Modern Logic Volume 8, Number 1/2 (January 1998–April 2000), pp. 142–153.
Review of EDMUND HUSSERL, EARLY WRITINGS IN THE PHILOSOPHY OF LOGIC AND MATHEMATICS
Translated by Dallas Willard (Edmund Husserl Collected Works, Volume 5) Dordrecht, Boston: Kluwer Academic Publishers, 1994 xlviii+505 pp. ISBN 0-7923-2262-2
CLAIRE ORTIZ HILL
The particular nature of Husserl intellectual crisis becomes clearer when we remember that at the time Husserl was keeping company with Georg Cantor who during Husserl’s tenure in Halle was hard at work exploring, mapping and defending the uncharted, rich and strange world of transfinite sets. The new numbers and countless infinities Cantor was creating at the time were certainly counter-intuitive and paradoxical enough to shake most almost anyone’s logical assumptions, and Cantor’s work could have easily inspired in Husserl an acute awareness of the logical questions the introduction of such new numbers might raise. Remember that it was in the late 1880s that Cantor did some of his strangest work with numbers, creating what Joseph Dauben has called "dinosaurs of his mental creation, fantastic creatures whose design was interesting, overwhelming, but impractical to the demands of mathematics in general" ([3, p. 159]).
Husserl was also on hand as Cantor began discovering the antinomies of set theory ([3, pp. 240–270]). So in contemplating the intellectual evolution chronicled in this collection of Husserl’s early writings, it is also helpful to remember that Husserl was not the only one whose logical assumptions were shaken upon coming into contact with Cantor’s ideas. The ideas of the founder of set theory played a role in rocking the ground upon which Bertrand Russell, Gottlob Frege, Richard Dedekind and many others had hoped to derive arithmetic too. For it was in studying Cantor’s 1891 proof by diagonal argument that there is no greatest cardinal number that Russell came upon the famous contradiction of the set of all sets that are not members of themselves which made him too call for important reforms in logic (, , [15, p. 1]).
Husserl’s early first-hand experience of inconsistent sets and some of the more logic defying aspects of Cantor’s theory of sets might actually have permanently innoculated the future founder of the phenomenological movement against any recourse to sets or classes. For Husserl would express grave doubts about extensional logic, by which he meant a calculus of classes (p. 443, for example), for the rest of his career. He would say that extensional logic was na¨ýve, risky, doubtful and the source of many a contradiction requiring every kind of artful device to make it safe for use in reasoning ([19, pp. 74, 76, 83]; [18, p. 153]), a wariness already evident in "The Deductive Calculus and the Logic of Contents" and related articles (pp. 92–114, 115–120, 121–130, 135–138, 443-451) in which we find Husserl intent upon laying bare the "the follies of extensional logic" (p. 199) which he would replace by a calculus of conceptual objects. In these texts he seeks to show "that the total formal basis upon which the class calculus rests is valid for the relationships between conceptual objects," and that one could solve logical problems without "the detour through classes" (p. 109), which he considered to be "totally superficial" (p. 123). In the Philosophy of Arithmetic Husserl had attacked certain of Gottlob Frege’s ideas about extensionality (), but in this volume of writings Husserl’s chief target is Ernst Schr¨oder, which brings us to another interesting matter...
Husserl ultimately concluded that "the profound diculties which are tied up with the opposition between the subjectivity of the act of knowledge and the objectivity of the content and object of knowledge" (p. 250) could only be resolved through what he began calling phenomenological analyses. According to his new theories, pure logic would not itself include anything mental, any reference to acts, subjects, or real people. He would not, however, develop a theory of logic independent of all intuition and experience in Frege’s sense. For it was Husserl’s abiding conviction that one can considerably advance logical understanding of the soundness of symbolic thought (and above all, of course, mathematical thought) without a more penetrating insight into the essence of those elementary processes of intuition and the Representation which everywhere make that thought possible. But without such insight one cannot obtain a full and truly satisfactory understanding of symbolic thought or of any logical process (pp. 168–169).
For modern logicians wary of talk of phenomenological analyses or of any preoccupation with what Husserl called "that peculiar kind of psychological foundation which truly is indispensable for the illumination of the sense of the pure concepts and the laws of logic" (p. 208), a look at the connections between Husserl’s ideas and those of David Hilbert can help set the issue into perspective and make Husserl’s ideas more comprehensible. Remember that Hilbert wrote on several occasions that "the eorts of Frege and Dedekind were bound to fail" because:
No more than any other science can mathematics be founded by logic alone; rather, as a condition for the use of logical inferences and the performance of logical operations, something must already be given to us in our faculty of representation (in der Vorstellung), certain extralogical concrete objects that are intuitively (anschaulich) present as immediate experience prior to all thought. If logical inference is to be reliable, it must be possible to survey these objects completely in all their parts, and the fact that they occur, that they dier from one another, and that they follow each other, or are concatenated, is immediately given intuitively, together with the objects, as something that neither can be reduced to anything else nor requires reduction. ([14, pp. 464–465]; also [13, pp. 376, 392]; [12, p. 162]).
This, Hilbert said, was the basic philosophical position that he regarded "as requisite for mathematics and, in general, for all scientific thinking, understanding and communication" (Ibid .). Now Husserl’s phenomenological analyses would perform precisely the task Hilbert described as being so necessary. Moreover, the philosophy of logic and mathematics which Husserl began developing in the early 1890s actually has a formalist flavor which was already making itself known in the reaction Husserl had in 1891 to Frege’s article "On Formalist Theories of Arithmetic". In the dispute between Hilbert and Frege over formalism, Husserl would side with Hilbert. Partial copies of letters Frege sent to Hilbert were even found among Husserl’s papers ().
Further support for Husserl’s conviction that "logic must not be a mere formal (mathematical) theory . . . but requires phenomenological and epistemological elucidations in virtue of which we not merely are completely certain of the validity of its concepts and theories, but also truly understand them" (p. 215) has come from Kurt G¨odel, a secret admirer of Husserl’s phenomenology. In a posthumously published paper called "The modern development of the foundations of mathematics" ([7, pp. 374–387]), G¨odel argues that the certainty of mathematics is to be secured not by proving certain properties by a projection onto material systems—namely the manipulation of physical symbols—but rather by cultivating (deepening) knowledge of the abstract concepts themselves which lead to the setting up of these mechanical systems, and further by seeking, according to the same procedures, to gain insights into the solvability, and the actual methods for the solution, of all meaningful mathematical problems (p. 383).
G¨odel thought that the procedure by which it might be possible to extend knowledge of the abstract concepts in question was most nearly supplied by the systematic method for clarifying meaning prescribed by Husserl’s phenomenology where, as G¨odel writes,
"clarification of meaning consists in focusing more sharply on the concepts concerned by directing our attention in a certain way, namely, onto our own acts in the use of these concepts, onto our powers in carrying out our acts, etc." (p. 383).
G¨odel viewed phenomenology as "a procedure or technique that should produce in us a new state of consciousness in which we describe in detail the basic concepts we use in our thought, or grasp other basic concepts hitherto unknown to us" (p. 383). According to G¨odel, Husserl’s theories could "safeguard for mathematics the certainty of its knowledge" and "uphold the belief that for clear questions posed by reason, reason can also find clear answers" (p. 381).
Finally, a word about the translation. Dallas Willard’s choice of terms to translate some of the notoriously ambiguous terminology of the late nineteenth century is excellent (pp. xlv–xlvi). And his translation does justice to the clear, readable style of these texts which were written at a time in Husserl’s life when he could still say: "I unfortunately do not have the gift of first coming to clarity in the process of writing and rewriting. But once I have come to a clear understanding, everything moves along rapidly" (p. 13).
For those interested in the development of symbolic logic and twentieth century logic in general, however, it is useful to add the following remarks. The extremely ambiguous German word "Vorstellung" was translated by Russell as "presentation", but has very often been translated into English by "idea" or "imagination". Willard uses "representation", a good choice. He has translated the uneigentlich of uneigentliche Vorstellungen as "inauthentic". It is helpful here to note that Husserl’s distinction between eigentliche Vorstellungen (what is known directly through perception, intuition, memory, etc.) and uneigentliche Vorstellungen or symbolische Vorstellungen (what can only be known indirectly through signs, concepts, descriptions, etc.) is closely related to Russell’s distinction between knowledge by acquaintance and knowledge by description as both men were influenced by Franz Brentano’s distinction between authentic and symbolic presentations ([15, pp. 58–66, 125–135]).
Another extremely dicult term to translate is "Inhalt". Willard has chosen to use "content", which is correct. However, in certain logical contexts when the word is used to refer to the content of a concept, or when Inhalt is contrasted with Umfang, extension, the issues become clearer when "intension" is used in the place of "content". This is particularly the case in Husserl’s discussions of an extensional logic of classes as opposed to an intensional logic of conceptual objects.
"I do not strive for honor and fame. My aim is not to be admired . . . Only one thing will fulfill me: I must come to clarity! Otherwise I cannot live. I cannot endure life without believing that I
shall attain it . . . " (p. 494).