15 Dynamic Structuralism-Cassirer and Method in Culture
Pierre Keller
Ernst Cassirer’s position from beginning to end, and with the philosophy of symbolic forms at its core, constitutes a form of structuralism. It is however a fundamentally dynamical conception of structuralism. It is easy to misconstrue this claim, however, as “structuralism” has been understood in a wide variety of ways. In particular, it is often thought of in a static way and in a way that involves exclusively objective, logical and mathematical structural relations between individuals. According to this static conception of “structural objectivity,” objects and subjects are taken as given prior to their relations and subjects are endowed with at least in principle private experiences that can only be bridged through objective logical and mathematical relations: “We can’t say any “thing” or concept is objective or objectively communicable, but only the relations among concepts or things that remain the same throughout all these languages” (Daston and Galison 2007, 290). The recent literature in the history of analytic philosophy almost universally views Cassirer’s structuralism as of the same form as the logical structuralism of Bertrand Russell, Moritz Schlick and Rudolf Carnap (cf. for instance Friedman 1987, 1992). The stucturalism of analytic philosophy is a very different conception of the structure of logic as logos than the dynamic structuralism of “continental” philosophy. For analytic philosophy is built on a fundamentally tenseless object-oriented basis, while continental philosophy takes process to be fundamental.
Michael Friedman ascribes a structural holism to Carnap, but importantly concedes that Carnap wishes to do away with the synthetic a priori and intuition a priori; Friedman is himself committed at least to the elimination of intuition a priori: “Scientific knowledge is objective solely in virtue of its formal or structural properties, and these properties are expressed through the "places" of items of knowledge within a single unified system of knowledge. The project is not strictly Kantian, of course, because the notion of form or structure in question here is a purely logical one, understood solely in terms of formal logic.” (Friedman 1987, 521-545). Friedman’s own reading of Kant turns out to be along the lines of a very sophisticated version of Russell, Carnap and Reichenbach. This is the idea of structural objectivity with which Cassirer’s early work is generally (and I would argue falsely) associated. It is the kind of logical structuralism to which Bertrand Russell, Moritz Schlick and Rudolf Carnap are committed. Cassirer’s position has also been assimilated to the position of structural realism. However structural realism is largely predicated on fundamental philosophical assumptions that Cassirer rejects as pre-Copernican.23 From the vantage-point of such a narrowly logical and mathematical conception of structural objectivity, the content of the humanities and even of the social sciences seems to be reduced to what can be rendered in narrowly physical and mathematical terms. From the outset, but especially beginning with Freiheit und Form (1916), Cassirer is wedded to a more inclusive conception of normatively intersubjective structure, form and pattern that includes the arts and the normative commitments of morality and of political and
23 For the assimilation of Cassirer to structural realism cf. Gower 2000. Gower has useful information on logical positivism.
Gower reads Poincaré on p. 73 and Cassirer as well as Duhem, Schlick, Russell, Carnap on p. 74 as structural realists. But structural realism is a relaxed version of scientific realism that is intended to respond to a problem in scientific realism concerning skepticism and in the conception of structural objectivity that proper understanding of the Copernican revolution renders moot because it does not assume objects or subjects independent of their structural relations.
16 constitutional structure. He engages with and rejects the narrowly logical structuralism of Bertrand Russell and Rudolf Carnap in the thirties. Cassirer works with a much more inclusive conception of logic, as dialogic social reasoning, logos that Martin Heidegger and Hans-Georg Gadamer take over from him and from his teacher, Paul Natorp. Thus when Cassirer and the Marburg School and Karl Jaspers and Martin Heidegger use the term “logic” it seems to have little in common with the conception of logic in the Vienna Circle and in analytic philosophy. But it is the same thing looked at in the different terms in which each tradition views logic and all of experience. Cassirer’s position is still of great significance because he is at the center of a controversy that raged throughout the 20th century and continues to rage.
The very status of the humanities and social sciences as distinct from purely mathematical sciences is at issue and also the relation between continental and analytic philosophy is at the heart of this debate.
Michael Friedman has articulated this position from the vantage point of analytic philosophy and the narrow notion of logical-structural objectivity. A static structural objectivity is usually in play both in discussions of structuralism in mathematical and physical science and in linguistics and anthropology, in the social sciences, and even in the humanities. In contrast, I will present Cassirer as developing a form of dynamic temporal-historical and social structuralism. Ernst Cassirer’s structuralism is appropriate to think of as the method of the cultural sciences, but also of the exact sciences, including mathematics and theoretical physics. Indeed, much of the debates between analytic and continental philosophy and between the humanities, the social sciences and the natural sciences has turned on a fundamental disagreement about the nature and role of structural relations in our understanding of the world about us and of our own points of view on the world.
1. Cassirer and the Structuralist Revolution
Cassirer is committed to structuralist views from early on in his career. But it is not until his works in the 1940s that he begins to use the term “structuralism” to characterize a general development with which he identifies and that he opposes to “positivism”. As he writes in 1944: “The former positivism was superseded by a new principle which we may call structuralism” (EM, 132). While Cassirer became familiar with the structuralist movement in linguistics in the thirties, he seems first to use the term
“structuralism” after meeting and having extensive discussions with Roman Jakobson during his trip over to the United States from Sweden in 194124. (Their ship traveled through the wreckage of the Bismarck after fortunately narrowly missing the battle; this was just before the United States entered the Second World War in the wake of Pearl Harbor). Cassirer then played a role in making the term “structuralism”
widely available through his posthumously published 1945 article, Structuralism in Modern Linguistics (ECW 24, 304-306), and his 1944 book, An Essay on Man (EM 132-139)25.
24 As we will see, Cassirer is in agreement with Roman Jakobson’s conception of structuralism as expressed in the following passage: “Were we to comprise the leading idea of present-day science in its most various manifestations, we could hardly find a more appropriate designation than structuralism. Any set of phenomena examined by contemporary science is treated not as a mechanical agglomeration but as a structural whole, and the basic task is to reveal the inner, whether static or developmental, laws of this system. What appears to be the focus of scientific preoccupations is no longer the outer stimulus, but the internal premises of the development; now the mechanical conception of processes yields to the question of their functions” (Jakobson 1971 [1929], 711). (For more on Jakobson, cf. Holenstein 1976).
25 Somewhat surprisingly, Cassirer does not discuss Ferdinand de Saussure’s “static” linguistic structuralism in his earlier work, e.g., in Vol. I of The Philosophy of Symbolic Forms. His first discussion of Saussure is in An Essay on Man. The critique of Saussure (at EM 132-133) is then in the context of a discussion of Jakobson’s and the Prague Linguistic Circle’s more “dynamic”
structuralism (and in line with Jakobson’s critique of Saussure).
17 Cassirer’s commitment to structuralist views first becomes evident in connection with mathematics, in his 1907 article, Kant und die moderne Mathematik, and in his 1910 book, Substance and Function. In 1944, he offers the following account of structuralism for numbers (from the natural to the complex numbers), which has remained unchanged from these earlier works: “A single number is only a single place in a general systematic order. It has no being of its own, no self-contained reality. Its meaning is defined by the position it occupies in the whole numerical theory. […] In our modern theories -- in the theories of Frege and Russell, of Peano and Dedekind -- number has lost all its ontological secrets” (EM, 228). Modern field theory in physics is another paradigmatic example of structuralism for Cassirer. In physics the “electron is no longer regarded as an independent entity with an existence of its own, it is defined as a limit-point in the field as a whole”. Cassirer understands quantum mechanics as well as relativity theory in such field-theoretic structuralist terms.
Beyond the mathematical sciences, another clear example of a theory understood structurally by him is Gestalt psychology. In the 1940s, Cassirer relates the holism of modern theoretical biology to structuralism as well: “To my mind this new holism or organicism bears a close relationship to linguistic structuralism; the methodological views and ideals that we find on both sides are very much akin. […]
The new holistic theories […] have insisted that in the organic world ‘the whole is prior to the part’
[Aristotle, Politica 1253a 19ff.]” (EM 132). In fact, he recognizes the same tendency already in the nineteenth-century comparative anatomy of Georges Cuvier: “We may exchange every biological term of Cuvier for a linguistic term. In this case we should have, before our very eyes, the program of modern linguistic structuralism” (ECW 24, 307). Cassirer takes Jacob von Uexküll, with his emphasis on the way in which the morphology, the static architectonic structure of an organism, determines its interactions with and perception of its environment, as “Cuvier’s modern successor” (EPW 4, 232). For von Uexküll,
“architectonic structure [der Bauplan] is not a material thing: it is the unity of immaterial relationships among the parts of an animal body.” Here morphology relates to the physical parts of an organism as plane geometry relates to physical triangles sketched on a blackboard (EPW 4, 232).
The unity of such immaterial relationships is a general feature of “structure” and, as we shall presently see, of what Cassirer calls “symbolic form.” Cassirer’s conception of structure and of symbolic form is not static, however, but dynamic and interactive. As a consequence, he is critical of the static aspect of von Uexküll’s conception of a “Bauplan”. More specifically, by failing to view the morphological structure of organisms as itself a result of dynamic interactions with the environment, von Uexküll is led to embrace vitalism, reject evolution, and embrace a kind of solipsistic relationship of the organism to its environment26. These aspects are often connected with Kant as well, especially the (methodological) solipsism. But as I will suggest below, they are not part of Cassirer’s conception of Kant, nor of Kant properly understood.
Concerning linguistics and the philosophy of language, Cassirer emphasizes that “[l]anguage is neither a mechanism nor an organism […]; it is no thing at all.” By using his favorite phrase from Wilhelm von Humboldt (cfr. especially PSF I, 104), he clarifies this as follows: “[As] a very specific human activity, language is not an ergon but an energeia. […] Language is ‘organic’, but […] it is not an ‘organism’. It is organic in the sense that it does not consist of detached, isolated, segregated facts. It forms a coherent whole in which all parts are interdependent upon each other. In this way we may even speak of a poem, of a work of art, of a philosophic system as ‘organic’. Dante’s Divina Commedia, a tragedy of Aeschylus, Kant’s Critique of Pure Reason are ‘organic’” (ECW 24, 310). For Cassirer, the being of things in general is
26 None of these aspects survive in the appropriation of von Uexküll in current biosemiotics; cf. Deely 2004.
18 to be understood structurally in this systematic sense. It is again a dynamic sense of what it means to “be”, which he will ultimately trace back to Kant and Copernicus.
By drawing on von Uexküll’s work, while leaving behind its static aspect, Cassirer develops the beginnings of “biosemiotics”, in which organisms are understood in terms of complex information- signaling systems that give those organisms functional unity and relate them to their environments.27 As human beings, we too are organisms “adapted to” and “fitted into” our environment. We belong to the circle and function-cycle of stimulus and response in our environment, and as such are signaling animals (EM, 29). We are, however, also “symbolic” animals, as Cassirer emphasizes. This means that we have further adapted ourselves to our environment by means of symbolic systems that always interrupt and complexly mediate the connection between stimulus and response. For Cassirer, it is “the world of human culture”, with its various “symbolic forms” (language, myth, religion, art, science, history, technology, etc.), that opens up the wider world, the “cosmos” and its significance, for us.
The wider world of human culture also has a “functional unity” to it. As Cassirer notes, “the world of human culture is not a mere aggregate”, but itself “a system, an organic whole” (EM 238).
Human culture is the functional unity of the symbolic forms constituting it: “Language, myth, religion, art, history are the constituents, the various sectors of this circle”. Moreover, the symbolic forms are constituted in their “organic unity” by the “work” of human beings: “It is this work, it is the system of human activities, which defines and determines the circle of ‘humanity’, human activities linked together, by a chain of function rather than of substance (“a ‘vinculum functionale’ rather than ‘substantiale’)” (EM 76).
Like language, whose words and rules “really only exist in the act of connected speech” rather than as
“separate entities”, culture as a whole is a shared historical and social work, i.e., an energeia and not an ergon; culture is “the ever-repeated labor of the human mind” (EM 131).
As this shows, Cassirer sees dynamic structuralism as the general paradigm in terms of which all of the symbolic forms are to be understood. This raises the question of how he proposes to distinguish the exact sciences, especially mathematics and physics, from the “cultural sciences”. Mathematics and physics are as much a part of culture for him as are the humanities and social sciences; but it is only in the latter that we are concerned with our own self-interpretations. Hence Cassirer understands the dividing line between them in terms of the (holistic) notion of meaning and self-interpretation that is intrinsic to the cultural sciences and extrinsic to the exact sciences. Here biology is on the cusp. The notion of a biological organism involves a minimal ability of the organism to orient itself in its environment and to determine the significance of that environment for what it is doing. What distinguishes us, as human beings, from lower organisms is our ability to orient ourselves within the wider world, far beyond the immediate environment, by developing and using the symbolic forms.
2. Dynamic Structuralism and Human Experience
The method of Cassirer’s The Philosophy of Symbolic Forms, published in the 1920s, is already a dynamic or genetic structuralism, at least implicitly. It is based especially, but not exclusively, in the human symbolic process; and the dynamic process of symbolization is grounded in our “natural world- conception” (PSF III, 13, 31, 342). At its core, Cassirer’s approach is structuralist in that it understands the very being of anything that can be understood by a culture (numbers, electrons, works of art, historical
27For a relatively comprehensive survey of the development of biosemiotics cf. Favareau 2010. For the connection to Cassirer, see Favareau 2010, 57, 229, 384, 424, 428. Arguably the most important figure in the rise of biosemiotics was Thomas Sebeok, a student of Charles Morris and Roman Jakobson, who explicitly references Morris and Jakobson, as well as Cassirer; cf. 229.
19 events, etc.) in terms of the structural relations implicit in the pattern of signs that constitute its significance for us. And Cassirer’s structuralism is dynamic insofar as no sign or entity is taken to have a significance independently of the changing pattern of cultural relations and cultural processes in which it is embedded. As the latter is crucial for our interpretation of Cassirer, its philosophical significance should be clarified further. One way to do so is by reflecting on Cassirer’s notion of experience.
In his mature philosophy, Cassirer develops a transcendental-genetic account of experience as a process in which both subjects and objects are gradually differentiated, rather than being taking for granted from the start. For Cassirer, experience arises from an initially undifferentiated felt sense of social significance, which acquires an intuitive-representational form through language and the other symbolic forms, and which finally rises to the level of systematic conceptual articulation. In this sense, the
“symbolic process” gives human life and “consciousness” its distinctive significance, its “continuity”, and its “constancy” (MSF, 45ff). Cassirer adds more generally: “Life, reality, being, existence are nothing but different terms referring to one and the same fundamental first. These terms do not describe a fixed, rigid, substantial thing. They are to be understood as names of a process. Man is the only being that is not only engaged in this process but who becomes conscious of it [.] Myth, religion, art [,] science are nothing other than the different steps made by the human being in his consciousness, in his reflective interpretation of life. Each of them is a mirror of our human experience which, as it were, possesses its own angle of refraction. Philosophy as the highest and most comprehensive mode of reflection strives to understand them all” (ECN 7, 183).
Throughout his philosophy of symbolic forms, Cassirer emphasizes the dynamic function of symbols, a point that is not often recognized in the literature. The core of his position is this: Symbolic forms are dynamic unities of sense in which general significance is fused with a particular sensible sign.
Or as Cassirer puts it himself: “In the creation and use of different groups and systems of symbolic signs […] a sensible individual content without ceasing to be such acquires the power to present something universal to consciousness” (PSF I, 45). Along such lines, the gulf between the intellectual and the sensible -- the specter of two-worlds that has haunted both Platonism and Kantianism -- has always already been overcome. The key is Cassirer’s conception of dynamic sensible significance as grounded in the process of life: In “the concrete basic form of life the dualistic opposition [between given individual and universal] is transformed and as such sublated [aufgehoben]” (PSF I, 45).
The opposition between individual and universal has not been completely eliminated here; rather, it is transformed into a dynamically mediated process (as it is also transformed in Hegel’s philosophy and earlier, on the Marburg reading, in Kant’s philosophy). An “image-world” emerges in this process, yet it is not a copy of anything outside of culture; it is “the form of its [spirit’s] own action [die Form seines eigenen Tuns]” (PSF I, 45-46). Overall, what we take to be fixed and identical is not completely independent of the flux of human life and experience; nevertheless, consciousness and human life are not without reference to things that we take to be unchanging either (PSF I, 38ff.; 44-45). Guiding Cassirer here, at bottom, is Kant’s conception of schematism, understood as the systematic unification of concept and sensible sign, and as a temporal procedure for establishing significance (ECW 17, 229-231).
3. Dynamic Structuralism and the Infinitesimal-Method
In Cassirer, the dynamic structure of each of the symbolic forms (from myth and religion to science and history) is understood in terms of the relation of dynamic part to dynamic whole of
20 significance. Put in more mathematical terminology, it is understood in terms of the relation of
“differential” to “integral”. As this terminology indicates, Cassirer’s general method, which underwrites his philosophy of symbolic forms, is an extension of the infinitesimal-method of his teacher Hermann Cohen. For Cohen (and for Paul Natorp, Cassirer’s other main teacher in the Marburg School), all of logic and all of science are grounded in the infinitesimal-method, science and even logic are tied intimately to the unity of culture and its historical-temporal process of developing relevant to distinctions for us in a general context of significance. Cohen and Natorp thought of the infinitesimal method as committed to the existence of infinitesimals; the method of investigating events in terms of infinitesimal differences and the integration of those differences in a dynamic whole is however compatible with (Cassirer’s) adoption of the modern (Weierstrassian) treatment of differentiation and integration that became standard in the late nineteenth and early twentieth centuries.
Cohen takes Kant’s “unity of consciousness”, construed by him as a process constituting the dynamic unity of all culture, to be implicit in the integral and differential calculus (understood in a way that preserves both the dynamic conception of Leibniz and the fundamentally temporal conception of Newton’s fluxions). Like Leibniz and Newton, Cohen sees the calculus as part of a dynamic conception of the cosmos that also includes an account of how we as citizens of the universe, as cosmopolitans, can understand things and retrieve our proper historical and cultural place in the universe. Cassirer, in his turn, uses the integration of “infinitesimal differences” as a fundamental methodological tool in his philosophy of symbolic forms, and in particular, in his notion of “symbolic pregnancy” (PSF III, 232).
Symbolic pregnancy is the embodiment, by symbols as perceived by human beings, of a significance that transcends direct experience. This transcendence is grounded in a cultural-historical conception of the temporality of significance for Cassirer, in the sense of the dynamic symbolic process. Again, significance is here understood as a dynamic differential-integral whole. Hence, “infinitesimal differences” in symbolic significance are always already embedded in an integrated dynamic whole of symbolic significance for Cassirer (PSF III, 232).
Beyond the connection to Cohen’s method, Cassirer’s conception of symbolic pregnancy is derived from Leibniz’s conception of the “pregnancy of the present with the future” (praegnans future, LS 265, EPW 2, 128, PSF III, 231-232). For Leibniz, the pregnancy of the present with the future expresses the manner in which the punctual present always already contains within itself the infinitesimal tendency of any state to proceed from the past towards the future. (Here the punctual present is an abstraction from the law-governed series of changes that makes each substance what it is; it is not something basic with which we can start.) From this point of view, an object or “substance” is a dynamic unity of differential changes and interactions with its world. Corporeal substance is conceived of as a dynamic function of motion expressed by a differential equation, rather than as an inert material body or mind- thing. (This Leibnizian perspective on time and being, as filtered through Kant’s Transcendental Philosophy and through Solomon Maimon’s modification of it, is also the background for Cohen’s infinitesimal method.)
Infinitesimals are not ultimately real for Leibniz, however. Appealing to them is a façon de parler that allows us to mark the connection of the momentarily and only phenomenally real with the process that constitutes each substance. As a consequence, Cassirer can interpret and appropriate Leibniz in a way that reconciles Leibniz’s conception of infinitesimals with the modern understanding of the calculus due to Karl Weierstrass, where infinitesimals are eliminated in favor of relations between arbitrarily small finite quantities (PSF III, 467). In this way, Cassirer attempts to mediate between the fundamental
21 commitment to a process-based, historical-social conception of science and culture, as represented by the infinitesimal-method of his teachers Cohen and Natorp, and the attacks on infinitesimals and, with them, on process-based approaches by Georg Cantor, Bertrand Russell, Leonard Nelson, and their followers.
Cassirer defends the infinitesimal-method of his Marburg teachers and rejects a fundamentally object based approach; he embraces their conception of “the analysis of the infinite, the mathematics of becoming” (LS 378), but not, as Natorp does, by emphasizing the possibility of a non-Archimedean continuum that rehabilitates infinitesimals (along the lines of Giuseppe Veronese); rather, he defends the significance of infinitesimals by emphasizing the methodological importance the infinitesimal-method assigns to “operations of thought” as opposed to “the being of objects” (EPW 4, 39ff especially n61).
That defense is derived from Cassirer’s interpretation of Leibniz in his 1902 book on him; it informs the shift from “substance concepts” to “function concepts” advocated in Substance and Function; and it reaches its mature form in his philosophy of symbolic forms, including the notion of symbolic pregnancy. It is also importantly influenced beginning in the late teens by the works of Hermann Weyl on the foundations of logic, mathematics and physics. Weyl shows Cassirer through his works on the continuum and general relativity that a conception of mathematics is possible that is fundamentally process-based and not object- based and that one can ground general relativity in a “purely infinitesimal geometry” without a fundamental commitment to infinitesimals (EPW 4, 39; 89ff.; ECW 10, ; PSF III, 407ff, 429ff.; ECW 19, 222ff. on Weyl and Cassirer on infinitesimal geometry, cf. Ryckman 2005). Cantor’s, Russell’s, and related works led to the dominance of set-theoretic and object-oriented positions in English-speaking philosophy of mathematics, and in analytic philosophy more generally. From the resulting orthodox objective structuralist point of view, the significance of process-based approaches to mathematics, such as that of Cassirer, is all but invisible. The fundamental time and agency constituting dimension of the logic and of the mathematics even of Brouwer and Weyl is also largely ignored as it is in contemporary philosophy of agency. From the perspective of Cassirer, the positions of Frege, Cantor, Russell and Carnap and their more recent followers are fundamentally static positions that already assume objects as given, even when they are pushed in a structuralist direction. They start from objects and subjects and develop the objective structural-logical-mathematical relations between objects.
4. Dynamic Structuralism and Transformation Groups
There is another related dimension to Cassirer’s dynamic structuralism that also engages with the work of Hermann Weyl. It also starts from the philosophy of mathematics and logic, but has broader implications. Cassirer criticizes Aristotelian logic from the beginning of his career, in particular its inadequate reliance on narrowly syllogistic reasoning in mathematics. More than Cohen and Natorp, he is open to adopting the new logic of Frege and Russell as a replacement for neo-Aristotelian term logic.
But unlike for Russell (and more recently for Michael Friedman 1985, who follows Russell in this respect), this does not mean replacing Aristotelian syllogistic by a logic that uses multiply nested quantifiers ranging over sets of objects. From the 1930s on, Cassirer argues -- with Henri Poincaré, Hermann Weyl, and partly with L. E. J. Brouwer -- that we cannot simply take quantification over sets for granted. More specifically, to avoid the well-known paradoxes “basic concepts of set theory have to be grounded in the intuition of iteration and of the natural number-series” (EPW 4, 88). Thus structure-inducing process comes before objects and constitutes the very significance of objects and of subjects.
As Cassirer notes, Poincaré grounds the construction of the natural numbers in an intuition that
22 is the synthetic a priori principle of mathematical induction (EPW 4, 90; ECW 24, 211-212). Cassirer follows the three distinguished mathematicians Poincaré, Brouwer and Weyl in taking the limitations of Aristotelian syllogistic not simply to be overcome by the multiple nesting of logical quantifiers. Rather the synthetic a priori principle of mathematical induction is grounded in our intuition a priori of time and space. The dynamically structural synthetic unity of our self-conscious intuition of space and time constitutes the very process through which multiply nested quantifiers are able to induce (and we are also able in principle to survey) an infinite sequence of objects. Cassirer is critical of appeals to intuition within the foundations of mathematics in his earlier writings based on a more narrow, psychologistic understanding of “intuition,” a rejection of the kind of epistemic immediacy often taken to be embodied in intuition and a dynamic and encompassing conception of logic; the work of Husserl, Brouwer and Weyl, as well as Hilbert, was instrumental in convincing Cassirer of the epistemic significance of intuition a priori, even if he continued to reject their quest for immediate certainty in the local and immediately given flow of thought. He now comes to see synthetic a priori principles grounded in intuition a priori as a general consequence of his conception of structuralism. For the dynamic, synthetic unity of thought also involves some implicit grasp of an infinite whole that we are tempted to treat as an object in its own right. The dynamic structure implicit in that whole seems to suggest that it is an object in its own, but that structure is never exhaustively defined or given. Beginning with his Kant-book, Cassirer understands that whole of structural relations as given up (aufgegeben) to us by a priori intuition.
Crucial to Cassirer’s conception of the synthetic unity of mathematical and other cultural conceptions is the notion of a dynamic concrete universal. Concrete universal is a Hegelian expression introduced in late nineteenth century philosophy of mathematics by the Neo-Kantian and student of Herbart, Moritz Drobisch (SF 6, 20). It is an idea already emphasized in its conceptual significance for mathematical concepts by Kant’s contemporary Johann Lambert, as Cassirer emphasizes (SF 19; EPW 2, 455). for Cassirer, structural objectivity lies in the final invariants of experience, but those invariants cannot be abstracted from the pattern of permutations that both are constituted by them and constitute them or from our competence in getting about in our natural and cultural world. This is key to his conception of concepts as concrete universals first developed in the first chapter of Substance and Function (it is later also displayed in his account of the subjectivity of perception). Cassirer maintains that when a mathematician makes his formula more general, this means that he is able to retain all the more special cases. If a general concept had been arrived at by Aristotelian abstraction, the special cases could not be recovered from it, because the particularities have been forgotten. By contrast, the mathematical or scientific concept, and for Cassirer, any bona fide concept whatever, seeks to explain and clarify the whole content of the particulars that belong to the concept’s domain by exhibiting their deeper systematic connections to the ceteris paribus clauses that make them the concepts that they are; every concepts involves a systematic pattern of permutations in the space of possibility. Thus from Descartes’s general equation Ax2 + By2 + Cx + Dy + E = 0, we can elicit the particular conic sections (circle, ellipse, hyperbola), presented in systematic interrelation. The serial order of conic sections exhibits a systematic unity in which there is “a true infinity of intermediate members between any two given kinds”. This allows us to describe the various orbits of bodies in the solar system that “can deviate from the circle through each of an infinity of intermediate degrees according to constant laws” (KrV, A 663 B 691 [601]).
Thus in serial order or sequence the discrete items in the sequence emerge from the very process in which they are distinguished in the sequence. They are not prior to the intervals and the whole form of spatial and temporal intuition, but presuppose that holistic form of strictly continuous (and not inherently
23 discrete) order as what constitutes the very being of those things. For Cassirer, “Space does not arise because we construct it out of points, nor time because we construct it out of instants, as though they were substantial elements; rather, points and instants (and hence indirectly all figures in space and time) can be posited only through a synthesis in which the form of coexistence or succession in general originates. Thus we do not locate these forms in space and time as given, but only produce them by means of "space" and "time," if both are understood as basic constructive acts of intuition…. Because the subject matters of geometry, arithmetic, and mechanics are arrived at in this fashion, because they are not physical things whose properties we must discover a posteriori, but rather limitations we place on the ideal wholes of extension and duration, all propositions implicit in these fundamental forms are necessarily and universally valid for them.” (ECW 8, 162)
Cassirer follows Poincaré, as well as Felix Klein, in regarding our capacity to identify and order entities in terms of the kind of invariances articulated by groups of transformations to be fundamental. Indeed, he now takes this capacity for the articulation of groups of invariances and their variations to be a central a priori capacity of our understanding (ECW 24, 211-212). Groups of transformations are operations rather than arithmetic or geometrical objects (EPW 4, 87, 50). As Cassirer notes: “Lie and Klein define the group as the totality of unique operations A, B, C… so that from the combination of any two operations A and B there results an operation C which also belongs to the totality A∙B= C” (ECW 24, 211). Cassirer is particularly interested in transformations that leave an object unchanged in certain respects or congruent to itself in some way. In mathematics, this is particularly relevant with respect to symmetry groups in geometry. As Klein has shown (in his “Erlangen program”): “Every system of geometry is characterized by its group: it deals only with such relations of space as remain unchanged through the transformation of its group” (ECW 24, 211). Here Cassirer is also influenced by Hermann Weyl. Weyl points the way to a group-theoretical approach both to general relativity and geometry and to quantum mechanics, as well as emphasizing the importance of groups of transformation in expressing symmetry principles and in the very notion of the a priori.
“The German mathematician Hermann Weyl captured this parallelism in a metaphor that has proved singularly tenacious: invariants under transformation. Attempting to explain Johann Gottlieb Fichte’s and Edmund Husserl’s notion of “the absolute ego [das absolute Ich],” Weyl reached for an analogy from projective geometry. The points stand for objects in the world; the ordered triples locate points in a coordinate system for subjects. If the ordered triples are regarded only as numbers, “the experience of a pure consciousness,” these numerical relation ships will be unaltered by a change of coordinate systems — that is, by any arbitrary linear transformation. Under such transformations, all the subjective egos “have equal rights” so long as only the objective relationships are considered, as opposed to the geometric points that preserve indelible individuality…For Weyl — as for Carnap and Cassirer — Einstein’s special theory of relativity provided inspiration for a new scientific philosophy, with structural objectivity as its centerpiece.” (Daston and Galison 2007, 302) I would disagree only with two points here. First it is general relativity and the notion of a purely infinitesimal geometry and for Weyl and Cassirer, a fundamentally tensed and dynamic conception of structural objectivity, that becomes paradigmatic (Weyl 1918b). And second Weyl and Cassirer endeavor to include all of human experience, in its full subjectivity and process in their account of the dynamics of reason.
For Cassirer, groups of transformations are not only fundamental to mathematics and theoretical physics, but to all the symbolic forms in terms of which we experience the world. Not only Weyl, but
24 also Jakobson is an important fellow traveler in this respect. Jakobson emphasizes both the temporal dynamics of structure and the importance of groups of transformations in expressing that structural dynamic. Jakobson’s conception (and that of Cassirer) influences not only linguistics and literary theory, but also anthropology and psychoanalysis and becomes the basis for the dynamic conception of structure in French post-structuralist thought. With respect to language and linguistics, Noam Chomsky’s generative and transformational grammar can be seen as a systematic development of Cassirer’s (and Jakobson’s) focus on structural relations in different languages as underwritten by groups of transformations. However, Cassirer would reject Chomsky’s influential views about the innate biological and psychological basis for language understanding and acquisition as they depend on methodological solipsism. Chomsky seems to owe his methodological solipsism and his combination of systematicity and empiricism to Carnap and that pushes him and those influenced by him in the direction of a fundamentally private, if also universal language of thought. Chomsky’s conception thus parts company with that of Cassirer and Jakobson in harkening back to the objective structuralism that links subjective experiences via objective logical-structural relations rather than taking our very conceptions of subjects and objects to be constituted by dynamic structural relations.
The very morning of his death, Cassirer returned to his discussion of the role of the theory of groups for an account of perception; the discussion was intended as a talk to be presented before the Columbia philosophy club (ECN 8, 181-204). At the close of his discussion, he connects the concept of group in mathematics explicitly to the notion of structure in mathematics and physics by appeal to one of Arthur Eddington’s last books. Cassirer quotes the following passage from Eddington: “Our knowledge of structure is communicable whereas much of our knowledge is incommunicable. Physical science consists of purely physical knowledge…. This is not a conjecture as to the nature of physical knowledge, it is precisely what physical knowledge as formulated in present-day theory states itself to be.
In fundamental investigations the conception of group-structure appears quite explicitly as the starting point; and nowhere in the subsequent development do we admit material not derived from group- structure.” (Eddington 1936, 142-143). Cassirer notes and endorses Eddington’s position that the notion of structure has been made more precise through the notion of group (ECN 8, 203-204).
Eddington mentions “selective subjectivism” and “Structuralism” as the best ways of describing his philosophy. Eddington takes structuralism to have been made mathematically precise through the notion of group (Eddington 1936, 7). Eddington’s “selective subjectivism” takes our knowledge of the physical world by experiment and theory to be a function of our selective (subjective) interests and to reflect those (subjective) interests (there is nothing inherently relativist about this as his defense of structuralism displays; we understand things systematically in terms of the structures in which we are selectively interested (Eddington 1936, 17). In taking up the contrast between incommunicable and communicable knowledge, Eddington is actually picking up on a contrast that goes back to Poincaré’s distinction between qualitative and structural knowledge. According to Poincaré, what is objective must be common to many minds and consequently intersubjectively transmissible; structural knowledge meets that demand of transmissibility. Developments in psychology and physiology had suggested to many that there are incommunicable qualitative features of our experience that by threatening communicability also threaten intersubjectivity. For thinkers such as Poincaré and Eddington, sensations and qualities are inherently subjective and private, while their relations are objective.
Cassirer emphasizes the way in which dynamic structural relations are constitutive even of perception and its object. Groups of transformation are fundamental for our ability to make sense of
25 what we perceive (EPW 4, 49; ECW 24, 211ff.). As Poincaré notes, our ability to distinguish spatial movements relative to our own body from qualitative changes in the body depends on our ability to move the body so as to restore the original perceptual image and correct the appearance of qualitative change. Poincaré also argues that making sense of this ultimately involves group-theoretic geometry.
Thus Cassirer is importantly influenced by the account provided by Poincaré. He is also at one with Weyl and Eddington on the importance of the group-theoretical approach to mathematics and physics.
However Cassirer rejects the very notion of an epistemically privileged private realm of experience. For Cassirer, intersubjective structure goes all the way down. Individual, isolated, subjects are not linked by their objective structural relations. Even the experience of individual subjects would have no significance for those subjects if it were not mediated by intersubjective dynamic structure. Subjects and objects emerge from a world that is simultaneously constituted in its objectivity and subjectivity.
According to Cassirer, it is Plato's position that there is no other access to the world of ideas than through questioning and answering each other in speech. In question and answer the "I" and "the you"
must be distinguished, not only for each one to be able to understand the other, but also for each of us to be able to understand ourselves. The thought of one partner is kindled by that of the other and through the medium of myth, language, art, logic and science and by virtue of our interaction we construct a common world of meaning for ourselves, a shared world that is only possible because we do not start from methodologically or even more substantially isolated individuals in order then to attempt to bridge the gap between the self and the world. Cassirer rejects the idea that the world of the "I" is a given and finished existence and that one only needs to communicate this givenness to another subject by bridging the divide between persons; for in that case the divide would be an abyss that one could not bridge. The world, the self, the other, numbers, things and organisms are constituted in a dynamic and systematic structural process of symbolic signification that is prior to any distinctions that we may draw between items, persons or things. The distinction between subjects and objects and a conception of subjects only emerge through the give and take with each other mediated by language and by our other cultural and biological activities (ECN 2, 147-149).
The Marburg school rejects “the myth of the given” and with that myth the notion of sensations as subjective givens in the sense presupposed by purely objective structuralism. For Cassirer and his teachers, sensations are never themselves given but only inferred and thus cannot have the kind of evidentiary role often assigned to them. There is for Cassirer nothing that is merely subjective or merely given in this sense. In consequence, Cassirer endorses Poincaré’s idea that making sense of our experience of objects and their changes presupposes our implicit grip on group theoretical relations. But Cassirer pushes this conception of structural objectivity a fundamental step further. He argues that even subjective perceptual experience is dependent on our grip on such structural and group theoretical notions, a grip that displays itself in the essentially dynamic symbolic pregnancy of perception (ECN 2, 97).
Following suggestions by Gestalt psychologists (especially by Karl Bühler who also helped familiarize Cassirer with the most recent developments in structural linguistics as well as psychology), Cassirer pushes the role of groups a fundamental step further than either do Weyl or Poincaré given their commitments to epistemic immediacy. Cassirer rejects Poincaré’s structural objectivism of private sensations connected by objective relations. Cassirer points out that we need the group-theoretic framework of theoretical space even to make sense of perceptual constancy and the qualitative appearance of objects perceived; there is thus no space for immediately given private experiences: “A piece of chalk […] would on a cloudy day present the same color as a piece of coal on a sunshiny day,
26 and in the course of one day it would display all possible colors intermediate between black and white”
(ECW 24, 250). A consequence of this is that we cannot take the supposedly immediate sense-data of
“sensationalism” at face value. Instead, a “datum of sense” is always an abstraction from a complex, dynamic systematic whole of significance for a perceiving agent (including even animals ECN 2, 84 ff.).
The constant “coordination” Zuordnung between perception (say the experience of color) and stimulus (say a light stimulus) presupposed by the logical positivists reveals itself to be a limit rather than the standard case (ECN 2, 101).
Kant’s doctrine of schematism has rightly been connected to perceptual constancy by Karl Bühler in his work on perception and language; Cassirer builds on that insight (ECW 24, 246; ECN 2, 105). For Cassirer, Kant’s schemata are sensible conditions of objectification. They allow us to form perceptually constant images of objects that make concepts of objects possible; and that is what allows us to make objectively valid claims (ECW 24, 248-249). We must start, as Kant insists, from a grip on the object of perception provided by general laws and invariances. We are not necessarily conscious of these laws and invariances as such; we also do not generally appreciate that perceptions are tied to groups that define systems of perspectival transformations. But these transformation groups are at work in the background of our experience. Basically, we have always already interpreted sensations in terms of their contextual significance within inter-subjective space, the space grounded in our public use of language.
Closely tied to the notion of schema, Cassirer attributes to Kant (and to Plato himself) a Platonism not of abstract ideas, but of “concrete universals”. Universals are significant insofar as they serve as the implicit standards for our social and cultural practices and for the traditions in which we stand (but also including thought understood, with the later Plato and the later Kant, as the dialogue of a person with others and with herself). It is for Cassirer important that we are enmeshed in culture practices, in a broadly social and ethical life (Sitte); it is from that social life that we derive our standards even when we succeed in transforming them. Thus as a putatively universal representation, the schema is always an abstraction from the temporality of our and also social life. In Kantian terms, the schema of a concept is prior to the concept understood as an abstract universal representation. In particular, we grasp concepts in terms of their schemata, i.e., the temporal and spatial procedural rule-guided but not explicitly rule- following competences involved in providing and recognizing instances of the concepts in intuition.
Cassirer thus connects conceptual understanding with our systematic ability to recognize things and events in terms of their differential significance within a certain domain. This involves the “infinite task”
[unendliche Aufgabe] of systematically relating everything that is in the contrastive domain of a corresponding “idea”. An idea is here conceived as providing the systematic unity of a historically grounded practice. As such, ideas have their own historical schemata. In this respect, their conception is very much of a piece with Hegel’s dynamic-systematic idealism and his conception of concepts as in process or motion. (For Hegel, there is a transformation in concepts [Begriffsbewegung] as one proceeds from the seemingly immediate and implicit to the fully expressed and articulated significance of any concept.) But while Cassirer interprets Kantian concepts in terms of an expression from Hegel and while there are other parallels, important differences remain. Crucially, he takes Kant always to ground the
“concrete universal” in our a priori intuition of space and time; and that intuition is constituted in its distinctive form (including its topology and metric) through processes of synthesis and schematism that give our concepts their distinctive significance.
Cassirer’s commitment to concrete universals as manifested in schematism leads him to reject the Windelband-Rickert identification of the distinction between the natural and the cultural sciences with
27 the distinction between nomographic and idiographic descriptions. For Cassirer, we have no competence in understanding natural law without a competence in the permutations and ceteris paribus conditions of such law; likewise for Cassirer, historical events are never simply individual events, they are all subject to general descriptions and even laws. The “Heraclitean flux is not the destruction of its [language’s] form, it is the condition of its form.” (ECN 2, 172) But that is no less true of mathematics, even the notion of group is not for Cassirer “the end”, but only “an indication of the direction that the road is going” and constants in geometry do not have the “character of static givenness” (ECN 2, 132-133). Every manifestation of culture is a dynamically concrete universal: “Thus there is a complete coming together [In-Einander-Aufgehen], a true concrescence, of the individual and universal in every true manifestation of culture.” (ECN 2, 173) Cassirer understands this concrescence in Platonic terms as the “parousia” or
“presence of the idea in appearance” (ECN 2, 175), a presence of the idea in appearance that he illustrates in its dynamic structural systematicity by noting the full cultural context, or intertext, that is presupposed for an adequate understanding of Goethe’s East-West Divan. Cassirer does not lose the very distinction between the cultural and the natural sciences. For there is for Cassirer a fundamental difference between the cultural and the natural sciences, we cannot approach even animal behavior without attempting to understand its significance, but this holds even more truly for expressions of the human spirit. The works of human culture strive to express the sense of that which is universal and timelessly true that those who produce those works have and that is how we must approach them; but even those who produce works of culture always grasp and express the universal and the very nature of the cosmos from their own particular changing standpoint within the cosmos. The difference between human beings and other organisms is that as human beings we are able to express our sense of the cosmos as a whole, and not just our sense of our local environment and its relevance to our adaptive set. Logic, mathematics and science are not juxtaposed to other expressions of culture as timeless truth to historical reason or logic, all reason and logic is ineliminably historical, but also constituted by the ambition to transcend the historical. The works of human life and spirit need to be approached from the perspective and meaning that they express, they have been approached hermeneutically. The behavior of bits of matter is the limit case for such significance, not the model in terms of which organisms and human culture are to be understood.
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