no evidence that he

changes his mind in 1878: he only emphasizes that these characteristics are broader than

the specific axiomatic structures that distinguish the three classical cases of such manifolds

from one another. I do not think that there can be any doubt about which characteristics are

5 Holder 1924, 371; 2013, 17. In 1924 Holder recalls that Poincare proposes a similar explanation: ‘‘Parmiles objets qui nous entourent, il y en a qui eprouvent frequemment des deplacements susceptibles d’etre […]corriges par un mouvement correlatif de notre propre corps, ce sont le corps solides’’ (Poincare 1902, 79).

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said to be general. So I addressed another question: does Helmholtz’s claim that space can

be transcendental without the axioms being so admit a Kantian interpretation? If this claim

means that the Kantian theory is valid only for intuitive space, then the general charac-

teristics of space might differ considerably from those established by Helmholtz, because

they might be indescribable qualities or qualities describable only as topological properties.

I argued against this reading because it overlooks the fact that Helmholtz’s considerations

in his discussion with Krause and Land are about spatial magnitudes and physical space in

the first place. One might object that Helmholtz does not distinguish between intuitive and

physical space. Nevertheless, my reconstruction of that discussion shows that Helmholtz’s

purpose is to defend the reliability of scientific measurements. The required argument for

the Kantian theory of space is that its characteristics can be proved to have a constitutive

function—namely, to provide us with general conditions of measurement. My interpreta-

tion is that Helmholtz deems space ‘‘transcendental’’ because such proof corresponds to a

transcendental argument in Kant’s sense, and I interpret Helmholtz’s proof that topogenous

factors influence the course of real processes as one such argument.

Helmholtz’s analysis of measurement is completed in his paper of 1887. So it is no

accident that, in that paper, Helmholtz resumes his considerations on transcendental intu-

itions and reformulates his argument as follows: the objectivity of measurements presup-

poses an interaction between inner and outer experience. This is because, on the one hand,

mathematical structures can only be defined independently of the objects we experience; on

the other hand, generally valid judgments about magnitudes presuppose a physical inter-

pretation of the same structures. Helmholtz apparently believes that his attempt to explain the

psychological origin of these structures is part of his argument. In this sense, he maintains that

geometrical axioms are related to space as arithmetical axioms are related to the form of

intuition of time. My point is that these considerations are not necessary for Helmholtz’s

argument, which can be interpreted as a Kantian one. The argument is that some structures

provide us with general conditions of experience. In addition to my own interpretation, I

considered two alternative formulations. The first is provided by Cassirer: Helmholtz’s

argument presupposes not so much his psychological explanation of the ordinal conception

of number, as the logical development of arithmetic. Similarly, congruence and other rela-

tions between spatial magnitudes should be analyzed group-theoretically. Following a

similar line of thought, Holder develops a more precise version of the argument—namely,

that magnitudes can be constructed as hypothetic-synthetic concepts. Holder’s argument

entails both the Kantian view that mathematics is synthetic and an empiricist conception of its

(arguably progressive) applicability, which is characteristic of Helmholtz’s approach.

Acknowledgments An earlier version of the paper was presented at the Max Planck Institute for theHistory of Science. I thank Vincenzo De Risi and his research team for their feedback on that occasion. Ialso wish to thank my colleague and friend Giovanni Gellera for stylistic suggestions. I am grateful to twoanonymous referees for their constructive criticisms.

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