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Chapter 6 Husserl's Legacy in the Philosophy of Mathematics: From Realism to Predicativism
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CHAPTER SIX Mathieu Marion HUSSERL'S LEGACY IN THE PHILOSOPHY OF MATHEMATICS:FROM REALISM TO PREDICATIVISM The list of mathematicians and philosophers ofmathematics who claimed to have been influenced by Husserl is rather impressive. It includes Weyl, who successively claimed that his own predicativist programme in his 1918 book TheContinuum, and a few years later, the intuitionism of Brouwer that he espoused at that stage were to be linked, as far as epistemology is concerned, to Husserlian phenomenology. In the preface to The Continuum, one reads: Concerning the epistemological side of logic, I agree with the conceptions which underlie Husserl's Logical Investigations. The reader should also consult the deepened presentation in Husserl's Ideas Pertaining to a Pure Phenomenology and a Phenomenological Philosophy which places the logical within the framework of a comprehensive philosophy.1 As is well known, Weyl abandoned his own programme in favour of Brouwer's intuitionism in the early 1920s. Although there are incompatibilities between his own predicativist programme and intuitionism, even in Weyl's own version, which differs on some not so trivial points from Brouwer's, he nevertheless saw some deep connections between intuitionism and phenomenology.2 In 129 HUSSERL AND THESCIENCES a paper in which he tried to adjudicate the Grundlagenstreit between Hilbert and Brouwer,Weyl wrote emphatically: If Hilbert's view prevails over intuitionism, as appears to be the case, then I see in this a decisive defeat ofthe philosophical attitude of pure phenomenology, which thus proves to be insufficient for the understanding of creativescience even in the area ofcognition that is most primal and most readily open toevidence - mathematics.3 Weyl was not the only one to see connections between intuitionism and phenomenology. So too, did one of Husserl's assistants , Becker, in his Mathematische Existenz* which appeared in the same 1927issue of Husserl's Jahrbuch as Heidegger's Sein und Zeit. Becker blended phenomenology and Heideggerian ideas. He influenced in turn Heyting, a student of Brouwer, who referred to Husserl's theory of intentionality in his well-known presentation of intuitionism to the Konigsberg conferencein 19305 and whose basic idea can be summarized as being that mathematical constructions are identified with fulfilled mathematical intentions.6 In addition, the Viennese philosopher Kaufmann claimed the legacy of Husserl in his 1930 book, The Infinite in Mathematics andIts Elimination,where he alsopropounded a form of mathematicalconstructivism- more precisely a form of finitism which is not equivalent to intuitionism, since he did not call for a rejection of the universal applicability of the Law of Excluded Middle, because he believed, erroneously, arithmetic tobe decidable .7 Codel's incompleteness results invalidated his standpoint a year later. On another note, Cavailles linked Husserl's phenomenology with the formalist programme of Hilbert, Brouwer's arch-rival, making heavy weather of the connection between Husserl's notion of 'definiteness' with Hilbert's axiom of completeness; a connection made by Husserl himself in Formal and Transcendental Logic,8 in connection with his Mannigfaltigkeitslehre.9 Cavailles thought that, although the notions of completeness involved here are different ones, Godel's incompleteness theorems dealt a fatal blow to Husserl's phenomenology.10 His idea that a 130 [3.238.162.113] Project MUSE (2024-03-29 10:54 GMT) PHILOSOPHY OF MATHEMATICS: FROMREALISM TO PREDICATIVISM Husserlian 'philosophy of consciousness' had to be replaced by a (structuralist) philosophy or 'dialectic' of concepts had a lasting influence on French philosophy ofmathematics- onethinks,for example, of Desanti, who developed this structuralist approach in Les idealites mathematiques.11 Linking phenomenology with Hilbert's programme contradicts Weyl's opinion in the passage just quoted. I shall not enter this debate; rather, I shall state without argument that both claims are wrong: Husserl's phenomenology should be linked neither with either Hilbert's programme, nor with Brouwerianintuitionism, nor, for that matter , with any constructivist programme. As a constructivist, I am far from being convinced that phenomenology can serve as the basis for a clear and convincing argument in favour of the abandonment of classical mathematics. In light of these constructivist and structuralist claims to Husserl's legacy, the fact that Godel, who is usually considered to have propounded the strongest form of realism about mathematical entities in the twentieth century, also claimed connections with phenomenology will appear at first blush as a rather wild one. In his Gibbs Lecturein 1951, Godel described his own realist position in the followingterms: I am under the impression that after sufficient clarification of the concepts in question it will be...