This is a preprint version of the paper that appeared in definitive form in Phenomenology World Wide, Anna-Teresa Tymieniecka (ed.), Kluwer, Dordrecht, 2002, pp. 76-92. The published version should be consulted for all citations.


On Husserl's Mathematical Apprenticeship and Philosophy of Mathematics

by Claire Ortiz Hill

Insight into the formative role that Edmund Husserl's early training in mathematics played in the development of his ideas is fundamental to understanding his philosophy as a whole. Besides shedding light on the genesis of phenomenology, which began to take shape in Husserl's reflections on the inability of the logic, psychology, mathematics, and philosophy of his time to respond to certain onerous questions raised by his earliest attempts to secure radical foundations for arithmetic, understanding Husserl's ideas about mathematics sheds needed light on a number of other dimensions of his thought that have puzzled and challenged philosophers in this century. For example, this is precisely where many of the clues are to be found that are needed to answer questions of a controversial nature about seemingly enigmatic aspects of his thought, among them questions regarding the nature and evolution of his views on psychologism, on Platonism, on realism, and the relationship between his formal and his transcendental logic.

Moreover, this is the only way there is to situate and evaluate Husserl's philosophy in relation to the ideas and innovations of the most eminent and influential mathematicians of his time, notably Karl Weierstrass, Georg Cantor, David Hilbert, and Kurt Gödel, or Gottlob Frege and Bertrand Russell, men who often shared Husserl's desire to discover secure, scientific foundations for mathematics and the theory of knowledge, his concern to reform logic, his intent to fight against psychologism, his desire to develop a theory of meaning, his questions as to the philosophical significance of the latest developments in mathematics, and so on.

Understanding the evolution of Husserl's views on mathematics is therefore essential to establishing Husserl's proper place in 20th century philosophy of logic and mathematics, a field with deep roots in Austro-German ideas about mathematics, logic and philosophy that flowered in English speaking countries in the twentieth century, but into which Husserl's ideas have never been properly integrated. Given the preeminent role that philosophy of logic and mathematics has played in shaping the way philosophy was done in English speaking countries in the twentieth century, investigations into Husserl's work in this area thus also supply the material essential for the building of any possible bridge between phenomenology and its principal rival, analytic philosophy. And such investigations afford the best possible explanation as to why so many of Husserl's ideas seem so close to those of that antagonistic school, while others remain so plainly diametrically opposed to it.

Under the Influence of Weierstrass

Husserl came to the decision to pursue mathematics as a career during his student years in Berlin, where he enthusiastically threw himself into the study of that most rigorous of disciplines. It was there that from1877-1881 he attended the courses of the great mathematician Karl Weierstrass (Schuhmann, 7; M. Husserl; Osborn, 12-14).

Weierstrass' thoroughgoing, systematic treatment, ab initio, of the theory of analytic functions had led him to profound investigations into the principles of arithmetic. His scrupulous manner of submitting the foundations of analytic functions to close scrutiny awoke in Husserl an interest in seeking radical foundations for mathematics. "I came to understand", Husserl recalled, "the pains he was taking to transform analysis from the mixture of reason and irrational instincts and know-how it was at the time into a pure rational theory. His aim was to expose its original roots, its elementary concepts and axioms on the basis of which the whole system of analysis might be deduced in a completely rigorous, perspicuous way" (Schuhmann, 7; Jourdain, 295-96).

In reaction to the Kantian psychologization of mathematics popular among his contemporaries, Weierstrass was preaching the arithmetization of analysis, the rigorous founding of analysis purely on the basis of the positive whole numbers.. Weierstrass was famous for teaching that once one had thus admitted the notion of whole number, arithmetic needed no further postulate, but then could be built up in a purely logical fashion. This would have the effect of depsychologizing and degeometrizing analysis, of liberating it from the insidious appeals to intuitions of space and time that had been imported into it since Kant had proclaimed that mathematical propositions were synthetic a priori (Coffa; Demopoulos 1994).

Husserl's encounter with Weierstrass had a deep and lasting effect on the future founder of the phenomenological movement. It was from Weierstrass, Husserl would say, that he acquired the ethos of his intellectual endeavors (Schuhmann, 7). Late in his career he would even say that he had sought to do for philosophy what Weierstrass had done for mathematics (Becker, 40-42; Schuhmann, 34). As Andrew Osborn, who actually consulted with Husserl about this, explained: "Through Weierstrass especially, too, the Berlin school placed enormous importance on the rigor of demonstration, a practice that seized hold on Husserl's imagination so that when later he turned to philosophy he sought to find there a strict science similar to that on which Weierstrass insisted, along with the certainty that follows from such strictness and such rigorous proof" (Osborn, 12).

Indeed, closely inspecting the course of Husserl's intellectual career one continually finds him reworking themes present in Weierstrass' work and striving to apply the very principles that underpinned the mathematician's efforts to rigorize analysis. This is, for example, evident not only in Husserl's early espousal of Weierstrass' conviction that the cardinal number was "the first and most underivative domain, the sole foundation of all remaining domains of numbers" (Husserl 1994, 2), but also in Husserl's struggles with psychologism, his lifelong search for radical foundations for knowledge, his striving to lay bare the original roots, the most primitive concepts and principles of knowledge, to uncover the fundamental building blocks on the basis of which his whole system of philosophy might rest, his ideas about phenomenology as a strict science, his efforts to extend the notion of the analyticity, and so on. The nature of his attraction to Weierstrass' work also explains much about the nature of Husserl's attraction to the work of Franz Brentano, Georg Cantor, Bernard Bolzano, David Hilbert, and even Gottlob Frege.

Husserl was, of course, not alone in being decisively influenced by Weierstrass' thoroughness and systematic approach. What we know of Husserl's reaction before Weierstrass' efforts to rigorize analysis is consonant with the impression that he left on much of the mathematical world of his time. "Mathematicians under the influence of Weierstrass," Bertrand Russell once noted, "have shown in modern times a care for accuracy, and an aversion to slipshod reasoning, such as had not been known among them previously since the time of the Greeks" (Russell 1917, 94).

In Berlin, Husserl was also influenced by Leopold Kronecker who also believed that: "Sometime we shall succeed in 'arithmetizing', --that is to say, in founding alone on the number-concept in the narrowest sense, and therefore in stripping away again all the modifications and extensions of this concept, which have mostly been caused by the applications to geometry and mechanics, --the whole of arithmetic" (cited Jourdain, 5). Osborn credited Kronecker with having "sown the first seeds of philosophical understanding" in Husserl "and fostering the interest so aroused". "Husserl found in him," Osborn recounted, "a depth of understanding that stirred an echo in his own nature. Kronecker's special field was the philosophy of mathematics and it was through contact with him accordingly that Husserl first came to any appreciation of the philosophic point of view. Reflective by nature, Husserl found a ready interest in the philosophy of mathematics which was for him, as it proved, a very big step in the direction of an interest in pure philosophy". Osborn speculates that Husserl's interest in Descartes may have first been awakened by Kronecker (Osborn, 12).

As happy as Husserl was in Berlin, acting upon his father's wishes, he left for Vienna to prepare his doctoral thesis on the calculus of variations. Summoned by Weierstrass to serve as his assistant, Husserl later returned to Berlin. However, he quickly took advantage of an opportunity to return to Vienna to indulge a growing interest in philosophy (M. Husserl; Osborne, 15).

Husserl Makes Philosophy His Life's Work

Although Husserl manifested little interest in philosophy during his time in Berlin, it became the minor subject for his doctorate in mathematics in Vienna During that time, when his interest in philosophy was growing and he was wondering whether to make mathematics or philosophy his life's work, Husserl began attending the courses of the philosopher Franz Brentano. At first he did so merely out of curiosity, but these courses finally proved to be the decisive factor encouraging him to dedicate himself entirely to philosophy. But for Brentano, Husserl would say, he would not have become a philosopher (Husserl 1919, 342; M. Husserl; Brück).

The specific reasons for admiring Brentano that Husserl gave actually quite resemble his reasons for admiring Weierstrass. The man in whom Weierstrass had awakened an interest in seeking radical foundations for knowledge was impressed by Brentano's clear, rigorous, insightful, objective, and precise philosophical analyses and ability to transform unclear beginnings into clear thoughts and insights, his "finely dialectical measuring of various possible arguments, his clarifying of equivocations, and retracing of every philosophical concept to its original intuitive sources". "Brentano relatively quickly moved from intuition to theory, to the delimitation of sharp concepts, to theoretical formulation of working problems", Husserl recalled. For Husserl, Brentano was someone entirely devoted to the austere ideal of a strict philosophical science, someone completely certain of his method who believed that his sharply polished concepts, his strongly constructed and systematically ordered theories, and his all round aporetic refutation of alternative interpretations, captured final truths. He "strove constantly to satisfy the highest claims of an almost mathematical strictness". "Sometimes it was the subject matter which overcame me," Husserl recalled, "other times the quite singular clearness and dialectical sharpness of his expositions, the cataleptic power as it were of his way of developing problems and of his theories". It was from Brentano, Husserl acknowledged, that he acquired the conviction that philosophy "was a serious discipline which could and must be dealt with in the spirit of the strictest science" (Husserl 1919, 343-44).

Georg Cantor and Husserl's Philosophy of Arithmetic

Having attended Brentano's lectures for two years, Husserl's next career move was to the University of Halle, to prepare his Habilitationsschrift under the direction of Carl Stumpf, a member of Brentano's circle (Smith, 21-24) convinced of the great need for cooperation between mathematicians or scientists and philosophers in the area of logic (Frege 1980a, 171).

Husserl would reside in Halle from 1886 to 1901. These were years during which his ideas were particularly malleable and changed considerably and definitively. In 1887, he completed On the Concept of Number. The Philosophy of Arithmetic was published in 1891. The better part of the subsequent years was spent in the throes of an intellectual struggle in the course of which he abandoned some of the main lessons he had learned from Weierstrass and Brentano and came to write the groundbreaking Logical Investigations, in which he began laying the foundations of the phenomenological movement that went on to shape the course of 20th century philosophy in Continental Europe.

Georg Cantor, the creator of set theory, taught at the University of Halle during those years and served on the Habilitationskommittee that judged Husserl's On the Concept of Number (Gerlach). The two became close friends. At the height of his creative powers in the 1880s and 1890s, Cantor had studied in Berlin from 1863 to 1869, where he too had come under the influence of Weierstrass, a fact which explains much of the initial intellectual kinship between Husserl and Cantor, whose ideas overlapped and crisscrossed in a number of respects (Hill 1997a; Hill 1999).

During Husserl's time in Halle, Cantor was particularly seeking philosophical justification for his theories. He wanted to show how his entire transfinite set theory rested upon sound principles and how the transfinite numbers might be regarded as consistent extensions of the finite reals. He had begun his Mannigfaltigkeitslehre explaining to his readers that he had come to a point of realizing that further work on set theory would require extending the concept of real whole numbers beyond previously set bounds and in a direction which as far as he knew no one had searched yet, and he offered this a justification or an excuse for introducing apparently strange ideas (Cantor 1883).

Cantor was one of the few mathematicians of his time intent upon wedding mathematics and philosophy. Over the years he had grown increasingly interested in philosophy and by the time of Husserl's arrival in Halle was primed to abandon mathematics for philosophy. In 1894 Cantor would write to the French mathematician Charles Hermite that "in the realm of the spirit" mathematics had no longer been "the essential love" of his soul for more than twenty years. Metaphysics and theology, Cantor "openly confessed", had so taken possession of his soul as to leave him relatively little time for, his "first flame", i.e., mathematics. He was now serving God better, he told Hermite, than, owing to his "apparently meager mathematical talents", he might have done through exclusively pursuing mathematics (Cantor 1991, 350).

Although older, and far less in a position to change course than Husserl was, this did not prevent Cantor from trying to teach philosophy (Cantor 1991, 210, 218) and from seasoning his writings with philosophical reflections and references. In 1883, Cantor had published the Grundlagen einer allgemeine Mannigfaltigkeitslehre, a work which, according to its original 1882 foreword, had been "written with two groups of reader mind --philosophers who have followed the developments in mathematics up to the present time, and mathematicians who are familiar with the most important older and newer publications in philosophy" (Hallett, 6-7). During Husserl's early years in Halle, Cantor published his theories in the Zeitschrift für Philosophie und philosophische Kritik because, as he said, he had grown disgusted with mathematical journals. He was in fact trying to integrate philosophy into his mathematical work to such an extent that colleagues warned him that this was liable to harm his reputation. (Dauben, 139, 336 n. 29).

During Husserl's years in Halle, Cantor persisted in clothing his theories about numbers in a metaphysical garb. And he left no doubts as to where his philosophical sympathies lie. In the Mannigfaltigkeitslehre he had emphasized that the idealist foundations of his reflections were essentially in agreement with the basic principles of Platonism according to which only conceptual knowledge in Plato's sense afforded true knowledge (Cantor 1883, 181, 206 n. 6). His own idealism being related to the Aristotelian-Platonic kind, Cantor wrote in an 1888 letter, he was just as much a realist as an idealist (Cantor 1991, 323). "I conceive of numbers", he informed Giuseppe Peano, "as 'forms' or 'species' (general concepts) of sets. In essentials this is the conception of the ancient geometry of Plato, Aristotle, Euclid etc." (Cantor 1991, 365). To Hermite he wrote that "the whole numbers both separately and in their actual infinite totality exist in that highest kind of reality as eternal ideas in the Divine Intellect" (cited Hallett, 149). Cantor considered his transfinite numbers to be but a special form of Plato's arithmoi noetoi or eidetikoi, which he thought probably even fully coincided with the whole real numbers (Cantor 1884, 84; Cantor 1887/8, 420). By "manifold" or a "set" he explained in the Mannigfaltigkeitslehre, he was defining something related to the Platonic eidos or idea, as also to what Plato called a mikton (Cantor 1883, 204 n. 1). For Cantor, the transfinite "presented a rich, ever growing field of ideal research" (Cantor 1887/8, 406).

Cantor considered that his technique for abstracting numbers from reality provided the only possible foundations for his Platonic conception of numbers (Cantor 1991, 363, 365; Cantor 1887/8, 380, 411). Abstraction was to show the way to that new, abstract realm of ideal mathematical objects that could not be directly perceived or intuited. It was a way of producing purely abstract arithmetical definitions, a properly arithmetical process as opposed to a geometrical one with appeals to intuitions of space and time (Cantor 1883, 191-92). He envisioned it as a technique for focussing on pure, abstract arithmetical properties and concepts which divorced them from any sensory apprehension of the particular characteristics of the objects figuring in the sets and freed mathematics from psychologism, empiricism, Kantianism and insidious appeals to intuitions of space and time to engage in strictly arithmetical forms of concept formation (ex. Cantor 1883, 191-92; Cantor 1885; Cantor 1887/8, 381 n. 1; Eccarius 1985, 19-20; Couturat, 325-41).

With his theory of abstraction Cantor believed that he was laying bare the roots from which the organism of transfinite numbers developed with logical necessity. In the "Mitteilungen," written during the late 1880s, the embattled mathematician was particularly intent upon proving that his theorems about transfinite numbers were firmly secured "through the logical power of proofs" which, proceeding from his definitions which were "neither arbitrary nor artificial, but originate naturally out of abstraction, have, with the help of syllogisms, attained their goal" (Cantor 1887/8, 418). He was hard at work demonstrating that the positive whole numbers formed the basis of all other mathematical conceptual formations inspired by Weierstrass' famous theory to that effect.

All this was part of his greater strategy aimed at providing his "strange" new transfinite numbers with secure foundations by demonstrating precisely how the transfinite number system might be built from the bottom up (Dauben 1979, Chapter 6). In so doing, he was acting on a conviction, spelled out in a 1884 letter to Gösta Mittag-Leffler, that the only correct way to proceed was "to go from what is most simple to that which is composite, to go from what already exists and is well-founded to what is more general and new by continually proceeding by way of transparent considerations, step by step without making any leaps" (Cantor 1991, 208).

Husserl's First Forays into Philosophy

Impressed by Karl Weierstrass' work to arithmetize analysis and armed with analytical tools learned from Brentano, Husserl embarked on a project to help supply radical foundations for mathematics by submitting the concept of number itself to closer scrutiny. On the Concept of Number and The Philosophy of Arithmetic were the result.

Husserl began On the Concept of Number writing of the need to examine the logic of the concepts and methods that mathematicians were introducing and using and for a logical clarification, precise analysis, and rigorous deduction of all of mathematics from the least number of self-evident principles. The definitive removal of the real and imaginary difficulties on the borderline between mathematics and philosophy, he deemed, would only come about by first analyzing the concepts and relations which were in themselves simpler and logically prior, and then analyzing the more complicated and more derivative ones (Husserl 1887, 92-95).

The natural and necessary starting point of any philosophy of mathematics, Husserl still believed, was the analysis of the concept of whole number (Husserl 1887, 94-95). He was confident that: "a rigorous and thoroughgoing development of higher analysis ... would have to emanate from elementary arithmetic alone in which analysis is grounded. But this elementary arithmetic has . . . its sole foundation . . . in that never-ending series of concepts which mathematicians call 'positive whole numbers'. All of the more complicated and artificial forms which are likewise called numbers the fractional and irrational, and negative and complex numbers have their origin and basis in the elementary number concepts and their interrelations" (Husserl 1887, 95).

As he undertook his project to provide a more detailled analysis of the concepts of arithmetic and a deeper foundation for its theorems, the still faithful student of Brentano also considered that psychology was the indispensable tool for analyzing the concept of number (Husserl 1913, 33; see Husserl 1891, l6). However, although the psychological analyses of On the Concept of Number were almost entirely incorporated into the first four chapters of The Philosophy of Arithmetic, the enthusiastic espousal of psychologism found in the earlier work is absent from the later one. And Husserl, who had not initially considered Brentano's teachings to be empirical and psychological in a pernicious sense, later confessed that there had been "connections in which such a psychological foundation never came to satisfy" him, that it could bring "no true continuity and unity", that he had grown "more and more disquieted by doubts of principle, as to how to reconcile the objectivity of mathematics, and of all of science in general, with a psychological foundation for logic" (Husserl 1900-01, 42; Husserl 1975, 34).

Husserl also soon abandoned Weierstrass' teaching on the primacy of the cardinal number. In a letter to Stumpf, written in 1890 or 1891, Husserl revealed that the theory that the concept of cardinal number forms the foundation of general arithmetic that he had tried to develop in On the Concept of Number had soon proved to be false. By no clever devices, he explained, "can one derive negative, rational, irrational, and the various sorts of complex numbers from the concept of cardinal number. The same is true of the ordinal concepts, of the concepts of magnitude, and so on. And these concepts themselves are not logical particularizations of the cardinal concept" (Husserl 1994, 13). Husserl's tergiversation in this regard also becomes apparent through a comparison of the forward and the introduction to The Philosophy of Arithmetic (Husserl 1891, VIII, 5 and note; Hill 1991, 81-85).

The lessons learned from his revered mentors had left him in the lurch. Husserl felt forced to embark upon an independent path. Ten years of hard, lonely work and struggle ensued. He felt that his efforts had brought him "close to the most obscure parts of the theory of knowledge", and that he was standing before "great unsolved puzzles" concerning the very possibility of knowledge in general. He described himself as having been "powerfully… gripped by deep, and by the deepest, problems" (Husserl 1975, 16-17; Husserl 1994, 167, 492-93). His search for answers that he did not believe his early training could provide eventually led him to adopt metaphysical and epistemological views that he had learned to consider odious and despicable (Hill 1998).

From Bolzano the Mathematician to Bolzano the Philosopher

By his own account, Husserl had always been well positioned to appreciate the work of Bernard Bolzano, who as a mathematician, had already come to his attention as a student of Weierstrass. Husserl had become further acquainted with Bolzano's ideas through Brentano's critical discussions of the paradoxes of infinity in his lectures, and then through Georg Cantor (Husserl 1975, 37).

Bolzano was a forerunner of the movement to rigorize analysis that would gain momentum later in the 19th century. His pioneering work to rebuild intuitively accepted proofs of theorems in a rigorous way solely on the basis of arithmetical and logical concepts prepared the way for much that Weierstrass would later advocate and undertake. And, as Weierstrass himself acknowledged, Bolzano actually developed much of the theory of real functions in much the same form that, inspired by him, Weierstrass would teach it in his inspiring courses forty years later (Sebestik, 17, 107 and note; Kline, 948, 950-55; Jourdain, 297; Føllesdal, 7-10; Coffa).

With so many of his mentors impressed by Bolzano's work, Husserl should have been primed to appreciate it. This was not, however, immediately the case. Once acquainted with Bolzano's thought, Husserl recalled, he had "made a point of looking through the long-forgotten Wissenschaftslehre of 1837 and of making use of it from time to time with the help of its copious index", but he originally misinterpreted Bolzano's original thoughts about ideas, propositions and truths in themselves as being about mythical entities, suspended somewhere between being and non-being (Husserl 1975, 37; Husserl 1994, 201-02).

This particular reaction on Husserl's part is understandable. For Brentano inculcated in his students a model of philosophy based on the natural sciences and trained them to despise metaphysical idealism. So, it is easy to see how Husserl, so completely under Brentano's influence in the beginning, might not have quickly warmed to philosophical ideas that Brentano taught his students to disdain (Husserl 1919, 344-45). It was only after Husserl had grown disillusioned with Brentano's empirical psychology that he became receptive to Bolzano's idealism.

Cantor's anti-naturalistic prejudices and deep pro-idealistic convictions must have had a hand in prying Husserl away from empirical psychology and steering him in the direction of Platonic idealism. But it was the study of Hermann Lotze's logic and his reflections on the interpretation of Plato's theory of ideas, Husserl maintained, that provided him with his first major insight and was responsible for his fully conscious, radical turn from psychologism and attendant espousal of Platonic idealism (Husserl 1975, 36; Husserl 1994, 201).

Lotze's work provided Husserl with the key to understanding the "curious conceptions" of Bolzano that had initially seemed so naive and unintelligible to him. Lotze's talk of truths in themselves gave Husserl the idea to transfer all of mathematics and a major part of traditional logic into the realm of the ideal. It then suddenly occurred to him that the first two volumes of Bolzano's Wissenschaftslehre on ideas in themselves and propositions in themselves were to be seen as a first attempt at a unified presentation of the area of pure ideal doctrines and that a complete plan of pure logic was already available there (Husserl 1994, 201-02; Husserl 1975, 36-38, 46-49).

Though Bolzano's propositions in themselves had originally seemed to Husserl to be metaphysical abstrusities, it then became clear to him that what Bolzano had in mind was basically something quite obvious. By proposition in itself, Husserl now understood what people ordinarily called the sense of a statement, what is explained as one and the same when, for example, different persons are said to have asserted the same thing. Or, again, propositions in themselves were simply what scientists called a theorem, for example the theorem about the sum of the angles in a triangle, which no one would think of considering the product of anyone's subjective experience of judging. This realization demystified Bolzano's teachings for Husserl (Husserl 1994, 201-02; Husserl 1905, 37).

It then further became clear to Husserl that this identical sense could be nothing other than the universal, the species, which belongs to a certain Moment present in all actual assertions with the same sense and makes that very identification possible, even when the descriptive content of the individual lived experiences of asserting varies considerably otherwise. Interpreted in this way, he found Bolzano's idea that propositions are objects that nonetheless have no existence quite intelligible. They had the ideal being or validity of objects which are universals, the being which is established, for example, in the existence proofs of mathematics (Husserl 1994, 201-02).

So, although Husserl had come to Halle free of Platonic idealism, he was to leave a committed Platonic idealist, who had come to believe that idealistic systems were of "the highest value", that entirely new and totally radical dimensions of philosophical problems were illuminated in them, that "the ultimate and highest goals of philosophy were opened up only when the philosophical method which these particular systems require is clarified and developed" (Husserl 1919, 345). Every possible effort, Husserl would write, had been made in the Logical Investigations "to dispose the reader to the recognition of this ideal sphere of being and knowledge . . . to side with 'the ideal in this truly Platonistic sense', 'to declare oneself for idealism' with the author" (Husserl 1975, 20). Phenomenology would be an "eidetic" discipline. The "whole approach whereby the overcoming of psychologism is phenomenologically accomplished", Husserl explained, "shows that what . . . was given as analyses of immanent consciousness must be considered as a pure a priori analysis of essence" (Husserl 1975, 42).

This transformation had been prepared, Husserl said, by the study of Leibniz and reflections on his distinction between vérités de raison and vérités de fait and on Hume's ideas about knowledge about matters of fact and relations of ideas. Husserl had become keenly aware of the contrast between Hume's distinction and Kant's distinction between analytic and synthetic judgments and this became crucial for the positions that he later adopted (Husserl 1975, 36).

The early 1890s thus found Husserl striving to develop the true concept of analyticity and to discover the basic philosophical line separating genuine analytical ontology from material, synthetic a priori, ontology, which he believed must be fundamentally distinct from it (Husserl 1975, 42-43). In the Logical Investigations, he would condemn Kant's logic as being utterly defective (Husserl 1900-01, Prolegomena, § 58). Kant, Husserl maintained, had not understood the nature and role of formal mathematics and the way in which he had defined the concept of analyticity was totally inadequate and even utterly wrong (Husserl 1906-07, § 23). "Not only", Husserl complained, did Kant "never see how little the laws of logic are all analytic propositions in the sense laid down by his own definition, but he failed to see how little his dragging in of an evident principle for analytic propositions really helped to clear up the achievements of analytic thinking" (Husserl 1900-01, Sixth Investigation, § 66).

Persuaded of the inadequacy of Kant's analytic-synthetic distinction, Husserl came to believe that Bolzano's more Leibnizian approach to analyticity and meaning harbored the insights logicians needed to prove their propositions by purely logical means. However, in Husserl's opinion, Bolzano never saw the internal equivalence between the analytic nature of both formal logic and formal mathematics made possible by developments in the field of mathematics that had only taken place after his death (Husserl 1929, § 26; Husserl 1975, 36-38).

By drawing the boundary line existing a priori between mathematics and natural sciences like psychology, Husserl believed that he was drawing the line of demarcation and expanding the domain of the analytical in keeping with the most recent discoveries in mathematics. Analytic logic, Husserl would ultimately explain in Formal and Transcendental Logic, is first of all valid as an absolute norm presupposed by any rational knowledge. His "war against logical psychologism, was in fact meant to serve no other end than the supremely important one of making the specific province of analytic logic visible in its purity and ideal particularity, freeing it from the psychologizing confusions and misinterpretations in which it had remained enmeshed with from the beginning". The value of his criticisms of logical psychologism lie, he believed, precisely in his drawing attention to a pure, analytic logic, distinct from any psychology, as being an independent field, like geometry or the natural sciences. Epistemological questions may well arise regarding this pure logic, he considered, but this must not interfere with its independent course, or involve delving into the concrete aspects of the logical life of the consciousness. For that would be psychology (Husserl 1929, § 67).

No psychologistic empiricism, Husserl had come to believe, "can change the fact that pure mathematics is a strictly self-contained system of doctrines which is to be cultivated using methods that are essentially different from those of natural science" (Husserl 1913, 29). "The empirical sciences --natural sciences", Husserl wrote to Brentano in 1905, "--are sciences of 'matters of fact'.... Pure Mathematics, the whole sphere of the genuine Apriori in general, is free of all matter of fact suppositions.... We stand not within the realm of nature, but within that of Ideas, not within the realm of empirical. . . generalities, but within that of the ideal, apodictic, general system of laws, not within the realm of causality, but within that of rationality.... Pure logical, mathematical laws are laws of essence…" (Husserl 1905, 37).

Husserl did, though, realize that not all the sciences are theoretical disciplines that, like mathematical physics, pure geometry or pure arithmetic, are characterized by the fact that their systemic principle is a purely analytical one. Sciences like psychology, history, the critique of reason and, notably, phenomenology, he believed, require that one go beyond the analytico-logical model. When they are formalized and one asks what it is that binds the propositional forms into a single system form, one finds oneself facing, Husserl maintained, nothing more than the empty general truth that there is an infinite number of propositions connected in objective ways that are compatible with one another in that they do not contradict each other analytically (Husserl 1917/18, § 54).

In Conflict with Gottlob Frege

Disproportionately more has been written about Husserl and Gottlob Frege than about Husserl and any other mathematician, a fact attributable to the enthusiasm so many feel about Frege's work and also to a genuine desire on the part of a smaller number of philosophers to see the legitimate ties recognized between Husserl's ideas and those that went into the making of analytic philosophy and of 20th century philosophy of logic and mathematics in general (ex. Haaparanta (ed.)).

Husserl's interest in Frege's work can be traced back to the late 1880s. There is no mention of it in On the Concept of Number. However, by 1887 Husserl had obtained a copy of the 1884 Foundations of Arithmetic (Schuhmann, 18), which is thoroughly examined in The Philosophy of Arithmetic, as a mere glance at the index of names reveals.

It should not be surprising that Husserl picked up and studied the book of that reputedly obscure thinker at that particular point, for members of Brentano's school knew Frege's work (Linke 1947). In particular, in 1882 Carl Stumpf had expressed to Frege his pleasure upon learning that he was "working on logical problems, an area where there is such a great need for cooperation between mathematicians or scientists and philosophers". Stumpf moreover suggested to Frege that his work might be more favorably received were he first to explain his ideas in ordinary language, advice that Frege scholars think that he might have taken in writing the Foundations. Stumpf also, importantly, expressed his conviction that arithmetical, algebraic and geometrical judgements were analytic (Frege 1980a, 171-72).

In 1885 Cantor had reviewed Frege's Foundations. In the review, Cantor decried Frege's recourse to extensions, but praised him for demanding that all psychological factors and intuitions of space and time be banned from arithmetical concepts and principles because that was the only way their strict logical purity and validity might be secured (Cantor 1932, 440-41). So two of the three members of Husserl's Habilitationskommittee knew Frege's work.

Husserl's reaction to the Foundations was mixed. In 1891, he wrote to Frege that of all the many writings he had before him as he worked on The Philosophy of Arithmetic he could not name another that he had studied with nearly as much enjoyment as he had Frege's. Husserl acknowledged "the large amount of stimulation and encouragement" that he had derived from the book. He had, he explained, derived constant pleasure from the originality of mind, clarity and honesty of Frege's investigations, "which nowhere stretch a point or hold back a doubt, to which all vagueness in thought and word is alien, and which everywhere try to penetrate to the ultimate foundations". Husserl did, however, admit that he had not been able on the whole to agree with Frege's theories and alluded to "fundamental divergences" (Frege 1980a, 64-65).

In The Philosophy of Arithmetic itself Husserl called Foundations an ingenious, insightful, remarkable attempt to analyze and define the concept of cardinal number that came close to providing the right answer only to move further off the mark. Husserl characterized Frege's ideal as being that of grounding arithmetic on a series of formal definitions out of which all the theorems of this science might be derived in a purely syllogistic manner. Husserl called Frege's goal chimerical and criticized him for wandering off into hyper-subtleties in a sterile way and concluding with no positive result (Husserl 1891, 129-31). Husserl eventually retracted those particular criticisms of Frege's work (Husserl 1900-01, Prolegomena § 45 n.).

In comments that Husserl never retracted, he complained that Frege's results were such that one could only be astonished that anyone might have taken them to be true, except for temporarily. All Frege's definitions, Husserl argued become true propositions when one substitutes the concepts to be defined with their extensions, but then they are absolutely self-evident propositions and worthless. Husserl also made some very perspicacious and timely criticisms of the fundamental role that Frege was according Leibniz's principle of substitutivity of identicals in his logical reconstruction of arithmetic (Husserl 1891, 133-35 and note, 104-05; Hill 1994b).

In case Husserl's judgement may seem unduly harsh, especially in light of the indulgent approach to Frege's ideas that is popular nowadays, it is important to realize that in Foundations Frege himself acknowledged that that his plan to use Leibniz's principle to obtain the concept of number by fixing the sense of a numerical identity (Frege 1884, §§ 62-65) was liable to produce nonsensical conclusions or be sterile and unproductive (Frege 1884, §§ 66-67, 105). As a remedy, he cautiously introduced the extensions (Frege 1884, §§ 68-69 and note, § 107) that by the time he published The Basic Laws of Arithmetic in 1893 had come to take on such "great fundamental importance" that he said that he could just no longer do without them (Frege 1893, ix-x; Hill 1997b, 61-68; Hill 1997c, 58-62).

Never one to accept criticism gracefully, Frege lashed back at Husserl in a cranky review of The Philosophy of Arithmetic (Frege 1894), in which a great deal is going on below the surface (Hill 1994a). For one thing, in the review Frege tellingly chose to attack Husserl's failure to appeal to extensions. Any view according to which a statement of number is not a statement about a concept or about the extension of a concept, Frege said, is naive for "when one first reflects on number, one is led by a certain necessity to such a conception" (Frege 1894, 197). Had Husserl used the term 'extension of a concept' in the way he, Frege, had, they "should hardly differ in opinion about the sense of a number statement" (Frege 1894, 201-02).

By criticizing Frege's use of extensions Husserl had surely struck a sensitive chord in Frege who in Basic Laws I, published two years after The Philosophy of Arithmetic, pinpointed this as the place where any decision about any defects or errors in his logic would ultimately be made (Frege 1893, vii; Hill 1997b, 68-72). Indeed, upon learning of the famous contradiction of the set of all sets that are not members of themselves derivable in Basic Laws, Frege immediately replied that it was the law about extensions that was to blame and that its collapse seemed to undermine his foundations for arithmetic (Frege 1980a, 130-32). At least, it was consoling to know, Frege wrote in the appendix to Basic Laws II, after Russell's discovery, that everybody who had made use of extensions of concepts, classes, sets in proofs was in the same position that he was (Frege 1980b, 214).

Frege never believed that his law about extensions recovered from the shock that it had sustained from Russell's paradox, and he came to rue having used the expression 'extension of concept' which, he finally concluded, easily "can get one into a morass" and leads "into a thicket of contradictions" (Frege 1980a, 55; Frege 1979, 269-70). "Only with difficulty", Frege confessed, "did I resolve to introduce classes (or extents of concepts), because the matter did not appear to me quite secure -and rightly so, as it turned out…. I was constrained to overcome my resistance …. I confess that by acting thus, I fell into the error of letting go too easily my initial doubts" (Frege 1980a, 191). "While I sometimes had slight doubts during the execution of the work, I paid no attention to them. And so it happened that after the completion of the Basic Laws of Arithmetic the whole edifice collapsed around me" (Frege 1980a, 55). Frege's review of Husserl actually contains some of the most forceful statements that Frege ever made in favor of extensions, a fact that, in retrospect, but reflects his insecurity about resorting to them.

Telling too is the fact that Frege directly incorporated into his review of Husserl's book criticisms and examples that he had previously used in reviewing Cantor's Mitteilungen (Frege 1892, 178-81 ; Frege 1979, 68-71). For example, Frege had complained that Cantor was "asking for impossible abstractions" (Frege 1892, 179). In his review of Husserl's book, Frege charged, that abstraction would "cleanse things of their peculiarities… in the wash-tub of the mind" where "we can easily change objects by directing our attention towards them or away from them… We attend less to a property and it disappears" (Frege 1894, 197). Suppose, Frege went on, "that there are a black and a white cat sitting side by side before us. We do not attend to their colour, and they become colourless -but they still sit side by side. We do not attend to their posture, and they cease to sit… but each of them is still in its place. We no longer attend to the place and they cease to occupy one…. By continued application of this procedure, each object is transformed into a more and more bloodless phantom" (Frege 1894, 197-98). It must surely be assumed, Frege maintained in the review, that the process of abstraction effects some change in the objects and that they become different from the original objects which are either transformed or actually created by the abstraction process (Frege 1894, 204).

Yet in Foundations § 34, Frege himself had acknowledged that in the abstraction process the things themselves do not lose any of their characteristics, that one might disregard the properties that distinguish a white cat and a black cat, but the cats would not thereby become colorless. What is more, despite the ridicule he heaped on the abstraction process in his review of Husserl's book, Frege admitted there that Husserl himself did not claim that the mind creates new objects or changes old ones. Frege actually acknowledged that Husserl "disputes this in the most vehement terms" (Frege 1894, 205), which is true (ex. Husserl 1891, 28-30, 42, 46, 139).

In his review, Frege also accused Husserl of taking "the road of magic rather than of science (Frege 1894, 205). In the posthumous draft of a review of Cantor's Mitteilungen (Frege 1979, 68-71), Frege writes of the "astounding magical properties", "magical effects" and "miraculous powers" in Cantor's work (Frege 1979, 69). Choosing mice instead of cats, Frege there complained that mathematicians like Cantor find a whole host of things in mice which are unworthy to form a part of number. So they begin abstracting. Ridiculing these mathematicians, Frege writes that for them "everything in the mice is out of place: the beadiness of their eyes no less than the length of their tails and the sharpness of their teeth. So one abstracts from the nature of the mice… one abstracts presumably from all their properties…." (Frege 1979, 70).

These are but a few reasons why Frege's review should not be considered the straightforward, objective evaluation of Husserl's book that it has popularly been believed to be. Much, in fact, still remains to be rectified regarding Frege's review, the uncritical reading of which has unfairly distorted philosophers' perception of a work that they do not know very well. Husserlians themselves have been only too willing to dismiss The Philosophy of Arithmetic as the product of an immature, pre-phenomenological phase of Husserl's life and have not risen to its defense.

Husserl and Russell's Paradox

A grasp of the nature of the exchange of ideas that took place among Husserl, Cantor and Frege naturally raises questions as to what exactly Husserl knew about the famous contradiction of the set of all not sets that are not members of themselves and when he knew it.

Bertrand Russell has received the lion's share of the credit for having discovered this "paradox". However, though he was certainly the first to publicize Frege's errors and to bring the point home to Frege himself, the finding was not at all as surprising to the circle of eminent mathematicians that Husserl frequented as has been assumed.

In Halle, Husserl had taken up residence with the antinomies of set theory themselves (Cantor 1991, 387-464; Dauben 240-70) and so was one of the very first to be on hand to witness the paradoxical consequences of Cantor's theories. Remember that it was in studying Cantor's theories that Russell discovered the famous antinomy that put an end to the logical honeymoon that he was having when he began writing the Principles of Mathematics (ex. Russell 1903, §§ 100, 344, 500; Russell 1959, 58-61; Grattan-Guinness 1978, 1980).

Of the mathematical world's reaction to Cantor's theories, Hilbert has written that it was violent and "took very dramatic forms… purely through the ways in which notions were formed and modes of inference used… contradictions appeared, sporadically at first, then ever more severely and ominously…. In particular, a contradiction discovered by Zermelo and Russell had, when it became known, a downright catastrophic effect in the world of mathematics. Confronted with these paradoxes Dedekind and Frege actually abandoned their standpoint and quit the field" (Hilbert 1925, 375). Hilbert's mention of Zermelo in this context brings us to the next point.

The paradoxes of set theory were a topic of lively discussion in Göttingen, where Husserl took up residence in 1901 (ex. Peckhaus, 168-95). In a November 7, 1903 letter, Hilbert told Frege that Russell's antinomy was already known to them in Göttingen. Hilbert added that he believed that Ernst Zermelo had found it three or four years earlier after having learned of other, even more convincing, contradictions from Hilbert himself as many as four or five years before. Hilbert further commented that the idea that "a concept is already there if one can state of any object whether or not it falls under it" did not seem adequate to him. What would be decisive, he states, "is the recognition that the axioms that define the concept are free from contradiction" (Frege 1980a, 51-52).

Concrete evidence corroborating Zermelo's finding of the paradox is to be found in a note that he sent to Husserl in April 1902, in which Zermelo conveyed his proof. Husserl recorded what Zermelo wrote and that record has survived (Husserl 1994, 442). The exchange turned upon certain remarks that Husserl had made in his 1891 review of Ernst Schröder's Vorlesungen über die Algebra der Logik (Husserl 1994, 52-91, 421-41). There Schröder had tried to show that bringing all possible objects of thought into a class gives rise to contradictions. In his review, Husserl had written that although Schröder's argument might at first sight appear astonishing, it was actually sophistical. However, importantly, Husserl conceded that:

in the cases where we simultaneously have, besides certain classes, also classes of those classes, the calculus may not be blindly applied. In the sense of the calculus of sets as such, any set ceases to have the status of a set as soon as it is considered as an element of another set; and this latter in turn has the status of a set only in relation to its primary and authentic elements, but not in relation to whatever elements of those elements there may be. If one does not keep this in mind, then actual errors in inference can arise (Husserl 1994, 84-85).

In Zermelo's opinion, however, Schröder had been basically right, but his reasoning had been faulty. According to Zermelo's argument as recorded by Husserl: given a set M which contains each of its sub-sets m, m'… as elements, and a set M0 which is the set of all sub-sets M, which do not contain themselves as elements, it can then be shown that M0 both does and does not contain itself (Husserl 1994, 442).

In reflecting on Husserl and "Russell's" paradox, it is also of interest to note that Husserl and Frege exchanged letters between October 1906 and January 1907. Copies of two of the letters that Frege wrote to Husserl have survived. The three letters that Husserl wrote to Frege were, however, been lost during World War II. We do know, however, that these letters dealt with, among other things "the paradoxes" (Frege 1980a, 70; Hill 1995a).

So, it is intriguing to note in this regard that preeminent Frege scholar, Michael Dummett has contended that "Frege's posthumous writings allow us with high probability to date almost exactly Frege's disillusionment over the attempt, to which he had devoted his life, to derive arithmetic from logic" in mid-1906 (Dummett, 21). Dummett cites the article that Frege had begun writing during that year about the paradoxes of set theory and the inadequacy of certain remedies that were being proposed (Frege 1979, 176-83). "Tantalizingly little of the article survives…", Dummett concludes, and "very probably it represents the very moment at which Frege came to realize that the attempt was hopeless" (Dummett, 22). Hans Sluga too has concluded that by 1906 Frege "was beginning to think that the theory of sets was undermined by the contradiction. He concluded that there was no use for sets or classes anymore" (Sluga, 169-70). Indeed, nothing that Frege ever wrote after 1906 indicates that he again tried to salvage the specific logical doctrines that he had concluded had led to the paradoxes of set theory. However, a distressing amount of material on this precise matter has been lost or destroyed (Hill 1995a).

A hundred pages of manuscripts at the Husserl Archives in Cologne show that Husserl worked directly on the paradoxes of set theory, and on Russell's paradox in particular, in1912 and the 192Os (Rosado Haddock). Unfortunately the manuscripts need further study. As someone who frequented Hilbert's school in Göttingen, and who pointed to the kinship existing between his own manifolds and Hilbert's axiomatic systems (Husserl 1913, § 72, Husserl 1929, § 31), one may hypothesize, however, that Husserl considered, as so many mathematicians have, that properly carried out, the axiomatization of set theory might neutralize the contradictions found in Cantorian set theory. Defined and regulated by a complete axiomatic system, sets would thus be apt to play their fruitful role in mathematics, which brings us to our next topic.

Situating Husserl With Regard to Hilbert and Formalist Theories of Mathematics

Although Husserl retracted the three pages of his criticism of Foundations in which he had denied that one could provide sound foundations for arithmetic by deriving theorems from a series of formal definitions in a purely logical fashion, it should be clear from what has been said here that he did not ultimately chose Frege's way. Rather, Husserl's mature philosophy would have a formalist flavor.

Once appointed to the University of Göttingen in 1901, Husserl was welcomed into David Hilbert's circle (Reid 1970, 1976; Husserl 1983, XIII). Frege and Hilbert had just been corresponding about truth and logical consistency. In their letters, Hilbert had put his finger on one of the main points dividing Frege and him when he wrote that he was particularly interested in Frege's statement that from the truth of the axioms it follows that they do not contradict one another, because for as long as he had been thinking, writing and lecturing on these things he had been saying exactly the opposite: "If the arbitrarily given axioms do not contradict one another, then they are true, and the things defined by the axioms exist. This for me is the criterion of truth and existence… only the whole structure of axioms yields a complete definition…. Every axiom contributes to the definition, and hence every new axiom changes the concept" (Frege 1980a, 42, 39-40).

Husserl had access to the Frege-Hilbert correspondence, and partial copies of it, along with notes that Husserl made on it, were found in his Nachlass. In his notes Husserl remarked that Frege had not really understood the meaning of Hilbert's axiomatic foundation for geometry, that it was a matter of a purely formal system of conventions that coincides formally with the Euclidean system (Frege 1980a, 34-51; Husserl 1970, 447-52).

Now, in the letter to Stumpf from the early 1890s discussed above, Husserl had expressed his frustrations regarding the inability of Brentano's methods to cope with imaginary numbers and his new faith in the arithmetica universalis as a part of formal logic understood as a symbolic technique and making up a special, important chapter in logic as technology of knowledge (Husserl 1994, 17). Husserl used the term 'imaginary' in the broadest way to include negative, irrational numbers, fractions, negative square roots (Husserl 1983, 244-49; Husserl 1970, 432) and he called infinite sets 'imaginary concepts' (Husserl 1891, 249). In the case of 'imaginary' numbers like v2 and v -1, he told Stumpf:

I first sought to get clear on how operations of thought with contradictory concepts could lead to correct theorems…. Finally I noticed that, through the calculation itself and its rules (as defined for those fictive numbers), the impossible falls away, and a genuine equation remains. Indeed, the procedure of calculation runs its course another time with the same signs, but now referred to valid concepts, and again the result is correct. Thus it is not a matter of the "possibility" or "impossibility" of concepts. Even if I mistakenly imagine that the contradictory exists -even if I hold the most absurd theories concerning the content of the corresponding concepts of number… the calculation remains correct, if it follows the rules. So it must be an accomplishment of the signs and their rules (Husserl 1994, 15-16).

And, in Husserl's epistolary exchange with Frege in 1891 (Frege 1980a, 61-66), we already find Husserl expressly entertaining a formalist solution to problems about imaginary numbers that were preoccupying him. This was, remember, the stumbling block that led him to abandon Brentano's empirical psychology and Weierstrass' thesis about the primacy of the cardinal number when he was writing the second volume of The Philosophy of Arithmetic.

Husserl wrote that he had only a rough idea of how Frege would justify the imaginary in arithmetic, since in "On Formal Theories of Arithmetic" (Frege 1885) he had rejected the path that Husserl himself had found after much searching. In the passage of the article cited in Husserl's letter (Frege 1980a, 65), Frege had criticized the procedure by which one which but sets down rules by which one passes from the equations given to new ones in the way one moves chess pieces. Unless an equation contains only positive numbers, it no more has a meaning than the position of chess pieces expresses a truth. Now in virtue of these rules, Frege continues his criticism, an equation of positive whole numbers may actually appear. And if the rules are such that true equations can never lead to false conclusions, then only two results are possible: either the final equation is meaningless, or it has a content about which we can pass judgement. The latter will always be the case if it contains only positive whole numbers, and then it must be true, for it cannot be false. If the rules contain no contradictions among themselves, and do not contradict the laws of positive whole numbers, then no matter how often they are applied, no contradiction can ever enter in. Consequently, if the final equation has any meaning at all, it must be non-contradictory, and hence be true. This is a mistake, Frege concludes, for a proposition may very well be non-contradictory without being true.

Husserl's reasons for believing that Frege's logic could not satisfactorily justify the imaginary in arithmetic surely involved Frege's well-known thesis that in a logically perfect language, expressions that do not denote objects are unfit for scientific use. The use of signs or combinations of signs without reference was at the heart of Frege's dispute with formalists who, he believed, only manipulated signs without any regard for what those signs might stand for. But, Frege insisted, "logic is not concerned with how thoughts, regardless of truth-value, follow from thoughts. . . we have to throw aside proper names that do not designate or name an object …." (Frege 1979, 122). In Foundations and "On Formal Theories of Arithmetic", he wrote, "I showed that for certain proofs it is far from being a matter of indifference whether a combination of signs - e.g. Ö-1 has a meaning or not, that, on the contrary, the whole cogency of the proof stands or falls with this" (Frege 1979, 123).

Invited by Hilbert and Felix Klein, Husserl addressed the Göttingen Mathematical Society in 1901 on the subject of imaginary numbers (Husserl 1970, 430-506). Questions regarding imaginary numbers, he explained, had come up in mathematical contexts in which formalization yielded constructions that, arithmetically speaking, were nonsense but which could nonetheless be used in calculations. When formal reasoning was carried out mechanically as if these symbols had meaning, if the ordinary rules were obeyed, and the results did not contain any imaginary components, it seemed that these symbols might be legitimately used. However this raised significant questions about the consistency of arithmetic and about how one was to account for the achievements of certain purely symbolic procedures of mathematics despite the use of apparently nonsensical combinations of symbols.

By the time Husserl gave his talk, he had concluded that the key to the only possible answer to his questions lie in the theory of complete manifolds that he first expounded in the first volume of the Logical Investigations (Husserl 1900-01, Prolegomena § 70), a conviction that he reaffirmed late in his career in Formal and Transcendental Logic (Husserl 1929, § 31).

For Husserl, the general theory of the manifolds, or science of theory forms, was a field of free, creative investigation that was made possible once the form of the mathematical system had been emancipated from its content. Discovering that deductions, series of deductions, continue be meaningful and to remain valid when one assigns another meaning to the symbols, actually freed one to liberate the mathematical system, he considered. Nothing more need be presupposed than the fact that the objects figuring in it were such that, for them, a certain connective supplied new objects and did so in such a way that the form determined is assuredly valid for them.

According to Husserl's theory, manifolds themselves were pure forms of possible theories which, like molds, remain totally undetermined as to their content, but to which thought must necessarily conform in order to be thought and known in a theoretical manner. By using axioms of such and such a form, theories of such and such a form may be developed. The objects are exclusively determined by the form of the interconnections assigned them, meaning, neither directly inasmuch as individuals, nor indirectly by their kind or species. The interconnections themselves are just as little determined in terms of content as are the objects. Only their form determines them by virtue of the form of the elementary laws admitted as valid for these interconnections, laws that also determine the theory to be constructed, the form of the theories.

One can operate freely within a manifold with imaginary concepts and be sure that what one deduces is correct when the axiomatic system completely and unequivocally determines the body of all the configurations possible in a domain by a purely analytical procedure. A domain is complete, Husserl held, when each grammatically constructed proposition exclusively using the language of this domain is, from the outset, determined to be true or false in virtue of the axioms. In that case, calculating with imaginary concepts can never lead to contradictions. It is the completeness of the axiomatic system that gives one the right to operate freely. It was formal constraints requiring that one not resort to any meaningless expression, no meaningless imaginary concept that were restricting us in our theoretical, deductive work. Husserl considered that the kinship between his conception of the completeness of axiomatic systems and Hilbert's was obvious (Husserl 1900-01, Prolegomena, § 70; Husserl 1913, §§ 71-72; Husserl 1906/07, §§ 18-19; Husserl 1917/18, Chapter 11).

Kurt Gödel's Secret Admiration for Husserl's Work

Husserl did not, of course, believe that formal logic alone could suffice. For one thing, he believed that once armed with the objective structures of formal logic, philosophical logicians were obliged to go further and come to terms with really hard epistemological questions about subjectivity, logic and mathematics themselves. According to him, philosophical logicians had to see that the logical sense of the formal sciences also includes a sphere of cognitive functioning and a sphere of possible applications. They could only submit to a logic that they had thought through, and thought through with insight, a fact that Husserl believed cried out for thorough epistemological investigations into the subjective and intersubjective processes and the ways in which they inevitably interact with the objective order.

In this respect too, Hilbert and Husserl were in agreement with one another and disagreed with Frege. Upon several occasions, Hilbert expressed his conviction that "counter to the earlier endeavors of Frege and Dedekind… if scientific knowledge is to be possible certain intuitive conceptions and insights are indispensable; logic alone does not suffice" (Hilbert 1925, 392). According to Hilbert, "the efforts of Frege and Dedekind were bound to fail" because "as a condition for the use of logical inferences and the performance of logical operations something must already be given in our faculty of representation, certain extralogical concrete objects that are intuitively 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 differ 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". This, Hilbert held, "is the basic philosophical position that I regard as requisite for mathematics and, in general, for all scientific thinking, understanding and communication" (Hilbert 1925, 376; see also Hilbert 1922, 202; Hilbert 1927, 464-65).

Now, new support for Husserl's conviction that "logic must not be a mere formal (mathematical) theory… but, as a philosophical logic… 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" (Husserl 1994, 215) has come from the famous discoverer of the incompleteness of formal systems, Kurt Gödel (Gödel 1995a; Gödel 1995b), who, the philosophical world was surprised to learn from Hao Wang in the 1980s (Wang 1986, 1987, 1996), had been a secret admirer of Husserl's work.

In a posthumous paper, Gödel contended 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" (Gödel 1995b, 383).

Gödel 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 Gödel wrote in his Nachlass, "clarification of meaning consists in focusing more sharply on the concepts concerned by directing our attention a certain way, namely, onto our own acts in the use of these concepts, onto our powers in carrying out our acts, etc." (Gödel 1995b, 383). Gödel 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" (Gödel 1995b, 383). According to Gödel, 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" (Gödel 1995b, 381).

Of additional interest is the fact that Wang has reported that Gödel was particularly interested in Husserl's ideas on axiomatization (ex. Wang 1996, 168, 334), a topic yet to be fully investigated.


In 1929 Husserl published Formal and Transcendental Logic, which was the product of decades of reflection upon the relationship between logic and mathematics, between mathematical logic and philosophical logic, between logic and psychology, and between psychologism and his own transcendental phenomenology. One of the stated goals of the book was to redraw the boundary line between logic and mathematics in light of the new investigations into the foundations of mathematics. A second goal was to examine the logical and epistemological issues such developments have raised (Husserl 1929, 10-17).

In Formal and Transcendental Logic, Husserl expressed his conviction that the formalization of large tracts of mathematics in the nineteenth century had laid bare the deep, significant connections obtaining between formal mathematics and formal logic, and had thus raised profound new questions about the deep underlying connections existing between the two fields. Logic and mathematics, he believed, had originally developed as separate fields because it had taken so long to elevate any particular branch of mathematics to the status of a purely formal discipline free of any reference to particular objects. Until that had been accomplished the important internal connections obtaining between the two fields were destined to remain hidden. However, once large tracts of mathematics had been formalized, the parallels existing between its structures and those of logic became apparent, and the abstract, ideal, objective dimension of logic could then be properly recognized, as it traditionally had been in mathematics. Developments in formalization had thus unmasked the close relationships between the propositions of logic and number statements making it possible for logicians to develop a genuine logical calculus which would enable them to calculate with propositions in the way mathematicians did with numbers, quantities and the like (Husserl 1929, Chapter 2).

Mathematics, Husserl deemed, has its own purity and legitimacy. Mathematicians are free to create arbitrary structures. They need not be concerned with questions regarding the actual existence of their formal constructs, nor with any application or relationship their constructs might have to possible experience, or to any transcendent reality. They are free to do ingenious things with thoughts or symbols that receive their meaning merely from the way in which they are combined, to pursue the necessary consequences of arbitrary axioms about meaningless things, restricted only by the need to be non-contradictory and coordinated to concepts previously introduced by precise definition. And the same, Husserl contended, was true for formal logic when it was actually developed with the radical purity that is necessary for its philosophical usefulness and gives it the highest philosophical importance. Severed from the physical world, it lacks everything that makes possible a differentiation of truths or, correlatively of evidences (Husserl 1929, 138, §§ 23, 40, 51).

However, as theoreticians of science in general, philosophical logicians are obliged to contend with the question of basic truths about a universe of objects existing outside of formal systems. They are called upon to seek solutions to the problems that come up when scientific discourse steps outside the purely formal domain and makes reference to specific objects or domains of objects. They are not free to sever their ties with nature and science, to accept a logic that tears itself entirely away from the idea of any possible application and becomes a mere ingenious playing with thoughts, or symbols that mere rules or conventions have invested with meaning. They must step out of the abstract world of pure analytic logic, with its ideal, abstract entities, and confront those more tangible objects that make up the material world of things. In addition, they are obliged to step back and investigate the theory of formal languages and systems themselves, and their interpretations (ex. Husserl 1929, §§ 40, 52).

So, Husserl believed that formal logic required a complement. Once liberated from things and psychologizing subjectivity, pure logic had to find its necessary complement in a transcendental logic that would take into account the connections that philosophical logic inevitably maintains with both knowing subjects and the concrete world. For Husserl, true philosophical logic could only develop in connection with a transcendental phenomenology by which logicians penetrate an objective realm which is entirely different from them (ex. Husserl 1929, §§ 40, 42).

However, Husserl always insisted on the primacy of the objective side of logic. He insisted that the subjective order could not be properly examined until the objective order had been, and until the objectivity of the structures girding scientific knowledge had been established and demonstrated. He maintained that pure logic with its abstract ideal structures had to be clearly seen and definitely apprehended as dealing with ideal objects before transcendental questions about them could be asked (Husserl 1929, §§ 8, 9, 11, 26, 42-44, 92, 98, 100).

It is knowledge of formal logic, he reminded readers in Formal and Transcendental Logic, that supplies the standards by which to measure the extent to which any presumed science meets the criteria of being a genuine science, the extent to which the particular findings of that science constitute genuine knowledge, the extent to which the methods it uses are genuine ones (Husserl 1929, § 7). The world constituted by transcendental subjectivity is a pre-given world, Husserl explained in Experience and Judgement. It is not a pure world of experience, but a world that is determined and determinable in itself with exactitude, a world within which any individual entity is given beforehand in an perfectly obvious way as in principle determinable in accordance with the methods of exact science and as being a world in itself in a sense originally deriving from the achievements of the physico-mathematical sciences of nature (ex. Husserl 1939, §11; Husserl 1929, 26b).

Husserl was perfectly conscious of the extraordinary difficulties that this dual orientation of logic involved. Since, according to his theories, the ideal, objective, dimension of logic and the actively constituting, subjective dimension interrelate and overlap, or exist side by side, logical phenomena thus seem to be suspended between subjectivity and objectivity in a confused way. In Formal and Transcendental Logic, he suggested that almost all that concerns the fundamental meaning of logic, the problems it deals with, its method, is laden with misunderstandings owing to the very fact that objectivity arises out of subjective activity. He even considered that it was due to these difficulties that, after centuries and centuries, logic had not attained the secure path of rational development (ex. Husserl 1929, § 8).


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