Husserl’s philosophy of mathematics: its origin and relevance

Abstract

This paper offers an exposition of Husserl's mature philosophy of mathematics, expounded for the first time in Logische Untersuchungen and maintained without any essential change throughout the rest of his life. It is shown that Husserl's views on mathematics were strongly influenced by Riemann, and had clear affinities with the much later Bourbaki school.

Appendix

This Appendix , based on the last chapter of my dissertation—see references—, discusses a still unpublished manuscript of Husserl, Manuscript AI 35, which I read in the Husserl Archives in Cologne in 1971 or 1972. I hereby thank the Husserl Archives in Cologne for having allowed me to read the manuscript while I was working on my dissertation. So far as I know, Claire O. Hill is the only other Husserl scholar that has referred in print—in her paper mentioned in the references—to this valuable manuscript.

In the recent literature on Husserl’s views on mathematics² the tendency already mentioned in footnote 40 to associate Husserl with different variants of constructivism, which, due to the apparent affinities of Husserl’s later trascendental phenomenology with Kant’s trascendental idealism would look very plausible, has experienced a sort of revitalization. However, as already mentioned, the fact of the matter is that both in his 1929 Formale und transzendentale Logik and other recently published courses contemporary with his trascendental turn or later,³ there is no sign of any constructivism. Husserl’s views on mathematics remained the same after the trascendental turn. Although this is difficult to swallow by traditional phenomenologists, it represents no problem for those like the present author, who coincide with Carnap in seeing the phenomenological reduction as a purely methodological device. Thus, Platonism survives the trascendental turn.⁴

In our dissertation of 1973, however, we discussed an extensive manuscript of Husserl, which we read in the Husserl Archives in Cologne, in which Husserl seriously considered constructivism. The manuscript, titled “The Paradoxes” and with the inscription A I 35, consists of two parts, namely, part α, dated 1912, and β, dated 1920. In the 1912 part of the manuscript Husserl is concerned with different ways to solve Russell’s paradox—or better: Zermelo-Russell’s—and similar paradoxes. Husserl bases his discussion on two important points, namely: (i) Not every meaning is fullfilable in a possible intuition; e.g., a round quadrangle can be thought, but cannot be intuited, there is no sensible or categorial intuition of it; (ii) One has to distinguish between different levels of language, thus, modifying a little Husserl’s example,⁵ a proposition (or name) of a proposition S is of a level immediatly higher than S. We have here the nucleus of a theory of types. As Husserl makes it clear,⁶ the Russell set would simply be a countersense. Moreover Husserl argues⁷ that membership in a set is an example of what he calls relations of essences, and in such relations the members cannot be identical. Beginning on p. 13 of the manuscript, Husserl speaks also about sets constructible from the axioms and definitions. As Husserl points out,⁸ one should not ascribe an extension (or set) to all general concepts. As an example he mentions⁹ that the mere something in general of formal ontology, on which all formal ontological fundamental concepts are based, does not have any extension. Husserl considers other less palatable solutions in the first part of the manuscript, like the possibility of considering the notion of set as a special case of the notion of a whole, but what is important is that, as he states on p. 25 of the manuscript,¹⁰ from the fact that we can speak about all sets does not follow that the totality of sets is a set, in the same sense that from the fact that we can speak about all possibilities does not follow that the totality of all possibilities is a possibility.

Contrary to what I sustained in my dissertation, and even though Husserl uses the expression “contructible” on pp. 12–13, the whole discussion of Husserl in part α of the manuscript is perfectly compatible with his philosophy of mathematics as presented in Logische Untersuchungen, especially if we consider his epistemology of mathematics of the second part of the Sixth Logical Investigation, in which he offers an iterative constitution of mathematical objects in categorial intuition. Such a view is clearly related to the views of his friends Cantor and Zermelo on the iterative notion of set, which is not to be related with constructivisms of Kantian or Brouwerian, or any other sort.

The case of part β of the manuscript is somewhat different. It dates from 1920, two years after the publication of Hermann Weyls Das Kontinuum. Weyl and his wife had been students of Husserl and were life-long friends of him. It seems that the publication of Weyl’s book, in which a mild constructivism was propounded, as well as the personal contact with his much younger friend, exerted a momentary influence on Husserl, which reflected itself in the second part of the manuscript. In this manuscript Husserl tries to show that the way to avoid the paradoxes consists in a constructive axiomatization of set theory. More explicitly, he stresses that a manifold is to be understood as a “constructively (definite) characterized region of objects, which remains (materially) undetermined, whose objects can be constructed by the iteration into infinity of definitely formed operations, and whose axioms must be so chosen as to found a priori such constructibility.”¹¹ Thus, for Husserl in part β of the manuscript, the doctrine of manifolds should be “the mathematical discipline of the possible constructible infinities” and its task should be “to construct a priori the possible forms of such infinities as constructive systems.”¹² With respect to Russell’s—better Zermelo-Russell’s—Paradox, Husserl says¹³ that it should not be assumed that concepts like that of the set of all sets that do not contain themselves as members have a totality, i.e., a set as extension, and that what such a paradox shows is that there is still no logic of sets in general. Moreover, Husserl adds¹⁴ that sets should be demonstrably constructible with respect to all its members, and that mathematics must furnish an existence proof of each and every set. Husserl is, however, not explicit enough with respect to his notion of constructibility. It is only clear that he requires an existence proof of each set. An interesting question here is whether Husserl’s theory of manifolds and, in general, of mathematical objects required some revision on the basis of these constructivistic leanings of 1920, since, as we have shown elsewhere,¹⁵ neither Russell’s nor Cantor’s sets can be obtained in the iterative hierarchy of mathematical objects propounded in the Sixth Logical Investigation. The fact of the matter is that in his later Formale und transzendentale Logik there is no explicit mention of such a restriction to constructible manifolds. Thus, either Weyl’s impact on Husserl was of short duration or he was convinced that his original philosophy of mathematics, together with his epistemology of mathematics, in which his iterative hierarchy of mathematical objects is inserted, were enough to prevent the paradoxes. These alternative explanations are by no means exclusive, and most probably both are correct.