According to Bertrand Russell, the sense in which every object is ‘one’ is a very shadowy sense because it is applicable to everything alike. However, Russell argues, the sense in which a class may be said to have one member is quite precise. “A class u has one member when u is not null, and ‘x and y are us’ implies ‘x is identical with y’.” In this case the one-ness is a property of a class and Russell calls this class a unit-class. Thus, Russell claims further, the number ‘one’ is not to be asserted of terms but of classes having one member in the above-defined sense. The same distinction between the different uses of ‘one’ was also made by Frege and Couturat. Frege says that the sense in which every object is ‘one’ is very imprecise, that is, every single object possesses this property. However, Frege argues that when one speaks of ‘the number one’, one indicates by means of the definite article a definite and unique object of scientific study. In his reply to Poincaré’s critique of the logicist programme, Couturat says that the confusion which exists in Poincaré’s mind arises from the double meaning of the word for ‘one’, that is, it is used both as a name of a number and as an indefinite article:
To sum up, it is not enough to conceive any one object to conceive the number one, nor to think of two objects together to have by that alone the idea of the number two.
According to Couturat, from the fact that the proposition “x and y are the elements of the class u” contains the symbols x and y one should not conclude that the number two is implied in this proposition. As a result, from the viewpoint of Russell, Couturat and Frege, the neo-Kantians are making here an elementary logical mistake. This awakens an interesting question. Why the neo-Kantians did not notice the mistake they had made? The answer is not that they would not have been aware of the opinion of the logicists. Both Cohn and Cassirer discuss the above-mentioned passage in Russell’s Principles. However, although Cohn and Cassirer were familiar with the distinction presented by Russell, it did not convince them. In Cohn’s view, Russell’s unit-class does not define ‘one’ but ‘only one’. As Cohn sees it, ‘only one’ means the limitation of a class to one object. Thus Russell’s ‘unit-class’ already presupposes that an object is seen as a unit. As a result, Russell’s definition of ‘one’ is unsuccessful since it already presupposes the number ‘one’. Cassirer, too, refers to Russell’s explanation, according to which it is naturally incontestable that every member of a class is in some sense one, but, Cassirer says, it does not follow from this that the concept of ‘one’ is presupposed. Cassirer mentions also Russell’s explanation according to which the meaning of the assertion that a class u possesses ‘one’ member is determined by the fact that this class is not null and that if x and y are u, then x is identical with y. According to Cassirer, the logical function of number is here not so much deduced as rather described by a technical circumlocution. Cassirer argues that in order to comprehend Russell’s explanation it is necessary that the term x is understood as identical with itself, and at the same time it is related to another term y and the former is judged as agreeing with or differing from the latter. In Cassirer’s view, if this process of positing and differentiation is accepted, then all that has been done will be to presuppose the number in the sense of the theory of ordinal number.
The neo-Kantian critique cannot be explained away as a mere logical error. The real reason why they did not accept the distinction is that to accept it would be to accept at least part of the logicist programme. As Warren Goldfarb has pointed out, Poincaré’s argument will be logically in error only if one simultaneously accepts the analysis of notions ‘in no case’ and ‘a class with one object’ that was first made available through modern mathematical logic. In other words, the logicists claim that the appearance of circularity is eliminated when one distinguishes uses of numerical expressions that can be replaced by purely quantificational devices from the full-blooded uses of such expressions that the formal definition is meant to underwrite. Hence the notions ‘in no case’ and ‘a class with one object’ do not presuppose any number theory if one simultaneously accepts the analysis which first-order quantificational logic provides for them. Poincaré does not accept this analysis, and, as result, he can bring the charge of petitio principii.
Like Poincaré, the neo-Kantians were not ready to accept Russell’s analysis of the expression ‘a class with one object’. As they see it, although the notion ‘a class with one object’ does not presuppose the number ‘one’ if one accepts the logicist definition of number, it will presuppose it if one advocates a neo-Kantian theory of number. According to Cassirer, the concept of number is the first and truest expression of rational method in general. Later Cassirer added that number is not merely a product of pure thought but its very prototype and source. It not only originates from the pure regularities of thought but designates the primary and original act to which these regularities ultimately go back. In Natorp’s view, number is the purest and simplest product of thought. Natorp claims that the first precondition for the logical understanding of number is the insight that number has nothing to do with the existing things but that number is only concerned with the pure regularities of thought. Natorp connects number to the fundamental logical function of quantity. In his view, the quantitative function of thought is produced when multiplicity is singled out from the fundamental relation between unity and multiplicity. Moreover, multiplicity is a plurality of distinguishable elements. Plurality, in turn, is necessarily a plurality of unities. Thus unity in the sense of numerical oneness is the unavoidable starting-point, the indispensable foundation of every quantitative positing of pure thought. According to Natorp, the quantitative positing of thought proceeds in three steps. First, pure thought posits something as one. What is posited as one is not important (it can be the world, an atom, and so on). It is only something to which the thought attaches the character of oneness. Second, the positing of the one can be repeated in the sense that while the one remains posited, we can posit always another in comparison with it. This is the way in which we attain plurality. Third and last, we collect the individual positings into a whole, that is, to a new unity in the sense of a unity of several. In this way we attain a definite plurality, that is, “so much” as distinguished from an indefinite set. In other words, one and one and one, and so forth, are here joined to new mental unities (duality, triplicity, and so forth).
According to Cohn, the natural numbers are the most abstract objects possible. Everything thinkable can be an object, and every object has two elements: the thinking-form and the objectivity. The thinking-form belongs to every object, and Cohn calls it “positing”. It can be described by saying that every object is identical with itself. This formal definiteness of an object has nothing to do with the determination of an object with regard to content. Since the thinking-form belongs to every object in the same way, it alone is not enough to form any specific object. The particularity of any individual object, or as Cohn puts it, the objectivity of any individual object, is something new and foreign when compared to the thinking-form of the object. In other words, Cohn argues that the necessary elements of every object are the thinking-form, and the objectivity. As a result, natural numbers are objects which have the thinking-form of identity and the minimum of objectivity, that is, the form of identity must be thought to be filled with something in some way or other. Moreover, Cohn says that his theory of natural numbers presupposes the possibility of arbitrary object-formation, that is, the possibility to construct arbitrarily many objects. On the basis of these two logical presuppositions, Cohn says that we are able to form arbitrarily many objects which are all equal with each other. According to Cohn, all of these objects can be described by the same symbol 1, and after this operation the fundamental equation 1 = 1 can be presented. Cohn says that the two separate symbols 1 in the equation signify different unities and the sign of equality means only that in any arithmetical relation any arbitrary unity can be replaced with any other unity. Moreover, Cohn says that we can collect an arbitrary number of objects into an aggregate, that is, into a new object. This is expressed by the repeated use of the word ‘and’. In arithmetic the combination of unities into a new unity has the form: 1 + 1 + 1 and so on (when ‘and’ is replaced by ‘+’). The most simple combination (1 + 1) can be described as 2, the following one (1 + 1 + 1) as 3, and so on. Thus a new number can always be attained by adding a new unity.