SOURCE: "Cassirer's Theory of Concept Formation," in The New Scholasticism, Vol. XLII, No. 1, Winter, 1968, pp. 91-102.

[In the following essay, Lindgren explores the formal logic that undergirds Cassirer's theory on how concepts are formed.]

In English speaking countries we have always been inclined to jump to the conclusion that the feature of Cassirer's thought which catches our notice must be the central and unifying theme of his work. In 1941, when he first came to America, his work was characterized as a philosophy of science and in 1943 it was thought to be a type of language philosophy. The only book-length interpretation of his work [Carl H. Hamburg, Symbol and Reality] considers it as a type of semantic philosophy. The discouragement which follows the discovery that none of these themes adequately organizes his theories into a coherent whole is amply witnessed by the few studies of his work which have appeared since his death in 1945.

Adequate understanding of the meaning of any one of Cassirer's doctrines awaits a grasp of the unity of his thought. This paper is intended to contribute to that understanding by clarifying the precise nature of his method. Both his statement and his demonstration of his central methodological principles will be indicated. It will also be contended that although these principles are stated in many contexts, their demonstration is available only in the opening chapter of Substance and Function. There it becomes clear that what characterizes the work of Cassirer most essentially is a new logic of the concept which, when applied to the various subjects of inquiry, yields awe-inspiring transformations which have always captured the fancy of his readers.

Perhaps the most widely read of Cassirer's books is his Essay on Man. In it the author focused attention upon the implications of his philosophical position for the study of myth, art, language, history, religion and science. The most widely quoted sentence of this work, however, does not bear upon any of these fields of inquiry. Instead, it is the remark which captures the peculiar vantage point from which each of these particular areas is considered and illumined. "Hence, instead of defining man as an animal rationale, we should define him as an animal symbolicum."

Cassirer attempted to demonstrate few of the points mentioned in the e y. He did not indicate the basis of his proposed definition of man. He advised those of his readers who were interested in discovering the demonstrations of this and other of the points merely posited in the Essay to turn to the more detailed discussions of the PSF [Philosophy of Symbolic Forms]. Indeed, any thoughtful attempt to understand any of his later thought must include this one at a very early stage.

Cassirer began the PSF by stating his purpose and his methodological assumption. The purpose was to continue the work which he began in S&F [Substance and Function] in which he laid bare the logical structure of the natural sciences by mounting a similar critique of the cultural sciences. "It seemed to me that the theory of scientific concepts and judgments which defines the natural 'object' by its constitutive traits, and apprehends the 'object' of cognition as contingent on the function of cognition must be amplified by an analogous specification of pure subjectivity."

Having stated his purpose he indicated his methodological assumption and then proceeded to the task of rejuvenating the cultural sciences by a general critique of all forms of human consciousness, concentrating mainly upon language, and myth and conceptual thinking, to which he devoted one volume each. Like the Essay, however, the PSF consists in the elaboration of the broad ramifications of a single principle, methodological in character, which is stated but not given explicit justification in that context. In the Essay that principle was the definition of man as a symbolical animal. In the PSF it is the basic epistemological postulate that relation holds primacy over substance or that function-concepts are logically prior to substance-concepts. The passage which is usually cited in this regard reads: "… [T]he fundamental principle of critical thinking [is] the principle of the 'primacy' of the function over the objects.…" This same principle was asserted with slightly greater precision in an earlier remark: "This postulate of a purely functional unity replaces the postulate of a unity of substance and origin, which lay at the core of the ancient concept of being."

Just as we were obliged to have reference to an earlier work in order to discover the demonstration of the fundamental principle of the Essay, so too we must look to an earlier work in order to discover the justification of the methodological principle of the PSF. Since the PSF was proposed as a continuation of the work begun in S&F it is not surprising that Cassirer's demonstration of the 'primacy of function' and so of the definition of man is to be found in the opening chapter of his first systematic work. That chapter develops his theory of the formation of concepts, which theory constitutes the continuing philosophical perspective from which he approached the remainder of his work. In what follows this theory will be shown to be the immediate basis of the 'primacy of function' doctrine as well as that of the definition of man. The theory of concept formation is the ultimate ground of Cassirer's definition of man just as it had been that of the rational-animal definition which he proposed to replace.

Cassirer's purpose in S&F was to assist in the transformation of formal logic by issuing a critique of the general doctrine of the construction of concepts. This project first occurred to him when he discovered traditional logic to be inadequate to account for some of the more advanced concepts of modern mathematics. He was quick to observe, however, that these concepts were not in any way limited to mathematics. The exact sciences are also founded upon them. Thus, while the task at hand was a purely logical one, it was conceived from the onset as pertinent to mathematics and the exact sciences as well.

At the heart of the traditional theory of concept formation lies the doctrine of the generic concept. This doctrine presupposes only the inexhaustible multiplicity of existing things and the power of the mind to select from that multiplicity those features which are common to several of them. Thought, by a process of comparison and differentiation, gathers objects characterized by common features into classes. As this same process is repeated upon the classes themselves, a determinate and hierarchical order begins to arise and finally achieves a classification of things which approaches completeness. The guiding principles of the generic concept follow from these foundations. The generic concept or class comprehends within itself all the determinations in which these objects agree. The species or subclasses which divide the genus at various levels are distinguished by features which are common to only a portion of those objects. The rule of inverse ratio of comprehension and extension follows immediately from this description.

An immediate difficulty with this account of concept formation would seem to be that in organizing the multiplicity of objects by similar features it clearly calls for the substitution of a portion of that multiplicity for the whole. Such an objection, however, would miss the whole point of concept formation. The negation of objects characterized by dissimilarity, it is held, is simply the reverse side of the act of discerning the essential moment of the whole. True, a part is substituted in thought for the sensuous whole, but this part is no random section. Rather, it is the ground, reason, or cause according to which that whole is determined. There is no difficulty here. But one soon appears, for the rule of "inverse ratio" contains in itself no guarantee that this end will actually be achieved. It is at this point that the metaphysical presupposition of Aristotle's logic becomes clear.

The only way to justify the employment of the generic concept is to assert that what is common in knowledge is at the same time "… the real Form which guarantees the causal and teleological connection of particular things." By achieving a generic knowledge of things, thought isolates the specific type which, as the active factor in individual things, gives the general pattern to the manifold of special forms. "The determination of the concept according to its next higher genus and its specific difference reproduces the process by which the real substance successively unfolds itself in its special forms of being. Thus it is this basic conception of substance to which the purely logical theories of Aristotle constantly have reference." Accordingly, Aristotle constantly proposes the primacy of the category of substance over all other categories, especially that of relation.

Modern empiricists have developed Aristotle's logical theory of concept formation by inquiring into the manner in which abstract objects arise. Of particular note in this regard is their "psychology of abstraction." Cassirer singled out the work of Mill for specific consideration. According to Mill the mind is possessed of the power of selecting common features from the sensuous manifold in virtue of its ability of reproducing any given content of presentation. In the course of time, sense determinations leave behind certain traces of themselves in the form of memory residues. These contents of past presentations are again aroused by newly occurring sense stimuli of a similar sort. On such an occasion the association of the past contents with the present one is established. As this process continues, a firm connection is established between the similar contents of successive presentations while the traces or residues of dissimilar ones—not being freshly aroused—tend to recede from explicit consciousness until they finally form only a shadowy background against which the similar features stand out more clearly. This progressive solidification of those features which agree into a unitary and indivisible whole constitutes the whole psychological nature of the concept.

By considering the empiricists and the realists as being within the same tradition Cassirer did not intend to gloss over their real metaphysical differences, but only to emphasize that in matters of logic they are at one. Both claim that the common feature captured in the concept is identical with the essential feature of the series of objects which confront the formative process, whether that is thought to be a series of things or their presentations. As regards the presupposition of a metaphysics of substance and the resultant logical doctrine of the primacy of the category of substance over relation there is no conflict between these traditions. Yet, it is precisely here, where there is no conflict within traditional logic, that the modern mathematical concept occasions profound difficulties.

The concepts employed in modern mathematics and the exact sciences cannot be adequately comprehended within the traditional doctrine of the generic concept. The crucial difficulty lies in the fact that the object of such a concept cannot be given either as a thing or as a presentation. On the one hand, such an object is not given at all but is posited by a progressive synthesis as a systematic connection of thought-construction (Denkgebilder). The object of such a concept is the product rather than the starting point of concept formation. On the other hand, such an object cannot even be represented in a presentation with accuracy. Not even a straight line is susceptible of sensuous representation. On both counts, then, the traditional doctrine of the generic concept along with the theory of concept formation which is founded upon it fails to account for the concepts of modern mathematics and the exact sciences.

That the traditional logical doctrine fails to admit of sufficient scope to accommodate mathematical and scientific concepts constitutes a limitation but not of itself a fundamental failure. There is still the possibility that it may be developed sufficiently to allow of such procedures. In order to show that this difficulty is more than a mere shortcoming, that it is symptomatic of a profound failure and so requires a radical transformation of the very foundations of logic itself, Cassirer proceeded to indicate the source of the inadequacy of the traditional doctrine of the concept.

Mill's doctrine of abstraction requires that concepts arise for us by the gradual predominance of the similarities of things over their differences. Memory residues are aroused when similar presentations occur. "The similarity of things, however, can manifestly only be effective and fruitful, if it is understood and judged as such." A presentation can trigger the arousal of memory residues only on the condition that this presentation is seen to be similar to those residues and vice versa, i.e., only on the condition that they are recognized as similar. The doctrine of abstraction glosses over this by allowing that the similarities between the contents of the several presentations are apprehended as themselves members of the presentational sequence. Once this is exposed to the light of day it falls by the wayside. "And yet even the most simply psychological reflection shows that the 'likeness' between any contents is not itself given as a further content; that similarity or dissimilarity does not appear as a special element of sensation side by side with colors and tones, with sensations of pressure and touch."

The generic concept is formed according to Aristotle's theory by progressive negation or neglect of objects characterized by dissimilarity. Accordingly, as thought proceeds concepts become more and more emptied of content. The whole idea of the generic concept thus stated seems to fly in the face of scientific thought, for the whole purpose of taking thought in the first place is to resolve an original indefiniteness and ambiguity and to achieve in its stead sharp lines of distinction. Aristotle's generic concept, on the contrary, works in exactly the reverse order. The further thought proceeds along generic lines the more effaced become the lines of distinction and so the greater become the confusion and ambiguity. This difficulty is fatal. The very law of generic concepts—the inverse ratio rule—spells the doom of thought. Universality can be achieved only at the expense of content and at that price the project is self-defeating.

Cassirer proceeded from the two objections just stated to formulate a more adequate theory of concept formation. From the observation that abstraction, of necessity, involves recognition he went on to show that the act of identification is the very foundation of abstraction. Objects in the sensuous manifold which are to be recognized as similar and so grouped together must first be identified as like one another. In order to do so some principle of comparison must be employed. By allowing that "likeness" and "difference" are themselves members of the presentational sequence Mill insisted that the principle of identification is obvious from mere inspection of that manifold. In fact, however, the sensuous manifold may be ordered according to any number of the most divergent principles, e.g., according to equality, inequality, number, magnitude, spatial or temporal relation, causal dependence, etc. There are a variety of ways in which the sensuous manifold may be ordered. Accordingly, it must be admitted that the sensuous manifold is indeterminate in respect of its order and that the principle by which the manifold is ordered is independent of sensation. The sensuous manifold as an ordered series must, therefore, be considered to be a composite. On the one hand, there is the manifold of members which is indeterminate in respect of its order, and on the other, there is the principle, function, or law of arrangement according to which that manifold is determined in respect of its order.

By thus defining the concept as a propositional function f(x), Cassirer indicated with utmost clarity the failures of both Aristotle and Mill. The latter attempted to seize the propositional function as if in his hands, while the former assigned it to a suprasensory abode while still regarding it as something substantial. On the contrary, the propositional function can neither be reduced to one of its own variables, a mere "x," nor be intelligible apart from all variables; it is always a function of some variable. Mill viewed it as one of the variables, while Aristotle made it into one of the absolute separate substances. Both erred in the same sense, however, by failing to distinguish between the members of an ordered series and the general law of arrangement by which this serial order is determined.

The advent of the concretely universal mathematical concept signals the collapse of the Aristotelian doctrine of substance, which had previously formed the implicit basis of formal logic. Both empiricists and realists agree that the common feature captured in the concept is identical with the essential feature of the series of objects which confront the formative process. The determination of the order of the members of the series is, according to that scheme, fixed independently of thought. Nature is here conceived as an already ordered series of substances which thought can only accept, achieving universality in its concepts by systematic forgetfulness. But when it is seen that the determination of the order of the members of the series is deduced from the concept itself, it must be concluded on the one hand that thought is the source of the ordering of the sensuous manifold, and on the other that the common feature achieved in the concept is the law of arrangement according to which that manifold is thus determined. "In opposition to the logic of the generic concept, which, as we saw, represents the point of view and influence of the concept of substance, there now appears the logic of the mathematical concept of function."

The transformation of formal logic is now all but complete. The primacy of function-concepts over substance-concepts is established. The primacy of function over object is clearly demonstrated. Still Cassirer had yet to draw his most original conclusion. All that had been said does not differ widely from the position of Kant. Indeed, Cassirer, correctly or not, everywhere attributes the doctrine of the primacy of function to Kant. It remained for him to break with his master and show that: "Instead of saying that the human intellect is an intellect which is 'in need of images' we should rather say that it is in need of symbols."

So far, in the argument of S&F, Cassirer had confined his argument to showing that the manifold in sensation is ordered by thought under its own laws. Yet even this will not suffice for the concepts of modern mathematics or even those of the natural sciences, for the objects of these concepts can not be given in sensation. Later in the PSF he indicated that non-Euclidean geometries make it necessary to depart from Kant on this point. It can no longer be required that mathematical concepts be grounded in sensuous intuition in order to assure their validity. Rather, all that is necessary is that they be represented in a set of marks which as variables substitute for the sensuous manifold.

This is usually the discussion cited when reference is made to Cassirer's departure from Kant. Yet, as was seen to be the case with the other cardinal methodological principles of Cassirer's synthesis, it was demonstrated only in the opening chapter of S&F. Lotze had criticized the traditional doctrine of abstraction on the score that in actual practice the universality of the concept is never advanced by neglecting the particular properties of objects without retaining an equivalent for them. Thus, when forming a concept of metal we may not ascribe to that abstract object the color of gold or the weight of copper, but the positive thought must be added that it is colored in some way and that it have some weight. It is not, as the traditional rule of abstraction would have it, that the peculiar "marks" p1p2, q1q2 are neglected, but that the general terms P and Q are substituted in their place, of which the particular "marks" are one of their values. "We represent this systematic totality (Inbegriff) when we substitute for the constant particular 'marks,' variable terms, such as stand for the total group of possible values which the different 'marks' can assume." What had seemed to be a falling aside of the peculiar "marks" of particular objects is only the substitution of another type of mark in their stead. Thus, concept formation proceeds according as variable terms, one value of which are the particular "marks," are substituted for them as the members ordered under a law. It is only in this way that thought may proceed, i.e., by the substitution of symbols for images.

In the opening chapter of S&F Cassirer accomplished a considerable transformation of formal logic. Such a change always has profound repercussions in all ranges of our knowledge of reality, whether of objectivity or subjectivity. "Whenever, in the history of philosophy, the question as to the relation of thought and being, of knowledge and reality, has been raised, it has been dominated from the first by certain logical presuppositions, by certain views about the nature of the concept and judgment. Every change in this fundamental view indirectly produces a complete change in the way in which the general question is stated. The system of knowledge tolerates no isolated 'formal' determination without consequences in all the problems and solutions of knowledge. The conception, therefore, that is formed of the fundamental nature of the concept is directly significant in judging the questions of fact which are generally considered under 'Critique of Knowledge,' (Erkenntniskritik) or 'Metaphysics.'"

In this paper we have been content to explore the changes in logical presuppositions which announced themselves to Cassirer at the onset of his professional career. It is no exaggeration to suggest that the remainder of that career was devoted to completing the cycle which he began in this early work by showing how reality must appear when viewed from this more profound logical perspective.