The Essence Of Number:Disputes Between Frege And Husserl

Since around the 1870’s in Germany, the research in the foundations of mathematics had gradually gone deeper into the logical and philosophical foundations of mathematics. Frege’s study of the foundations of arithmetic was not affected by the Berlin school’s "arithmetization of analysis," but deeply by the mathematical style of Gottingen, one of whose characteristics was good at using the "creative definition." Husserl’s philosophy of arithmetic is the philosophical extension of the programme of arithmetizing analysis, and his initial goal is to found the analysis on number theory. The separation between analytic philosophy and phenomenology originated in the differences between Berlin school and Gottingen’s tradition of mathematics. Frege and Husserl happened to coincide with founding analysis on arithmetic, thus clarifying the essence of number becomes the key to solving the problem. At the same time, according to the requirement of rigorous analysis, the investigations involved must determinate the sense of number, so that a clear-cut system of concepts and propositions can be established in arithmetic, whose precondition is to grasp the essence of number.Locke highly appreciated the simplicity, clarity, rigor and universality of the ideas of numbers, but if we want to explore the nature of number, the difficulty and complexity far exceeds our imagination. A basic problem is that it is difficult to find a reliable starting point to study it. Pythagoras’significant contribution is the creation of concrete number, this kind of number remains the core foundation of mathematics at least to the 19th century. Modern mathematics can not deny a 4cm long segment is four unit-segments (i.e. specific number), but the philosophy of mathematics can not explain the ground of this kind of number. Plato had long ago found that concrete number is not reliable, so mathematical number must be pure number, arithmetic should be built into a deductive theory of science. Plato’s this ideal is so lofty that until Frege times only some breakthroughs were made, but it cannot be fully completed today.The basic flaw of Kant’s philosophy of mathematics consists in having no illumination of any specific meaning of all mathematical concepts, so it could not meet the needs of modern mathematics’rigor. But Kant’s achievement is remarkable, he first proposed the formal number, which results from arranging sense data on the base of intuitional form. In this sense, number is not a physical object, nor the physical properties of things, and it does not exist in the physical world, nor in the mental world as certain psychological entity. Affected by Kant, Frege’s number and Husserl’s number are also formal. Frege’s number is a priori logical form, and his Grundlagen had already anticipated the logical structure of the world. Husserl’s cardinal number is pure form as well, i.e. category, an objective form which bases on phenomena, but it is not the Kantian a priori form, nor is it logical form governing the physical laws. This kind of form neither pre-exists in subjectivity nor in the phenomenon, but generates in the subject’s inner experience. Thus disputes inevitably generated between the logicist number theory and the phenomenological number theory.Given the above properties, Frege and Husserl both recognized that number is immutable, non-temporal and non-spatial entity. Such entity is independent of the process of our knowing, and can only be grasped in rational way. Specifically, since number is not the property of any object, it can only be abstracted from concepts. But concepts themselves are abstractive, numbers cannot be directly abstracted from concepts, so they all started from the objects under certain concept. Frege considered that the concepts involved could be thought of as equinumerous by means of a reciprocal one-to-one correspondence of the objects falling under the one concept and those falling under the other, and thus number could be defined as an object (i.e. the extension of concept) from equinumerosity by logical abstraction. Husserl thought that the extension of concept is multiplicity, and that number is the "how many" of a multiplicity (i.e. a set), so he could grasp the intension of the concept of number through abstracting from a concrete multiplicity by way of phenomenology. This difference reflects the ambiguity of the concept of extension in traditional logic.Accordingly, a difference also emerged about the issue of the ultimate foundation of number. Frege thought the foundation of arithmetic is logic, because only the truth of laws of logic is really indisputable. In the case of which the intension of the number cannot be grasped by logic, Frege tried to give number an extensional definition, and was committed to building an extensionalist number theory. Husserl believed that number, as the primitive concept, is impossible to define, and that the essence of number could be clarified only by tracing out the phenomenon on which number is based. He appealed to the experience or the representations about the number to reveal the intension of number, and thus established the intensionalist number theory.The core of logicist project is one-to-one correspondence with the objects, by means of which Frege defines the concept of equinumerosity as an extension, and further number is defined as the extension of the concept "equinumerous to certain concept". At this time Frege thought that the concept of extension had long been used in logic, and paid no special attention to it. Subsequently Frege was increasingly aware that in his second-order definition the transition from concept to the extension (i.e. object) is a logical fault, but he thought of it as a logically technical question for a long time and tried to fill the gap technically. But the nature of this problem is whether it is reasonable to think of extension as object in philosophy, Russell thought it is difficult to understand in a letter to Frege.Husserl attempted tracing back to the origin and contents of the concept of multiplicity and number to grasp the essence of number. The contents of the concept of multiplicity and number is identical, and the number is determinate multiplicity, but the "how many" of multiplicity is indeterminate. The extension of the multiplicity and the corresponding number is also identical, which is the concrete phenomenon directly given to us. Therefore, the analysis of number may begin with analyzing multiplicity. The discrete objects contained in a representation of the multiplicity must be linked as a collection by a particular relation, which is collective combination. But this collective combination does not actually exist in the first-order representation, and cannot be given together with the objects. collective combination is the representation generated in an act of reflection and can be aware of, so it is not a subjective thing. Through collective combination, a concrete multiplicity can be constituted, that is to say, a concrete multiplicity is given to our consciousness. This collective relation must possess its relational terms, and such kind of logical relation indicates the contents of a concrete multiplicity are collective combination and objects. Husserl thought that "something" is the concept appropriate to any object, and accordingly, the contents of multiplicity and number consist of the concepts of collective combination and something. More analyses show that collective combination and something are unanalysable, so both are the ultimate origin of the concept of multiplicity and number. Further psychological analyses can determine the "how many" of multiplicities, thus the individual numbers are obtained.Husserl criticized Frege’s definition of numbers is useless, because Frege based number on one-to-one correspondence, but this relation is just the necessary and sufficient conditions of the concept of number, and it isn’t the intension of number. Husserl believed number originates from multiplicity, so the grounds of number is collective combination. Frege thought Husserl confused "A" with "the representation of A", and that regarding number as collective representation is a psychologistic error. The starting point of Frege and Husserl is mathematical realism, the key of their debate is whether the number itself, being independent of us, can be really grasped. Husserl pointed out that the extension of the concept "equinumerous to the concept F" is actually the entirety of infinitely many equinumerous concepts insofar as Frege was concerned, and that it is not an object but a multiplicity, so the number itself is not defined. Frege thought a representation is individual and unshareable, so Husserl had transformed the objective concept into a subjective mental entity, and Husserl had not captured the number itself in consciousness. Although Frege’s criticism was very profound and original, but he in fact misunderstood Husserl.Husserl’s phenomenological-constitutional research on cardinals is reliable in Arithmetic philosophy, but he could not further constitute the other numbers besides cardinals, which made him fail to base analysis on the number theory. A second program was aborted soon afterwards. In order to solve the problem of the foundations of mathematics, Husserl proposed a third solution in 1901. This program suggested that analysis is an instance of manifold theory, and analysis is essentially a formal system of axioms. Husserl thus turned to an axiomatic theory of number from the initial opposition to Frege’s axiomatic number theory.