The Containment

Cassirer’s points can be elaborated by considering the distinction that Kant drew between conceptual and intuitive representations. He ac-cepted a notion of concept that was, by and large, in accordance with the Aristotelian model. In the passage quoted above, Cassirer men-tions two salient features of this model.¹⁷ Firstly, there is the explana-tion of the origin of concepts. On the tradiexplana-tional view, concepts have their origin in the process of “universalising abstraction”.¹⁸ That is, a

17 These are explained in greater detail in Cassirer (1910, Ch. 1).

18 I borrow the term “universalising abstraction” from Gambra (1996, p.

285). As Gambra points out, philosophical tradition distinguished two vari-ants of abstraction: firstly, universalising abstraction, or the “operation by which we produce general ideas on the basis of other ideas which are similar to each other” (ibid.); secondly, separating abstraction, by which was meant “the operation of distinguishing or discerning some ideas from others” (ibid.) In the Aristotelian tradition abstraction in both of the above senses was impor-tant, because it had a substantial explanatory role. Philosophers appealed to abstraction to render intelligible how entities which cannot exist on their own – universals and mathematical entities in particular – can still be legiti-mately conceived independently of individual things, which are the primary existents (for some details, see Weinberg 1965, pp. 6-12). This same pattern was exploited by nominalists/empiricists as well for their own purposes.

Philosophers in either of these camps (Aristotelians and nominal-ists/empiricists) could argue that two kinds of abstraction are needed. Uni-versalising abstraction accounts for universals either in the Aristotelian manner, according to which universals exist only in primary substances, or in the nominalist/empiricist manner, according to which universals exist only as “general names” (for the latter account, see Locke’s Essay Concerning Hu-man Understanding, bk. II, Ch. xi, §2, where it is explained that in abstraction the mind separates particular ideas from the “circumstances of real exis-tence” and thereby converts them into general names; without this possibil-ity, every object distinguished in experience would have to possess a name

concept is supposed to result from the power of the mind to reflect on objects which it encounters in experience and compare them with one another so as to select or abstract from them a feature with respect to which they are similar. Secondly, this account of origin leads to the description of a concept as a general representation; a concept classi-fies objects by dividing them into similarity-classes, and a concept itself is a mark of such a similarity shared by objects falling under it.

This explanation is complemented by the further assumption that the abstractionist process of concept-formation always results in de-terminate hierarchies of concepts. These hierarchies organise them-selves according to the notions of genus, differentia and species.¹⁹ On of its own and there would have to be endless names). Separating abstrac-tion in turn is needed to account for the prima facie problematic character of mathematical entities (for Aristotle’s views, see Gambra (op. cit) and Heath (1949)). It is the sort of abstraction that enables us to separate, in our intel-lect, those aspects of things that cannot exist separately in reality. By elimi-nating more and more of a thing’s perceptible qualities, separating abstrac-tion yields the most ‘abstract’ properties, namely continuous and discrete quantities, which are then available for systematic study in geometry and arithmetic. At the time of Frege and Russell, this latter variety of abstraction was still widely appealed to by philosophers and mathematicians alike and Frege in particular spent some time in disposing the idea (see Frege 1884,

§§29-44, 1899; cf. Dummett 1991, pp. 83-85). This variety of abstraction does not figure in Kant’s discussion (although it is clear from his conception of mathematical concepts that he would have rejected separating abstraction as an explanation for mathematical entities); the concepts of traditional logic the terms of propositions which figure in syllogistic reasoning are ones to which universalising abstraction was taken to apply, hence the focus of Kant’s criticisms.

19 As Cassirer (1910, Ch. 1) points out, the hierarchical picture of the structure of concepts and the abstractionist explanation of their origin are independent of each other. The process of abstraction, as traditionally de-scribed, does not put any constraints on its possible outcomes. These straints come from a different source, when the rudimentary picture of con-cept-formation is backed up by substantial metaphysical assumptions. As Cassirer puts it, “[i]n the system of Aristotle [...] the gaps that are left in logic [sc. concept-formation] are filled and made good by the Aristotelian meta-physics” (id. p. 7). That is, “[t]he determination of concepts according to its next highest genus and its specific difference reproduces the process by

this view, concepts form a hierarchy, on the top of which lies the su-preme, or most general concept, which applies to all objects of com-parison. Transition to a new, more specific concept (species) is ef-fected by adding to the higher concept (genus) a new feature (differ-entia), which characterises only some of the objects belonging to the original similarity-class; conversely, from a given species-concept a more general one is formed by disregarding some feature, and bring-ing more objects into a similarity-class. For example, if the concept material body is taken as a genus, a species falling under it is distin-guished by adding the concept animate, which serves to divide the class of material bodies into two non-overlapping classes (animate and inanimate material bodies). The former class is further divided by dint of the concepts animal and vegetable; the concept animal has under it the concept human, which is arrived at by adding the differentiating feature of rationality, and so on. There are thus two sides to the struc-ture of concepts, corresponding to whether we move downwards or upwards in the hierarchy: the extension of a concept consists in all those concepts that are lower to it in the hierarchy; the intension or content of a concept includes all those concepts that are contained in it – ascending the hierarchy, we see that the concept human being contains (at least) the concepts rational, animate and material body.

From the standpoint of traditional logic, a complex concept is essentially a conjunction of its characteristics or partial concepts (those that together form its extension), and concept-formation is simply an act whereby from a given concept a new, more specific one is formed by adding a differentia. Thus, exhibiting the content of a concept in accordance with its logical form or structure is to exhibit it as a conjunction of its partial concepts.

which the substance successively unfolds itself in the special forms of being”

(ibid.) This independence, it may be added, is demonstrated by such exam-ples as that provided by Locke. His doctrine of general ideas was thoroughly abstractionist; at the same time, he was thoroughly sceptical of Aristotelian natural kinds.