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Philosophy of Mathematics

The Neo-Kantians and the 'Logicist' Definition of Number

Jarmo Pulkkinen
University of Oulu

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ABSTRACT: The publication of Russell's The Principles of Mathematics (1903) and Couturat's Les principes des mathematiques (1905) incited several prominent neo-Kantians to make up their mind about the logicist program. In this paper, I shall discuss the critiques presented by the following neo-Kantians: Paul Natorp, Ernst Cassirer and Jonas Cohn. They argued that Russell's attempt to deduce the number concept from the class concept is a petitio principii. Russell replied that the sense in which every object is 'one' must be distinguished from the sense in which 'one' is a number. I claim that Russell was wrong in dismissing the neo-Kantian argument as an elementary logical error. To accept Russell's distinction would be to accept at least part of Russell's logicist program. The expression 'a class with one member' would presuppose the number 'one' only if one simultaneously accepted the analysis which mathematical logic provides for it (the class u has one member when u is not null and 'x and y are us' implies 'x and y are identical'). My point is that the aforementioned analysis provided by mathematical logic was something that the neo-Kantians were not ready to accept.

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Although Frege published the first informal exposition of his 'logicist' programme in Die Grundlagen der Arithmetik (1884), his thesis that all mathematics follows from logic was almost completely neglected in Germany for a long time. Frege remained an isolated figure whose works were either strongly criticised or completely neglected by German philosophers. Frege's ideas started to have an impact in Germany only in the first decade of the twentieth century. In particular, the publication of Bertand Russell's The Principles of Mathematics (1903) and Louis Couturat's Les principes des mathématiques (1905) incited several prominent German philosophers to state their opinion about mathematical logic and the logicist programme. In this paper I shall discuss how the neo-Kantians Paul Natorp (1854-1924), Ernst Cassirer (1874-1945) and Jonas Cohn (1869-1947) criticised Russell's and Frege's theories of number. The study of their criticism will also throw some light on the historical origins of the current situation in philosophy, that is, on the split between analytic and Continental philosophy.

1. The 'logicist' definition of number as a class of classes

According to Russell, the goal of the logicist programme is to show that

all pure mathematics deals exclusively with concepts definable in terms of a very small number of fundamental logical concepts, and that all its propositions are deducible from a very small number of fundamental logical principles (Russell 1903: v).

That is to say, pure mathematics is defined as a class of propositions asserting formal implications and containing only logical constants. The logical constants are: implication, the relation of a term to a class of which it is a member, the notion of such that, the notion of relation, and the notions that are involved in formal implication, that is, truth, propositional function, class, denoting, and any or every term (Russell 1903: 106). According to Russell, the above apparatus of general logical notions is sufficient to establish "the whole theory of cardinal integers as a special branch of logic" (Russell 1903: 111). In his view, the "irreproachable" definition of number in purely logical terms is to define number as a class of classes. Two classes have the same number when their terms can be correlated one to one so that any one term of either class corresponds to one and only one term of the other class. When the two classes have the same number, Russell calls them similar. The number of a class is the class of all classes similar to the given class. The Membership of this class of classes is a common property of all the similar classes and no others (Russell 1903: 115).

2. The neo-Kantian critiques of Russell's definition of number

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 (Cohn 1908: 168-169).

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 (Cassirer 1990: 65; Cassirer 1953: 50).

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 (Cohn 1908: 167). 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 (Einzahl). 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 (Natorp 1910: 114-115). 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" (1906) Poincaré, too, argues that the logicist definition of number already presupposes the number concept.

3. The distinction between the different uses of 'one'

Russell anticipates the argument employed by the neo-Kantians and attempts to overcome it in The Principles of Mathematics. According to 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 (Russell 1903: 132-133). 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 (Frege 1974: 40). 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 (Frege 1974: 49). 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" (Couturat 1912: 505). 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. Couturat argues that the situation here is the same as with the proposition "Peter and Paul are wise". According to Couturat, when one says this proposition, one means to say that "Peter is wise and Paul is wise". The fact that Peter and Paul are "two wise men" is not taken into consideration (Couturat 1912: 505).

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' (Cohn 1908: 167). 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 (Setzung) 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 (Cassirer 1990: 65-66; Cassirer 1953: 50-51).

In my view, 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 in his paper on Poincaré, 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 (Goldfarb 1988: 66-67).

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. In order to show this I shall briefly discuss the different number theories of the neo-Kantians. According to Cassirer, the concept of number is the first and truest expression of rational method in general (Cassirer 1990: 35: Cassirer 1953: 27). 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 (Cassirer 1963: 346). 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 1910: 98). 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 (Natorp 1910: 53-54). 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 (Einsheit). 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, that is, a series (Natorp 1910: 54-55). This series is yet indefinite and has only plurality in general. 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" (So viel) 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). Natorp says that here is presented the general method on which the formation of the number depends, and on which the scientific expression of number is based (Natorp 1910: 56-59).

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 (Denkform) and the objectivity (Gegenständlichkeit). The thinking-form belongs to every object, and Cohn calls it "positing" (Setzung). 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 (Cohn 1908: 82-83). 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. Cohn describes this objectivity which every object must possess as a denkfremd element or Denkfremdheit (Cohn 1908: 106). In other words, Cohn argues that the necessary elements of every object are the thinking-form, that is, the denkerzeugt element, and the objectivity, that is, the denkfremd element. 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 (Cohn 1908: 108-109). 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 (Cohn 1908: 170-173).

4. Conclusion

It is clear that from the point of view of the above-mentioned theories of number, the class concept already presupposes the number concept. While Cassirer and Natorp see number as a product of pure thought, Cohn, in turn, sees it as the most abstract object possible. In both cases the number concept is seen to be something logically more fundamental than the class concept. Thus, Russell's and the neo-Kantians' views were at this point irreconcilable. Although the current split between the analytic and the Continental traditions in philosophy did not yet exist at the turn of the twentieth century, the disagreements between the neo-Kantians and Russell can be seen as one of the first steps on the path that gradually led to the current situation in philosophy. It was certainly not going to be the last time that German philosophers did not appreciate Russell's ideas.

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Cassirer, Ernst (1990), Substanzbegriff und Funktionbegriff, 6. unverändert Auflage, wissenschaftliche Buchgesellschaft, Darmstadt (originally published in 1910). English translation in Ernst Cassirer (1953), Substance and Function & Einstein's Theory of Relativity, translated by William Curtis Swabey and Marie Collins Swabey, Dover Publications, New York.

Cassirer, Ernst (1963), The Philosophy of Symbolic Forms, Volume three: The Phenomenology of Knowledge, 2nd edition, translated by Ralph Manheim, Yale University Press, New Haven and London (originally published in 1929).

Cohn, Jonas (1908), Voraussetzungen und die Ziele des Erkennes. Untersuchungen über die Grundfragen der Logik, Leipzig.

Couturat, Louis (1912), "For Logistics", The Monist 22, 483-523.

Frege, Gottlob (1974), Die Grundlagen der Arithmetik / The Foundations of Arithmetic, reprinted and translated by J. L. Austin, 2nd edition, Basil Blackwell, Oxford (originally published in 1884).

Goldfarb, Warren (1988), "Poincaré against the Logicists", in William Aspray and Philip Kitcher (eds.), History and Philosophy of modern mathematics, University of Minnesota Press, Minneapolis, 61-81.

Natorp, Paul, Die logischen Grundlagen der exakten Wissenschaften, Druck und Verlag von B.G. Teubner, Leipzig und Berlin, 1910.

Russell, Bertrand (1903a), The Principles of Mathematics, Vol. I, Cambridge University Press, Cambridge.

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