At the beginning of the twentieth century, neo-Kantianism was the dominant force in German academic philosophy. Its most important schools were Marburg and Southwestern (or Baden). The Marburg school concentrated on logical, methodological and epistemological themes. Its founder and leader was Hermann Cohen (1842-1918), a professor of philosophy at Marburg between 1876 and 1912. Cohen’s most famous disciples were Paul Natorp (1854-1924) and Ernst Cassirer (1874-1945). The Southwestern school emphasised the theory of values. Its founder and leader was Wilhelm Windelband (1848-1915). Windelband’s student Heinrich Rickert (1863-1936) was the great system-builder of the Southwestern school. Among the members of the Southwestern school were Jonas Cohn (1869-1947) and Bruno Bauch (1877-1942). At the beginning of the twentieth century, the philosophy of mathematics in general and the nature of number in particular were subjects of lively discussion among the neo-Kantians. Natorp, Cassirer and Cohn, among others, constructed their own theories of number which also formed the basis of their critiques of Russell and Frege. The neo-Kantians, too, supported the idea that mathematics should be based on a logical foundation. However, their conception of the logical foundation differs greatly from that of Russell and Frege. The main difference is that although the neo-Kantians argue that mathematics should be based on a logical foundation, these two sciences must be strictly separated from one another. Consequently, they argued that if the logicist programme were carried out, there would not exist any line of demarcation between logic and mathematics. Cohn’s distinction between two possible ways to found the number concept on logic brings forward the main difference between Russell’s and the neo-Kantians’ viewpoint. According to Cohn, there exist two possible ways to found the number concept on logic. Either the number concept is reduced to a logical concept or it is shown that the number concept itself is a fundamental logical concept. Cohn says that while Russell’s theory of number is founded on logic in the first sense, his own theory of number is logical in the latter sense.
According to the neo-Kantians, deducing the number concept from the class concept is a petitio principii. In other words, Cassirer, Natorp and Cohn all argue that the class concept already presupposes the number concept. In Cassirer’s own words,
What it means to apprehend an object as “one” is here assumed to be known from the very beginning; for the numerical equality of two classes is known solely by the fact that we coordinate with each element of the first class one and only one of the second.
According to Cohn, the number concept is something that logically precedes the class concept. As Cohn sees it, Russell’s definitions often contain such expressions as “an object” and Russell himself admits that the sense in which every object is one is always involved when speaking of an object. Consequently, Cohn argues that Russell’s definition of number already presupposes the number concept. In Natorp’s view, Frege’s definition of number presupposes the use of such propositions as ‘X falls under the concept A’. As Natorp sees it, in this proposition an individual is presupposed in the sense of a singular number. Thus Frege’s definition of number already presupposes the singular number. According to Natorp, this mistake, consisting of a simple petitio principii, is shared by all attempts to derive the number concept from the concept of objects belonging to a class (or sets or aggregates). It is inevitable that these objects are thought of as individuals. It is noteworthy that Henri Poincaré presents a similar argument in his critique of the logicist programme. In his paper “Les mathématiques et la logique” Poincaré, too, argues that the logicist definition of number already presupposes the number concept.