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INFINITESIMALS AS AN ISSUE OF NEO-KANTIAN

PHILOSOPHY OF SCIENCE

THOMAS MORMANN AND MIKHAIL G. KATZ

Abstract. We seek to elucidate the philosophical context in which
one of the most important conceptual transformations of modern
mathematics took place, namely the so-called revolution in rigor
in infinitesimal calculus and mathematical analysis. Some of the
protagonists of the said revolution were Cauchy, Cantor, Dedekind,
and Weierstrass. The dominant current of philosophy in Germany
at the time was neo-Kantianism. Among its various currents, the
Marburg school (Cohen, Natorp, Cassirer, and others) was the one
most interested in matters scientific and mathematical. Our main
thesis is that Marburg neo-Kantian philosophy formulated a so-
phisticated position towards the problems raised by the concepts
of limits and infinitesimals. The Marburg school neither clung to
the traditional approach of logically and metaphysically dubious in-
finitesimals, nor whiggishly subscribed to the new orthodoxy of the
“great triumvirate” of Cantor, Dedekind, and Weierstrass that de-
clared infinitesimals conceptus nongrati in mathematical discourse.
Rather, following Cohen’s lead, the Marburg philosophers sought
to clarify Leibniz’s principle of continuity, and to exploit it in mak-
ing sense of infinitesimals and related concepts.

Keywords: Infinitesimals; Marburg neo-Kantianism; princi-
ple of continuity; Cantor-Dedekind-Weierstrass; Hermann Cohen;
Cassirer; Natorp; Leibniz.

Contents

1. Introduction 2
2. Neo-Kantian Philosophy of Science and Mathematics 4
2.1. The Transcendental Method 5
2.2. Concepts in Mathematics and Science 7
2.3. Substantive versus Relational Concepts 9
3. Two Guiding Metaphors of neo-Kantian philosophy of

science 13
3.1. Natorp’s Knowledge Equation 13

2000 Mathematics Subject Classification. Primary 01A60, Secondary 03A05,
01A85, 26E35 .

1

http://arxiv.org/abs/1304.1027v1


2 THOMAS MORMANN AND MIKHAIL G. KATZ

3.2. Cassirer’s Convergent Series 15
3.3. Idealisations, completions, and infinitesimals 17
4. Three attempts to make sense of Cohen 19
4.1. The Point of departure: Cohen’s Prinzip 20
4.2. Cassirer’s Leibniz’ System 24
4.3. The Continuity principle 26
4.4. Gawronsky’s The Judgment of Reality 32
4.5. Natorp’s The Logical Foundations of the Exact Sciences 37
5. From infinitesimals to functional concepts 42
Acknowledgments 47
References 47

1. Introduction

The traditional historical narrative concerning infinitesimals runs as
follows. The idea of infinitesimals has been with us since antiquity.
Mathematicians have used one or another variety of infinitesimals or
indivisibles without really understanding what they were doing. Even-
tually, infinitesimals fell into disrepute for logical and philosophical
reasons, as enunciated by Berkeley and others.

Despite Berkeley’s devastating criticism, mathematicians continued
to use them until the 19th century with more or less good intellectual
conscience. Finally (according to the traditional narrative) Cauchy,
followed by Cantor, Dedekind, and Weierstrass, succeeded in formulat-
ing a rigorous foundation for the calculus in terms of the epsilon-delta
approach. Thereupon infinitesimals were “officially” expelled from the
realm of legitimate mathematics once and for all. Or so it seemed.

This traditional narrative is, however, seriously incomplete. Some
80 years after mathematics had allegedly dismissed “infinitely small
magnitudes” and related concepts as pseudo-concepts once and for all,
in 1960 the mathematician Abraham Robinson claimed to have saved
infinitesmals from the bin of pseudo-concepts. Thereby, an improved
version of the traditional narrative goes, Robinson restored the repu-
tation of infinitesimals as legitimate mathematical entities by means of
his non-standard analysis. From the perspectives of mathematics and
history of mathematics, such a completed version of the traditional nar-
rative is certainly to be considered as an improvement. Nevertheless it
still suffers from serious shortcomings. For centuries, the infinitesimal
and related concepts were discussed not only by mathematicians and
scientists but also by philosophers like Leibniz, Newton, Malebranche,
and Berkeley. While the thought of these classic 17th and 18th century



INFINITESIMALS AS AN ISSUE OF NEO-KANTIAN PHILOSOPHY 3

authors has been extensively studied by historians of philosophy and
mathematics, not much is known about the 19th century philosoph-
ical context in which the “great revolution in rigor” and the alleged
dismissal of infinitesimals took place. This is hardly acceptable. A
philosophically satisfying account has to take into consideration the
historical fact that the “great revolution in rigor” in mathematical
analysis, led by the German mathematicians Cantor, Dedekind, and
Weierstrass (henceforth, CDW), took place when philosophy in Ger-
many was dominated by various currents of neo-Kantian philosophy.
Some of the neo-Kantian philosophers had a keen interest in science
and mathematics. Indeed, the issue of the infinitesimal was vigorously
debated in neo-Kantian quarters, as we will discuss in detail in this
text.

Our main thesis is that the Marburg neo-Kantians elaborated a philo-
sophically sophisticated approach towards the problems raised by the
concepts of limits and infinitesimals. They neither clung to the obsolete
traditional approach of logically and metaphysically dubious infinites-
imals,1 nor whiggishly subscribed to the new orthodoxy of the “great
triumvirate” (Cantor, Dedekind, Weierstrass) that insisted on the elim-
ination of infinitesimals from any respectable mathematical discourse
in favor of a new approach based on the epsilontic doctrine. Instead,
the Marburg school developed a complex array of sophisticated, al-
beit not always crystal-clear, positions that sought to make sense of
both infinitesimals and limit concepts. With the hindsight enabled
by Robinson’s non-standard analysis, the Marburg stance seems wiser
than that of Russell, Carnap, and Quine who unconditionally accepted
the orthodox epsilontic doctrine, along with its simplistic philosophical
ramifications stemming from a strawman characterisation of infinitesi-
mals as a pseudo-concept.

The outline of this paper is as follows. In Section 2, we recall the
basics of the Marburg school’s neo-Kantian philosophy of science and
mathematics. In particular, we dwell on some crucial features of the
Marburg account that distinguishes it from the Kantian orthodoxy.
We also recall the basic ingredients of Cassirer’s philosophy of science

1Actually, there are good reasons to contend that the infinitesimals of the tra-
ditional approach were neither logically nor metaphysically dubious, though they
were attacked as such by George Berkeley. Sherry [63] dissected Berkeley’s criticism
(of Leibnizian infinitesimal calculus) into its logical and metaphysical components,
and a closer inspection thereof reveals that the Leibnizian system for differential
calculus was both more firmly grounded than the Berkeleyan criticism thereof, and
free of logical contradictions (Katz & Sherry 2012 [38]), (Katz & Sherry 2013 [39]),
(Sherry & Katz 2013 [64]).



4 THOMAS MORMANN AND MIKHAIL G. KATZ

as a theory of the formation of scientific concepts, pointing out the
salient differences between ‘Aristotelian’ substantial concepts of com-
mon sense and the functional or relational concepts of modern science.
In Section 3, we consider two neo-Kantian attempts to elucidate the
progressive conceptual evolution in science with the help of mathemat-
ical metaphors, to wit, Natorp’s equational metaphor and Cassirer’s
Cauchy metaphor. In Section 4, we discuss several attempts by the
members of the Marburg school to elucidate some of Cohen’s notori-
ously obscure theses on the “essence” of the infinitesimal. The issue of
the closing Section 5 is Cassirer’s road from infinitesimal to functional
concepts in his mature philosophy of science and mathematics.

2. Neo-Kantian Philosophy of Science and Mathematics

To set the stage, let us review some historical and philosophical back-
ground. Following the collapse of German idealism after Hegel’s death
in 1831, German philosophy once again returned to Kant (cf. Coffa 1991
[12]). Such a reorientation did not, however, result in a new Kantian
orthodoxy. Rather, the emerging neo-Kantian philosophy, subscribing
to the maxim “With Kant Beyond Kant” (Otto Liebmann 1865 [45])
adopted some of Kant’s ideas and at the same time came to criticize
the master. The most important currents of neo-Kantianism were the
so-called Marburg school founded by Hermann Cohen, and the South-
west or Baden school founded by Wilhelm Windelband and Heinrich
Rickert. With some oversimplification one may say that the Southwest
school was mainly interested in matters of Geisteswissenschaften, while
the members of the Marburg school were mainly engaged in the task
of a philosophical understanding of mathematics and the sciences. We
will therefore concentrate on the neo-Kantian doctrines of the Mar-
burg school and its contributions to a philosophical understanding of
the problems posed by infinitesimals, limits, and related concepts.

Baldly characterizing an account in epistemology or philosophy of
science as neo-Kantian may suggest that such an approach is rather
similar to Kant’s. This would be an error. All neo-Kantians agreed that
Kant’s philosophy was a promising starting point for modern episte-
mology and philosophy of science, but not a doctrine that had to be
followed literally. Not surprisingly, they vigorously disagreed concern-
ing the best way to go “beyond Kant”. In this paper, however, we
will not dwell on the issue of whether or not the Marburg neo-Kantian
interpretation of Kant did true justice to Kant (cf. Friedman 2000 [23]
and M. Kühn 2010 [40]), as our main topic is neo-Kantian rather than
Kantian philosophy of science and mathematics.



INFINITESIMALS AS AN ISSUE OF NEO-KANTIAN PHILOSOPHY 5

An authoritative survey of the essence of the Marburg neo-Kantianism
was published by Natorp on the occasion of Cohen’s seventieth birth-
day in 1912, in the prestigious journal Kant-Studien. The article Kant
und die Marburger Schule [54] can be considered as a kind of official
position paper of the Marburg school. More precisely, Natorp sought to
present the Marburg current as the true heir, although not an epigone,
of Kant’s original philosophy. This endeavor had two parts. On the
one hand, he emphasized the salient differences among the Marburg
school’s neo-Kantianism, Kantian orthodoxy, and rival contemporary
philosophical currents such as the Baden school and neo-Hegelianism;
on the other hand, he pointed out that the Marburg school preserved
the true essence of Kant’s doctrines.

2.1. The Transcendental Method. Natorp emphasized that for the
Marburg school, the true core of Kant’s philosophy was the transcen-
dental method ([54, p. 194f]).2 Everything in Kant’s system that did
not fit well with this method had to be given up by true Kantians.
The transcendental method3 deals with the problem of the possibility
of scientific experience. More precisely, pursuing the ‘transcendental
method’ as the universal method of philosophy is contingent upon two
requirements:

The first is a solid contact with the established facts of
science, ethics, arts, and religion. Philosophy cannot
breathe in empty space of pure thought, where reason
aims to fly high only on the wings of speculative ideas.
. . . The place of philosophy . . . is the fertile lowlands of
experience in a broad sense, i.e., it seeks to take roots in
the entire creative work of culture (science, politics, art,
religion) . . .

The second, decisive requirement of the transcenden-
tal method is to provide for these cultural facts (sci-
ence, ethics, art, . . . ) their conditions of possibility. In
other words, philosophy, by following the transcenden-
tal method, has to exhibit and to elaborate the lawful4

2Whether or not the ‘transcendental method’ à la Natorp was also the core of
Kant’s philosophy need not concern us here. Indeed, many renowned Kant scholars
deny this.

3The method was called “transcendental” since it went beyond the cognition
that is immediately directed onto the objects. A more detailed discussion appears
in the main text at footnote 8.

4The original German “gesetzmässig” is a key term of neo-Kantian philosophy
of science and difficult to translate. Its meanings range from order-generating to
exhibiting regularity.



6 THOMAS MORMANN AND MIKHAIL G. KATZ

ground (“Gesetzesgrund”) or logos of those creative acts
of culture (Natorp [54, p. 197]).

Restricting our attention to the cultural fact of science, we may re-
state Natorp’s thesis in more modern terms by saying that philosophy
of science in the transcendental mode has the task of rationally recon-
structing the evolution of scientific knowledge and the conditions of its
possibility.

The task of revealing the conditions of the possibility of scientific
experience binds the Marburg ‘transcendental method’ to Kant’s orig-
inal ‘transcendental logic’, which, by definition, investigates how it is
possible that our concepts are related to real objects. More precisely,
transcendental logic in Kant’s sense is concerned with the origin, the
content, and the limits of experiential knowledge.

Pursuing the transcendental method, the “critical idealism” of the
Marburg school is led to a genetic epistemology and theory of science
that regards the ongoing process of scientific creativity as its essen-
tial feature, more so than its temporary results. Natorp put it as
follows: Knowledge is always in the state of “becoming”, it is never
“closed” or “finished”. Something non-conceptually “given”, in par-
ticular something allegedly intuitively “given” cannot be accepted as
such. A “given” is just another name for a problem to be solved.5

In other words, for the philosophers of the Marburg school the fact
of science was a “fact of becoming” (Werdefaktum). Accordingly, the
basic task of a truly “Kantian” philosophy of science was to make ex-
plicit the method of science as “the method of an infinite and unending
creative evolution of reason” (Natorp [54, p. 200]).

The rejection of a non-conceptual given in any form brought the Mar-
burg neo-Kantianism in open conflict with one of the cornerstones of
Kant’s epistemology, to wit, the dualism of conceptual understanding
and intuition, as a “non-conceptually given”. Indeed, as the Marburg
neo-Kantians argued against Kant’s original position, if one really fol-
lows the transcendental method in its true sense, then

it is virtually impossible, as Kant does, to maintain this
dualism of epistemic factors if one takes seriously the
core idea of the transcendental method. ([54], 201)

5As a then popular neo-Kantian pun put it: An object is not gegeben but
aufgegeben. This pun loses its effectiveness in English: it claims that an object
is not “given” (gegeben) but “posed” (aufgegeben) as a problem. The Russian
equivalent dan/zadan appears in a recent collection of essays on neo-Kantianism,
where it is mentioned in an essay by Sebastian Luft as translated by N. Dmitrieva
[30, p. 121], and in an essay by T. B. Dlugach [30, p. 224].



INFINITESIMALS AS AN ISSUE OF NEO-KANTIAN PHILOSOPHY 7

For the Marburg neo-Kantians, in contrast to Kant, both the Kan-
tian categories and his forms of sensibility of space and time were purely
conceptual. Kant’s sharp separation of understanding and sensibility
as two complementary faculties of the mind had to be given up.

Since Kant’s original transcendental logic as a logic of the possibility
of experience was closely related to the forms of sensibility, the Marburg
neo-Kantians were led to give up the distinction between formal (de-
ductive) logic and transcendental logic. For them, there was only one
logic, namely, the logic of the transcendental method as the compre-
hensive logic of the conditions of the possibility of scientific experience
(cf. Heis 2010 [33, p. 389]).

2.2. Concepts in Mathematics and Science. The emphasis on the
evolving character of scientific knowledge gave the issue of the evolu-
tion of scientific concepts a central position in the Marburg philoso-
phy of science. In particular, to Cassirer, this meant that philosophy
of science had to investigate the common evolution of scientific and
mathematical concepts firmly planted in the course of their historical
development. From the Marburg perspective, these two conceptual de-
velopments were two aspects of the same problem. In Substanzbegriff
und Funktionsbegriff 1910 [7] (henceforth, SF ) Cassirer wrote:

[We should consider] physical concepts no longer for them-
selves but, as it were, in their natural genealogy, in con-
nection with the mathematical concepts. In fact, the
physical concepts only carry forward the process that is
begun in the mathematical concepts, and which here [in
mathematics–the authors] gains full clarity. The mean-
ing of the mathematical concept cannot be comprehended,
as long as we seek any sort of presentational correlate for
it in the given; the meaning only appears when we rec-
ognize the concept as the expression of a pure relation,
upon which rests the unity and continuous connection of
the members of a manifold. The function of a physical
concept also is first evident in this interpretation. (SF
1910 [7, pp. 219-220]; p. 166 in the 1953 edition)

Note that Cassirer employed terms such as “continuous”, “connec-
tion”, and “manifold” not only in their strict mathematical sense, but
also in a metaphorical sense. Such usage of philosophical and scien-
tific concepts was typical of Cassirer’s thought throughout his entire
philosophical career (see Orth 1996 [55]). Following the Husserl scholar
Eugen Fink, Orth refers to concepts used in such a “metaphorical” way,
as “operative concepts” as opposed to “thematic concepts”. Operative



8 THOMAS MORMANN AND MIKHAIL G. KATZ

concepts are concepts based on “intellectual schemata” without being
fully explicit. They serve as “orientations in a conceptual field” and
are of a metaphorical character. As Orth points out, Cassirer had at
his disposal a particularly rich supply of operative concepts stemming
from the (Kantian) philosophical tradition and from contemporary sci-
ence, in particular physics and mathematics (cf. Orth, ibid. 111).
Continuity and its relatives were among Cassirer’s favored operative
concepts.

In contrast to many currents of contemporary philosophy of science,
the philosophers of the Marburg school regarded philosophy of the sci-
ences and philosophy of mathematics as being of the same ilk, namely,
as a study of relational concepts. It may be considered as a pleasing
confirmation of Cassirer’s unified approach to mathematical and phys-
ical concepts that one of the great mathematicians of the 20th century,
Hermann Weyl, subscribed to a similar view, quite independently of
Cassirer and apparently unaware of the similarity of convictions.

Weyl sought to overcome the deficiencies of a purely formal concep-
tion of mathematics such as Hilbert’s without being forced to build
mathematics on a restricted base of a Brouwerian intuitionism. Weyl
therefore proposed to seek help from physics. More precisely, he consid-
ered theoretical physics as the guiding example of a kind of knowledge
endowed with a meaning completely dfferent from that of the common
sense or phenomenal meaning. Thus, in order to endow the symbols of
mathematics with a meaning, Weyl saw only one possibility:

. . . [to] completely fuse mathematics with physics and as-
sume that the mathematical concepts of number, func-
tion, etc. (or Hilbert’s symbols) generally partake in the
theoretical construction of reality in the same way as the
concepts of energy, gravitation, electron, etc. (Weyl 1925
[66, p. 30])

Weyl clung to this thesis of the essential similarity of conceptual-
ization in mathematics and the sciences till the end of his life. The
conclusion he reached in his late essay A Half Century of Mathematics
is fully in line with the neo-Kantian approach:

It is pretty clear that our theory of the physical world is
not a description of the phenomena as we perceive them,
but [rather] is a bold symbolic construction. However,
one may be surprised to learn that even mathematics
shares this character. (Weyl 1951 [67, p. 553])

This agreement between Weyl and the Marburg school is all the more
remarkable since Weyl arrived at it from a rather different philosophical



INFINITESIMALS AS AN ISSUE OF NEO-KANTIAN PHILOSOPHY 9

background: he was influenced mainly by Husserl’s phenomenology,
and never had shown in his mature age any affinity to Kantian or neo-
Kantian philosophy.

2.3. Substantive versus Relational Concepts. After these general
remarks on the Marburg account of mathematical and scientific con-
cepts, it may be expedient to consider some concrete examples in detail.
Thereby we can hope to illustrate what the Marburg philosophers, in
particular Cassirer, intended to convey by their thesis of the relational
(or functional) character of scientific concepts. Let us start with an
elementary example. At first view, which is sometimes called the Aris-
totelian one, there appears to be a close analogy between common-sense
concepts such as rock and mathematical concepts such as number or
triangle, in that the concept rock corresponds to the class of all empir-
ical entities that are rocks, i.e. the class of all entities that have all the
properties a rock is assumed to have; and similarly, the mathematical
concept number is said to correspond to the class of all mathematical
“entities” that are numbers, i.e. the class of entities that have all the
properties numbers are assumed to have.

Cassirer rejected such an analogy. According to him, the unity of
mathematical and scientific concepts was not to be found in any fixed
group of properties, but rather in the rules,6 which represented, in a
lawful way, the mere diversity of objects that “fall under the concepts”
as their cases (i.e., instantiations). Elementary examples of relational
scientific concepts in this sense are mathematical formulas that de-
scribe arithmetic series (i.e., sequences) such as 1, 3, 6, 10, . . .. For such
a series, the “construction of unity” is provided by a formula that de-
scribes their generation according to some general law. For instance,
the series 1, 3, 6, 10, . . . is characterized by the law that the difference
of the differences of its consecutive members is always 1. This fact is
succinctly expressed by the formula a(n) = n(n + 1)/2, n ∈ N. The
members of such a series do not have a common property (in any or-
dinary sense of property) but appear as cases of a common functional
law.

More generally, Cassirer considered the formulas of mathematics,
physics, and chemistry as paradigmatic examples of relational scien-
tific concepts since they brought singular facts into a lawful context.
Algebraic equations of geometric curves provide somewhat less elemen-
tary examples. Such equations can be used to describe the movement of

6Mathematically speaking, the dichotomy is between properties, i.e., unary rela-
tions, and binary (and higher) relations.



10 THOMAS MORMANN AND MIKHAIL G. KATZ

material bodies. More precisely, they are conceptual devices for embed-
ding the individual perceived positions of a body in a continuous, even,
smooth trajectory. Continuity, smoothness and other concepts of the
infinitesimal calculus, are, however, highly theoretical ‘ideal’ concepts.
The embedding of singular data into a continuous or smooth trajectory
is anchored in a complex web of crucial idealizing assumptions.

For the Marburg neo-Kantians, the indispensable role of idealizations
such as continuity and smoothness for modern science demonstrated
that the real could be understood only through the ideal. To Co-
hen, this translated into a statement that the infinitesimal was a core
concept of any truly modern logic of science. Cohen appears to have
assumed that the notion of the infinitesimal necessarily underlies the
concepts of continuity and smoothness (which is technically speaking
not the case from the viewpoint of modern ǫ, δ definitions of continuity
and smoothness). In the opening chapter Infinitesimal-Analysis of his
Logik der reinen Erkenntniss he explicitly contended:

If logic is to be a logic of science, i.e., a logic of the math-
ematized natural sciences, then it must be primarily the
logic of the principle of the infinitesimal. If this is not
the case and this core principle does not occupy cen-
tre stage, then logic itself still hasn’t gained its proper
centre, it still belongs to the past. The new scientific
thought is that which since Galileo, Leibniz, and New-
ton has become systematically efficient [and for which
the infinitesimal does play a fundamental role–the au-
thors]. (Cohen 1902 [14, p. 31])

According to Cohen, the Leibnizian principle of continuity was the
key that had opened the gate toward such a truly modern logic of the
infinitesimal (see Section 4). Regrettably, however, later generations
of philosophers and scientists had not faithfully followed Leibniz’ lead.
Therefore, a logic of the infinitesimal was still in its infancy. It was
incumbent upon the Marburg school to develop the foundations of a
working logic of the infinitesimal.

A paradigmatic example of a relational concept in physics was for
Cassirer the concept of energy. The utility of the concept of energy is
not to describe any new class of objects, alongside the already known
physical objects such as light and heat, electricity and magnetism.
Rather, it signifies only an objective lawful correlation, in which all
these “objects” stand. The meaning of the concept of energy resides
in the equations that it establishes among different kinds of events and
processes. Energy in the sense of modern science is not an object in



INFINITESIMALS AS AN ISSUE OF NEO-KANTIAN PHILOSOPHY 11

the traditional sense, but a unifying perspective that sheds light on a
manifold of experiences.

This is rendered most evident by the functional identity of potential
and kinetic energy through which states are identified with temporal
processes:

The two [moments of kinetic and potential energy] are
“the same” not because they share any objective prop-
erty, but because they occur as members of the same
causal equation, and thus can be substituted for each
other from the standpoint of pure magnitude (SF 1910
[7, p. 264-265], (1953, p. 199))

Energy cannot be understood as the conceptual counterpart of some-
thing empirical out there. Rather, it is to be understood as an order-
generating principle. In this respect it resembles the notion of num-
ber by which we make the sensuous manifold unitary and uniform in
conception (cf. SF (1910, 252), (1953, 189)). In contrast to the con-
cept of number, the concept of energy is a genuine concept of the
empirical sciences. Hence, since “number” and “energy” both served
as order-generating principles in essentially the same manner, this was
considered as another argument in favor of the Marburg thesis that
mathematics and mathematized empirical sciences followed the same
rules of one and the same transcendental logic. The concept of en-
ergy shows that in modern science the allegedly objective “things” of
common sense and traditional metaphysics are replaced by a web of
mathematically formulated relations that yield objectivity to scientific
knowledge. Thereby the notorious Kantian “things-in-themselves” can
be dispensed with:

We need, not the objectivity of absolute things, but rather
the objective determinateness of the method of experi-
ence. [SF, (1910, 428), (1953, 322)]

Characterizing scientific knowledge by idealizing functional relations
reveals that it does not aim at a description of how the world “really
is”. The concepts of modern science are not the mental images of cer-
tain pre-existing objects; rather they are tools that offer new unifying
perspectives as he elaborated in full detail in his magnum opus Philoso-
phy of Symbolic Forms (Cassirer 1923–1929, 1953–1957 [9]) (henceforth
PSF), in particular in the third volume that takes up many issues of
SF.

Ideal gases, ideal fluids, etc. are not limiting cases approximated
by the more or less homogenous gases or the more or less ideal fluids
found in nature. Rather, idealizing concepts such as perfect gases or



12 THOMAS MORMANN AND MIKHAIL G. KATZ

perfect fluids have an epistemological role. They provide conceptual
perspectives that allow the formulation of general relational laws and
thereby they help to make sense of reality as a manifold of experiences.

Cassirer described this theoretical unification of the scattered data
of sensations as an embedding of an incomplete empirical manifold of
sensations in a completed conceptual manifold. Typically, such embed-
dings can be carried out in a variety of ways. In contrast to Kant, for
the neo-Kantian Cassirer there were no fixed forms that determined
how this process was to be carried out. Rather, the ever-growing va-
riety of conceptual completions of our experiences is revealed in the
historical evolution of science itself. For Cassirer, the paradigmatic
example of such a conceptual completion was Dedekind’s completion
of the rational numbers Q to the real numbers R. The essential point
of this completion was not that some “ideal” numbers were “added”
to the already existing rational numbers, but that the relational sys-
tem R of real numbers provided us with a new conceptual perspective
to “see” more clearly the conceptual essence of the rational numbers Q
themselves (cf. PSF III, 392).

Although the processes of concept formation in mathematics and
physics are similar, they are not identical. After all, there is a differ-
ence between mathematics and physics, and philosophy of science has
the task of elucidating this difference. Roughly, Cassirer considered
conceptualization in mathematics as a simplified version of conceptu-
alization in physics:

In contrast to the mathematical concept, however, in em-
pirical science the characteristic difference emerges that
the construction which within mathematics arrives at
a fixed end,7 remains in principle incompletable within
experience. But no matter how many “strata” of rela-
tions we may superimpose on each other, and however
close we may come to all particular circumstances of the
real process, nevertheless there is always the possibil-
ity that some relevant factor in the total result has not

7Here Cassirer appears to express the view that in mathematics the construction
arrives at a fixed end. This reading seems hardly compatible with his general neo-
Kantian outlook according to which the essence of science resides in its unending
evolution. A more plausible reading of Cassirer here would be that he contended
that mathematical concepts are relatively fixed with respect to empirical concepts,
just as the elements of an infinite convergent series of numbers are fixed although
the series itself may not reach a fixed limit point in finitely many steps.



INFINITESIMALS AS AN ISSUE OF NEO-KANTIAN PHILOSOPHY 13

been calculated and will only be discovered with the fur-
ther progress of experimental analysis (Cassirer 1910 [7,
p. 337], (1953, 254)).

Factual and theoretical components of scientific knowledge cannot
be neatly separated. In a scientific theory “real” and “non-real” com-
ponents are inextricably interwoven. Not a single concept is confronted
with reality but a whole system of concepts.

3. Two Guiding Metaphors of neo-Kantian philosophy of
science

In line with the essence of the transcendental method that conceived
of science “as an infinite and unending creative evolution of reason”,
the philosophers of the Marburg school considered empirical or mathe-
matical concepts (or theories as systems of concepts) as stages in an on-
going process of an unending conceptual approximation. Accordingly,
the task of philosophy of science was to investigate the conditions of
possibility for such an evolution. Remarkably, for this endeavor the
Marburg neo-Kantians heavily relied on that science whose concep-
tual evolution they sought to elucidate, to wit, mathematics. In other
words, both Natorp and Cassirer, each in his own way, sought to tap
the resources of mathematics to elucidate the structure of the concep-
tual evolution of the sciences. To this end, they introduced certain
mathematical metaphors, by exploiting the mathematical concepts of
approximation and convergence for which the concept of the infinite
played an essential role.

3.1. Natorp’s Knowledge Equation. Perhaps the best-known of
these metaphors in the hightime of neo-Kantian philosophy was Na-
torp’s “equational metaphor” that compared the evolution of science
with the solution of a numerical equation. According to it, coming to
know an object (“Erkenntnisobjekt”) was analogous to the process of
solving a numerical equation. To be specific, the reader may keep in
mind a specific equation such as x3 − x2 + x − 1 = 0.

In line with Natorp’s didactic intentions this equation has been cho-
sen to convey several ideas concerning knowledge and its objects. First,
the fact that it has several different solutions indicates that the process
of research may not lead to unique results. Furthermore, the fact that
two of its solutions are imaginary reflects the fact that the research
process may lead to an expansion of the original fundamental concepts
one started with. Note that the admission of complex numbers as
solutions transcends the conceptual space in which the equation was
formulated, since its coefficients are all integers. What is still missing



14 THOMAS MORMANN AND MIKHAIL G. KATZ

in this metaphor is the “infinite character” of the knowledge equation.
Natorp was aware of this shortcoming and tried to remedy it (see be-
low).

According to Natorp, the object of knowledge may be considered as
an “unknown x of the knowledge equation”:

If the object is to be the x of the equation of inquiry,
then it must be possible to determine the meaning of
this x by the nature of this equation (i.e., the inquiry
itself) in relation to its known factors (our fundamental
concepts). From this it must follow whether and in what
sense the solution of this problem is possible for us. This
is the very idea of the transcendental or critical method
(Natorp 1903 [52, p. 10]).

Natorp added the following further elucidations. The transcendental
method did not aim to extend our knowledge beyond the limits of the
scientific method. Rather, it sought to clarify the limits of scientific
knowledge. It was called “transcendental” since it went beyond the
cognition that is immediately directed onto the objects, but aimed
to obtain information about the general direction of the path to be
taken.8 It did not provide us with any specific knowledge about an
object beyond experience. Hence, following the established Kantian
terminology it was transcendental, but not transcendent.

Both Natorp and all his fellow philosophers of the Marburg school
viewed the object of knowledge, not as an unproblematic starting point
of the ongoing process of scientific investigation, but rather as its limit.
This object was a problem to be solved. In its various versions, this
equational account of knowledge can be found in virtually all of Na-
torp’s epistemological writings. One might object that Natorp’s equa-
tional model of scientific cognition is far too simple in the sense that the
empirical objects hardly ever show up as solutions of a finite equation
such as the one considered above. It seems hardly plausible that physi-
cal entities such as “proton” or “quark” fit in the conceptual framework
of one physical theory without remainder.

Natorp did not ignore this difficulty and complemented his equa-
tional account so as to counter this objection. Elaborating the equa-
tional model, he pointed out that the object of knowledge was not
simply a problem (“aufgegeben”) but an infinite problem that could

8Here we elaborate further on the term “transcendental” as discussed in foot-
note 3.



INFINITESIMALS AS AN ISSUE OF NEO-KANTIAN PHILOSOPHY 15

be solved in finite time only approximately by finite creatures like our-
selves. He thus sought to elude the trap of an overstated Hegelian
rationalism:

Although we conceive of the object of knowledge (= x),
similarly9 as Hegel does, only in relation to the functions
of knowledge itself, and consider it . . . as the x of the
equation of knowledge, . . . we have understood that this
“equation” is of such a kind that it leads to an infinite cal-
culation. This means that the x is never fully determined
by the parameters a, b, c . . . of the equation. Moreover,
the sequence of the parameters . . . is to be thought of
as being not “closed” but rather extendable further and
further. (Natorp 1912 [54, pp. 211-212])

3.2. Cassirer’s Convergent Series. As a second example of the us-
age of mathematical metaphors for the elucidation of the meaning of
philosophical ideas, let us now have a closer look at how Cassirer con-
ceptualized the guiding idea of the Marburg school, to the effect that
the evolution of scientific knowledge could be understood as a concep-
tual approximation process. In contrast to Natorp, Cassirer insisted
that this approximation was not assumed to converge to an externally
given limit, but rather as defined by a general internal rule, as is done
in the arithmetic of rational numbers. To give a mathematical model
for such a conception of the evolution of science, we can characterize a
rational sequence such as 3.1, 3.14, 3.141, 3.1415, . . . as being conver-
gent, obviously without relying on an assumed existence of an element
in Q to which it would converge.

Cassirer used this elementary mathematical insight to illustrate his
thesis that one may meaningfully speak of the convergence of scien-
tific theories without assuming that there is a fixed reality “out there”
to which the sequence of our theories is expected to converge. To be
specific, consider a sequence of positive numbers such as (1/n) that
converges to 0. This may apparently suggest that a convergent series10

of concepts (or theories) converges to some ultimate external entity
(or “reality”), just as the arithmetical series (1/n) converges to the
real number 0 external to it, with 0 itself not being a member of the

9A common criticism of the Marburg school’s epistemology was that it was dan-
gerously close to an overstated Hegelian rationalism. Hence Natorp, although he
had to admit some similarity with Hegel, was at pains to distance the Marburg
neo-Kantianism from any sort of Hegelian rationalism.

10Cassirer used the term equivalent to “series” for the mathematical entity usu-
ally referred to as a “sequence”. We have retained Cassirer’s terminology.



16 THOMAS MORMANN AND MIKHAIL G. KATZ

series (1/n). Cassirer vigorously rejected such a realist “exterior” in-
terpretation:

The system (Zusammenhang) and the convergence of the
series take the place of an external standard of reality.
Both system and convergence can be established and de-
termined, analogously to arithmetic, entirely by compar-
ison of the serial members and by the general rule, which
they follow in their progress. (Cassirer 1910 [7, p. 426],
(1953, 321))

As already 19th century mathematics had taught us, in order to be
able to speak meaningfully about a convergent sequence of numbers,
it is not necessary to assume that there “really” is a number to which
the sequence converges. Rather, an arithmetical series can be defined
as convergent if it satisfies an internal requirement that can be formu-
lated without reference to a possibly inexistent external limit. Such an
internal requirement is provided by Cauchy’s criterion.

Mathematicians have pushed this “internalization” of the concept of
convergence even further. As is well known, the “gappiness” of the
rational numbers Q (residing in the absence of limit points of certain
Cauchy-convergent sequences) may be overcome by completing Q in
an appropiate way. More precisely, one can embed the rationals Q
into a set C(Q) of appropriately defined equivalence classes of Cauchy
sequences. Thereby it can be ensured that in the new completed
realm C(Q) that englobes the rational numbers Q as a part, every
Cauchy sequence has a limit point. From this viewpoint, a real num-
ber is an encapsulation, or reification, of the concept of convergence of
a Cauchy sequence.

Cassirer took these mathematical constructions to be more than mere
technicalities. He considered them as the pattern for his internally
defined account of the continuous evolution of scientific knowledge:

No single astronomical system, the Copernican any more
than the Ptolemaic, can be taken as the expression of the
true cosmic order, but only the totality of these systems
as they unfold continuously according to a definite con-
nection. (Cassirer 1910 [7, p. 427], (SF, 322))

In other words, for Cassirer the “true cosmic order” was not given
by a single theory but by a convergent series of theories. He did not
assert that our theories ontologically converge to a mind-independent
realm of substantial things as the substrate of a “final” theory. His
notion of theoretical convergence was epistemological rather than on-
tological. He understood the approximation of theories essentially as



INFINITESIMALS AS AN ISSUE OF NEO-KANTIAN PHILOSOPHY 17

an epistemological progress within which the historical progression of
our theories continually approximates, but never reaches, any ideally
complete mathematical representation of the phenomena. Such an ideal
representation is not waiting “out there” to be approximated; rather,
it resides in the reification of the approximation process that comes
into being through this very process itself. In other words, in the
course of its history science completes itself, so to speak, analogously
as a convergent sequence without limit gives rise to a corresponding
sequence with limit point in a suitably completed domain. For Cas-
sirer, the paradigmatic example of such a completion was Dedekind’s
completion of the rationals to the real numbers. But for him, the sig-
nificance of Dedekind’s construction went beyond developing a more
abstract version of unending decimal expansions. According to him,
idealizing completions were the essence of the modern empirical and
mathematical sciences (cf. Mormann 2008 [50]).

3.3. Idealisations, completions, and infinitesimals. The “completion-
friendly” perspective of the Marburg school on the conceptual evolution
of science had important consequences for matters infinitesimal. With
respect to completions of number systems by infinitesimals, the thesis
of the “incompletability” of the conceptual evolution in science and
mathematics suggested that Cassirer’s account (and that of Marburg
neo-Kantianism in general) had no built-in source of resistance to fur-
ther ontological extensions beyond the rational and the real numbers.
On the contrary, according to its own rules, the neo-Kantian approach
would have welcomed the advent of the hyperreals à la Edwin Hewitt
[35] and Abraham Robinson [58, 59] and related developments. As we
shall see in the following sections, the insistence on the openness of
the evolution of mathematical and physical concepts brought the Mar-
burg philosophers in conflict with mathematicians and logicians such
as Cantor and Russell, who adhered to more realist accounts that con-
sidered concepts as more or less direct descriptions of what there is
(in the empirical or in an ideal realm), instead of conceiving of them
as epistemological tools for progressing in the task of making sense of
some aspects of the world. According to Cassirer, both empiricism



18 THOMAS MORMANN AND MIKHAIL G. KATZ

à la Bacon and Berkeley, and naive realism11 with respect to “ideal”
mathematical entities, à la Cantor and Russell, must be rejected:

For the existence of the ideal, which can alone be criti-
cally affirmed and advocated, means nothing more than
the objective logical necessity of idealization. (SF 1910
[7, p. 170], (1953, 129))

And again:

The relation beween the theoretical and factual elements
at the basis of physics cannot be described in this simple
way. It is a much more complex relation, it is a peculiar
interweaving and mutual interpenetration of these two
elements, that prevails in the actual structure of science
and calls for clearer expression logically of the relation
between principle and fact. (Cassirer (1910, 172), (1953,
130))

According to Cassirer, no scientific theory directly relates to the facts
of perception. Rather, such a theory relates to the ideal limits, which
we substitute for the facts of perception. Thus, we investigate the
impact of bodies by regarding the masses, which affect each other, as
perfectly elastic or inelastic. We study perfect fluids even though no
such are to be found. In other words, Cassirer sought to present his
“Critical Idealism” as a theoretical framework that overcame both a
naive empiricism and a misled platonist idealism.

For the Marburg neo-Kantians, who had always emphasized the es-
sential unity of mathematics and empirical science, the new relational
logic (Frege, Peano, Russell and others; see e.g., Gillies [28]) was part
and parcel of a single comprehensive transcendental logic of science
that was emerging in the course of the history of science. For them,
it was a fundamental philosophical mistake of the ‘logicism’ of Frege
and Russell to ignore its transcendental character in conceiving of it
as a purely formal device neatly separated from the empirical realm.

11“Realism” is taken here as an unsaturated term, i.e., questions of realism arise
with respect to a certain subject matter, e.g., realism with respect to atoms, values,
mathematical objects, possible worlds, causality, or macroscopic material objects.
Cantor and Russell were realists with respect to mathematical entities. Other
names for this sort of realism are “platonism”, “platonist realism”, or even “pla-
tonist idealism”. Cassirer, a self-proclaimed “critical idealist” was not a partisan
of platonist idealism. On the contrary, he vigorously criticized it.



INFINITESIMALS AS AN ISSUE OF NEO-KANTIAN PHILOSOPHY 19

For the neo-Kantians, this was just another example of a philosoph-
ically untenable dualism, analogous to the Kantian dualism between
the conceptual and the sensual.12

One of Cassirer’s criticisms of Russell’s philosophy of logic was that
it succumbed to a naive “platonizing idealism” (see also J. Heis [33,
p. 386]) since it insisted on a strict separation between the logico-
mathematical conceptual realm, on the one hand, and the empirical
realm, on the other. According to Cassirer, this stance expressed itself
in an outdated dualistic metaphysics that was bound to lead into un-
solvable, self-inflicted pseudo-problems (cf. SF 1910 [7, pp. 313, 359],
(1959, pp. 237, 271)). For instance, the applicability of mathematics in
the mathematized natural sciences became an unfathomable mystery.13

4. Three attempts to make sense of Cohen

The philosophers of Marburg neo-Kantianism considered themselves
consciously as members of a well-defined school under the leadership of
Hermann Cohen as the school’s founder. Even when they held differing
opinions concerning a philosophical question, they sought to minimize
their differences vis-à-vis outsiders. An important example of such
“school discipline” concerns the concept of infinitesimals and how it
was dealt with in the treatises of the school’s leader Cohen.

Even for sympathetic readers, it is often difficult to make sense of Co-
hen’s writings. Such a difficulty is not limited to contemporary readers
accustomed to doing philosophy in a more analytic style. Already the
young Husserl in 1886 complained in a letter to Brentano that Cohen’s
allegedly “scientific philosophy” was actually nothing but “nonsensical
profundity” or “profound nonsense” (see Mohanty 2008 [47, p. 3]). As
evidence for this strong claim Husserl took Cohen’s theory of the prin-
ciple of continuity; see Das Prinzip der Infinitesimalmethode und seine
Geschichte (The Principle of the infinitesimal method and its history)
(henceforth Prinzip) 1883 [13, §40ff].

We argue that Husserl’s radical verdict was not entirely justified.
For this purpose we will rely on three different attempts to make sense
of Cohen’s “scientific philosophy” undertaken by three members of the
Marburg school, namely Ernst Cassirer, Dimitry Gawronsky, and Paul
Natorp. The works involved are Cassirer’s Leibniz’ System (1902), Sub-
stanzbegriff and Funktionsbegriff, SF (1910), Gawronsky’s Das Urteil

12See Section 2.
13Perhaps Wigner put forward the most influential plea for this attitude in terms

of “the unreasonable effectiveness of mathematics in the natural sciences” (cf. [68]
and [29]).



20 THOMAS MORMANN AND MIKHAIL G. KATZ

der Realität (1910), and Natorp’s Die logischen Grundlagen der exak-
ten Wissenschaft (1910). As we will see, it reveals that the evolving
thought of the Marburg school about infinitesimals and related con-
cepts was not monolithic and without cracks. This is not surprising,
since Cohen’s Prinzip and Cassirer’s SF are separated in time by al-
most thirty years, not to mention the rather different technical styles
and the scientific background of their authors. Thus, Cohen’s edu-
cation in logic did not correspond to the state of the art at the be-
ginning of the 20th century. He apparently never took proper notice
of Frege, Russell, or any other contemporary logician. On the other
hand, Cassirer, Gawronsky, and Natorp were aware at least partially of
the new developments in logic and mathematics, and sought to adapt
the Marburg school’s philosophical stance to the new circumstances
(cf. Cassirer 1907 [6]).

4.1. The Point of departure: Cohen’s Prinzip. Cohen treated
the issue of the infinitesimal first in Das Prinzip der Infinitesimalmeth-
ode und seine Geschichte and later, in Logik der reinen Erkenntnis.
The Logik may be considered as a continuation and philosophical elab-
oration of Prinzip. The most significant difference is that the last traces
of any sort of Kantian ‘intuition’ in the constitution of infinitesimals
are eliminated. Pure thought,14 and pure thought alone, takes care of
matters infinitesimal. Cohen even went so far as to contend that the
infinitesimal was to be considered as the most important and most
typical issue of pure thought, überhaupt.15 In his Logik, Cohen appears
to assume that the reader has read and digested the argumentation of
Prinzip. That is to say, mathematically there is nothing new in Logik
that cannot be found already (usually more fully elaborated) in Prinzip.
The main purpose of Logik is rather to explicate Cohen’s account of
the philosophical or metaphysical presuppositions and ramifications of
pure thought, centering on the notion of the infinitesimal.

In Prinzip, Cohen still sought to establish the existence of infinites-
imals with the aid of an intuition in the sense of Kant, while in Logik
any vestige of Kantian intuition had vanished. According to mature
Marburg neo-Kantianism, intuitions played no role in scientific knowl-
edge. This must not be misunderstood: Cohen’s target was Kantian

14The expression pure thought (“reines Denken”) was the technical term de-
signed by the neo-Kantians as the successor concept of the two separate components
of Kantian epistemology, to wit, “concepts” and “intuitions”.

15According to Cohen transcendental logic was the logic of the infinitesimal
(see Subsection 2.3). Infinitesimals provided the key example of how the real was
clarified by the ideal.



INFINITESIMALS AS AN ISSUE OF NEO-KANTIAN PHILOSOPHY 21

intuition, not intuition in the everyday understanding of the term. Re-
call that Kant distinguished between two basic components of human
representations, to wit, concepts and intuitions. These correspond to
two essentially different cognitive faculties: understanding and sensibil-
ity. Only if these faculties are united can knowledge be achieved. The
neo-Kantians argued against this two-tiered Kantian epistemology, not
against the common-sense claim that for the individual scientist, “in-
tuitions” may play an important role in the context of discovery.

Though this may not be crystal-clear already from the first pages of
Prinzip, in later works, in particular in Logik, Cohen repeatedly em-
phasized this difference between orthodox Kantianism and Marburg
neo-Kantianism. The rejection of the significance of “Kantian intu-
ition” for scientific knowledge was, as is well-known, a common trait
of all members of the school. Hence, critizising Cohen for relating the
concept of the infinitesimal to some sort of Kantian intuition would
be a gross misunderstanding. Another more subtle, albeit quite com-
mon misconception would be to ascribe to Cohen the claim that he
sought to locate the “problem of the infinitesimal” in the realm of a
psychologistically conceived epistemology (Erkenntnistheorie).

Already in Prinzip, Cohen had insisted that the problem of the in-
finitesimal could be properly treated only in what he referred to as the
logic of science a.k.a. transcendental logic. According to him “the logic
of science must be the logic of the principle of infinitesimal calculus”
(cf. Logik, already quoted in Section 2.3). He was eager to point out
that a glance at the literature revealed that the logic of his day had
not yet recognized the decisive logical significance of the infinitesimal
principle. In other words, and to give it a more personal twist, in
Logik he admitted that his Prinzip had not yet found the recognition
it deserved.

This situation did not change in the ensuing decades. Thus, in the
otherwise rather comprehensive survey by Haaparanta, The Relation
between Logic and Philosophy, 1874 – 1931 (Haaparanta 2009 [31]),
Cohen’s and, more generally, the Marburg account of logic as tran-
scendental logic of science was completely ignored.

Cohen blamed Kant, to some extent, for this myopic conception of
logic. According to him, Kant had failed properly to understand the
role of the infinitesimal for a true critique of pure reason. Instead,
he introduced pure sensitivity as a second ingredient of knowledge,
whereby the independence of pure thought had been compromised (cf.
Cohen (1902) p. 32)).

Let us now take a closer look at Cohen’s general conception of logic.
Cohen’s conception incorporates both modern and obsolete ideas in a



22 THOMAS MORMANN AND MIKHAIL G. KATZ

peculiar mixture. The first thing to note is that for Cohen logic is inde-
pendent of anything else: logic is neither a branch of psychology nor a
branch of linguistics. The laws of logic, i.e., the laws of “pure thought”
are neither psychological laws nor grammatical laws. Rather, accord-
ing to Cohen, the laws of logic constitute the core of pure thought.
Thoughts in Cohen’s sense should not be confused with “sensations”
(Empfindungen) or individual mental representations (Vorstellungen).
Cohen vigorously rejected any dependence of logic on another realm.
For him, logic described the activity of pure thought that took place in
the ongoing conceptual evolution of science: “The thought of logic is
the thought of science. Thought constitutes the foundation of being”
(Cohen 1902 [14, pp. 17, 18]).

Cohen’s conception of logic is a far cry from any modern post-Fregean
or post-Russellian conception of logic. There are no axioms, inferential
rules or anything of that sort. Rather, his logic follows, at least super-
ficially, quite closely the patterns of a Kantian (or even pre-Kantian)
Urteilslogik (judgment logic). Thereby Cohen’s Logik reveals a some-
what paradoxical relation to Kant. On the one hand, it can be read
as a definitive parting of ways with Kantian orthodoxy, in particular
by giving up the basic structure of the Kantian philosophical system,
namely the distinction between the two pillars of pure logic and pure
sensitivity. On the other hand, Cohen formally followed the architec-
tonics of the transcendental logic of the Critique of Pure Reason when
he mimicked in his Logik Kant’s system of categories and judgments.
In close analogy to Kant’s pair of 4×3 schemata of categories and judg-
ments, Cohen set up the following 4 × 3 schema distinguishing between
four classes of judgments each consisting of three types of judgments:

• The judgments of laws of thought
(Origin, Identity, Contradiction)

• The judgments of mathematics
(Reality, Majority, Totality)

• The judgments of mathematized natural sciences
(Substance, Law, Concept)

• The judgments of methodology
(Possibility, Contingency, Necessity)

For the purposes of the present text, it is unnecessary to dwell upon
Cohen’s schema in full detail. But the following remarks may be in
order. Superficially, Cohen’s and Kant’s tables of judgments are quite
similar. Both exhibit the 4 × 3 schema, and, moreover, Cohen’s “judg-
ments of methodology” and virtually identical with Kant’s “judgments
of modality”. Behind this formal similarity, however, are lurking deep



INFINITESIMALS AS AN ISSUE OF NEO-KANTIAN PHILOSOPHY 23

conceptual differences. The first is that Kant’s schema is based on
what he called general logic, based on the standard formal logic of
his time. In contrast, Cohen’s table is deeply soaked with contentious
assumptions of his account of transcendental logic. Moreover, it is
not concerned with knowledge in general, but with knowledge of the
mathematized natural sciences. In this respect, Cohen’s epistemologi-
cal perspective was considerably narrower than Kant’s. Perhaps even
more important is the difference between Kant and Cohen concern-
ing the problem of how to conceive of the relation between the table
of judgments and the corresponding table of categories. In Cohen’s
Logik, there is nothing that even remotely resembles Kant’s famous
transcendental deduction of the categories.

Kant invested immense efforts in the task of this deduction that
resulted in a strict and rigid 1-1 correspondence between the 4 × 3
items of the table of judgments and the 4 × 3 items of the resulting a
priori categories. In contrast, Cohen was content with the vague as-
sertion that there was a mutual “correlation” between judgments and
categories relying on the bland metaphorical explication that “the cat-
egory is the aim of the judgment, and the judgment is the road to
the category” (Logik, p. 47). In particular he gave up the 1-1 corre-
spondence between categories and judgments, and allowed that every
category might be contained in several judgments and every judgment
might be contained in several categories. Actually, this vagueness and
indecision is no co-incidence. Kant’s categories were intended to be
valid a priori, once and for all. There was no change or evolution in
the categorical schema that Kant had set up in the Kritik. In contrast,
Cohen repeatedly emphasized the evolving character (“Werdecharak-
ter”) of scientific knowledge, for instance, when he contended that the
“truly creative elements of scientific thought reveal themselves in the
history of scientific thought” (Logik, 46), but, obviously, this dynamic
character of scientific knowledge was hardly compatible with a Kantian
schema of fixed a priori categories.

For contemporary philosophy of science, Cohen’s half-baked proposal
of how to reconcile the categorical structure of scientific knowledge and
its historical character can be of historical interest at best.

Nevertheless one may note that Cohen’s problem, as we may call it,
has remained on the agenda of virtually all accounts of philosophy of
science that have been inspired by Kant up to this very day, as exem-
plified, for instance, by Reichenbach’s reformulation of the Kantian a
priori in the 1920s, up to Michael Friedman’s neo-neo-Kantian Dynam-
ics of Reason (Friedman 1999 [22]) where the author seeks to reconcile



24 THOMAS MORMANN AND MIKHAIL G. KATZ

historical and the a priori aspects of scientific knowledge as a synthesis
of ideas taken from Kuhn, Cassirer, and Carnap.

As was already mentioned at the beginning of this section, we do
not aim at an exhaustive treatment of Cohen’s logic of judgments.
Instead, we aim to shed some light on Cohen’s often obscure analyses by
consulting the writings of other members of the Marburg school, to wit,
Ernst Cassirer, Dimitry Gawronsky, and Paul Natorp. These authors
may be helpful in elucidating their master’s thoughts, as all of them
intended not to deviate from Cohen’s ideas unless absolutely necessary.
This does not mean, of course, that they actually provided faithful and
accurate interpretations of Cohen’s account. Nonetheless, the works of
these three philosophers can be read as sympathetic readings that try
to make the best out of Cohen.

4.2. Cassirer’s Leibniz’ System. Cassirer’s first philosophical works
were his 1899 dissertation Descartes’ Kritik der Mathematischen und
Naturwissenschaftlichen Erkenntnis16 and his Leibniz’ System in seinen
wissenschaftlichen Grundlagen (Cassirer 1902 [5]).

Cassirer eventually published his Descartes and Leibniz texts to-
gether as one book proposing that Descartes may be conceived of as a
forerunner of Leibniz. More precisely, according to Cassirer, Descartes
and Leibniz may be considered as two stations of the long and wind-
ing road toward an idealistic conception of science. Provisionally, this
was achieved in Kant’s account; after Kant it found its contemporary
expression in the philosophy of science, or the scientific philosophy,
of the Marburg school. Cassirer’s Leibniz’ System is a 400 page long
text. In its ten chapters Cassirer seeks to treat Leibniz’ philosophi-
cal and scientific achievements within logic, mathematics, mechanics
and metaphysics. The latter is understood in a broad sense, including
Leibniz’ reflections on issues such as “the problem of consciousness”,
“the problem of the individual”, and “the concept of the individual
in the system of Geisteswissenschaften.” By far the largest (and for
our purposes most interesting) chapter is the fourth, dealing with The
Problem of Continuity. It comprises not less than 70 pages. In this
section we will mainly concentrate on this chapter of Leibniz’ System.
Leibniz’ System is engaged in the ambitious task of presenting the

conceptual evolution of modern science by presenting the achievements
of the two geniuses of Descartes and Leibniz. The starting point was
Descartes’ overcoming of the medieval conception of science. According
to Cassirer’s fundamental thesis, the philosophical systems of these
men could not be understood by separating them from their scientific

16Descartes’ Kritik was published as the first part of Leibniz’ System.



INFINITESIMALS AS AN ISSUE OF NEO-KANTIAN PHILOSOPHY 25

achievements, to wit, analytical geometry in the case of Descartes, and
the infinitesimal calculus in the case of Leibniz:

Through the discovery of analytic geometry Descartes
lays the foundations for the modern way of scientific
thinking, which finds its mature expression in the in-
finitesimal calculus. . . . The synthesis of philosophy and
science which is carried out thereby must not be con-
ceived of as a mere juxtaposition. . . . One must attempt
to identify a common basis of these thoughts (Cassirer
1902 [5, pp. 1-2]).

Identifying such a common base will lead to a more profound under-
standing of the role of Descartes’ system in the historical evolution of
critical epistemological idealism and its continuation and completion
in Leibniz and Kant and, one may add, in the idealism of the Marburg
school. So much for Descartes. In the remainder of this section, we will
concentrate on Cassirer’s study of Leibniz as one of the most important
early sources for the philosophy of science of the Marburg school.

According to Cassirer, for Leibniz mathematics was primarily an in-
strument of scientific research and a presupposition for the discovery
of a new concept of nature, rather than an aim in itself (Cassirer 1902,
99). This became fully evident through the “new mathematics”, i.e.,
the infinitesimal calculus. Following Cohen, Cassirer contended that
for Leibniz’ thought the concept of the infinitesimal was to be consid-
ered of fundamental importance, not only with respect to mathematics,
but much more generally, also for Leibniz’ philosophical understand-
ing of the mathematized empirical sciences, a new concept of nature
in general, and his metaphysics in general. Indeed, Leibniz’ System
is to be considered only as Cassirer’s first attempt to contribute to
this overall programme of a genuine Marburg philosophy of science in-
augurated by Cohen’s Das Prinzip der Infinitesimalmethode. While
in Leibniz’ System, Cassirer concentrated on the historical figures of
Leibniz and Descartes, a few years later, he widened his perspective.
In his monumental Das Erkenntnisproblem in der Neuzeit (The Prob-
lem of Knowledge) (Cassirer (1906–1950)) he became engaged in the
huge project of writing a comprehensive intellectual history of ideas
(Ideengeschichte) of Western thought of the modern period that he
pursued during his entire lifetime in various forms: The last volume of
Das Erkenntnisproblem, which eventually comprised four bulky tomes,
was published only postumously in 1950.



26 THOMAS MORMANN AND MIKHAIL G. KATZ

Although in Leibniz’ System young Cassirer still followed Cohen’s
lead in emphasizing the crucial importance of the infinitesimal for mod-
ern science and mathematics, the reader may notice an inclination to-
ward relativizing its central role. This tendency gained momentum
in later works such as Das Erkenntnisproblem (Cassirer 1906 - 1950),
Substance and Function (Cassirer 1910 [7]), and The Philosophy of
Symbolic Forms (Cassirer 1923 - 29). But already in Leibniz’ System
we find the sweeping thesis that the central concept of modern science
is a concept of function17 although it is not made clear how this asser-
tion fits with the alleged primacy of the concept of the infinitesimal.
These tensions became more evident in Substance and Function. Cas-
sirer’s failure to toe the party-line on the primacy of the concept of
the infinitesimal over the concept of function was explicitly noted by
Cohen; see Section 5.

For Cassirer the really modern character of Leibniz’ thought was
encapsulated in the thesis that “the real is conditioned by the ideal.”
According to the Marburg idealism, this thesis was the key that opened
the possibility of a truly modern philosophy of science, mathematics,
and logic:

From this perspective we can really understand Leibniz’
tendency to equate logic and mathematics in its true
significance. This equation does not aim to constrict
the rich content of mathematics in the form of tradi-
tional logic. Rather, it intends to bring about funda-
mental reformation for logic. Instead of being a theory of
“thought forms”, logic is to become a science of objective
knowledge (gegenständliche Erkenntnis). This transfor-
mation is essentially due to its relation to mathematics:
Mathematics turns out to be the necessary mediation
between the ideal logical principles and the reality of na-
ture. (Cassirer 1902 [5, p. 113])

For Cassirer and Cohen, Leibniz was the one who opened the gate
for an idealist “transcendental logic” that later in the hands of Kant
and the Marburg neo-Kantians was to become a “transcendental logic
of objective knowledge”.

4.3. The Continuity principle. The essential means for overcoming
the traditional Aristotelian conception of logic as a theory of abstract
“thought forms” toward a contentful theory of objective knowledge was

17Here Cassirer is not referring to the notion of function in a narrow mathemati-
cal sense, but rather to the functional (or relational) account of science according to
which entities are secondary and functions or relations primary (see Subsection 2.2).



INFINITESIMALS AS AN ISSUE OF NEO-KANTIAN PHILOSOPHY 27

said to be Leibniz’ famous continuity principle. The continuity princi-
ple was considered as the most important device to unfold the general
thesis that “the real is conditioned by the ideal”, mentioned in Subsec-
tion 4.2. Consequently, the bulk of Leibniz’ System was dedicated to
the task of explicating this principle.

It should be kept in mind that for Cassirer and the Marburg philoso-
phers in general, continuity was not a property that some things (or
processes) have and others do not. Cassirer described its significance
in the following terms:18

For the philosophy before Leibniz, continuity was essen-
tially nothing but the property of a thing or an attribute
of a ready-made concept. When it was understood in this
way, one could attempt to refute or to prove the claim
that a certain thing or concept possessed it or lacked
it. This holds true of the synechés of the Eleates till
Descartes’ concept of continuous space. Leibniz over-
comes this stance. For him, the problem of continuity
dissolves in the problem of “continuation”. Continuity is
no longer a characteristic of a thing, but rather that of a
development; not of a concept, but of a method. (ibid.,
153).

Therefore, it would perhaps have been more appropriate to refer to
this principle as the principle of continuation rather than the princi-
ple of continuity. In order to be understood as a general conceptual
achievement, continuation needs to be elaborated in the framework of
a scientific methodology (wissenschaftliches Verfahren). At this point
the infinitesimal and related concepts enter the stage. As Cassirer was
eager to point out, the method of continuation first obtained its deeper
“scientific” meaning in the domain of geometry,

where it designates the transition from point to line, from
the line to the area and so on.19 Similarly, in mechanics
and dynamics the method of continuation describes the

18Note that here Cassirer is using the concept of “continuity” as an “operative
concept”, as explained in Subsection 2.2.

19From a purely mathematical viewpoint, Cassirer’s formulation is a bit unfor-
tunate here because it sounds as if one is still dealing with indivisibles rather than
infinitesimals. The difference between them is that indivisibles were thought of as
codimenion-1 entities whereas infinitesimals were of the same dimension as the fig-
ure composed of them. This was the content of the major advance as accomplished
by Roberval, Torricelli, Wallis, Leibniz, and others, as compared to earlier work by
Archimedes and Cavalieri.



28 THOMAS MORMANN AND MIKHAIL G. KATZ

relation of the material point to the structures of higher
dimension.

In its true scientific generality the relation between an
element and the structure that results from its contin-
uation corresponds to the relation of a differential and
its integral. In other words, the “continuation” is the
methodical expression of the integration as a continuous
summation of infinitesimal moments. (ibid., 153 - 154)

To a modern reader, Cassirer’s “logical” thesis may sound utterly
“metaphysical”. Cassirer himself recognized that this general charac-
terization of the relation between the infinitesimal and the real was in
need of further clarification and scrutiny. For this task he proposed
to have a closer look at Leibniz’ foundations of the infinitesimal calcu-
lus. According to Cassirer, the key to understanding the true novelty
of Leibniz’ account resided in the Leibnizian concept of motion. The
crucial point was not to view motion as an empirical concept stemming
from the realm of empirical experience. For Leibniz, motion was always
continuous motion, i.e., the expression of a unifying principle of con-
ceptual construction. Leibnizian motion was not something empirically
given, but something conceptually constructed (cf. ibid., 156). Invok-
ing the idealist principle that “the real is clarified by the ideal”, still
another way of expressing this may be that, the concept of continuity,
in Leibniz’s view, belongs to the realm of the ideal.

Probably the most famous expression of this view can be found in
Leibniz’s letter to Varignon from which Cassirer quoted in his Leibniz’
System (ibid., p. 188/189) and elsewhere:

. . . one can say in general that, though continuity is
something ideal and there is never anything in nature
with perfectly uniform parts, the real in turn, never
ceases to be governed perfectly by the ideal and the
abstract . . . (Leibniz 1702 [42])

Later, in the same letter, Leibniz explicitly stated that not only
continuity but also infinitesimals have the capacity to “govern the real
perfectly”:

So it can be said that infinites and infinitesimals are
grounded in such a way that everything in geometry,
even in nature, takes place af if they were perfect reali-
ties. (ibid.)

On the other hand, in the very same letter he asserted:



INFINITESIMALS AS AN ISSUE OF NEO-KANTIAN PHILOSOPHY 29

I’m not myself persuaded that it is necessary to con-
sider our infinities and infinitesimals as something other
than ideal things (choses ideales) or wellfounded fictions
(fictions bien fondées). (ibid.)

It is far from clear, however, how precisely the relation between “ideal
things” and “(wellfounded) fictions” is to be thought and how infinites-
imals as “fictions” could have this power of governing perfectly. This
difficulty was already observed by Cassirer, who even contended that
Leibniz’s notion of a “fiction bien fondée” had an air of paradox (Cas-
sirer 1902 [5, p. 187ff]). For a contemporary survey of the debate
concerning this issue of Leibnizian scholarship the reader may consult
(Sherry and Katz 2013 [64]).

Cassirer himself contended that the doctrines of the Marburg school,
in particular his account of the role of idealizations in science and
mathematics, might help to overcome the remaining obscurities that
still beset Leibniz’s account. According to him, for this endeavor it
was essential to properly understand Leibniz’ concept of motion that
underlied his dynamical conception of geometry. From the viewpoint
of classical Euclidean geometry, one may suspect that introducing the
concept of motion into geometry amounts to an illicit confusion of pure
geometry and empirical science. Cassirer vigorously argued against this
interpretation:

It is not a systematical infringement to integrate the con-
cept of motion into geometry. The concept that is dealt
with here, is not from physics, but from logic: It de-
notes the conceptual continuation of the “principle” that
was expressed in the concept of continuation. Thereby
the concept of motion is separated from its empirical
context and allocated in the area of pure and eternal
“forms”. . . . The general achievement of the concept of
motion resides in the formulation of the thought that
the extensional being has to be constituted from an orig-
inal lawful determination that preceeds it as its logical
prius.20 (Cassirer 1902 [5, pp. 156-157])

In other words, motion in the realm of science is always lawful mo-
tion. The purely conceptual character of the concept of motion, its
non-extensionality is shown by the concept of the differential (ibid.
157). This is not to say that Cassirer was not aware of the existence

20A prius is something that comes before or preceeds something else in some
respect. The term was used in the philosophical jargon of the 19th century. Today
it seems to be an outmoded Latinism.



30 THOMAS MORMANN AND MIKHAIL G. KATZ

of non-continuous and perhaps even continuous but non-differentiable
functions which had been vigorously discussed among mathematicians
since Cauchy in the 1820s (cf. Hankel 1882 [32]). Indeed, he explicitly
pointed out that a more general notion of a function à la Dirichlet,
perfectly made sense from a purely mathematical point of view.21 He
only objected that such general functions were not meaningful for the
determination of real processes of nature (cf. 217). He went on to de-
clare that the meaningfulness of the limiting processes of the calculus
“demonstrated” that the concept of (lawful) motion in nature was a
non-empirical, logical notion:

If the transition to the quantitative zero does not elim-
inate the lawful character of the magnitude this is evi-
dence that it (i.e., lawfulness) is not grounded in a quan-
titative principle. The magnitude must first disappear
from our sensual perception before we can recognize its
determinateness in the pure concept (Cassirer 1902, 157).

Here Cassirer does not distinguish between continuity and differentia-
bility. As we shall see in a moment, this conflation enabled him to com-
bine Cohen’s “infinitesimal-centered” account with his own “function-
oriented” one in an elegant but somewhat dubious way.

For the contemporary reader this passage may sound opaque, to put
it mildly. We propose the following interpretation. Scientifically mean-
ingful magnitudes obey continuous motions, i.e., motions the law of
which could be described by a differentiable function f. In calculat-

ing the derivative of f, expressions such as lim
f(x)−f(x′)

x−x′
occur. These

contain “quantitative zeroes” (if x′ approximated x). This means that
such expressions have no direct quantitative meaning. In particular
they could not be perceived or experienced in any reasonable way.
Nevertheless, conceptually, the calculation of the derivative makes per-
fect sense. Hence, the magnitude could be recognized as a meaningful
and determinate magnitude, only after it had been submitted to a con-
ceptual process (derivation) that eliminated all its sensual qualities.
Continuity was thereby asserted to be a necessary presupposition for
the constitution of nature as a possible object of rational investigation:

Continuity is a necessary presupposition for the existence
of a mutually 1-1 relation between two series of change
(Veränderungsreihen). This strand of thought is first for-
mulated in Leibniz’ best known formulation of the prin-
ciple of continuity: “Datis ordinatis etiam quaesita sunt

21Compare footnote 8.



INFINITESIMALS AS AN ISSUE OF NEO-KANTIAN PHILOSOPHY 31

ordinata”. The “data” denote the hypothetical condi-
tions from which we start; the “quaesita” are the series
of the conditioned that we look for. The order is thought
as a law that determines the transition inside the two se-
ries in a continuous fashion. (ibid., 211 - 212)

Cassirer went on to contend that the standard “epsilon-delta” defini-
tion of continuity was merely a mild reformulation of Leibniz’ original
characterization of continuity “Datis ordinatis etiam quaesita sunt or-
dinata” (cf. ibid., 215). Thereby he could conclude that there was
an intimate relation between Leibniz’ continuity principle, the mod-
ern epsilon-delta definition of continuity, and Cohen’s infinitesimal-
centered account. Modern mathematics has shown that the relation is
more complicated than Cassirer might have thought.

Nevertheless, despite its allegedly close relation to the modern math-
ematical concepts, according to Cassirer, Leibniz’ continuity principle
should not be understood as a mere mathematical definition. Here
Cassirer is treading on somewhat dangerous ground. Leibniz’s law of
continuity had several meanings. It was not a single concept but rather
a family of concepts; see for example (Jorgensen 2009 [36]). The prob-
lem is not merely the fact that the mathematical concept of continuity
is not general enough to encompass Leibniz’s concept. Rather, the law
of continuity invokes several related concepts in a chain where the con-
cepts at the extrema may be unrelated to each other at all. Perhaps
one may say that Leibniz used this concept as an ‘operative’ one (see
Subsection 2.2).

It would be a misunderstanding to read it simply as the claim that
the processes of nature should be viewed as ready-made entities that
were to be described in terms of continuous (or differentiable) functions:

The requirement of conceiving of nature ultimately as a
complex of continuous functions (Inbegriff stetiger Funk-
tionen) would not make sense, if the task of knowledge
would be to reproduce a ready-made material descrip-
tively. Continuity obtains its meaning only if it is con-
ceived of as a basic act of the mind through which the
subject conditions the object. (ibid., 218)

The principle of continuity should be understood as a guiding maxim
for the evolution of scientific concepts that urges us to seek ever more
profound systematic connections among them. Concepts should be
connected in a uniform conceptual system and each concept should be
transformable into each other continuously. For the Marburg school
Leibniz’ principle of continuity was the philosophical expression of one



32 THOMAS MORMANN AND MIKHAIL G. KATZ

of the basic moments of modern science. It asserted that the well-
defined and determinate character of scientific concepts did not reside
in their isolation but in the lawfulness of their transitions (cf. (ibid.
201)).

Although Cassirer, faithfully following in the footsteps of his mas-
ter Cohen, hailed Leibniz as the genius who made explicit for the first
time the principle of continuity as a fundamental principle of modern
science, already in Leibniz’ System his assessment of Leibniz’s achieve-
ments went in directions other than that of Cohen. The emerging differ-
ences between Cohen and Cassirer concerned the relation between the
concepts of the infinitesimal and function. While Cohen emphasized
that the “infinitesimal calculus (of Leibniz) had placed the concept of
function, conceived of as a law of interdependency between two variable
magnitudes, in the center of the methodology of mathematics” (Cohen
1902 [14, p. 239]), Cassirer put less emphasis on the role of the infini-
tesimal as a conceptual base for the concept of function. He generally
praised Leibniz as an early partisan of a “functional” or “relational”
worldview without mentioning infinitesimals at all:

If one understands by “substantial” worldview the con-
ception according to which all beings and occurrences can
be traced back to ultimate, rigid, absolute “things”, then
Leibniz’ philosophy is strictly opposed to this standpoint.
The tendency of Leibniz’ philosophy that from now on
will prevail in the ongoing progress of idealism points at a
replacement of the older concept of being by the concept
of function. (ibid., 486) [emphasis added–the authors]

In the evolution of Cassirer’s own thought, this functional interpre-
tation of the principle of continuity gained ever greater momentum
and superseded the infinitesimal interpretation eventually leading to
certain discrepancies with Cohen that surfaced in Cohen’s letter to
Cassirer dating august 24, 1910 (see Section 5).

4.4. Gawronsky’s The Judgment of Reality. The second sus-
tained effort to make sense of Cohen’s approach is due to Dimitry
Gawronsky (1883 - 1955). His dissertation under Cohen and Natorp
was entitled Das Urteil der Realität (Gawronsky 1910 [24]). Although
he was a close friend of Cassirer’s, in the emerging discrepancies be-
tween Cohen and Cassirer on the relation between Cohen’s ‘infinitesi-
mal’ and Cassirer’s ‘functional’ approach, he sought to find a mediating
position between the two but eventually sided rather with Cohen than
Cassirer. For this issue, two works of Gawronsky are relevant. Besides
his already mentioned dissertation Das Urteil der Realität (henceforth



INFINITESIMALS AS AN ISSUE OF NEO-KANTIAN PHILOSOPHY 33

Urteil), we have also his contribution Das Kontinuitätsprinzip bei Pon-
celet (Gawronsky 1912 [25]) to a Festschrift dedicated to Cohen on the
occasion of his 70th anniversary in 1912. Today, Gawronsky’s philo-
sophical work has fallen into almost complete oblivion. Yet he was
an important figure in the internal debate that took place within the
Marburg school on matters infinitesimal between Cassirer, Natorp, and
Cohen in the early years of the 20th century.

Gawronsky took upon himself the difficult task of updating Cohen’s
infinitesimal account, defending it against the less than orthodox ac-
counts of Cassirer and Natorp. Probably his best known work among
Cassirer scholars is the biographical article Ernst Cassirer: His Life
and His Work (Gawronsky 1949 [26]) that appeared as a contribution
to the Schilpp volume dedicated to Cassirer. As far as we know, the
only contemporary discussion of Gawronsky’s work and his role as a
vigorous (although not uncritical) defender of Cohen’s position against
Cassirer (and, to a lesser extent, Natorp) is Massimo Ferrari’s paper
Dimitry Gawronsky and Ernst Cassirer: On the History of the Marburg
School between Germany and Russia (Ferrari 2010 [20]) published in
Russian.

In contrast to Cohen, Gawronsky was fully competent in matters of
contemporary mathematics. He discussed the achievements of Bolzano,
Grassmann, Cantor, Weierstrass, Veronese, and Dedekind with evident
expertise. Moreover, Gawronsky expressed a positive appreciation of
the limit method (cf. Ferrari [35, p. 249]).

Nevertheless, in contrast to Cassirer, Gawronsky sought to leave the
philosophical core of Cohen’s ‘infinitesimal-centered’ account intact.
Hence, with respect to the infinitesimal approach he, rather than Cas-
sirer or Natorp, may be considered as Cohen’s true heir. His disserta-
tion Das Urteil der Realität und seine mathematischen Voraussetzun-
gen, literally ‘The judgment of reality and its mathematical premises’,
may be regarded as the only serious attempt of amending Cohen’s
rather obscure pseudo-Kantian table of judgments.

In order to arrive at a better understanding of Gawronsky’s Urteil,
we have to recall that, as Gawronsky explains at the end of Urteil, the
‘judgment of reality’ refers to a distinction already made by Kant (and
later modifed by Cohen) that can only be translated with difficulty
into English. This is the distinction between Wirklichkeit and Realität
that are both usually translated as reality. Roughly, reality in the sense
of Realität is to mean ‘the systematic knowledge of nature as it arises
from the chaos of the immediately sensed’ (unmittelbares Empfinden)
(Gawronsky 1910 [24, p. 107]). Then the main thesis of Urteil is that



34 THOMAS MORMANN AND MIKHAIL G. KATZ

the infinitesimal calculus plays a crucial role in the systematic knowl-
edge of nature (or empirical reality) insofar as

the basic problem of objective empirical knowledge is
the problem of change. But we only understand change
if we obtain complete knowledge of the law that gener-
ates change, i.e., only if we can pursue the effect of the
generating law in every infinitely small element of this
change. . . And exactly this is achieved by the infinitesi-
mal analysis. (Gawronsky 1910, [24, p. 105]).

In line with Cohen, Gawronsky asserted that “there is no other way
to formulate and to justify the laws of nature than the infinitely small.”
(ibid.)

By identifying “reality” with the systematized knowledge of nature,
Gawronsky saw “reality” as a “logical method” whose essence “was
the assumption of the existence of a generating law”. This “logical
method” came along in two different ways, namely “by the method of
number and by the method of the infinitesimal” (108). Both methods
are carried out in the same three steps:

(1) Positing (Setzung)
(2) Infinite repetition (unendliche Wiederholung)
(3) Actual synthesis in a higher unity (aktuale Zusammenfassung

in einer höheren Allheit)

Gawronsky’s attempt to construe an analogy between the two meth-
ods is apparently based on the idea, which he shared with Dedekind and
Cassirer, that the essence of numbers resided in their ordinal structure.
More precisely, according to Gawronsky the conceptual generation of
the natural numbers proceeded by first positing the unit ‘1’ and then
applying the generating principle of the successor function, thereby
constituting the other natural numbers. This construction, however,
had to be ‘completed’ by an “actual synthesis resulting in a higher
unity”. Or, formulated negatively, Gawronsky was not content with
simply asserting that this repetition could go on and on leading to ever
larger natural numbers. Rather, one had to look for a higher synthesis.

This was achieved, Gawronsky contended, by Cantor’s theory of infi-
nite ordinals. More precisely, Gawronsky conceived of Cantor’s positing
of the first infinite ordinal ω as the sought-for completion or synthe-
sis. This completion of the natural numbers in terms of the first in-
finite ordinal ω, however, was not simply the end of the constitution
process of pure thought. On the contrary, it was just the beginning
of a new stage in that it gave rise to a new infinite series generated
by a new generating principle ω, ω + 1, ω + 2, . . .. This new series,



INFINITESIMALS AS AN ISSUE OF NEO-KANTIAN PHILOSOPHY 35

then, had to be conceptually completed by positing 2ω, which served
as the starting point for a new series 2ω, 2ω + 1, 2ω + 2, . . ..and so on.
The determination of the limit of an infinite arithmetical series such
as 3.1, 3.14, 3.141, 3.1415, . . . converging to π had a similar conceptual
structure, and even for the calculation of differentials and derivations
Gawronsky assumed an analogous conceptual structure. According to
him, they all followed the three-tiered pattern of ‘positing’, ‘repetition’,
and ‘actual synthesis’.

A modern mathematician may view Gawronsky’s contention merely
as the recognition that the construction of both the real numbers and
infinitesimals involves infinitary constructions, a point made much later
also by Robinson. But Gawronsky, following his master Cohen, made
much more of it. According to him, the usage of infinitary constructions
revealed the very essence of both (empirical) science and mathematics
as being based on “pure thought”. The transcendental analysis of
science revealed that the origins of these methods were to be found in
the ‘judgment of reality’. Something that could not be counted22 or
differentiated, was not “real” in the sense that it could not possibly be
the object of scientific knowledge.23 Differentiating and counting were
the two basic methods of scientific knowledge. From a modern point of
view, this may be a somewhat narrow and outdated characterization
of the conceptual apparatus used in science and mathematics, but it
certainly makes sense.

Let us now examine Das Kontinuitätsprinzip bei Poncelet (Gawron-
sky 1912). The main aim of this work was to elucidate Cohen’s dictum
that continuity is a basic law of thought (Denkgesetz ) (Cohen 1902
[14, p. 76]). In Logik, Cohen traced the principle of continuity back
to Leibniz. Furthermore, he offered the following high-sounding expli-
cation of the role of continuity in the ongoing process of philosophical
and scientific thought:

Continuity is a law of thought. It is the law of thought
of the connection which enables the generation of the
unity of knowledge and thereby the unity of the object
of knowledge. Continuity as a law of thought garan-
tees the connection of all methods and disciplines of

22It should be noticed that Gawronsky here relied on a rather broad concept
of counting that not only included “ordinary” counting but also various kinds of
“infinite completion of counting”.

23This has the apparently paradoxical consequence that there may be something
real - in the sense of wirklich - that is not real. This apparent contradiction does
not threaten in the original German and is avoided if one carefully distinguishes
between the two meanings of “real” in Kant’s language.



36 THOMAS MORMANN AND MIKHAIL G. KATZ

mathematized empirical science (mathematische Natur-
wissenschaft). This law is therefore of crucial importance
for the thinking of knowledge. Continuity is the law of
knowledge. Continuity characterizes the basic feature of
thought (Cohen 1902, 76, 77).

Gawronsky’s aim in Das Kontinuitätsprinzip was to confirm and eluci-
date Cohen’s global thesis on the central role of the principle of conti-
nuity for the evolution of scientific thought, by studying its role in the
development of 19th century geometry exemplified in the work mainly
of French geometers like Poncelet, Chasles, Carnot, and others. In
line with Cohen, Gawronsky points out that this principle is not an
achievement of 19th century science but a basic feature of all scientific
thought. What is new, according to Gawronsky, is the way Poncelet
applied the principle. The crucial point is not that new objects are
subsumed under the known theorems and relations (cf. Gawronsky
1912, 69) but that entire systems of theorems and relations themselves
are ‘continuously’ modified and generalized (ibid. 71). This new inter-
pretation of the principle entails that it must not be understood as an
argument that generates mathematically secure results but rather as
a heuristic principle that helps one find novel concepts whose relevant
connections have yet to be secured by other means:

We see that the formation of concepts that is determined
and guided by the principle of continuity cannot be com-
pletely justified by that principle alone. Rather, a subse-
quent check has to be carried out in order to determine
the value of every newly introduced concept. (Gawron-
sky 1912 [25, p. 73])

For Gawronsky, the anticipatory and heuristic character of the princi-
ple of continuity in mathematics, as it was employed by Poncelet and
others, was essential for discovering its true logical base that comes to
the fore when we compare it with its “prototype” as it appears in the
realm of pure thought:

Since the discovery of the principle of continuity, Her-
mann Cohen in his Logik der reinen Erkenntnis was the
first who sought not only to evaluate the achievements
of this principle in a comprehensive way, but also to de-
termine its systematic significance, to introduce it in the
system of pure thought and to render precise its position.
He was the one who recognized this priniciple as a rather



INFINITESIMALS AS AN ISSUE OF NEO-KANTIAN PHILOSOPHY 37

general and basic method of scientific thought, identify-
ing it as a law of thought of knowledge. (Gawronsky
1912, 74)

From the viewpoint of Cohen’s neo-Kantian approach, it is then the
task of philosophy of science to integrate the issue of a purely mathe-
matical evaluation of the principle of continuity, as it was understood
by Poncelet and his contemporaries, into the general agenda of the
transcendental logic of science that treats the basic methods of scien-
tific knowledge Überhaupt (ibid. 76). For this purpose the philosopher
must not rely on idle metaphysical speculations but has to know how
this principle is actually applied in scientific practice. As an exam-
ple of how this may be achieved in the case of geometry, Gawronsky
discusses in detail two principles that Poncelet introduced in modern
geometry, namely, the principles of central projections and his “théorie
des polaires reciproques” (ibid., 76ff). This leads him to the conclu-
sion that also Klein’s Erlanger Programm, which proposes to define the
essential properties of geometrical objects as invariants of certain trans-
formations groups, can be unterstood as a realization of the principle
of continuity (78). In sum, one may contend that Gawronsky’s Das
Kontinuitätsprinzip bei Poncelet offers a knowledgeable and not im-
plausible narrative of the development and significance of the principle
of continuity in 19th century geometry. This can be taken as indirect
evidence that Husserl overstated his case when he summarily dismissed
Cohen’s account of the principle of continuity as a basic law of thought
as “profound nonsense”. Admittedly, it often takes considerable effort
to distill some meaning out of Cohen’s obscure prose, but the attempt
to rescue at least some parts of Cohen’s transcendental logic as pre-
sented in Die Logik der reinen Erkenntniss cannot be bluntly dismissed
in Husserl’s fashion.

4.5. Natorp’s The Logical Foundations of the Exact Sciences.
In his Die logischen Grundlagen der exakten Wissenschaften (Natorp
1910 [53]) the second leader of the Marburg school, Cohen’s friend and
colleague Paul Natorp also sought to come to terms with the problem of
infinitesimals. He sought to develop a philosophically founded synthesis
of two antagonistic, or at least very different, mathematical programs
for the foundations of analysis. These are, on the one hand, the CDW
program, which in Natorp’s day had nearly achieved the status of a rul-
ing orthodoxy, and on the other, the maverick, “infinitesimal-friendly”



38 THOMAS MORMANN AND MIKHAIL G. KATZ

program of the Italian mathematician Giuseppe Veronese, put forward
in his Fondamenti di geometria (Veronese 1892).24

Natorp’s contribution to the Marburg neo-Kantian philosophy in
general, and to mathematics in particular, has been usually neglected
compared to the better known works of Cohen and Cassirer. This may
be considered, historically speaking, as an injustice, in particular with
regard to the issue of infinitesimals. In Natorp’s Die logischen Grund-
lagen der exakten Wissenschaften (Natorp 1910) we find the most elab-
orate and most complete discussion of limits and infinitesimals that any
neo-Kantian philosopher ever published.25

Indeed, Natorp dedicated two chapters (namely, chapter III, 98 - 159,
and chapter IV, 160 - 224) of Logische Grundlagen to a detailed criti-
cism of the accounts of the various concepts of number as put forward
by Frege, Dedekind, Cantor, Weierstrass, Pasch, and Veronese. Fur-
thermore, Natorp believed that the accounts of Cantor and Veronese
are compatible, and viewed Veronese as the “most eminent succes-
sor of Cantor” [53, p. 171]. He appears to have held that the differ-
ences between them were only technical differences of no conceptual
and philosophical relevance. This was certainly an error, as Cantor
and Veronese were well aware of the fact that their accounts differed
in essential ways. Cantor rejected Veronese’s numbers. Veronese was
more tolerant, seeing Cantor’s transfinite numbers and Veronese’s own
transarchimedean26 infinitely large numbers as two admissible, but nev-
ertheless quite different types of mathematical entities.

An analysis of Veronese’s account would go beyond the scope of the
present text. What we wish to explore is Natorp’s philosophical moti-
vation that inspired him to engage in the risky endeavor of sketching
an all-embracing panoramic view of the landscape of the various kinds
of numbers27 and their calculi (cf. Natorp [53]).

24See P. Cantù [4] for an extensive bibliography on Veronese.
25Die Logischen Grundlagen has never been translated into English. Even in

German-speaking philosophy Natorp always remained in the shadow of the more
brilliant Cassirer.

26This somewhat unusual coinage was utilized by Laugwitz [41, p. 104] and
Peiffer-Reuter [56, p. 124f.].

27In retrospect, one can assert that Natorp’s all-embracing vision was not merely
a philosopher’s pipe dream. It gained some mathematical substance over the
decades. Thus, Kanovei and Reeken proved that there is a certain (class-size)
structure ∗V which is κ+-saturated for every cardinal κ, together with an elemen-
tary embedding of the ZFC set universe V into ∗V (Kanovei and Reeken 2004 [37,
Theorem 4.3.17, p. 151]). (Ehrlich 2012 [19]), working in Von Neumann–Bernays–
Gödel set theory with Global Choice, showed that a maximal (class) hyperreal field
is isomorphic to the maximal surreal field.



INFINITESIMALS AS AN ISSUE OF NEO-KANTIAN PHILOSOPHY 39

In line with all neo-Kantians, Natorp contended that intuition alone
could not provide a foundation for our knowledge about infinitesimals,
limits, and continuity. As he unequivocally put it: “Intuition cannot
be a foundation for continuity, neither for space nor for time” [53,
chapter IV, §4, p. 175].

According to him, the only relevant factor was “pure thought”, i.e.
the “transcendental method” or “transcendental logic” (see Subsec-
tion 3.1). Indeed, Natorp’s insistence on the crucial role of an unfet-
tered investigation of infinitary objects in applying the “transcenden-
tal method” gave his interpretation of infinitesimals in particular and
numbers in general its specific flavor:

Numbers must not have any other basis than the laws
of pure thought. (Natorp [53], ch. IV. §4, p.176).

His insistence on the infinitary character of the never-ending road of
the “transcendental method” led him to criticize attempts to base the
concept of number on something “finite” that only at a later stage of the
conceptual evolution was overcome in favor of something “infinite”.28

For him, from the very start, the concept of number was soaked with
the infinite. Hence, for him, any attempt to conceive of the “finite”
rational numbers as a more solid base for allegedly the more elusive
real numbers was philosophically mistaken. He appears even to have
blamed Dedekind for having succumbed to such a temptation to some
extent.

Natorp also came to formulate a perceptive criticism of a reductive
aspect of Dedekind’s approach of introducing new numbers as “cuts” of
the set of rational numbers (cf. Dedekind 1872). Natorp pointed out
that Dedekind assumed without justification that every cut29 corre-
sponds to exactly one non-rational number, which according to Natorp
was a petitio principii. Of course, two distinct non-rational numbers
which correspond to one and the same cut, cannot differ by a finite
number. However, they could still differ by an infinitesimal.30 Thus,

28Natorp’s insistence on the thoroughly infinite character of numbers, and the
resulting emphasis on infinitesimals, was noted not only by his fellow philosophers
but also by philosophically inclined mathematicians, regardless of whether they
belonged to the CDW camp or were sympathetic to infinitesimals; see Fraenkel [21,
footnote on pp. 50-51] and Robinson [59, p. 278].

29We will ignore the technical issue of cuts defined by the rationals themselves.
30This is indeed a mathematically coherent possibility as is shown already by the

Levi-Civita fields [43] developed at about the same time. In fact, any proper or-
dered field extension of the reals will have this property (for example, the hyperreal
numbers).



40 THOMAS MORMANN AND MIKHAIL G. KATZ

Dedekind’s axiom that every cut corresponds to a unique number is an
assumption that can be challenged.

Indeed, Natorp was the only neo-Kantian philosopher who ever ex-
plicitly evoked a suspension of the Archimedean axiom as an essential
ingredient for any infinitesimal account (cf. [53, p. 169ff]). He evoked
the Archimedean principle in his criticism of Cantor’s alleged “proof”
of the non-existence of infinitesimals (ibid.).31

Natorp further points out that already the classical founding fathers
of the concept of the infinitesimal, namely, Leibniz and Newton were
well aware of the special status of the infinitesimal, often referred to
by means of the modifier “intensive”. The latter term seems to have
had a meaning close to “non-Archimedean” to Natorp:

Already Galileo speaks of infinita non quanta; Leib-
niz contends the infinitesimal as praeter extensionem,
imo extensione prius, for Newton the infinitesimal “mo-
ments” are not quantitates finitae, but principia iamiam
nascentia finitarum magnitudinum; and Kant explains
the infinitesimal through the intensive magnitude that
include the base (Grund) for the extensive magnitudes,
but is itself not extensive ([53, p. 170]).

Natorp further criticized Dedekind’s assumption that the totality of
all cuts is ‘given’ somehow although the converging rational series are
to be considered just as procedures that allow us to approximate the
irrational limit numbers if they exist.

Natorp’s criticism of the idea that the set of Dedekind cuts is “given”
could appear to be related to a criticism often voiced at the time, and
that can be expressed in modern mathematical terms as follows. To say
that the set of cuts is “given” is to make certain foundational assump-
tions, such as the axiom of infinity, usually accompanied by the classical
interpretation of the existence quantifier, typically involving the law of
excluded middle. If so, Natorp’s criticism of Dedekind seems to echo
criticisms of mathematicians like Kronecker and Brouwer. This, how-
ever, would be a misinterpretation. Natorp was neither an intuitionist
à la Brouwer nor a platonist realist. Rather, Natorp was a critical
idealist. For him, the ideal was neither something given “out there”
nor did he require that it could be intuited in some way or other. For

31Natorp claimed that Cantor’s “proof” was flawed for rather trivial reasons.
His remarks are, however, too sketchy to be properly evaluated. Cantor was a com-
petent mathematician and although he did make some mistakes, they were rather
subtle ones. On the other hand, Natorp’s competence in matters mathematical was
that of an educated layman. Cantor’s errors are analyzed in detail by Ehrlich [18]
and Moore [48]; see also (Proietti 2008 [57]).



INFINITESIMALS AS AN ISSUE OF NEO-KANTIAN PHILOSOPHY 41

him, as well as for Cassirer, the existence of the ideal resided only in
its function. This led him to perceive similarities between Dedekind’s
approach and that of du Bois-Reymond:

Apparently this kind of argumentation [i.e., Dedekind’s—
the authors] is based on a way of thought that is manifest
in P. du Bois-Reymond’s Allgemeine Funktionentheorie
(General Theory of Functions). This author introduces
infinity and continuity by nothing short of an assumption
(which he himself calls “idealist” but which is actually
“realistic” in the sense of medieval scholasticism) that
can often be seen in arithmeticians: namely, that the ob-
jects of mathematics exist in-themselves, and that these
objects may have properties which our - always finite -
human thinking cannot fully grasp. (Natorp 1910, 180)

Natorp insisted on the thesis that the existence of mathematical ob-
jects can reasonably only mean that they are based on the mathemat-
ical thought (ibid.). Something that escapes mathematical thought,
does not exist mathematically. In other words, du Bois-Reymond’s
conception of the continuum as something “that cannot be thought”
did not make sense:32

What cannot be justified by mathematical thought, must
not be posited by mathematics” (ibid. 180).

Despite certain alleged shortcomings in Dedekind’s cut approach,
Natorp saw Dedekind as being on the right track. The merit of having
revealed the true kernel of Dedekind’s method, is ascribed by Natorp
to Weierstrass, Cantor, Pasch, and Veronese. He sees the basic flaw
in Dedekind in the fact that Dedekind started his construction with
the “finite”, fully understood rational numbers, whereas the irrational
numbers were considered as something derived.33 According to him,
Weierstrass and Cantor made the decisive conceptual step of taking
the infinite convergent “series” itself (Cantor’s Fundamentalreihe) as a
proper mathematical object to be considered in its own right. Then
there is no longer any reason to distinguish between a “series” and its
limit (cf. Natorp 1910 [53, p. 182]).

32At about the same time, a similar, but more elaborate critique of du Bois-
Reymond’s “empirist-idealist dialectics” was put forward in (Cassirer 1910 [7,
p. 122ff]).

33This criticism of Natorp’s is related to Cassirer’s insistence on the “ontological
equality” for the new entities (in this case, numbers) being introduced.



42 THOMAS MORMANN AND MIKHAIL G. KATZ

5. From infinitesimals to functional concepts

Cassirer sought to imbed Cohen’s infinitesimals in a larger framework
built upon his functional approach (cf. Cassirer 1912 [8]), but faithfully
followed the general doctrine of the Marburg school not to dismiss
infinitesimals as inconsistent pseudo-concepts. Unlike Cohen, however,
he was interested not so much in infinitesimals per se as in a more
profound and more precise understanding of the conceptual evolution
as it took place in the modern sciences, in particular in physics and
mathematics. For this purpose, he was led to a substantial recasting of
the neo-Kantian framework of philosophy of science and mathematics
as it had been designed by Cohen in Prinzip and Die Logik der reinen
Erkenntnis.

In a nutshell, this amounted to placing the concept of function or
relation center stage, rather than that of the infinitesimal. This project
of Cassirer’s began with his dissertation Leibniz’ System in seinen wis-
senschaftlichen Grundlagen (Cassirer 1902 [5]). The project was suc-
cinctly presented in Kant und die moderne Mathematik (Cassirer 1907
[6]), and eventually culminated in Substanzbegriff und Funktionsbegriff
(Cassirer 1910) and Die Philosophie der symbolischen Formen III (Cas-
sirer 1929 [9]).

Cohen was not entirely happy with this development as shown by
his letter to Cassirer dating from August 24 of 1910. Here Cohen first
heaps lavish praise on his most brilliant disciple:

I heartily congratulate you and our entire community
on your new and great achievement [i.e. the publication
of SF—the authors]. If I shall not be able to write the
second part of my Logik, no harm will be done to our
common cause.34

Cohen then continues with the second half of his comment:

Yet, after my first reading of your book I still cannot
discard as wrong what I told you in Marburg: you put
the center of gravity upon the concept of relation and
you believe that you have accomplished with the help
of this concept the idealization of all materiality. You
let even slip the remark that [the concept of relation] is
a category. (. . . ) Yet it is a category only insofar as
it is a function, and function unavoidably demands the
infinitesimal element in which alone the root of the ideal
reality can be found.

34Cohen never published a second part of Logik.



INFINITESIMALS AS AN ISSUE OF NEO-KANTIAN PHILOSOPHY 43

Among the recent interpretations of the relationship between Co-
hen’s infinitesimals and Cassirer’s relational approach, one can find
conflicting views, both of which take this letter as their main piece
of evidence. In his 2003 article Hermann Cohen’s Das Prinzip der
Infinitesimalmethode, Ernst Cassirer, and the Politics of Science in
Wilhelmine Germany, Moynahan [51] put forward the thesis that Cas-
sirer’s relational account in SF should be understood as a more or less
straightforward clarification of Cohen’s Prinzip. As evidence for his
claim, he quotes Cohen’s 1910 letter to Cassirer. However, Moynahan
only quotes the first half. The second half of the letter, in which Cohen
pointed out the profound differences between SF and Prinzip, is not
reproduced by Moynahan; see [51, p. 40].

In a more subtle and indirect way than Moynahan, recently Sei-
dengart (2012) also argued that Cohen and Cassirer essentially agreed
on the primordial role of the concept of the infinitesimal for modern
science and its proper philosophical understanding.

To bring home his point Seidengart first reminded the reader that
for Cohen the concept of the infinitesimal had to occupy center stage
in any logic of modern science deserving of its name, since “infinitesi-
mal analysis was the legitimate device of the mathematical science of
nature.” (Cohen, 1902, p. 30). According to Cohen, Leibniz, as the
inventor of the infinitesimal calculus, was the one who brought about
a situation where “mathematics became the mathematics of mathema-
tized science of nature” (Cohen 1914 [15, p. 22]). As Seidengart rightly
observes,

. . . independently of Kant, it was Leibniz who led Cohen
along the pathway of his “logic of origin” (“Logik des Ur-
sprungs”), which . . . is the logic of pure thought (“Logik
des reinen Denkens”). (Seidengart 2012 [62, p. 131])

However, Cohen’s assessment of Leibniz had not always been thus
positive. Only for Cohen’s later thought, from Logik (1902) onwards,
did Leibniz’s philosophy play a pre-eminent role. In contrast, for Cas-
sirer, Leibniz had always been the philosophical hero from the start of
his philosophical career, as is evidenced by his Leibniz’ System. Even-
tually, however, Seidengart concludes,

. . . inspite of the many innovations he was able to de-
rive from Leibniz’s infinitesimal analysis, Cohen aligned
in the end the interpretation that young Cassirer laid
out in his Leibniz’ System and in his Erkenntnisprob-
lem, both of which were explicitely cited by the founder



44 THOMAS MORMANN AND MIKHAIL G. KATZ

of the Marburg School in 1914 [i.e., in (Cohen 1914 [15,
p. 24])–the authors].

Since Cohen (1914) was the last time that Cohen dealt with Leibniz’s
infinitesimal analysis and its philosophical implications, this would ap-
pear to suggest that Cohen and Cassirer agreed on matters Leibnizian
from the beginning of the 20th century until 1914 and perhaps even
later, until Cohen’s death in 1918.

This, however, is not quite true as is already shown by Cohen’s
letter of August 1910 where Cohen complained that Cassirer in SF had
deviated from the party line as he no longer recognized the primacy of
the concept of the infinitesimal.

Seidengart does not take into account the 1910 letter, and concen-
trates on (Cohen 1914 [15]). However, a closer look at Cohen (1914)
reveals that this discrepancy of 1910, had not disappeared in 1914.
True enough, in 1914 Cohen praised Cassirer’s Leibniz’ System (1902)
and his Erkenntnisproblem (1906) as congenial elaborations of his own
account of Leibniz and the role of the infinitesimal. More telling, how-
ever, is the fact that in 1914 Cohen did not cite Cassirer’s Substance
and Function (Cassirer 1910)! This omission suggests that the differ-
ences of 1910 between the two philosophers had not been resolved in
the meantime. Rather, Cohen implicitely recognized in 1914 that his
interpretation of Leibniz and that of Cassirer essentially differed.

Pursuing the opposite path, Skidelsky in his recent book Ernst Cas-
sirer. The Last Philosopher of Culture [65] seeks to emphasize the
differences between Cassirer and Cohen, as well as the alleged obso-
leteness of the latter’s infinitesimal account. He too invokes the 1910
letter, but leaves out the sentence in which Cohen characterized SF as
a possible substitute for the second part of his Logik (cf. [65, p. 64]).
Skidelsky seeks to drive home his case against Cohen by contending
that “Cohen’s theory of infinitesimals is in fact mistaken even from
a purely mathematical point of view, being based on an outmoded
interpretation of calculus” ([65, p. 65]).

The crown witness to Skidelsky’s sweeping claim is, predictably, Rus-
sell’s The Principles of Mathematics. Describing an infinitesimalist ap-
proach to the calculus as “outmoded” amounts merely to toeing the
line on the CDW approach to the formalisation of analysis. The al-
leged uniqueness of such an approach is being increasingly challenged
in the current literature. See, e.g., (B laszczyk et al. 1013 [3]); (Bair et
al. 2013 [1]).

Cohen concluded his criticisms by urging Cassirer to “take these
thoughts into intimate consideration in the new edition” (of SF), but



INFINITESIMALS AS AN ISSUE OF NEO-KANTIAN PHILOSOPHY 45

such an edition never appeared. Hence, Cassirer had to find another
opportunity where he could pay due respects to Cohen’s philosophi-
cal interpretation of infinitesimals. He appears to have attempted to
overcome the clash with Cohen on matters infinitesimal in the article
Hermann Cohen und die Erneuerung der Kantischen Philosophie (Cas-
sirer 1912 [8]), dedicated to Cohen on the occasion of his 70th birthday.
In this paper he hailed Cohen as the innovator and true heir of Kant’s
philosophy who had brought to the fore the fundamental principle of
mathematized natural science in terms of the infinitesimal:

Matter and movement, force and mass may be conceptu-
alized in this respect as instruments of knowledge. The
high point of this development is not, however, reached
before we come back to the basic mathematical motif un-
derlying all specific conceptual formations of the natural
sciences. This motif presents itself to us in the concep-
tual methodology of the “infinitesimal”. (Cassirer 1912
[8, p. 260])

This appears to be a stronger endorsement of the infinitesimal ap-
proach than it really was. One should note that Cassirer spoke of
the “methodology of the infinitesimal” rather than the “infinitesimal”
itself. The methodology of the infinitesimal is something more gen-
eral than the infinitesimal itself, and Cassirer seemed to have been
well aware of this. Indeed, he sought to employ this greater gener-
ality to bind Cohen’s infinitesimal approach with his own “relational
approach”. Actually he did not go beyond what he had already offered
in SF some years earlier, when he praised the infinitesimal calculus as
the first and most important example of the many calculi developed in
modern mathematics, e.g. Grassmann’s Ausdehnungslehre, Hamilton’s
theory of quaternions, the projective calculus of distances, and many
others (cf. SF, 95). From this general perspective, then, it is easy
to see that all these calculi are “relational” or “functional” in a broad
sense. Meanwhile, the only example of a calculus that Cohen ever men-
tioned was the infinitesimal calculus. Superficially, Cassirer’s remarks
may appear to be a reconciliation of the infinitesimal approach and
the relational approach, but they fail to convince. For instance, Cohen
explicitly asserted that he was not primarily interested in the infinites-
imal calculus, but rather in the specific philosophical ramifications of
the concept of the infinitesimal which for Cohen represented the tri-
umph of pure thought (cf. [13], p. 32). Hence, a general comparison
of the infinitesimal calculus with other calculi probably did not overly



46 THOMAS MORMANN AND MIKHAIL G. KATZ

impress Cohen. Cassirer might have felt this inadequacy and offered a
further argument:

Without [the mathematical leitmotif of the infinitesi-
mal], it would not be possible even to characterize rigor-
ously the concept of movement,35 as it is presupposed by
the mathematized natural science, to say nothing of the
task of fully comprehending the lawfulness of movements.
Thereby the circle of the critical investigations is clos-
ing. Since without doubt, the concept of the infinitely
small does not denote a “Being” that can be captured
by the senses, but a peculiar way and basic direction
of thought:36 but this basic direction is now revealed as
the necessary presupposition of the scientific object itself.
(Cassirer 1912 [8, p. 260-261])37

In fact, Cassirer never elaborated on the connection between Cohen’s
“infinitesimal analysis” and his own “relational analysis”. Neither in
SF nor in PSF did the concept of the infinitesimal play as prominent a
role as in Cohen’s Prinzip or his Logik der reinen Erkenntnis. Rather,
Cassirer used Cohen’s “methodology of the infinitesimal” only as a
launch pad to develop his own “methodology of the relational”.

Even more revealing is the fact that in the posthumously published
fourth volume of The Problem of Knowledge ([11]), written in the late
thirties during his Swedish exile, he had completely abandoned the
infinitesimal standpoint; Cohen is not even mentioned once.

Cassirer’s attempts in SF and elsewhere to connect his relational ac-
count with the infinitesimal account of Cohen are to be judged as less
than fully convincing. Partisans of the infinitesimal approach should
not blame Cassirer for this shortcoming too harshly, however. In his
day, the effectiveness of an infinitesimal approach compared with one
based on epsilontics was not too compelling. Moreover, the concep-
tualization of infinitesimals as an idealizing completion was not suffi-
ciently understood to undertake a reasonable comparison with other
idealizing completions. The advent of the various versions of modern

35Strictly speaking Cassirer’s claim is inaccurate. It is possible to get by in
mathematics without infinitesimals, as CDW had impressively shown.

36This cryptic remark is meant to emphasize the non-empirical character of the
infinitesimal. Unlike ideal realists as well as empiricists, the Marburg school held
that the infinitesimal was not to be found “out there” in some empirical or ideal
domain independent of the cognizing subject; rather the infinitesimal was a way of
thinking or conceptualizing the world.

37Similar remarks appear already in (SF, 1910 (130ff), 1953 (99ff)).



INFINITESIMALS AS AN ISSUE OF NEO-KANTIAN PHILOSOPHY 47

infinitesimal-enriched continua has changed the conceptual landscape
dramatically.

Acknowledgments

T. Mormann was partially funded by the research project FF2012-
33550 of the Spanish government. M. Katz was partially funded by the
Israel Science Foundation grant no. 1517/12. We are grateful to the
anonymous referees for a number of helpful suggestions.

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INFINITESIMALS AS AN ISSUE OF NEO-KANTIAN PHILOSOPHY 51

T. Mormann, Department of Logic and Philosophy of Science, Uni-
versity of the Basque Country UPV/EHU, 20080 Donostia San Sebas-
tian, Spain

E-mail address: ylxmomot@sf.ehu.es

M. Katz, Department of Mathematics, Bar Ilan University, Ramat
Gan 52900 Israel

E-mail address: katzmik@macs.biu.ac.il


	1. Introduction
	2. Neo-Kantian Philosophy of Science and Mathematics
	2.1. The Transcendental Method
	2.2. Concepts in Mathematics and Science
	2.3. Substantive versus Relational Concepts

	3. Two Guiding Metaphors of neo-Kantian philosophy of science
	3.1. Natorp's Knowledge Equation
	3.2. Cassirer's Convergent Series
	3.3. Idealisations, completions, and infinitesimals

	4. Three attempts to make sense of Cohen
	4.1. The Point of departure: Cohen's Prinzip
	4.2. Cassirer's Leibniz' System
	4.3. The Continuity principle
	4.4. Gawronsky's The Judgment of Reality
	4.5. Natorp's The Logical Foundations of the Exact Sciences

	5. From infinitesimals to functional concepts
	Acknowledgments
	References