Michael Friedman
Cassirer began his serious study of philosophy as a student of Hermann Cohen’s at
Marburg. He had learned of Cohen as an undergraduate from his teacher Georg Simmel at Berlin, who recommended Cohen’s writings to Cassirer as a way of improving his understanding of Kant. After devouring Cohen’s writings (at the age of nineteen),
Cassirer resolved to move to Marburg for graduate work; and he worked with Cohen from 1896 to 1899, when he completed his doctorate with a dissertation on Descartes’s analysis of mathematical and natural scientific knowledge. This then appeared as the
Introduction to Cassirer’s first published work, Leibniz’ System in seinen wissenschaftlichen Grundlagen , appearing in 1902. Upon returning to Berlin in 1903,
Cassirer further developed these themes while working out his monumental interpretation of modern philosophy and science from the Renaissance through Kant, Das
Erkenntnisproblem in der Philosophie und Wissenschaft der neuren Zeit , appearing in two volumes in 1906 and 1907. The first volume served as his habilitation at the
University of Berlin, where he taught as Privatdozent from 1906 to 1919.
Cassirer thus began his career as an intellectual historian, one of the very greatest of the twentieth century. Das Erkenntnisproblem , in particular, is a magisterial and deeply original contribution to both the history of philosophy and the history of science. It is the first work, in fact, to develop a detailed reading of the scientific revolution as a whole in terms of the “Platonic” idea that the thoroughgoing application of mathematics to nature
(the so-called mathematization of nature) is the central and overarching achievement of this revolution. And Cassirer’s work is acknowledged as such by the seminal historians,
Edwin Burtt, E. J. Dijksterhuis, and Alexandre Koyré, who developed this theme later in the century in the course of establishing the discipline of history of science as we know it today. Cassirer, for his part, simultaneously articulates an interpretation of the history of modern philosophy as the development and eventual triumph of what he calls “modern philosophical idealism.” This tradition takes its inspiration, according to Cassirer, from idealism in the Platonic sense, from an appreciation of the “ideal” formal structures paradigmatically studied in mathematics, and it is distinctively modern in recognizing the fundamental importance of the systematic application of such structures to empirically given nature in modern mathematical physics—a progressive and synthetic process wherein mathematical models of nature are successively refined and corrected without

2 limit. For Cassirer, it is Galileo, above all, in opposition to both sterile Aristotelian-
Scholastic formal logic and sterile Aristotelian-Scholastic empirical induction, who first grasped the essential structure of this synthetic process; and the development of “modern philosophical idealism” in the work of Descartes, Spinoza, Gassendi, Hobbes, Leibniz, and Kant then consists in its increasingly self-conscious philosophical articulation and elaboration.
In both the Leibniz book and Das Erkenntnisproblem Cassirer thereby interprets the development of modern thought as a whole (embracing both philosophy and the sciences) from the perspective of the basic philosophical principles of Marburg neo-Kantianism.
These principles, as initially articulated by Cohen, can be briefly outlined as follows. To begin with, Kant’s transcendental method is interpreted as beginning with the “fact of science”—the existence of the sciences in their modern, post-seventeenth-century form— as its ultimate given datum. The task of transcendental philosophy is to take the modern mathematical sciences as they are actually given, and then to seek, by a regressive argument, their ultimate presuppositions or preconditions. Kant had performed this task to perfection for the case of the fundamentally Newtonian mathematical sciences of the seventeenth and eighteenth centuries; and our task, at the end of the nineteenth century, is to generalize and extend Kant’s approach so as to embrace the main developments in the mathematical sciences that have occurred since Kant’s time. When we do this, the chief alteration that is necessary in the Kantian system is a rejection of Kant’s own sharp separation between two independent faculties of the mind: a passive or receptive faculty of sensibility and an active or intellectual faculty of understanding. And in this way, in particular, we avoid the idea that the pure forms of sensibility, space and time, have their own independent structure—the structure, basically, of Euclidean geometry plus
Newtonian absolute space and time—given independently of the synthesizing activity of the understanding. Other geometries than Euclid’s, and other structures for space and time than Newton’s, are therefore perfectly possible products of the a priori synthesizing activity of thought.
The crucial question, at this point, concerns how we are now to conceive the a priori synthesizing activity of thought—the activity Kant himself had called “productive synthesis.” For Cohen and the Marburg School there is no longer an a priori faculty of sensibility, embracing, as in Kant’s original system, the basic structures of Euclidean geometry and Newtonian mathematical physics. Indeed, for Cohen and the Marburg
School, there is no longer an independent contribution of a posteriori sensibility either— there is no independent “manifold of sensations” which is simply given, entirely independently of the activity of thought, within the pure spatio-temporal form of our

3 sensibility. What there is, instead, is an essentially dynamical or temporal procedure of active generation [ Erzeugung ], as the mind successively characterizes or determines the
“real” that is to be the object of mathematical natural science in a continuous serial process. The “real” itself—the true empirical object of mathematical natural science—is in no way independently given as something separate and distinct from this “productive synthesis” of thought; it is to be conceived, rather, as the necessary endpoint or limit towards which the continuous serial process exemplified in modern mathematical natural scientific knowledge is converging. This “genetic conception of knowledge” is the most characteristic contribution of the Marburg School.
For Cohen, the epistemological process in question is modelled, more specifically, on the methods of the infinitesimal calculus. Beginning with the idea of a continuous series or function, our problem is to see how such a series can be a priori generated stepby-step. The modern mathematical concept of a differential (in contemporary terminology, the concept of a tangent vector) shows us how this can be done, for the differential at a point in the domain of a given function literally points us towards its values on the succeeding points. The differential therefore infinitesimally captures the rule of the series as a whole, and thus expresses, at a given moment, the general form of the series valid for all times. It is in this way, in particular, that Cohen interprets the
Kantian Anticipations of Perception—the principles governing the categories of reality, negation, and limitation by which Kant himself initiates the determination of the “real” by the understanding. The difference is that Cohen now elevates the Anticipations of
Perception so that they, by themselves, contain the sole and entire key to the synthesizing activity of the understanding in general. For this activity is now completely expressed in the step-wise development of a continuous temporal series, representing the methodological progress of the modern mathematical sciences, whose a priori “general form” is now most aptly expressed by the differential of a continuous series or function.
In neither the Leibniz book nor Das Erkenntnisproblem does Cassirer diverge in any essential way from the fundamentals of Cohen’s point of view. Indeed, Cohen’s overriding emphasis on the differential and the methods of the infinitesimal calculus is especially prominent in the Leibniz book, where Leibniz’s great advance on Descartes is explained in terms of the priority of the Leibnizean calculus in relation to Cartesian analytic geometry. It is in Cassirer’s next great book, Substanzbegriff und
Funktionsbegriff , published in 1910, that Cassirer first takes an essential philosophical step beyond Cohen; and it is this book, accordingly, that is perhaps most characteristic of
Cassirer’s own particular approach to the philosophy of science. For it is here, in particular, than Cassirer, unlike Cohen, engages with the modern developments in the

4 foundations of mathematics and mathematical logic that exerted an overwhelming influence on twentieth-century philosophy of science—as represented, above all, by the philosophy of logical empiricism. And it is for this reason, in fact, that Cassirer, as a philosopher of science, counts as a compatriot, and, at times, fellow traveler, of logical empiricism.
Substanzbegriff begins by discussing the problem of concept formation, and by criticizing, in particular, the “abstractionist” theory according to which general concepts are arrived at by ascending inductively from sensory particulars. This theory, for
Cassirer, is an artifact of traditional Aristotelian logic, wherein the only logical relations governing concepts are those of superordination and subordination, genus and species— abstractionism then views the formation of such concepts as an inductively driven ascent from the sensory particulars to ever higher species and genera. Moreover, by this commitment to traditional subject-predicate logic, we are also committed, according to
Cassirer, to the traditional metaphysical conception of substance as the fixed and ultimate substratum of changeable qualities. A metaphysical “copy” theory of knowledge, according to which the truth of our sensory representations consists in a (forever unverifiable) relation of pictorial similarity between them and the ultimate “things” or substances lying behind our representations, is then the natural and inevitable result.
Cassirer is himself concerned, above all, to replace this “copy” theory of knowledge with what he calls the “critical” theory. Our sensory representations achieve truth and
“relation to an object,” not by matching or picturing a realm of metaphysical “things” or substances constituting the stable and enduring substrate of the empirical phenomena, but rather in virtue of an embedding of the empirical phenomena themselves into an ideal formal structure of mathematical relations—wherein the stability of mathematically formulated universal laws takes the place of an enduring substrate of ultimate substantial
“things.” Developments in modern formal logic (the mathematical theory of relations) and in the foundations of mathematics contribute to securing this “critical” theory of knowledge in two closely related respects. The modern axiomatic conception of mathematics, as exemplified especially in Dedekind’s work on the foundations of arithmetic and Hilbert’s work on the foundations of geometry, has shown that mathematics itself has a purely formal and ideal, non-sensory and thus non-intuitive meaning. Pure mathematics describes abstract “systems of order”—what we would now call relational structures—whose concepts can in no way be accommodated within the abstractionist conception. In addition, modern scientific epistemology, as exemplified especially in Helmholtz’s celebrated Zeichentheorie , has shown ever more clearly that scientific theories do not provide “copies [ Abbilder ]” or “pictures [ Bilder ]” of a world of

5 substantial “things” subsisting behind the flux of phenomena. They rather provide mere formal systems of “signs [ Zeichen ]” corresponding via a non-pictorial relation of
“coordination [ Zuordnung ]” to the universal law-like relations subsisting within the phenomena themselves.
In explicitly embracing these ideas from the work of Dedekind, Hilbert, and
Helmholtz, Cassirer is a compatriot and fellow traveler, as I have said, of early twentiethcentury logical empiricists such as Moritz Schlick and Rudolf Carnap. Indeed, Cassirer takes the modern logic implicit in the work of Dedekind and Hilbert, and explicit in the work of Frege and early Russell (the Russell of The Principles of Mathematics ), as providing us with our primary tool for moving beyond the abstractionism of Aristotelian syllogistic. The modern “theory of the concept,” accordingly, is based on the fundamental notions of function, series, and order (or order-structure)—where these notions, from the point of view of pure mathematics and pure logic, are entirely formal and abstract, having no intuitive relation, in particular, to either space or time.
Nevertheless, and here is where Cassirer diverges from logical empiricism, this modern theory of the concept only provides us with a genuine and complete alternative to
Aristotelian abstractionism when it is embedded within the genetic conception of knowledge. What is primary, once again, is the generative historical process by which modern mathematical natural science successively develops or evolves; and pure mathematics and pure logic only have philosophical significance as elements of or abstractions from this more fundamental process of “productive synthesis” aimed at the application of such pure formal structures in empirical knowledge.
It is for this reason, more specifically, that Cassirer, for his part, decisively rejects the logicist theory of the nature of mathematics—where this theory is understood (as it was by the logical empiricists) as providing a definitive refutation of the original Kantian conception of the synthetic a priori character of mathematical knowledge. Mathematics, according to this logicist view, is completely representable within pure formal logic (that is, within modern mathematical logic), and it is therefore analytic not synthetic. Without in any way rejecting the purely mathematical achievements of modern mathematical logic, Cassirer nonetheless denies that they can possibly show that mathematics is merely analytic in the philosophical sense. For philosophy’s distinctive task (the task of epistemology) is developing what Cassirer calls a “logic of objective knowledge”:
Thus a new task begins at that point where logistic ends. What the critical philosophy seeks and what it must require is a logic of objective knowledge. Only from the standpoint of this question can

6 the opposition between analytic and synthetic judgements be completely understood and evaluated. . . . Only when we have understood that the same fundamental syntheses on which logic and mathematics rest also govern the scientific construction of empirical knowledge, that they first make it possible for us to speak of a fixed lawful order among appearances and thus of their objective meaning— only then is the true justification of the principles [of logic and mathematics] achieved. (“Kant und die moderne Mathematik,” 1907)
Since pure formal logic, from a philosophical point of view, is merely an abstraction from the fundamentally synthetic constitution of mathematical natural scientific knowledge in general, developments in pure formal logic and mathematics, by themselves, can have no independent philosophical significance. They do not and cannot undermine the essentially Kantian insight into the priority and centrality of “productive synthesis.”
In now articulating the nature of this “productive synthesis” more precisely,
Cassirer no longer follows Cohen in giving overriding importance to the methods of the infinitesimal calculus. Instead, as already suggested, Cassirer employs his more abstract understanding of modern, late-nineteenth-century mathematics to craft a similarly abstract version of the genetic conception of knowledge. What we are concerned with, as before, is the progression of theories produced by modern mathematical natural science in its factual historical development. But we now conceive this progression as a series or sequence of abstract formal structures (“systems of order”), which is itself ordered by the abstract mathematical relation of approximate backwards-directed inclusion—as, for example, the new non-Euclidean geometries contain the older geometry of Euclid as a continuously approximated limiting case. In this way, in particular, we can conceive all the theories in our sequence as continuously converging, as it were, on a final or limit theory, such that all previous theories in the sequence are approximate special or limiting cases of this final theory. This final theory, of course, is only a regulative ideal in the
Kantian sense—it is only progressively approximated but never in fact actually realized.
Nevertheless, the idea of such a continuous progression towards an ideal limit constitutes the characteristic “general serial form” of our properly empirical mathematical theorizing, and, at the same time, it bestows on this theorizing its characteristic form of objectivity: namely, despite all historical variation and contingency, there is, nonetheless, a continuously converging progression of abstract mathematical structures framing, and making possible, all our empirical knowledge. Finally, in accordance with the “critical” theory of knowledge, as opposed to the “copy” theory, convergence, on this view, does not take place towards a mind- or theory-independent “reality” of ultimate

7 substantial “things.” Rather, the convergence in question occurs entirely within the series of our historically developed models or structures. “Reality,” on this view, is simply the purely ideal limit or endpoint towards which the sequence of such structures is mathematically converging—or, to put it another way, it is simply the series itself, taken as a whole.
This same view, considered from a slightly different perspective, also provides
Cassirer with a new interpretation of the synthetic a priori. Contrary to the original
Kantian conception of the a priori, even the most fundamental principles of Newtonian mechanics “do not need to hold for us as absolutely unalterable dogmas.” Such temporarily “highest” principles of experience—at a given stage of scientific theorizing—may evolve into others, and, in this case, even our most general “functional form” for the laws of nature would undergo a change. Yet such a transition would never entail that “the one fundamental form absolutely disappears, while another arises absolutely new in its place.” On the contrary:
The change must leave a determinate stock of principles unaffected; for it is solely for the sake of securing this stock that it is undertaken in the first place, and this shows it its proper goal. Since we never compare the totality of hypotheses in themselves with the naked facts in themselves, but can only oppose one hypothetical system of principles to another, more comprehensive and radical [system], we require for this progressive comparison an ultimate constant measure in highest principles, which hold for all experience in general. The identity of this logical system of measure throughout all change in that which is measured is what thought requires. In this sense, the critical theory of experience actually aims to construct a universal invariant theory of experience and thereby to fulfill a demand towards which the character of the inductive procedure itself ever more clearly presses.
( Substanzbegriff und Funktionsbegriff , 1910)
In other words, since induction is essentially a process of generalization, aiming to subsume individual facts under ever more universal laws, it does not rest content with any particular set of laws, but attempts to subsume even the most general laws at a given stage of theorizing (e.g., Newton’s laws) under still more general laws, in such a way that the most general laws at an earlier stage are exhibited as approximate special cases of the still more general laws at a later stage. And this implies, for Cassirer, that we must form the idea of an ultimate or limiting set of laws, such that all previous stages are approximate special cases of these ultimate laws. It is at this point—and only at this

8 point—that we can actually specify the content of the “universal invariant theory of experience.”
It then follows, as Cassirer immediately goes on to point out, that there is no way to determine the specific content of such ultimate principles in advance:
The goal of critical analysis would be attained if it succeeded in establishing in this way what is ultimately common to all possible forms of scientific experience, that is, in conceptually fixing those elements that are preserved in the progress from theory to theory, because they are the conditions of each and every theory. This goal may never be completely attained at any given stage of knowledge; nevertheless, it remains as a demand and determines a fixed direction in the continual unfolding and development of the system of experience itself.
The strictly limited objective meaning of the “a priori” appears clearly from this point of view. We can only call those ultimate logical invariant s a priori that lie at the basis of every determination of a lawlike interconnection of nature in general. A cognition is called a priori, not because it lies in any sense before experience, but rather because, and in so far as, it is contained in every valid judgement about facts as a necessary premise . ( Substanzbegriff und
Funktionsbegriff , 1910)
Just as the object of natural scientific knowledge, on the genetic or “critical” theory, is the never fully realized ideal mathematical structure towards which the entire historical development of science is converging, so the a priori form of scientific knowledge, for
Cassirer, can only be determined as that stock of “categorial” principles which, viewed from the perspective of the ideally completed developmental process, are seen
(retrospectively, as it were) to hold at every stage. So we do not know, at any given stage, what the particular content of spatial geometry, for example, must be, but we can now venture the well-supported conjecture that some or another spatial-geometrical structure must be present.
This example of spatial-geometrical structure turns out to be a particularly apt and revealing one. For, although Substanzbegriff was published in 1910, and thus prior to the formulation of Einstein’s general theory of relativity, Cassirer’s next important contribution to scientific epistemology, Zur Einsteinschen Relativitätstheorie , published in 1921, is devoted precisely to this revolutionary new theory. The main burden of
Cassirer’s book is to argue that Einstein’s theory actually stands as a brilliant confirmation of a purified and generalized version of the “critical” conception of

9 knowledge. For the increasing use of abstract mathematical representations in Einstein’s theory entirely supports the idea that “the reality of the physicist stands opposed to the reality of immediate perception as a thoroughly mediated [reality]: as a totality, not of existing things or properties, but rather of abstract symbols of thought that serve as the expression for determinate relations of magnitude and measure, for determinate functional coordinations and dependencies in the phenomena.” And it follows that
Einstein’s theory of relativity can be incorporated within the “critical” conception of knowledge “without difficulty, for this theory [i.e., Einstein’s] is characterized from a general epistemological point of view precisely by the circumstance that in it, more consciously and more clearly than ever before, the advance from the copy theory of knowledge to the functional theory is completed.”
In particular, the fact that Einstein now replaces the geometry of Euclid with a much more general geometry (a geometry of variable curvature depending on the distribution of mass and energy) in no way implies the collapse of a properly-understood
“critical” conception of the a priori:
For the “a priori” of space, which [physics] asserts as the condition of every physical theory, does not include, as has been shown, any assertion about a determinate particular structure of space, but is concerned only with the function of “spatiality in general,” which is already expressed in the general concept of the line-element ds as such—entirely without regard to its more particular determination.
( Zur Einsteinschen Relativitätstheorie , 1921)
According to Einstein’s generalized conception of the “line-element”—a conception derived from Riemann’s general theory of “n-fold extended manifolds”—we postulate only that space is infinitesimally Euclidean (continuously approximating to Euclidean geometry as the regions under consideration grow smaller and smaller); and what
Cassirer is saying is that it is precisely this new conception of the “line-element” that constitutes our current best candidate for an ultimate geometrical invariant.
Now Einstein’s general theory of relativity was also of crucial philosophical importance, of course, to the principal founders of logical empiricism. In sharp contrast to Cassirer, however, they used this theory (alongside the new mathematical logic) as a conclusive refutation of the original Kantian conception of the a priori. It is especially striking, in particular, that Schlick published a review of Cassirer’s relativity book in
1921, immediately after its first appearance. Schlick’s main complaint is that Cassirer’s talk of “spatiality in general” is much too vague and unspecific, and he challenges

10
Cassirer, accordingly, to specify exactly what definite principles or axioms are supposed to be comprised in it:
[A]nyone who asserts the critical theory must, if we are to give him credence, actually indicate the a priori principles that must constitute the firm ground of all exact science. . . . We must therefore require a declaration of the cognitions, for example, whose source is space. The critical idealist must designate them with the same determinateness and clarity with which Kant could refer to the geometry and ‘general doctrine of motion’ that were alone known and recognized at his time.
(“Kritizistische oder empiristische Deutung der modernen Physik?”,
1921)
Schlick then presents Cassirer with a dilemma: either he specifies the content of
“spatiality in general” as something like Einstein’s generalized “line-element,” in which case there is absolutely no guarantee that even this very general geometry might not itself be revised in the future, or he leaves the content entirely unspecified, in which case he has said nothing at all.
What is happening here is that Schlick is attempting to hold Cassirer to Kant’s original conception of the synthetic a priori, whereas Cassirer himself is articulating a quite different conception. For Kant, there are two distinct types of a priori principles in the realm of our theoretical cognition: constitutive principles (principles of Euclidean geometry and Newtonian mechanics, for example) due to the faculties of understanding and sensibility, and merely regulative principles (principles of systematicity and maximal simplicity, for example) due to the faculties of reason and judgement. The former must be realized in the phenomenal world of human sensible experience, for they are necessary conditions of its comprehensibility and intersubjective validity. The latter, by contrast, can never be fully realized in experience, but rather present us with ideal ends or goals for seeking a never fully attainable complete science of nature. And, by the same token, whereas the content of Kantian constitutive principles can be fully specified in advance, that of the regulative principles remains forever indeterminate (at least so far as theoretical cognition is concerned), because we can never know in advance what the content of the ideal complete science of nature might be. Now Cassirer’s conception of the a priori, it is clear, is modelled on the Kantian conception of regulative principles.
Indeed, precisely because Cassirer has self-consciously rejected the original Kantian distinction between the faculties of understanding and sensibility, there is also no room, on his view, for Kant’s distinction between constitutive and regulative principles.
Constitutive principles, for Kant, arise from applying the intellectual faculties

11
(understanding and reason) to the distinct faculty of sensibility, whereas regulative principles arise from the intellectual faculties themselves independently of such application to our particular forms of sensibility. By rejecting Kant’s original conception of the application of the understanding to a distinct faculty of sensibility in favor of the developmentally-oriented genetic conception of knowledge, Cassirer has thereby replaced Kant’s constitutive a priori with a purely regulative ideal.
Cassirer’s assimilation of Einstein’s general theory of relativity marked a watershed in the development of his thought. It not only gave him the opportunity, as we have just seen, to reinterpret the Kantian theory of the a priori conditions of objective experience in terms of what Cassirer calls a “universal invariant theory of experience,” but it also provided him with an occasion to generalize and extend the original Marburg genetic conception of knowledge in such a way that modern mathematical scientific knowledge in general is now seen as just one possible “symbolic form” among other equally valid such forms. Indeed, Cassirer here, in his 1921 relativity book, first announces the project of a general “philosophy of symbolic forms,” conceived, in this context, as a philosophical extension of “the general postulate of relativity.” Just as, according to the general postulate of relativity, all possible reference frames and coordinate systems are viewed as equally good representations of physical reality, and, as a totality, are together interrelated and embraced by precisely this postulate, similarly the totality of “symbolic forms”—aesthetic, ethical, religious, scientific—are here envisioned by Cassirer as standing in a closely analogous relationship. So it is no wonder, then, that, subsequent to taking up a professorship at Hamburg in 1919, Cassirer devotes the rest of his career to this new philosophy of symbolic forms.
The philosophy of symbolic forms is of the greatest interest and importance in determining Cassirer’s relationship to the later development of what we now call the continental tradition—and, more specifically, for understanding his remarkably close relationship with Martin Heidegger. Here, however, I will break off my discussion of the evolution of Cassirer’s philosophy at this point, and I will turn, instead, to the question of the relevance of Cassirer’s reinterpretation of Kantian epistemology to our contemporary situation in the philosophy of science. It might seem, in particular, that Cassirer’s vision of a “universal invariant theory of experience,” based on the idea of a continuously converging sequence of mathematical natural scientific theories, has very little relevance to our contemporary philosophical predicament. For one of the central points of Thomas
Kuhn’s theory of scientific revolutions is that all talk of inter-theoretical convergence, of the approximate containment of earlier scientific paradigms in later ones, must be rejected. Indeed, one of Kuhn’s primary examples is precisely the relationship between

12
Einstein’s theory of relativity and Newtonian mechanics, where Kuhn famously denies that Newtonian mechanics can be mathematically derived from relativity theory as an approximate special case.
Yet the situation is not as simple as it first appears, and the best way to see this is to take a very brief look at the historical background to the development of Kuhn’s own historiography. In the Preface to The Structure of Scientific Revolutions , first published in 1962, Kuhn portrays how he shifted his career plans from physics to the history of science, and, in explaining his initial intensive work in the subject, he states that he
“continued to study the writings of Alexandre Koyré and first encountered those of Emile
Meyerson, Hélène Metzger, and Anneliese Maier [; more] clearly than most other recent scholars, this group has shown what it was like to think scientifically in a period when the canons of scientific thought were very different from those current today.” Then, in the introductory first chapter on “A Role for History,” Kuhn explains the background to his rejection of what he calls the “development-by-accumulation” model, especially as represented by a naïve form of empiricism according to which science linearly progresses via the continuous accumulation of more and more observable facts. The background to his rejection of such views, Kuhn explains, lies in what he here calls “the new historiography,” as represented, especially, by such thinkers as Alexandre Koyré and
Emil Meyerson. “By implication [Kuhn concludes], these historical studies suggest the possibility of a new image of science[; t]his essay aims to delineate that image by making explicit some of the new historiography’s implications.”
In a survey article on the development of the history of science, first published in
1968, Kuhn again explains the new historiography’s break with the development-byaccumulation model, which began, according to Kuhn, with “the influence, beginning in the late nineteenth century, of the history of philosophy.” We here learned an “attitude towards past thinkers,” Kuhn explains, which “came to the history of science from philosophy”:
Partly it was learned from men like Lange and Cassirer who dealt historically with people or ideas that were also important for scientific development . . . . And partly it was learned from a small group of neo-Kantian epistemologists, particularly Brunschvicg and Meyerson, whose search for quasi-absolute categories of thought in older scientific ideas produced brilliant genetic analyses of concepts which the main tradition in the history of science had misunderstood or dismissed. (“The History of Science,” 1968)

13
In the same pages Kuhn cites the work of Edwin Burtt and Arthur Lovejoy and refers to
“the modern historiography of science” founded by “E. J. Dijksterhuis, Anneliese Maier, and especially Alexandre Koyré.”
These statements, from our present point of view, are of course especially intriguing. For, not only does Kuhn himself explicitly refer to Cassirer and what he calls
“neo-Kantian epistemology” as important influences on his own thought, but we also know, as we pointed out at the beginning, that Cassirer was an important influence, in turn, on some of the main representatives of the new historiography Kuhn cites— specifically, on Burtt, Dijksterhuis, and Koyré. It turns out, however, that perhaps the most important philosophical influence on Kuhn was Emil Meyerson, who was himself the principal philosophical influence on Alexandre Koyré. Moreover, not only was
Koyré, as Kuhn himself makes perfectly clear, by far the most important representative, for him, of the new historiography, but the philosophical perspective shared by both
Meyerson and Koyré is diametrically opposed, in most essential respects, to that articulated by Cassirer.
Meyerson, in particular, is vehemently opposed to all attempts to assimilate scientific understanding to the formulation of universal laws governing phenomena.
Indeed, the central thought of his Identité et Réalité , first published in 1908, is that genuine scientific knowledge and understanding can never be the result of mere lawfulness ( légalité ) but must instead answer to the mind’s a priori logical demand for identity ( identité ). And the primary requirement resulting from this demand is precisely that some underlying substance be conserved as absolutely unchanging and self-identical in all sensible alterations of nature. Thus, the triumph of the scientific revolution, for
Meyerson, is represented by the rise of mechanistic atomism, wherein elementary corpuscles preserve their sizes, shapes, and masses while merely changing their mutual positions in uniform and homogeneous space via motion; and this same demand for transtemporal substantial identity is also represented, in more recent times, by both Lavoisier’s use of the principle of the conservation of matter in his new chemistry and by the discovery of the conservation of energy. Yet, in the even more recent discovery of what we now know as the second law of thermodynamics (“Carnot’s principle”), which governs the temporally irreversible process of “degradation” or “dissipation” of energy, we encounter nature’s complementary and unavoidable resistance to our a priori logical demands. In the end, therefore, Meyerson views the development of modern natural science as progressing via a perpetual dialectical opposition between the mind’s a priori demand for substantiality and thus absolute identity through time, on the one side, and nature’s “irrational” a posteriori resistance to this demand, on the other.

14
In the work of Cassirer and Meyerson, then, we find two sharply diverging visions of the history of modern science. For Cassirer, this history is seen as a process of evolving rational purification of our view of nature, as we progress from naively realistic
“substantialistic” conceptions, focussing on underlying substances, causes, and mechanisms subsisting behind the observable phenomena, to increasingly abstract purely
“functional” conceptions, where we finally abandon the search for underlying ontology in favor of ever more precise mathematical representations of phenomena in terms of exactly formulated universal laws. For Meyerson, by contrast, this same history is seen as a necessarily dialectical progression (in something like the Hegelian sense), wherein reason perpetually seeks to enforce precisely the “substantialistic” impulse, and nature continually offers her resistance in the ultimate irrationality of temporal succession. It is by no means surprising, therefore, that Meyerson, in the course of considering, and rejecting, what he calls “anti-substantialistic conceptions of science,” explicitly takes issue with Cassirer’s claim (in Das Erkenntnisproblem ) that “[m]athematical physics turns aside from the essence of things and their inner substantiality in order to turn towards their numerical order and connection, their functional and mathematical structure.” And it is also no wonder, similarly, that Cassirer, in the course of his own discussion of “identity and diversity, constancy and change” (in Substanzbegriff ), explicitly takes issue with Meyerson’s views by asserting that “[t]he identity towards which thought progressively strives is not the identity of ultimate substantial things but the identity of functional orders and coordinations.” Indeed, this passage continues with a typical statement of Cassirer’s version of the genetic conception of knowledge:
But these [viz., “functional orders and coordinations”] do not exclude the moments of difference and change but only achieve determination in and with them. It is not manifoldness as such that is annulled
[ aufgehoben ] but [we attain] only a manifold of another dimension: the mathematical manifold takes the place of the sensible manifold in scientific explanation. What thought requires is thus not the dissolution of diversity and change as such, but rather their mastery in virtue of the mathematical continuity of serial laws and serial forms.
( Substanzbegriff und Funktionsbegriff , 1910)
Thus, in explicit opposition to the view of Meyerson (which is fully shared, as he himself makes perfectly clear, by Koyré), Cassirer’s whole point is that thought does not require a
“substantialistic” or “ontological” identity over time of permanent “things,” but rather a purely mathematical continuity over time of successively articulated mathematical structures.

15
If I am not mistaken, this deep philosophical opposition between Meyerson and
Cassirer receives a very clear echo in Kuhn’s theory of scientific revolutions, particularly with regard to the question of continuity over time at the theoretical level. Here Kuhn shows himself, in this respect, to be a follower of the Meyersonian viewpoint, for he consistently gives the question an ontological rather than a mathematical interpretation.
Thus, for example, when Kuhn famously considers the relationship between relativistic and Newtonian mechanics, he rejects the notion of a fundamental continuity between the two theories on the grounds that the “physical reference” of their terms is essentially different, and he nowhere considers the contrasting idea, characteristic of Cassirer’s work, that continuity of purely mathematical structures is quite sufficient. (Kuhn does not and cannot deny, of course, that laws of the same mathematical form as Newton’s can be mathematically derived as special cases of relativistic laws.) Moreover, Kuhn consistently gives an ontological rather than a mathematical interpretation to the question of theoretical convergence over time: the question is always whether our theories can be said to converge to an independently existing “truth” about reality, to a theoryindependent external world. By contrast, as we have seen, Cassirer definitively rejects this realistic construal of convergence at the very outset: our theories do not
(ontologically) converge to a mind-independent realm of substantial things; they
(mathematically) converge within the historical progression of our theories as they continually approximate, but never actually reach, an ideally complete mathematical representation of the phenomena.
It follows, then, that Kuhn’s remarks about continuity and convergence can in no way be taken as a straightforward refutation of Cassirer’s position. For Kuhn simply assumes, in harmony with the Meyersonian viewpoint, that there is rational continuity over time in the traditional sense only if there is also substantial identity. Since, as Kuhn argues, the “physical referents” of Newtonian and relativistic mechanics, for example, cannot be taken to be the same, we are therefore faced with the problem of theoretical incommensurability together with all its “irrationalistic” implications. Yet Cassirer, as we have seen, is just as opposed to all forms of naïve realism (as well as naïve empiricism) as is Kuhn. He instead proposes a generalized Kantian conception, emblematic of what he himself calls “modern philosophical idealism,” according to which scientific rationality and objectivity are secured in virtue of the way in which our empirical knowledge of nature is framed, and thereby made possible, by a continuously evolving sequence of abstract mathematical structures. In order to determine whether, and to what extent, such an extended Kantian conception is tenable, Kuhnian historiography of science, by itself, is clearly insufficient. We instead need a more

16 comprehensive historiography (as particularly well exemplified in all of Cassirer’s work), in which both the development of modern science and the parallel development of the modern philosophical tradition receive equal, and complementary, historical and philosophical attention. When we do this, I believe, we will find that a satisfactory philosophical interpretation of the development of modern mathematical science requires a mixture of Kuhnian ideas with ideas drawn from both the logical empiricist tradition and from Cassirer—but this is a story for another occasion.