There is evidence in Kant of the idea that concepts of particular numbers, such as the number 5, are derived from the representation of units, and in particular pure units, that is, units that are qualitatively indistinguishable. Frege, in contrast, rejects any attempt to derive concepts of number from the representation of units. In the Foundations of Arithmetic, he softens up his reader for his groundbreaking and unintuitive analysis of number by attacking alternative views, and he devotes the majority of this attack to the units view, with particular attention to pure units.Footnote ¹ Since Frege, the units view has been all but abandoned. Nevertheless, the idea that concepts of number are derived from the representation of units has a long history, beginning with the ancient Greeks, and was prevalent among Frege's contemporaries. I am not interested in resurrecting the units view or in righting wrongs in Frege's criticisms of his contemporaries. Rather, I am interested in the program of deriving concepts of number from pure units and its history from Kant to Frege. An examination of that history helps us understand the units view in a way that Frege's criticisms do not, and in the process uncovers important features of both Kant's and Frege's views. I will argue that, although they had deep differences, Kant and Frege share assumptions about what such a view would require and about the limits of conceptual representation. I will also argue that they would have rejected the accounts given by some of Frege's contemporaries for the same reasons. Despite these agreements, however, there is evidence that Kant thinks that space and time play a role in overcoming the limitations of conceptual representation, while Frege argues that they do not.

Because this paper examines views from Kant to Frege, in other respects it must remain fairly focused. For example, it concentrates on accounts of number concepts themselves, and only indirectly considers accounts of arithmetical operations. Furthermore, Kant thought that temporal succession and the representation of order are central to arithmetical cognition, but I will highlight the role of space and the representation of collections. This paper is an exploration of one theme found in Kant's statements about arithmetical cognition; a complete account of Kant's views will substantially qualify the view I attribute to him here.

This paper will begin with some background concerning Kant's philosophy of mathematics and Greek mathematics before turning to Frege's contemporaries and how Kant would have evaluated them. It will then take up Frege's criticisms of his contemporaries and the relation between Frege's and Kant's views. Finally, it will consider what each has to say about the possibility of a role for space or time.

1. Kant and concepts of number from pure units

Kant's philosophy of mathematics rests on his theory of magnitudes. Before the nineteenth-century arithmetization of mathematics placed natural numbers at its foundation, it was common to describe mathematics as the science of magnitudes, a view that can be traced to the Greek mathematical tradition, in particular to the Eudoxian theory of proportions found in Euclid's Books V and VII.Footnote ² The Eudoxian theory provides a general treatment of magnitudes that does not presuppose numbers and yet covers both continuous magnitudes such as lines, planes, and volumes, and discrete magnitudes, such as numbers.Footnote ³ Magnitudes are homogeneous within their kinds; that is, magnitudes that are homogeneous can be composed to constitute more of exactly the same kind—lines with lines, areas with areas, and so on.

Kant follows the Greek conception when he defines a magnitude as a homogeneous manifold, but he moves beyond it to provide a deeper analysis of homogeneity and a theory of our cognition of homogeneous manifolds. In Kant's view a homogeneous manifold exhibits numerical difference without qualitative difference. In other words, a homogenous manifold consists of a manifold of parts that are qualitatively identical.Footnote ⁴ This fact about homogeneous manifolds has important consequences for our cognition of them. Kant states in the Amphiboly of the Critique of Pure Reason that Leibniz's Principle of the Identity of Indiscernibles (PII) could not be disputed if objects were cognized solely by means of the understanding, and hence by means of concepts alone, without the aid of intuition (A263–4/B319–20).Footnote ⁵ This important counterfactual asserts a fundamental limitation of conceptual representation—our inability to represent numerical diversity with qualitative identity by means of concepts alone. Kant emphasizes this limitation by asserting that an attempt to use concepts alone to represent a multiplicity of identical things inevitably collapses into the representation of just one thing.Footnote ⁶ Consequently, concepts on their own cannot represent a homogeneous manifold.

In contrast, Kant denies that PII is true of appearances, because they are objects of intuition, and intuition allows us to represent numerical difference with qualitative identity.Footnote ⁷ Since intuition can represent numerical difference without qualitative difference, it allows us to represent a homogeneous manifold, which I have argued elsewhere is in Kant's view a fundamental role for intuition in mathematical cognition.Footnote ⁸

It is more difficult to see what role intuition should play in Kant's philosophy of arithmetic than in his geometry. The representation of succession is undoubtedly fundamental, but the role of intuition in representing discrete magnitudes is also central. When Kant discusses arithmetical cognition in the Critique of Pure Reason, he refers to discrete magnitudes such as fingers or dots on a page, which are both discrete and spatially disconnected (B15-6). They are related to our representation of units in some way similar to the way that a drawn triangle is related to our a priori construction of a triangle.Footnote ⁹

Kant's treatment of discrete magnitudes suggests another important connection between Kant and Greek mathematics. Reconstructing the Greek conception of number is a formidable challenge; I will limit myself to highlighting a few relevant features.Footnote ¹⁰

The Greek term for number, ‘arithmos’, had different senses. The thinnest sense seems to have been nothing more than a particular collection of things, such as the number of sheep in a particular field or the number of shoes in my closet, in roughly the way we think of a particular set. This notion of arithmos presupposes a choice of unit, such as a shoe or pairs of shoes. Another closely related sense of arithmos was the number that resulted from enumerating the members of a collection, which is sometimes called the ‘counting-number’ of the counted collection. It, too, presupposes the choice of a unit, which is often called a ‘counting-concept.’Footnote ¹¹ There is also a further conception of number apart from a kind of thing, one that is based on collections of ‘pure’ units. The motive for pure units may arise from the desire for a general representation of number apart from any kind of thing counted.Footnote ¹² The close connection of arithmos to particular collections of things and the notion of number as a collection of units were influential into the nineteenth century.

We can distinguish two sorts of purity that are included in the generality to which pure units aspire. Consider the seven samurai. In the first sort of purity, we would like the units to represent any seven samurai, not the seven masterless samurai who defended a small village in the sixteenth century. This requires that the units of the collection are ‘equal and not in the least different from each other,’ as Plato puts it.Footnote ¹³ The units could still have the characteristics common to all the things counted (that is, belonging to the Japanese warrior class) but the units would not be distinguished from each other by any further characteristics (such as being old, or short, or hot-headed). In the second sort of purity, we would like to represent seven of anything, be they seven samurai, seven swans-a-swimming, or the seven habits of highly effective people. This requires that the units also have no characteristics common to any particular kind of thing, such as being a samurai. Of course, if a pure unit has no characteristics at all other than being a unit, then both kinds of generality would be achieved, but the first kind of generality raises particular challenges for a conception of number based on units. How can we represent the units of a collection as distinct from each other without representing them with distinguishing characteristics? If we do not represent any distinguishing characteristics, then we are only left with the representation of a samurai, for example, with nothing to distinguish seven of them. But if we represent each samurai with distinguishing properties, then we lose the generality of the representation, and it will only represent a collection of samurai with those particular distinguishing characteristics. There is, then, a prima facie tension between a completely general representation of number by means of units and the particular characteristics required to distinguish those units. I will call this the ‘pure plurality problem’.

The idea that number is based on pure units bears a striking resemblance to what I have said about Kant's appeal to a homogeneous manifold of discrete units. In Kant's view, concepts on their own could never represent a collection of units as distinct and yet without distinguishing characteristics, since concepts on their own represent by means of qualities, which are distinguishing characteristics. Intuition, on the other hand, allows us to represent pure numerical difference without specific difference, which is just what is required for the representation of pure units.

Before saying more, I would like to lay to rest a possible confusion. First, the units I have described would be pure in the sense of being completely free of qualitative differences. This is not to be confused with Kant's notion of purity in relation to the a priori, which requires that there be no admixture of anything empirical. Any sense of purity associated with the a priori is in addition to purity in the sense of being qualitatively identical, and unless I explicitly indicate otherwise, I will mean the latter. For all I have said, someone might think that we could in some way rely on empirical experience, perhaps in combination with abstraction, to represent pure units. Second, Kant indeed holds that arithmetic is a priori and hence not dependent on particular empirical experience for its justification, even if we use empirical intuitions such as fingers as an aid. He holds that, despite the aid of empirical intuition in their representation, number concepts themselves are pure in the sense of a priori. In that case, however, what is it that the intuition of the fingers aids us in representing when ‘seeing the number 12 arise’ (B16)? It is plausible that in Kant's account, at least part of the answer is the pure intuition of different locations in space underlying the empirical intuition of the fingers. The resulting representation of units would be pure in both senses—qualitatively identical and free of empirical content.

What textual evidence do we have that Kant thinks that arithmetical cognition in particular rests on the representation of pure—that is, qualitatively identical—units? In addition to referring to fingers and dots on a page in his discussion of the role of intuition in arithmetical cognition (B16), Kant also refers to strokes and beads of an abacus:

Kant states explicitly that these empirical representations aid us in forming an a priori representation of the synthesis of units. It would be natural to suppose that an a priori and fully general representation of numbers based on units would have no admixture of the empirical in it, and hence would retain none of the distinguishing empirical characteristics of fingers or dots or beads or strokes, representing instead qualitatively identical units.

Comparison to Kant's comments on the relation between a drawn triangle and the a priori representation of a triangle is helpful on this point. Kant states that

A similar attention to construction in the case of arithmetic and hence a similar abstraction from the ‘entirely indifferent’ properties of units would lead to a representation of units as qualitatively identical. In other words, our fingers would be an aid to the a priori representation of units distinguished only by their difference in spatial location.Footnote ¹⁴

Kant's descriptions of arithmetical cognition as cognition of magnitudes provide more direct evidence of this view. Any reference to magnitudes is a reference to a homogeneous manifold in intuition, and as I mentioned above, in Kant's view a homogeneous manifold consists of qualitatively identical elements. Moreover, Kant repeatedly describes arithmetical cognition as a synthesis of homogeneous units in particular. In the Schematism of the Critique of Pure Reason, for example, Kant states that ‘the pure schema of magnitude (quantitas) is a representation that summarizes the successive addition of one to one (homogeneous)’ (A142–3/B182; see also A164/B205, A242/B300).

In Kant's time, however, there was also another, less stringent notion of homogeneity that was often mentioned as a condition of counting. Counting was said to require that one employs a counting-concept under which all the things to be counted fall; to count a collection of jaguar and tapir, for example, one would have to ascend at least to the concept of mammal, say, and count the items under this concept. As mentioned above, the counting-concept requirement is found in ancient Greek arithmetic, but the role of counting-concepts should not be restricted to acts of counting; any representation of a collection was thought to require a common concept under which the elements fall. In Kant's time, the things that fall under such a concept were said to be homogeneous with respect to that concept. Clearly, this sort of homogeneity does not require that the units be qualitatively identical.Footnote ¹⁵

I believe, however, that it is precisely here that Kant, following his deeper analysis of the Greek notion of homogeneity, introduces a further notion of homogeneity whose representation is required for a priori knowledge in geometry and arithmetic. This homogeneity is the property of numerical difference without specific difference, a property of magnitudes that intuition allows us to represent. The fact that intuition allows us to represent this further kind of homogeneity constitutes a fundamental role for intuition in all mathematical cognition.Footnote ¹⁶ This account of homogeneity derives further support from, on the one hand, the conception of early modern algebra as a science of magnitude common to geometry and arithmetic, and on the other, the connection between early modern algebra and the Eudoxian theory of proportions.Footnote ¹⁷ Kant's deeper notion of homogeneity does not displace the common-concept sense of homogeneity; the representation of a collection of seven samurai still requires the common concept of samurai. Nevertheless, a full account of our representation of seven samurai as a collection of seven also requires the deeper notion of homogeneity.Footnote ¹⁸

There is another interpretive possibility that seems to obviate the representation of pure units. Kant distinguishes between images and schemata. As an example of an image of the number five, Kant refers to five points set alongside one another. The schema, in contrast, allows us to think of a number in general and is a universal procedure of imagination for providing an image for a number concept. In the case of arithmetic, Kant states

These passages can leave the impression that arithmetical cognition is based only on a rule of successive synthesis in time, and hence suggest that the representation of pure units in space, which might seem to fall on the side of images, are strictly speaking unnecessary. Indeed, these passages may be taken as evidence of a rather abstract view of arithmetic as mere rules or techniques of calculation for finding the magnitudes of objects.Footnote ¹⁹ Most importantly, however, is the point Kant makes explicit in the geometrical case: the schema of a triangle is a rule of the synthesis of imagination for the construction of pure figures in space, and its being a rule rather than an image accounts for the generality of the conclusions we base on geometrical constructions (A140–1/B179–80). There are important differences between the generality of geometrical claims and arithmetical claims such as 5 + 7 = 12, as Kant himself emphasized. Yet, if Kant appeals to schemata to explain the generality of geometrical claims, it seems possible that he also appeals to schemata to explain any generality underlying particular arithmetical claims or particular numerical concepts. In short, if the generality of mathematical claims rests on schemata (understood as rules for construction) rather than images, then it seems there is no need to represent pure units to achieve a fully general representation of number.Footnote ²⁰ If successful, Kant's appeal to rules to explain generality would circumvent the pure plurality problem that arises from an attempt to explain the generality of our representation of five, for example, through the representation of pure units.

This shows that Kant's view is more sophisticated and complex than the simple pure units view outlined above. While Kant does appeal to schema to explain the generality of mathematical claims, the representation of pure units still plays an essential role in our cognition of fundamental a priori arithmetical truths, such as 5 + 7 = 12. At the very least, Kant seems to think that the representation of pure units is required for us to acquire the concepts of number (to ‘see the number 12 arise’) or to establish the objective validity of our basic arithmetical concepts, and to do so in a manner that preserves the generality achieved by appealing to schemata. In short, what Kant claims in the Schematism is compatible with the further claim that when we take things like fingers as an aid in our pure a priori cognition of 5 + 7 = 12, our taking them as representations of pure units is part of the schema of number itself. In fact, this last possibility is suggested by the Schematism passage just quoted, which refers to the successive addition of ‘(one to one) homogeneous’, that is, homogeneous units.

I have given textual and interpretive grounds for thinking that in Kant's view, arithmetical cognition requires the representation of pure units. As with so many of Kant's views, one could wish for clearer expressions of it in the Critique. Kant comes close to making the qualitative identity of units explicit in the Discipline of Pure Reason, where he states:

On my interpretation, the explicit mention of multiple instantiations of ‘one and the same in time and space’ is a reference to units distinguished only by their location in time and space and to Kant's discussion of numerically distinct indiscernibles in the Amphiboly.

The clearest statement of the pure units view is not found in Kant himself, but in his foremost defender and expositor Johann Schultz. His Prüfung der Kantischen Critik der reinen Vernunft, published in 1789, is an examination and defense of Kant's philosophy with a particular emphasis on his philosophy of mathematics. Schultz's work and the correspondence between them are valuable resources for insights into Kant's views.Footnote ²¹

As is well known, Schultz disagrees with Kant in thinking arithmetic is grounded in axioms, in particular, the axioms of commutivity and associativity.Footnote ²² Schultz states that the fact that mathematics rests on intuition can be shown both from the nature of its axioms and postulates and from the concept of counting itself. We can set aside Schultz's views on the axioms and postulates, for it is what he says about the concept of counting that is of particular interest to us. Schultz states:

This passage reflects Kant's own claims about the limits of conceptual representation and provide strong indirect evidence that Kant thought that our representation of numbers depends in some way on the representation of pure units.Footnote ²³

More would be required to defend and elaborate this interpretation of Kant. In particular, one would need to fill out the account of the Schematism and explain the relation between the representations of temporal succession and of pure units, but I will not attempt to do so here. Instead, I would like to concentrate on the idea that our concepts of number depend on the representation of units—its historical influence and its fate—from Kant to Frege. When discussing Kant in what follows, I will focus on his views on the limitations on conceptually representing pure units and the role of intuition in overcoming those limitations.

2. Numbers from units in Frege's contemporaries

Those of Frege's contemporaries who appealled to units were inclined to empiricism and, in so far as they considered the basis of mathematical cognition, leaned toward empirical psychology. Several espoused versions of formalism. In general, they do not appear to be particularly interested in Kant. In considering how Kant would react to them, I will set aside his most obvious criticism of empiricism: that no account that relies on empirical experience for justification of mathematical claims can explain the a priori status of mathematics. Likewise, in considering Frege's criticisms I will set aside arguments against the psychologism and subjectivism of mental abstraction as well as his arguments against formalism. I will instead focus on the way in which Frege's contemporaries thought that concepts of number were derived from the representation of units, and how Kant and Frege would or did respond to that view in particular.

Before I begin, a few points about terminology will be helpful. First, the German term ‘Anzahl’ was used in rough correspondence to the Greek notion of arithmoi, that is for a collection or a counting-number of a particular collection. The German term ‘Zahl’ was also used in this way, but ‘Zahl’ was first and foremost used for the notion of an individual number, such as the number 5, apart from any collection; some who wrote on mathematical foundations reserved it for the latter. Usage was by no means settled, however, and these terms were used in different ways, sometimes even interchangeably.Footnote ²⁴ To help avoid confusion, I will use ‘number’ only to translate ‘Zahl’ and will leave ‘Anzahl’ untranslated.

Second, in its most general use the German term ‘gleich’ was and remains highly ambiguous. It is often translated ‘equal,’ though it is also often translated ‘identical’, which is itself ambiguous. In many contexts, it is probably best translated by the correspondingly ambiguous term ‘same’. When describing a relation between two or more things, ‘gleich’ has at least five distinct senses, three of which correlate with three senses of identity:

1. 1. Similar, that is, the same with respect to some property, relation, or circumstance; e.g. the same in color;

2. 2. Equal, that is, the same with respect to particular quantitative properties; e.g. the same in length or volume;

3. 3. Practically Identical, that is, the same with respect to all readily noticeable intrinsic properties; e.g. the sameness of some identical twins;

4. 4. Qualitatively Identical, that is, completely the same with regard to all intrinsic properties; e.g. the sameness of two identical rain drops (supposing that they in fact share all intrinsic properties);

5. 5. Strictly or Numerically Identical, that is, one and the same; e.g. the sameness of Cicero and Tully.

Only the fifth sense of equal and identical are absolute; the first four senses are all notions of equal or identical in some respect, that is, in respect to some property or kind of property.Footnote ²⁵ One must keep these possible senses of equality and identity in mind when reading Frege's contemporaries.

There is, however, a further use of ‘gleich’ that is crucial to take into account:

1. 6. Equipollent, that is, the elements of one collection stand in a one–to–one correspondence with the elements of another collection.

This is also a notion of ‘equality in some respect,’ but this sort of equality asserts a one–to–one correspondence between the elements of one collection and the elements of another without further conditions on the elements themselves. This sense of equality between collections must be sharply distinguished from any requirement of sameness among elements of a collection. ‘Gleich’ is used for both. Moreover, it would be easy to run them together, since the relation of equipollence and the representation of pure units might both be thought to play a role in the representation of the natural numbers. As we will see in the next section, failing to carefully distinguish them can lead to misunderstandings.

With these preliminaries aside, let us turn to Frege's contemporaries. Rudolph Lipschitz opens his 1877 Lehrbuch der Analysis with a discussion of number comprising just two short paragraphs.Footnote ²⁶ He states that the concept of Anzahl arises from regarding separated things and neglecting their characteristics, adding that one who wishes to obtain an overview of certain given things will begin with a certain thing and add a new thing to the earlier ones. This procedure is called counting and results in a fully determinate concept of number [Zahl] for a particular collection of given things.

Lipschitz is so brief that it is hard to be sure of his position. In fairness, it should be noted that some of those whom Frege attacks, and in particular Lipschitz, do not show great interest in the foundations of arithmetic, and seem eager to get on to analysis, analytic functions, or other topics. Regardless, Lipschitz does not acknowledge the need for generality. From the Kantian perspective, Lipschitz simply fails to see what would be required of a general concept of number based on units and hence fails to address the pure plurality problem.Footnote ²⁷

I turn next to Ernst Schröder. His 1873 Lehrbuch der Arithmetik und Algebra sets out to give an account of the natural numbers, or natürlichen Zahlen, but immediately begins with the conditions of counting, and instead discusses Anzahl, where that is understood as a number of things.Footnote ²⁸

Schröder calls anything that is to be counted a unit [Einheit] (5), and says that the conditions of counting require that units be distinguished from each other in our representation of them through some characteristic (3). Counting also requires that the objects appear to us to be similar [ähnlich] to each other, though this similarity condition is extremely minimal, for it can be satisfied by any accidental relation, such as being counted by us. Schröder slides from similarity [ähnlich] to equality [gleich] when he states that the question of the Anzahl of things arises only when the things are viewed as equal to each other; clearly, he has in mind the minimal understanding of ‘gleich’ as similarity, corresponding to sense 1 listed above.

Schröder distinguishes between the equality among units of a collection and equality between collections. The latter is defined as an equipollence between two collections, that is, two collections are said to occur ‘in equal Anzahl [in gleicher Anzahl]’ when there is a 1–1 correspondence between their members (8). This, however, imposes no further conditions on the elements themselves.

Since Schröder's account of the notion of equality among units is extremely minimal and does not rule out distinguishing characteristics, his units are not pure. From a Kantian perspective, Schröder's treatment of Anzahl fails to acknowledge the need for a general representation of units.

This is only half of Schröder's account, however, for he has a different characterization of number itself, that is, Zahl. (It is noteworthy that Frege does not refer to this second characterization in his criticisms). Schröder states that number arises by means of drawing a stroke [Strich] corresponding to each element in a collection. He calls the strokes ‘ones’ or ‘oners,’ which are represented in his text by the numeral ‘1’.Footnote ²⁹ He states that a natural number is a sum of oners, where by ‘sum’ he appears to simply mean a collection.Footnote ³⁰ Thus, a number is a special case of a collection in which the units are oners, which are used to illustrate a particular collection of units in respect of their plurality.

Oners, like any other collection of units, can also stand in relations of equipollency. The convenience of numbers in determining equipollency relations among collections recommends them and leads to their use as a measure of frequency or plurality.

After saying that number is an illustration of units in respect of their frequency, Schröder states that with collections of oners, the very least of things is illustrated—‘so little, that if the one is left out, nothing at all would remain’. He concludes that ‘the abstraction in the illustration–process of counting is the greatest,’ and entails the greatest simplification of representations (6). While numbers are identified with the collections of oners themselves, the maximal abstraction involved in illustrating units by means of them leaves no distinguishing properties. Hence in this context, Schröder means qualitative identity, which corresponds to sense 4 of ‘gleich’ listed above, and we are left with a general representation of pure units.Footnote ³¹

It is important to note that this maximal abstraction is not the result of the traditional understanding of abstraction that might, for example, proceed from the concept of a Clydesdale to the concepts of horse, animal, extended body, and being. One of the most interesting features of Frege's predecessors is that they depart from the traditional conception of abstraction to make room for a numerical kind of abstraction that suits the needs of mathematics. Unfortunately, Schröder does not further describe or defend the nature of this numerical abstraction. For the sake of argument let us grant that through some sort of numerical abstraction, we can get to a collection of oners.

Schröder's account lays claim to a completely general representation of units while simply assuming that those units remain distinguished. Since the maximal abstraction is intended to leave behind all distinguishing properties, however, there is nothing left to render the units distinct. Hence, Schröder does not provide a solution to the pure plurality problem. As Kant might put it, Schröder attempts to have his cake and eat it too.Footnote ³²

Stanley Jevons presents a quite different approach in The Principles of Science of 1874.Footnote ³³ Jevons states that ‘with difference arises plurality’ and that ‘[N]umber is but another name for diversity’ (175). Jevons distinguishes between concrete number and abstract number; the former corresponds to arithmoi and Anzahl—for example, three horses—while the latter refers to numbers simpliciter, like the number three.

Unlike Schröder, Jevons explicitly distinguishes numerical abstraction from logical abstraction. The latter gives us a conceptual representation of an abstract quality, as when we abstract from several objects that are red to the conception of the quality redness (33). In this sort of abstraction, we ‘drop out of notice the very existence of difference and plurality’ among the red objects (178). In contrast, numerical abstraction consists in abstracting from ‘the character of the difference from which plurality arises, retaining merely the fact’ (177; see also 178). In keeping with this conception of numerical abstraction, he describes abstract number as merely ‘the empty form of difference (177)’. Jevons holds that leaving out both the kinds of objects numbered and any specification of their differences while retaining the fact of their differences accounts for the generality of number. He is explicit that the notions of both concrete and abstract number specify that the units counted are to be distinguished, while leaving unspecified what characteristics will distinguish them (177–8). In short, Jevons explicitly embraces a concept of number that appeals to indeterminate distinguishing characteristics, and he singles out a special form of abstraction that results in such a concept. We seem to have found a solution to the difficulty reconciling the generality of number with the need for distinguishing characteristics, a solution that does not require the representation of pure units, and hence avoids the pure plurality problem.

Jevons' solution is not without its own difficulties, however. It is essential to the generality of concepts that they represent indeterminately, that is, that they need not specify all the properties an object might have. But this is a different sort of indeterminacy from that which Jevons proposes. For example, the concept of being a shape requires that an object have specific characteristics, being completely bounded in space, say, while it simply leaves out the characteristics corresponding to a particular shape. In contrast, a concept that results from Jevons' numerical abstraction retains the fact of difference, which requires that each unit must have the characteristic of having (or lacking) one or more characteristics that, in coordination with the one or more characteristics that each of the other units have (or lack), is sufficient to distinguish it from each of the other units. The traditional understanding of conceptual representation, according to which characteristics simply represent particular qualities, simply does not make room for this sort of indeterminacy.

One might well respond: so much for the traditional understanding of a concept, which is also inadequate for other sorts of conceptual representation, and I would agree. What we are interested in, however, is how matters were viewed by Kant and others up to and including Frege. In general, Kant's discussions of concepts reflect a traditional understanding of the constitution of concepts. It seems likely that Kant at least implicitly rules out Jevons' approach, and hence Jevons' solution to the pure plurality problem.

The last text I will consider is Johannes Thomae's 1880 edition of Elementare Theorie der Analytische Functionen, which begins with the assumption that we can count.Footnote ³⁴ He states that counting rests on our ability to abstract from the peculiarity [Eigentumlickheit] of the individuals of a collection of objects and then give successive names to different collections of such objects. He thinks that we can abstract from all peculiarity of objects without collapsing the collection. As a result, each unit is represented as equal to every other, and the concept of number arises from the counting of such units (1). In his view, the equality of units consists in their substitutability, which in turn consists in their lack of distinguishing properties after abstraction. Significantly, however, Thomae also states that abstraction does not remove the differences in spatial or temporal location of the units (1). Thus, Thomae holds the view I have attributed to Kant and is found in Schultz, according to which spatial and temporal location allow us to represent otherwise indistinguishable units and thereby contribute to solving the pure plurality problem.Footnote ³⁵

3. Frege and Kant on the limits of conceptual representation

I would now like to turn to Frege's arguments in the Foundations of Arithmetic against the possibility of representing units in a way that allows concepts of number to be based on them.Footnote ³⁶ There are two terminological obstacles to understanding him. First, Frege rejects any conception of a number n as simply a collection of n things or as a property of a particular collection of n things.Footnote ³⁷ It is therefore not surprising that Frege uses ‘Zahl’ only for individual numbers. As we saw above, ‘Anzahl’ was commonly used for collections of things rather than individual numbers, but Frege departs from common usage and often uses ‘Anzahl’ as a synonym for ‘Zahl’, which widens the distance between him and those he attacks.Footnote ³⁸

Second, Frege's use of ‘gleich’ and its cognates raises particular difficulties. As noted in the previous section, ‘gleich’ is multiply ambiguous. Because Frege treats numbers as singular objects and rejects a notion of a number n as a collection of n things or a property of a particular collection of n things, by his lights, it strictly speaking makes no sense to say that the number of planets is equal to the number of coins in my pocket; if it is true, then the number of planets is one and the same number as the number of coins in my pocket. Both senses, however, can be expressed by ‘gleich’. Similarly, ‘Gleichung’ is usually translated as ‘equation,’ which connotes an equality, but Frege maintains that an equation expresses a strict identity. In order to make Frege's views clear, Austin translates ‘gleich’ as identical and ‘Gleichung’ as identity.Footnote ³⁹

Unfortunately, this leads to misunderstandings. Austin translates ‘gleich’ as ‘identical’ even where Frege is quoting or referring to the views of others, when they certainly do not mean one and the same, but mean something less than strict or numerical identity.Footnote ⁴⁰ Any translation of these terms is beset with perils, of course. Nevertheless, translating ‘gleich’ as ‘identical’ can lead to a misleading portrayal of the philosophers Frege discusses, if one has in mind strict or numerical identity.

There is a further issue about the meaning of ‘gleich’ that is of even greater importance. As noted in the previous section, ‘gleich’ is used for both equality between collections of things (i.e., equipollence) and for the purported requirement of equality or sameness among the distinct elements of a collection. It is crucial to distinguish between them. In the latter case as in the former, if we translate ‘gleich’ as ‘identical,’ we cannot understand ‘identical’ to mean one and the same, or we will again misconstrue what Frege's contemporaries had in mind. In fact, in the latter case, we will make complete nonsense of their claims, because it is an obvious contradiction for elements of a collection to be both distinct and one and the same.

Now, ‘identical’ is itself ambiguous (as is reflected in the list of senses of ‘gleich’ in the previous section). It can mean one and the same, that is, strict or numerical identity, which is probably the primary way a philosopher hears it today. But in a looser sense it can also mean that two or more things are qualitatively identical, that is, they share all of the same inner properties while remaining distinct—for example, the possibility of two identical rain drops, corresponding to sense 4 of ‘gleich’ listed above. In this context, identical units are pure units. Thus, Austin's translation of ‘gleich’ as ‘identical’ allows Frege's contemporaries to mean qualitatively identical when discussing the identity among distinct units of a collection.Footnote ⁴¹

What is also important, however, is to allow that Frege himself uses ‘identical’ in the looser sense when he describes the views of his contemporaries, and not in the sense of strict identity, which he intends for equations or the number belonging to the concepts of equinumerous collections. For if we read him as meaning strictly identical, then he will both misrepresent them and make their views manifestly absurd. It is true that Frege relishes making his opponents' views appear absurd, but to do so by misreading their use of ‘gleich’ as strictly identical would be willfully tendentious and weaken his attack. Indeed, if he understood ‘gleich’ as strictly and numerically identical, the question that composes the heading to §34—‘Are units identical [gleich] with one another?’—would have an immediate answer: it is logically impossible for a plurality of units to be one and the same unit. It is far better to understand Frege as allowing the looser, qualitative sense of identity in his attack and as arguing that his opponents' views lead to absurdity.Footnote ⁴²

Frege's argument strategy is to articulate what would be required for the unit view to succeed and to argue that those requirements cannot be met. He holds that if one is going to appeal to units, then the units must be pure, and that at least tacit recognition of this requirement motivates both the use of the maximally neutral word ‘unit’ and the appeal to identity (§34). Thus Frege and Kant are in complete agreement on the purity requirement. Moreover, both would agree that if someone, for example Lipschitz, does not acknowledge the need for pure units, then he is overlooking the requirements of a completely general representation of number by means of units.

Frege next argues that the units would also need to be distinct from each other, for where there is no diversity, there is no number (§35). Finally, he asserts that you cannot have the required sameness of units and a diverse collection of units at the same time. He cites Jevons, who refers to tokens of the numeral ‘1’ with hash marks to insure that each refers to a distinct unit, and then states in his characteristically crisp polemic:

Frege asserts that if we were to derive the concept of number from units, they would have to be identical and diverse at the same time, which is impossible. The reference to impossibility might suggest that Frege has strict identity in mind. But in that case, when Frege states that we ‘must have identity’, he is certainly not speaking for Jevons or any other adherent of the pure units view, who would only claim that we require qualitative identity. In fact, as mentioned above, this would make Jevons' view patent nonsense. If, on the other hand, Frege instead means qualitative identity, then he appears to hold that qualitative identity would lead to strict identity, which in turn excludes diversity.

This is similar to the position that Kant adopts when he considers objects as conceived by the understanding alone: you cannot represent the required units in their purity and represent them as diverse if you represent them merely by means of concepts. Thus, if we focus on Kant's views on representing by means of concepts alone, Frege and Kant agree that the requirements for representing pure units leave one in a bind (I will consider what each would think about appealing to space or time below). Both would claim that Schröder implicitly appeals to a notion of pure units without paying the price of abstracting from all distinguishing characteristics; hence, they would agree that Schröder is trying to have his cake and eat it too, and has not solved the pure plurality problem.

Let us look more closely at why Frege thinks that the units view is untenable. He describes the core of the difficulty with indistinguishable units in §36, where he says that units deprived of their distinguishing characteristics would collapse into one and be numerically identical (§39). Frege appears to be assuming some version of Leibniz's principle of the identity of indiscernibles. Section 34 provides evidence that he does; Frege asks why, in the view he is attacking, we ascribe identity to objects that are to be numbered. Frege seems to allow that identity comes in degrees and can be incomplete, which suggests that he is referring to qualitative identity. He then rules out the complete identity of two objects. Hence, Frege appears to assert that complete qualitative identity entails strict identity—the principle of the identity of indiscernibles.

Further evidence is found in §65, in which Frege states that all the laws of identity are contained in universal substitutability. The context makes it clear that Frege is here talking about strict, numerical identity. But if substitutability is the mark of strict identity, then it seems that qualitative identity would entail numerical identity, and hence some version of the principle of the identity of indiscernibles.Footnote ⁴³

This strengthens the parallel to Kant's views. As noted above, Kant states that if we only represented objects by means of concepts, then the principle of the identity of indiscernibles could not be disputed. And Kant, like Frege, describes the consequences as a kind of collapsing of the distinct units into one (§39).

There is one more important point of agreement between them. At several places in the Foundations, Frege describes traditional abstraction resulting in a general concept. In §34, for example, he states that if we disregard the distinguishing properties of a white cat and a black cat, our abstraction arrives at the concept ‘cat’ (see also §§44, 45, and 48). There are disagreements about Frege's views on abstraction, but we need not resolve whether Frege endorses traditional conceptual abstraction or merely concedes it for the sake of argument. The important point is that, even if he introduces traditional abstraction only for the sake of argument, he does so to contrast it with and to rule out a special form of abstraction that would result in number (e.g., §44). Moreover, Frege explicitly rejects the conception of numerical abstraction put forth by Jevons, that is, the possibility of abstracting to a concept which, in Jevons' terms, ‘retains the fact of difference,’ and ‘implies the existence of the requisite differences’ without specifying them.

Why does Frege reject Jevons' suggestion? He quotes Jevons' claim that numerical abstraction ‘consists in abstracting the character of the difference from which plurality arises, retaining merely the fact’. Frege then states that if we attempt to abstract from the distinguishing characteristics ‘(…) we should never get so far as to distinguish the things at all, and consequently could not retain the fact of the existence of the differences either (…)’ (§44).Footnote ⁴⁴ Frege is reasserting his view that abstracting from distinguishing properties would lead to a collapse of the units into one. In other words, he rules out the possibility of characteristics that simply ‘retain the fact of difference’.

Frege might be read as objecting to the process of numerical abstraction itself. Even if it is only for the sake of argument, however, Frege allows traditional conceptual abstraction in order to distinguish it from numerical abstraction, so what rules out numerical abstraction in particular? It seems to me unlikely that Frege has a worked out theory of mental activity that would distinguish between the former and the latter. More importantly and decisively, his earlier objection in §39 against units that are pure and yet distinct does not mention abstraction at all; nor do his discussions of Jevons in §36 and §38, where he focuses on the impossibility of units being pure yet distinguishable. He seems to think that the problem is not with the means of attaining a representation of pure units, but with the purported pure units themselves; in arguing against Jevons, he simply does not allow for Jevons' kind of characteristics. Thus, Frege seems to agree with Kant in rejecting characteristics that specify further characteristics indeterminately. It would seem that both Frege and Kant reject Jevons' solution to the problem for the same reason: a shared assumption concerning the limits of conceptual representation.Footnote ⁴⁵

4. Frege and Kant on the role of space and time

I would now like to turn to Frege's main argument that we cannot overcome the limits of conceptual representation by appealing to space or time to represent pure units. His argument presses a dilemma (§41). The first horn concerns the possibility of representing points of space as qualitatively identical. Frege states that only in themselves, apart from their relations, are points of space qualitatively identical. But, he adds, ‘if I am to think of them together, I am bound then to consider them in their collocation in space, or else they fuse irretrievably together into one’. Thus, points of space in themselves may be qualitatively identical, but they cannot be represented as distinct.

It seems that the homogeneity of points of space might support the claim that, in themselves, they are qualitatively identical. What, however, is the basis for Frege's claim that these qualitatively identical points will fuse into one? It appears that Frege either implicitly appeals to the principle of identity of indiscernibles, or thinks that this is a self-evident property of points of space, one that could be described as a case of the identity of indiscernibles.Footnote ⁴⁶ Regardless, Frege simply seems to assume that there is no possibility of a primitive distinctness among points of space in themselves. One might think, for example, that points of space, though qualitatively identical, have haecceities (or something analogous) that distinguish them one from another. This would be a contentious claim, of course. Nevertheless, the success of his main line of argument against the units view depends on his ability to close off the possibility of appealing to space and time, and without further argument for his assumption, his main line of attack is threatened.Footnote ⁴⁷

Despite these worries, however, the heart of Frege's argument appears on the second horn of the dilemma, where he claims that points of space can only be distinguished by their relations.Footnote ⁴⁸ He adds:

In short, Frege argues that the spatial relations that distinguish pure units will undermine the generality of the representation of number, because this spatiality will have to be represented in the concept of number.Footnote ⁴⁹

How would Kant respond? First, I think that Kant agrees with Frege that qualitatively identical parts and points of space are distinguished by their relations to one another.Footnote ⁵⁰ Thus, despite qualms about Frege's handling of the first horn of the dilemma, I believe Kant would find himself on the second horn. Second, Kant would deny that the relations distinguishing qualitatively identical parts of space must be represented in the concept. Kant makes room for the diversity of pure units despite their indistinguishability by, in effect, drawing a distinction between conceptual and intuitive distinguishability. The former is all that is required for the purity of pure units: conceptual indistinguishability gives us the required generality, while intuitive distinguishability accounts for the diversity of these units.

Frege might reject Kant's attempt to avoid importing a ‘special element’ into the concept of number by appealing to intuition. He holds that any means sufficient to distinguish units, whether it be spatial relations or something else, would have to be specified in the concept of their collection, and hence in the concept of number. The nature of the distinguishing property is immaterial; either it is represented in the concept or it is not. On the one hand, Kant appeals to spatial relations among parts of space to distinguish units, and on the other, he simply omits this difference from the concept. In short, Frege might say that Kant himself is trying to have his cake and eat it too.

This criticism simply underscores the differences between Frege and Kant's understanding of conceptual representation and its limits. On the criticism proffered on Frege's behalf, anything representable is in principle conceptually representable, while for Kant this is not the case, and intuitive representation can supplement conceptual representation. There is a further point brought out by the criticism, however. For in Kant's view, the generality of the representation of number is not undermined by supplementing conceptual representation with intuition. Whether Kant is entitled to this claim depends on a deeper analysis of the role of schemata in accounting for the generality of mathematical claims, as well as the role of pure units in allowing particular number concepts to arise and in establishing their objective validity in a way that does not undermine that generality. An answer to the questions raised on Frege's behalf will constrain an acceptable Kantian account of arithmetical cognition.

5. Summary

The idea that arithmetical concepts require the representation of units, and pure units in particular, arose as long ago as Plato, but its history effectively ends with Frege. Barring an overthrow of more than a century of mathematical logic, that is as it should be. Nevertheless, interest in the history of the foundations of mathematics requires that we understand this approach, the pure plurality problem, and attempts to overcome this problem. There is evidence that Kant assigns a role to the representation of pure units in the origin of particular number concepts and in establishing the objective validity of arithmetical claims. His account, however, is not subject to the pure plurality problem because he appeals to pure intuition. Understanding the views of Frege's contemporaries requires untangling ambiguous notions of equality and identity, but doing so reveals attempts to make the units view work by developing the notion of numerical abstraction and loosening the traditional understanding of conceptual representation. Clarifying their views and Frege's own notions of equality and identity allow us to better understand his arguments against them.

A comparison of Kant and Frege on pure units also uncovers unexpected features of their views. They are in closer agreement on the units view than one might have expected, and would criticize it on grounds of the limits of conceptual representation. Yet they deeply diverge on whether an appeal to space and time could make the representation of pure units possible. The core issue dividing them is whether space and time make it possible to distinguish units in a way that does not undermine the generality of the concept of number. Frege claims that they will not, but he assumes that space and time lack primitive distinctness, and his claim requires further argument. Kant thinks that they will, but whether he is entitled to this claim depends on his fuller account of the generality of mathematical cognition.Footnote ⁵¹