This is a preprint version of the paper which appeared in definitive form in Synthese 113, October 1997, 145-70 and Husserl or Frege, Meaning, Objectivity and Mathematics, by Claire Ortiz Hill and Guillermo Rosado Haddock, Chicago: Open Court, 2000. The published version should be consulted for all citations.
Few have entertained the idea that the creator of set theory may have influenced the founder of the phenomenological movement,1 and not a few would consider the mere suggestion utterly preposterous, if not downright blasphemous. Yet Georg Cantor and Edmund Husserl were close friends at the University of Halle2 during the last fourteen years of the nineteenth century, when Cantor was at the height of his creative powers and Husserl in the throes of an intellectual struggle during which his ideas were particularly malleable and changed considerably and definitively (Husserl 1913). It was there and then that Husserl wrote On the Concept of Number (Husserl 1887), the Philosophy of Arithmetic (Husserl 1891) and the groundbreaking Logical Investigations (Husserl 1900-01), where he lay the foundations of his phenomenology that went on to shape the course of 20th century philosophy in Continental Europe.
Although the relationship between Husserl's and Cantor's ideas has gone all but unnoticed, there is much information to be gleaned from a close examination of their writings. So in the following pages I turn to that source to show how Husserl's and Cantor's ideas overlapped and crisscrossed during those years in the areas of philosophy and mathematics, arithmetization, abstraction, consciousness and pure logic, psychologism, metaphysical idealism, new numbers, and sets and manifolds. In so doing I hope to shed some needed light on the evolution of Husserl's thought during that crucial time in Halle.
Before turning to the examination of these questions, however, it is important to bear in mind that influence may be negative or positive, temporary or permanent, and it may show through in different areas in different ways. Moreover, in the case of an original thinker of Husserl's stature it could not be a matter of the simple imitation of someone else's ideas. For example, few question that Franz Brentano influenced Husserl, but Husserl actually eradicated the most distinctive features of Brentano's teaching from phenomenology and embraced metaphysical and epistemological views which Brentano considered odious and despicable (Hill 1998).
1. Marrying Philosophy and Mathematics
Recently converted to philosophy by Franz Brentano (Husserl 1919), Husserl took a front row seat at the creation of set theory and the transfinite number system when he arrived in Halle in 1886 to prepare his Habilitationsschrift, On the Concept of Number, of which his Philosophy of Arithmetic would be a revised and much enlarged version. Cantor served on Husserl's Habilitationskommittee and approved the mathematical portion of Husserl's work (Gerlach and Sepp 1994). The two became close friends.
In the late 1880s Husserl and Cantor figured among the small number of their contemporaries intent upon wedding mathematics and philosophy (Frege being of course another). Over the years Cantor's thoughts had been increasingly turning to philosophy and in 1883 he published the Grundlagen einer allgemeine Mannigfaltigkeitslehre (Cantor 1883), a work, according to its 1882 foreword, "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". 3
As earnest as Cantor's forays into philosophy were, they hardly met with universal acclaim. In 1883 Gösta Mittag-Leffler deemed it fit to warn Cantor that his work would be much more easily appreciated in the mathematical world "without the philosophical and historical explanations" (Cantor 1991, 118). And in 1885 he warned him that he risked shocking most mathematicians and damaging his reputation with his new terminology and philosophical way of expressing himself in the "Principien einer Theorie der Ordnungstypen" (Cantor l991, 244), a posthumously published work which opened with a strong statement in favor of an admixture of philosophy, metaphysics and mathematics (Cantor 1884, 83-84).
By the time Husserl arrived in Halle, Cantor was practically ready to abandon mathematics for philosophy. He was much less in a position than Husserl was to change course in mid-stream, but this did not keep him from trying to teach philosophy (Cantor 1991, 210, 218) and from seasoning his writings with philosophical reflections and references to philosophers like Democritus, Plato, Aristotle, Augustine, Boethius, Aquinas, Descartes, Nicolas von Cusa, Spinoza, Leibniz, Kant, Comte, Francis Bacon, Locke and so on. In 1894 Cantor wrote to the French mathematician 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, he "openly confessed", had so taken possession of his soul as to leave him relatively little time for his "first flame". It might have been otherwise, he said, had his wishes for a position in a university where his teaching might have had a greater effect been fulfilled fifteen, or even eight years earlier, but he was now serving God better than, "owing to his apparently meager mathematical talents", he might have done through exclusively pursuing mathematics (Cantor 1991, 350).
During the late 1880s, the embattled creator of set theory was hard at work trying to put the new numbers he was inventing on solid foundations and philosophically justifying the claims he was making about them (Dauben 1979, Chapter 6). Those attempts found expression in his correspondence (Cantor 1991) and most particularly in the "Mitteilungen zur Lehre vom Transfiniten" (Cantor 1887/8), which he said he published in the Zeitschrift für Philosophie und philosophische Kritik because he had grown disgusted with the mathematical journals (Dauben 1979, 139, 336 note 29).
Much of the initial intellectual kinship between Husserl and Cantor can be explained by the mighty influence the great mathematician Karl Weierstrass exercised on both of them. "Mathematicians under the influence Weierstrass", Bertrand Russell once wrote, "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". Analytical Geometry and the Infinitesimal Calculus, he went on to explain, had produced so many fruitful results since their development in the seventeenth century that mathematicians had neither taken the time, nor been inclined to examine their foundations. Philosophers who might have taken up the task were lacking in mathematical ability. Mathematicians were only finally awakened from their "dogmatic slumbers" by Weierstrass and his followers in the latter half of the nineteenth century (Russell 1917, 94).
Cantor4 and Husserl had both fallen under the influence of Weierstrass and Husserl had even been his assistant.5 It was Weierstrass, Husserl has said, who awakened his interest in seeking radical foundations for mathematics. Weierstrass' aim, Husserl once recalled, was to expose the original roots of analysis, its elementary concepts and axioms on the basis of which the whole system of analysis might be deduced in a completely rigorous, perspicuous way. Those efforts made a lasting impression on Husserl who would say that it was from Weierstrass that he had acquired the ethos of his intellectual endeavors (Schuhmann 1977, 7). Late in his philosophical career Husserl would say that he had sought to do for philosophy what Weierstrass had done for mathematics (Becker 1930, 40-42; Schuhmann 1977, 345).
With "respect to the starting point and the germinal core of our developments toward the construction of a general arithmetic," Husserl wrote as he was carrying out his plan to help supply radical foundations for mathematics, "we are in agreement with mathematicians that are among the most important and progressive ones of our times: above all with Weierstrass, but not less with Dedekind, Georg Cantor and many others" (Husserl 1994, 1).
Husserl began On the Concept of Number writing of the exhilaratingly creative period mathematics had known over the previous hundred years, during which new and very far-reaching instruments of investigation had been found and an almost boundless profusion of important pieces of knowledge won. Mathematicians, however, he explained, had neglected to examine the logic of the concepts and methods they were introducing and using and there was now a need for logical clarification, precise analyses, and a 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 are 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 considered, was the analysis of the concept of whole number (Husserl 1887, 94-95). For the still faithful disciple of Weierstrass then believed the "domain of 'positive whole numbers' to be the first and most underivative domain, the sole foundation of all remaining domains of numbers" (Husserl 1994, 2). "Today there is a general persuasion," he wrote in "On the Concept of Number",
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).
So in On the Concept of Number and Philosophy of Arithmetic Husserl set out to carry Weierstrass's work to arithmetize analysis a step further by submitting the concept of number itself to closer scrutiny than Weierstrass had.
As for Cantor, after working for several years to clear up theoretical obscurities regarding infinity and the real number system raised but not fully answered by Weierstrass's lectures on analytic functions, the late 1880s found him hard at work laying the foundations of his set theory and reconnoitering, conquering, colonizing and defending the world of transfinite numbers (Jourdain 1908-14; Hilbert 1925; Hallett 1984; Dauben 1979).
Like Husserl, Cantor was immersed in a project aimed at demonstrating that the positive whole numbers formed the basis of all other mathematical conceptual formations inspired by Weierstrass's famous theory to that effect. Any further progress of the work on set theory, Cantor explained in the beginning of his Mannigfaltigkeitslehre was absolutely dependent upon the expansion of the concept of real whole numbers beyond the present boundaries and in a direction which, as far as he knew, no one had yet searched. He had, he claimed, burst the confines of the conceptual formation of real whole numbers and broken through into a new realm of transfinite numbers. Initially he had not been clearly conscious of the fact that these new numbers possessed the same concrete reality the whole numbers did. He had, however, become persuaded that they did. As strange and daring as his ideas might now seem, he was convinced that they would one day be deemed completely simple, appropriate and natural (Cantor 1883, 165-66). Much of the work he was doing was aimed at showing they were.
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), Cantor began devising a strategy as to how to provide his "strange" new transfinite numbers with secure foundations by demonstrating precisely how the transfinite number system might be built from the bottom up. He was in possession of principles by which, he claimed, one might break through any barrier in the conceptual formation of the real, whole numbers and with the greatest confidence and self-evidence arrive at ever new number classes and numbers having the same concrete definiteness and reality as objects as the previous ones (Cantor 1883, 166-67, 199).
Cantor developed a theory of abstraction which he believed was the distinctive feature of his number theory and represented an entirely different method for providing the foundations of the finite numbers than was to be found in the theories of his contemporaries (Cantor 1991, 365, 363). He accorded his theory a key role in his efforts to provide solid foundations for transfinite arithmetic, believing that with it he was laying bare the roots from which the organism of transfinite numbers develop with logical necessity (Cantor 1887/8, 380).
For Cantor, cardinal numbers were the general concepts assigned to sets which one may obtain by abstracting both from the properties of the elements and from the order in which they are given (Cantor 1991, 302) and the transfinite numbers could be "produced through abstraction from reality with the same necessity as the ordinary finite whole numbers by which alone all other mathematical conceptual formations have thus far been produced" (Cantor 1991, 136; Kreiser l979).
The earliest accounts he gave of his abstraction process appear in his correspondence (Cantor 1991, 178-80) and in the posthumously published "Principien einer Theorie der Ordnungstypen" (Cantor 1884). The theory did not, however, make its way into print until it appeared in the "Mitteilungen" (Cantor 1887/8), into which he directly incorporated letters he had written and where he stressed that he had advocated and repeatedly taught the theory in his courses as many as four years earlier (Cantor 1887/8, 378-79, 387 n., 411 n.).
In the "Mitteilungen" Cantor 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). And much of the work is devoted to explaining precisely how one might procure numbers by abstraction from reality and, in particular, how the actual infinite number concept might be formed through appropriate natural abstractions in the way the finite number concepts are won through abstraction from finite sets (Cantor 1887/8, 411; also Cantor 1991, 329, 330).
In the work Cantor repeatedly gave essentially the same recipe for extracting cardinal numbers from reality through abstraction (Cantor 1887/8, 379, 387, 411, 418 note1). In a text, which Husserl approvingly cites in the Philosophy of Arithmetic (Husserl 1891, 126 n.) Cantor explained that for "the formation of the general concept 'five' one needs only a set (for example all the fingers of my right hand) which corresponds to this cardinal number; the act of abstraction with respect to both the properties and the order in which I encounter these wholly distinct things, produces or rather awakens the concept 'five' in my mind" (Cantor 1887/8, 418 note 1).
"By the power or cardinal number of a set M (which is made up of distinct, conceptually separate elements m, m', . . . and is to this extent determined and limited)," Cantor said he understood
the general concept or species concept (universal) which one obtains by abstracting from the properties of the elements of the set, as well as from all the relations which the elements may have, whether themselves or to other things, but especially from the order reigning among the elements and only reflects upon what is common to all sets equivalent to M (Cantor 1887/8, 387; also Cantor 1991,178).
In the "Principien" Cantor had written that the cardinal number of a set seemed to him "to be the most primitive, psychologically, as well as methodologically simplest root concept, arisen through abstraction, from all particular characteristics which a set of a specific class may display, both with respect to the nature of its elements, as well as with regard to the relations and order in which the elements are to each other or can stand to things lying outside the set" (Cantor 1884, 86).
Cantor was propounding this theory of abstraction just as Husserl was writing On the Concept of Number and the Philosophy of Arithmetic, where a similar theory is espoused. Husserl actually had offprints of the "Mitteilungen" and marked and underlined precisely those passages (and almost exclusively those passages) in which Cantor explained the abstraction process.6 In On the Concept of Number, Husserl wrote:
It is easy to characterize the abstraction which must be exercised upon a concretely given multiplicity in order to attain to the number concepts under which it falls. One considers each of the particular objects merely insofar as it is a something or a one herewith fixing the collective combination; and in this manner there is obtained the corresponding general form of multiplicity, one and one and . . . and one, with which a number name is associated. In this process there is total abstraction from the specific characteristics of the particular objects.... To abstract from something merely means to pay no special attention to it. Thus in our case at hand, no special interest is directed upon the particularities of content in the separate individuals (Husserl l887, ll6-17). 7
In the Philosophy of Arithmetic Husserl studied this same process in greater detail (Chapter 4 especially), particularly underscoring the uniqueness of the abstraction process which yields the number concept. According to the traditional theory of concept formation through abstraction, he pointed out there, one is to disregard the properties which distinguish the objects while focussing on the properties they have in common and out of which the general concept will be made. But Husserl thought it absurd to expect that the concept of set or number might result from any comparison of the individual contents making up the sets which may be utterly heterogeneous. He emphasized that his analyses had led to the important, securely substantiated finding that it is impossible to elucidate the formation of the concept of number in the same way one elucidates the concepts of color, form, etc. And he pointedly disassociated his theory from the better known abstraction theories of Locke and Aristotle, something it is important to call attention to because philosophers have been all too wont to assimilate the process Husserl advocated to theories more familiar to them (Husserl 1891, 12-16, 50, 84-93, 182-86).
As similar as Husserl's and Cantor's theories of abstraction appear to have been in the late 1880s, there was, nonetheless, a major difference between them which reflected deep contradictions within both men's philosophies of arithmetic and soon brought Husserl to rethink and profoundly modify his entire approach to philosophy.
This major difference between them lie in the fact that Husserl began writing On the Concept of Number as a committed empirical psychologist à la Brentano, whose philosophical ideal was most nearly realized in the exact natural sciences (Husserl 1919, 344- 45). "In view of my entire training", Husserl wrote of his early work on the foundations of arithmetic, "it was obvious to me when I started that what mattered most for a philosophy of mathematics was a radical analysis of the 'psychological origin' of the basic mathematical concepts" (Husserl 1913, 33; see Husserl 1891, l6). In On the Concept of Number and the Philosophy of Arithmetic he set out to anchor arithmetical concepts in direct experience by analyzing the actual psychological processes to which, he believed, the concept of number owed its genesis. He considered the analysis of elementary concepts to be one of the more essential tasks of the psychology. "Not only is psychology indispensable for the analysis of the concept of number," he declared in On the Concept of Number, "but rather this even belongs within psychology" (Husserl 1887, 95).
Given his early training, it is easy to see how Husserl might have considered Cantor's philosophy of arithmetic ripe for the kind of analyses Brentano trained his students to undertake, --how Cantor's talk of intuitions and presentations might grow in sophistication through analysis in accordance with Brentano's teachings about presentation and intentionality. Cantor's appeals to inner intuition and talk of things like the fingers of his right hand helping produce or awaken concepts in his mind do sound unabashedly empiricistic or psychologistic. Frege certainly saw them as such as he decried the "psychological and hence empirical turn" in the "Mitteilungen" (Frege 1892, 180, 181).
However, psychology from the empirical standpoint was strictly incompatible with the ideas of Cantor, who considered himself to be an adversary of psychologism, empiricism, positivism, naturalism and related trends. However psychologistic his mysterious references to inner intuition (ex. Cantor 1883, 168, 170, 201) or to experiences helping produce concepts in his mind (Cantor 1887/8, 418 note 1) may appear, he opposed the new empiricism, sensualism, skepticism and Kantianism which, he argued, mistakenly located the sources of knowledge and certainty in the senses or in the "supposedly pure forms of intuition of the world of presentation". He maintained that certain knowledge could "only be obtained through concepts and ideas which, at most stimulated by external experience, are on the whole formed through inner induction and deduction as something which in a way already lay within us and was only awakened and brought to consciousness" (Cantor 1883, 207 note 6). In his 1885 review of the Foundations of Arithmetic, he praised Frege 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 1885, 440).
For Cantor, the transfinite "presented a rich, ever growing field of ideal research" (Cantor 1887/8, 406) and abstraction was to show the way to that new, abstract realm of ideal mathematical objects which could not be directly perceived or intuited. For him 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, 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 note 1; Eccarius 1985, 19-20; Couturat 1896, 325-41).
In this, Cantor was in step with Weierstrass' aim to arithmetize analysis (Jourdain 1908-14; Kline 1972). And herein lies a contradiction in Husserl's analyses. For Husserl too shared those particular aims and long sections of On the Concept of Number and the Philosophy of Arithmetic were devoted to discussions aimed at obtaining pure arithmetical concepts by detaching the concept of number from any spatio-temporal intuitions and so also from any taint of Kantianism in keeping with the goals Weierstrass had set (Husserl 1887, Section 2; Husserl 1891, Chapter 2).
Husserl did not initially see Brentano's empirical psychology as empirical and psychological in a pernicious sense and continued to resort to psychological analyses in the Philosophy of Arithmetic. However, the confidence he expressed in On the Concept of Number eroded quickly and the enthusiastic espousal of psychologism of On the Concept of Number does not even appear in the later work, --a sign that change was on the horizon.
There were respects, Husserl would confess in the foreword to the Logical Investigations, in which psychological foundations had never come to satisfy him. The psychological analyses of his earliest work on the foundations of arithmetic, he explained, left him deeply dissatisfied and he "became more and more disquieted by doubts of principle, as to how to reconcile the objectivity of mathematics, and of all science in general, with a psychological foundation for logic". His whole method by which he had hoped to illuminate mathematics through psychological analyses became shaken and he felt himself "more and more pushed towards general critical reflections on the essence of logic, and on the relationship, in particular, between the subjectivity of knowing and the objectivity of the content known" (Husserl 1900-01, 42)
5. Actual Consciousness and Pure Logic
In the foreword to the Logical Investigations Husserl also tells something of the evolution of his thought during his time in Halle. After working for many years to bring philosophical clarity to pure mathematics, he had found himself up against difficulties connected with developments in mathematics which defied his efforts at logical clarification. The problems were so compelling as to make him set aside his investigations into philosophico-mathematical matters until he had "succeeded in reaching a certain clearness on the basic questions of epistemology and in the critical understanding of logic as a science" (Husserl 1900-01,41-43).
Husserl wrote of having felt "tormented by those incredibly strange realms: the world of the purely logical and the world of actual consciousness" which he saw opening up all around him while he was laboring to understand the logic of mathematical thought and calculation. He believed that the two spheres had "to interrelate and form an intrinsic unity", but had no idea as to how to bring them together (Husserl 1994, 490-91; Husserl 1913, 20-22).
Into the strange world of pure logic whose interaction with the world of consciousness puzzled him, Husserl placed "all of the pure 'analytical' doctrines of mathematics (arithmetic, number theory, algebra, etc)" (Husserl 1913, 28), the pure theory of cardinal numbers, the pure theory of ordinal numbers, the traditional syllogistic and the pure mathematical theory of probability. He also put Cantorian sets, "the Mannigfaltigkeitslehre in the broadest sense" into the category of pure logic (Husserl 1994, 250; Husserl 1913, 28).
Now the new numbers and countless infinities the emancipated Cantor was producing proved counter-intuitive and paradoxical enough to challenge accepted logical assumptions (Hilbert 1925, 375). And Cantor himself was the first to admit that his theories were strange. He saw himself as exploring terra incognita and referred to his new number classes as "strange things", writing that he had been logically compelled to introduce them almost against his will, but did not see how he might proceed further with set theory and function theory without them (Cantor 1883, 165, 175; Cantor 1991, 95).
So the nature of the problems which beset Husserl becomes clearer when one takes into account that Cantor was busy exploring, mapping and inventing the strange world of transfinite sets at the very time Husserl broke out in crisis, making it easy to believe that the crisis was aggravated, if not actually induced by Cantor's bold experiments with mathematics and epistemology.
In this respect the section on infinite sets in the Philosophy of Arithmetic (Husserl 1891, 246-50) yields insight into the nature of the problems tormenting Husserl as he tried to complete the book. There was, Husserl noted there, a particularly odd way of extending the original concept of multiplicity or quantity, which by its very nature reaches beyond the necessary bounds of human cognition, and so wins for itself an essentially new content. These were the infinite sets, multiplicities or collections. "Infinite", he wrote, calling to mind some of Cantor's main preoccupations, "are the extensions of most general concepts. Infinite is the sequence of numbers extended by symbolic means, infinite is the set of points on a line, and in general that of the limits of a continuum" (Husserl 1891, 246-47).
Signs, symbolic presentation à la Brentano, Husserl explained, might aid the mind in reasoning in regions of thought beyond what could be known through direct cognitive processes like perception or intuition. The repeated application of operations permitting the collecting together of a multitude of objects one after the other into a set could take the place of the direct cognitive grasp of sets with hundreds, thousands or millions of members, and he was satisfied that this was a way of actually representing collections in an ideal sense and essentially unproblematic from a logical point of view (Husserl 1891, 246).
However, Husserl considered, this became impossible in the case of infinite totalities, multiplicities or collections since the very principle by which they are formed or symbolized itself immediately makes collecting of all their members together one by one a logical impossibility. By no extension of our cognitive faculties could we conceivably cognitively grasp or even successively collect such sets, he points out. So the logical problems connected with infinite sets were of a completely different order. With them we had reached the limits of idealization (Husserl 1891, 24--6-47).
With infinite sets, he reasoned, it is always a case of the symbolic presentation of a never-ending process of concept formation. We are in possession of a clear principle by which we can transform any already formed concept of a certain given species into a new concept which is plainly distinct from the previous one. And we can do this over and over in such a way as to be certain a priori of never coming back to the original concept and to previously generated concepts. Repeated application of this process yields successive presentations of continually expanding sets, and if the generating principle is really determinate, then it is determined a priori whether or not any given object can belong to the concept of the expanding set of concepts (Husserl 1891, 247).
In the case of the concept of the infinite set of numbers we begin with a direct presentation, he explained. The other natural numbers can be reached by repeated additions of 1, and nothing prevents us from advancing indefinitely in this way to new cardinals; we have a method for adding one to a previously given number, an operation that necessarily generates one new number after another without return and limitation, each new number being determined by the process (Husserl 1891, 247-48).
It is easy, Husserl continues, to see why mathematicians have tried to transpose the concept of quantity to such constructions, which are, however, of an essentially different logical nature. In the usual cases, the process by which the sets were generated was finite, there was always a last stage, it was sometimes possible actually to bring the process to a halt, and also to construct the corresponding set. However, this is quite absurd in the case of infinite sets. The process used to generate them is non-terminating, and the idea of a last stage, of a last member of the set is meaningless. And this constitutes an essential logical difference (Husserl 1891, 248).
Nonetheless, Husserl points out, despite the absurdity of the idea, the analogies pertaining foster a tendency to transpose the idea of constructing a corresponding collection for infinite sets, thereby creating what he calls a kind of "imaginary" concept whose anti-logical nature is harmless in everyday contexts precisely because its inherent contradictoriness is never obvious in life. This is, he explains, the case when "All S" is treated as a closed set (Husserl 1891, 249).
However, Husserl warns, the situation changes when this imaginary construct is actually carried over into reasoning and influences judgements. It is clear, he concludes, that from a strictly logical point of view we must not ascribe anything more to the concept of infinite sets than is actually logically permissible, and above all not the absurd idea of constructing the actual set (Husserl 1891, 249).
As for those other "strange worlds", the strange worlds of pure consciousness, the naive epistemological theorizing which Cantor was so earnestly engaging in while Husserl was grappling with analogous questions could easily have impressed upon him an urgent need to develop the more sophisticated logical and epistemological tools needed to attain a deeper, clearer understanding of how the human mind interacted with the world of numbers. For the "world" of consciousness and the "world" of pure mathematics mingle together in Cantor's work in confusing and frustrating ways which cry out for clarification.
For instance, one might fairly wonder in just what way the cardinal number belonging to a set is an abstract image in our intellect (Cantor 1887/8, 416), exactly how the act of abstraction awakened the number concepts in Cantor's mind (Cantor 1887/8, 418 note 1), or how concepts and ideas are formed through inner induction like something already lying within us which is merely awakened and brought to consciousness (Cantor 1883, 207 note 6). Frege was perfectly justified in qualifying Cantor's appeals to direct inner intuition (Cantor 1883, 168, 170, 201) as "rather mysterious" (Frege 1884, §86).
So given the particular nature of Cantor's experiments it is not surprising to find Husserl asking in those years how rational insight was possible in science (Husserl 1994, 167), how the mathematical in itself as given in the medium of the psychical could be valid (Husserl 1913, 35), how logicians penetrated an objective realm entirely different from themselves (Husserl 1913, 222), how objective, mathematical and logical relations constituted themselves in subjectivity (Husserl 1913, 35), how symbolic thinking was possible (Husserl 1913, 35), how abandoning oneself completely to thought that is merely symbolic and removed from intuition could lead to empirically true results (Husserl 1994,167), or how mechanical operations with mere written characters could vastly expand our actual knowledge concerning number concepts (Husserl 1994, 50). These are all questions Cantor's theories raise.
Facing only what he variously called riddles, tensions, puzzles and mysteries about consciousness and pure logic, and seeing all around him only unclear, undeveloped, ambiguous, confused ideas, but no "full and truly satisfactory understanding of symbolic thought or of any logical process" (Husserl 1994, 169), Husserl set out on his own to solve the problems his investigations into the foundations of mathematics had raised, concluding after a "decade of solitary, arduous labor" that the puzzles surrounding the being in itself of the ideal sphere and its relationship to consciousness would only be solved through the pure phenomenological elucidation of knowledge he developed (Husserl 1994, 251; Husserl 1900-01, 42-43, 223-24; Husserl 1913).
6. Metaphysical Idealism
To the chagrin of his contemporaries, as we saw in the preceding chapter, (Cantor 1991, 110, 113, 118, 178, 227, 241) Cantor persisted in clothing his theories about numbers in a metaphysical garb. In the Mannigfaltigkeitslehre he 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 note 6). His own idealism being related to the Aristotelian-Platonic kind, he wrote in an 1888 letter, he was just as much a realist as an idealist (Cantor 1991, 323).
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 note 1). "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 Charles 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 1984, 149).
For Cantor the realm of the transfinite was "a rich, ever growing field of ideal research" (Cantor 1887/8, 406). He 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). And he considered that his technique for abstracting numbers from reality provided the only possible foundations for that Platonic conception of numbers (Cantor 1991, 363, 365; Cantor 1887/8, 380, 411). Abstracting from both the characteristics of the elements of the set and the order in which they are given, one obtained the cardinal numbers; abstracting only from the characteristics of the elements and leaving their order intact, one obtained the ideal numbers or eidetikoi (Cantor 1887/8, 379-80; Cantor 1883, 180-81). His talk of awakening and bringing to consciousness the knowledge, concepts and numbers slumbering in us (Cantor 1883, 207 notes 6, 7, 8; Cantor 1887/8, 418 note 1) is, of course, an unmistakable allusion to Plato's theory of recollection and Socratic theories of concept formation (ex. the Meno 81C-86C; Phaedo 72E, 75E-76A)
A good measure of the freedom Cantor felt he possessed as a mathematician in fact derived from his distinguishing between an empirical treatment of numbers and Plato's pure, ideal arithmoi eidetikoi which by their very nature are detached from things perceptible by the senses. In 1890 he wrote to Veronese that contradictions he had found in Cantor's theories were but apparent and that one must distinguish between the numbers which we are able to grasp in our limited ways and "numbers as they are in and for themselves, and in and for the Absolute intelligence" (Cantor 1991, 326; see also 267, 282).
In sharp contrast to this, Husserl came to Halle free of Platonic idealism. For Franz Brentano inculcated in his students a model of philosophy based on the natural sciences and taught them to despise metaphysical idealism. So completely under Brentano's influence in the beginning, Husserl was, initially quite disinclined to traffic in the kind of idealism that so pervaded Cantor's writings (Husserl 1919, 344-45) and any such considerations are conspicuously absent from both On the Concept of Number and Philosophy of Arithmetic.
Notwithstanding, Husserl left Halle a committed Platonic idealist persuaded that pure mathematics was a strictly self-contained system of doctrines to be cultivated by using methods that are essentially different from those of natural science (Husserl 1913, 29). "The empirical sciences --natural sciences", he 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).
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 1913, 42). He came to consider idealistic systems to be of "the highest value", that entirely new and totally radical dimensions of philosophical problems were illuminated in them, and "the ultimate and highest goals of philosophy are opened up only when the philosophical method which these particular systems require is clarified and developed" (Husserl 1919, 345). Every possible effort, he wrote, 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, 1913, 20).
While Cantor's experiments very likely acted to pry Husserl away from psychologism and to steer him in the direction of Platonic idealism, Husserl always maintained that it was Hermann Lotze's work, we saw in the previous chapter, which was responsible for the fully conscious and radical turn from psychologism and the Platonism that came with it. Lotze's interpretation of Plato's doctrine of Ideas, Husserl said, provided him with his first major insight, became a determining factor in all his further studies, and gave him the idea to transfer all of the mathematical and a major part of the traditionally logical into the realm of the ideal. Through Lotze's work Husserl also found the key to understanding what he called Bolzano's "curious conceptions", which had originally seemed naive and unintelligible to him, but in which he was to discover a complete plan of a pure logic and an initial attempt to provide a unified presentation of the domain of pure ideal doctrines (Husserl 1994, 201-02; Husserl 1913, 36-38, 46-49).
7. Imaginary Numbers
In the foreword to the Logical Investigations Husserl specifically alluded to having been troubled by the theory of manifolds, the Mannigfaltigkeitslehre, with its expansion into special forms of numbers and extensions. The fact, he explained, that one could obviously generalize, produce variations of formal arithmetic which could lead outside the quantitative domain without essentially altering formal arithmetic's theoretical nature and calculational methods had brought him to realize that there was more to the mathematical or formal sciences, or the mathematical method of calculation than would ever be captured in purely quantitative analyses (Husserl 1900-01, 41-42; Husserl 1913, 35).
And he always said that it was particularly difficulties he experienced in trying to answers questions raised by "imaginary" numbers that arose while trying to complete the Philosophy of Arithmetic which had marked the turning point in his thinking. He used the term "imaginary" in a very broad sense to cover negative numbers, negative square roots, fractions and irrational numbers, and so on (Husserl 1994,13-16; Husserl 1913, 33; Husserl 1970, 430-51; Husserl 1929, §31; Hill 1991, 81-86). As we saw above he called infinite sets "imaginary concepts" (Husserl 1891, 249) 8
In On the Concept of Number Husserl had maintained that all the more complicated and artificial forms of numbers had their origin and basis in the concept of positive whole numbers and their interrelations and were derivable from them in a strictly logical way. However, hints of a change of mind were already to be found in the very first pages of the Philosophy of Arithmetic, where Weierstrass' thesis that cardinal numbers form the sole basis for arithmetic is never endorsed in the confident way it was in On the Concept of Number (Husserl 1887, 94-95).
Conspicuously apparent in the later work is Husserl's tergiversation regarding Weierstrass' teaching. In the Philosophy of Arithmetic we find Husserl explaining that he will provisionally use it as a springboard for his own analyses. However he warns readers that, although cardinal numbers in a certain way seem to be the basic numbers involved in arithmetic because the signs for them figure in expressions for positive, negative, rational, irrational, real, imaginary, alternative, ideal numbers, quaternions etc. the analyses of the second volume would perhaps show that thesis to be untenable (Husserl 1891, 5-6). In the April 1891 preface to the book he went further to state that the analyses of the second volume would actually show that in no way does a single kind of concept, whether that of cardinal or ordinal numbers, form the basis of general arithmetic (Husserl 1891, VIII).
Finally, before the Philosophy of Arithmetic had even made its way into print, Husserl wrote to Carl Stumpf that the opinion by which had been guided in writing On the Concept of Number that the concept of cardinal number formed the foundation of general arithmetic had soon proved to be false, that through no manner of cunning could negative, rational, irrational and the various kinds of complex numbers be derived from the concept of the cardinal number (Husserl 1994, 13).
By 1891 Husserl had become keenly aware that "a utilization of symbols for scientific purposes, and with scientific success, is still not therefore a logical utilization" (Husserl 1994, 48). As he explained:
General arithmetic, with its negative, irrational, and imaginary (impossible) numbers, was invented and applied for centuries before it was understood. Concerning the signification of these numbers the most contradictory and incredible theories have been held; but that has not hindered their use. One could quite certainly convince oneself of the correctness of any sentence deduced by means of them through an easy verification. And after innumerable experiences of this sort, one naturally comes to trust in the unrestricted applicability of these modes of procedure, expanding and refining them more and more --all without the slightest insight into the logic…. (Husserl 1994, 48).
Alluding to "the endless controversies over negative and imaginary numbers, over the infinitely small and the infinitely large, over the paradoxes of divergent series, and so on", Husserl lamented the mental energy "wasted on this route, more governed by chance than logic" (Husserl 1994, 49). Arithmetic, he contended, would have made quicker and surer progress had there been clarity about the logic of its methods upon their development instead. He believed that insight into the logical character of arithmetic would have to play a decisive role in the future development of the field (Husserl 1994, 48-51).
A lecture Husserl gave before the Göttingen Mathematical Society in 1901 (Husserl 1970, 430-51) affords some insight into the steps in his reasoning which led to the discovery of the solution he would ultimately deem satisfactory. Questions regarding imaginary numbers, he explained to his listeners there, had come up in mathematical contexts in which formalization yielded constructions which arithmetically speaking were nonsense but which, astonishingly, could nevertheless be used in calculations. It became apparent that when formal reasoning was carried out mechanically as if these symbols had meaning, if the ordinary rules were observed, and the results did not contain any imaginary components, then these symbols might be legitimately used. And this could be empirically verified (Husserl 1970, 432).
However, this fact 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. Faced with these problems, Husserl said that his main questions were: (1) Under what conditions can one freely operate within a formally defined deductive system with concepts which according to the definition of the system are imaginary and have no real meaning? (2) When can one be sure of the validity of one's reasoning, that the conclusions arrived at have been correctly derived from the axioms one has, when one has appealed imaginary concepts? And (3) To what extent is it permissible to enlarge a well-defined deductive system to make a new one that contains the old one as a part? (Husserl 1970, 433; Husserl 1929, §31).
Again, the work on numbers Cantor was doing in the late 1880s makes such questioning on Husserl's part completely understandable. Those were years during which Cantor was particularly engaged in what Grattan-Guinness has called "rather strange work on theory of numbers" (Grattan-Guinness 1971,369), producing what Dauben has called "dinosaurs of his mental creation, fantastic creatures whose design was interesting, overwhelming, but impractical to the demands of mathematicians in general" (Dauben 1979, 158 -59).
Husserl found the answers he was looking for in his own theory of complete manifolds (Hill 1995), the definite Mannigfaltigkeiten of the "Prolegomena to Pure Logic" in the Logical Investigations (§§69-70), in which believed he had discovered "the key to the only possible solution of the problem" as to "how in the field of numbers impossible (essenceless) concepts can be methodically treated like real ones" (Husserl 1900-01, 242).
8. Sets and Manifolds
In On the Concept of Number and the Philosophy of Arithmetic Husserl made set theory the basis of mathematics. Using the terms 'multiplicity' and 'set' interchangeably to neutralize any differences in meaning among the terms, and citing Euclid's classical definition of the concept of number as "a multiplicity of units", he began the analyses of the Philosophy of Arithmetic by affirming that "the analysis of the concept of number presupposes the concept of multiplicity" (Husserl 1891, 8 and note; also Husserl 1887, 96).
The most primitive concepts involved, he explained in the work, are the general concepts of set and number which are grounded in the concrete sets of specific objects of any kind whatsoever and to which particular numbers are assigned. There could be no doubt, he maintained, that multiplicities or sets of determinate objects were the concrete phenomena which formed the basis for the abstraction of the concepts in question. He supposed that everyone knew what the terms 'multiplicity' and 'set' meant, that the concept itself was well-defined and that there was no doubt as to its extension, which might be taken as given (Husserl 1891, 9 10,13; Husserl 1887, 96-97, 111).
In the Philosophy of Arithmetic, number was the general form of a set under which the set of objects a, b, c fell. To obtain the concept of number of a concrete set of like objects, for example A, A, and A, one abstracted from the particular characteristics of the individual contents collected, only considering and retaining each one of them insofar as it was a something or a one, and thus obtaining the general form of the set belonging to the set in question: one and one, etc. and . . . and one, to which a number name was assigned (Husserl 1891, 88, 165-66; also Husserl 1887, 116-17).
However, Husserl confessed to having been disturbed, and even tormented, by doubts about sets right from the very beginning (Husserl 1913, 35). In the Logical Investigations, we have seen, he specifically mentioned having been troubled by the theory of manifolds (Husserl 1900-01,41 42; Husserl 1913, 35) and he specifically put Cantorian sets, "the Mannigfaltigkeitslehre in the broadest sense" into the category of pure logic which was a source of torment to him (Husserl 1994, 250; Husserl 1913, 28).
The concept of collection in Brentano's sense, he explained in 1913, was to arise through reflection on the concept of collecting. Sets, he had reasoned, arose out of collective combination, in being conceived as one. This combining process involved when objects are brought together to make a whole only consisted in that one thought of them "together" and was obviously not grounded in the content of the disparate items collected into the set. It could not be physical, so it must be psychological, a unique kind of mental act connecting the contents of a whole. But then, he began asking, was "the concept of number not something basically different from the concept of collecting which is all that can result from the reflection on acts?" (Husserl 1913, 34-35; re. Husserl 1891, 14-15; Husserl 1887, 97-98).
Now it is not surprising to find that Husserl had doubts about sets when one considers that he was on hand as Cantor began discovering the antinomies of set theory (Cantor 1991, 387-465; Dauben 1979, 240-70). Others certainly had their logical assumptions shaken upon contact with transfinite set theory. David Hilbert has described the reaction to it as having been dramatic and violent (Hilbert 1925, 375). And it was in studying Cantor's proof that there can be no greatest cardinal number that Bertrand Russell was led to the famous contradiction of the set of all sets that are not members of themselves (Russell 1903, 100, 344, 500; Russell 1959, 58-61; Frege 1980, 133-34, 147; Grattan-Guinness 1978; 1980) which Hilbert has described as having had a "downright catastrophic effect in the world of mathematics", making Dedekind and Frege abandon "their standpoint and quit the field" (Hilbert 1925, 375).
That first-hand experience of inconsistent sets may have actually permanently inoculated Husserl against any recourse to sets or classes. In the early 1890s he was already expressing grave doubts about extensional logic, by which he meant a calculus of classes (Husserl 1994, ex. p. 121). His chief target then was Ernst Schröder (Husserl 1994, 52-91, 421-41). But his antipathy is evident in several articles of the period (ex. Husserl 1994, 92-114, 115-20, 121-30, 135-38, 443-51) in which he was intent upon laying bare the "the follies of extensional logic" (Husserl 1994,199), 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" (Husserl 1994, 109), which he considered to be "totally superfluous" (Husserl 1994, 123). Late in his life Husserl was still denouncing extensional logic as naive, risky, doubtful and the source of many a contradiction requiring every kind of artful device to make it safe for use in reasoning (Husserl 1929, §§ 23, 26).
While Husserl used the various terms for set interchangeably in the late 1880s, he rarely resorted to use of the more Cantorian term 'manifold', 'Mannigfaltigkeit'. In the 1890s, though, he did began studying manifolds, but in the Riemannian, rather than the Cantorian sense (Husserl 1983, 92-106, 408-11; Husserl 1970, 475-78, 493-500). Those investigations culminated in the above-mentioned theory of complete Mannigfaltigkeiten expounded in the Logical Investigations (Husserl 1900-01, §70). He always considered that theory to represent the highest task of formal logic and the formulation of it in the Prolegomena to have been definitive (Husserl 1929, §28).
According to that theory, logic defying creations like those flowing from Cantor's pen could be shown to have redeeming scientific value when they were integrated into a whole which was greater than the sum of its parts, where the anti-logical character of imaginary constructions would become neutralized within the context of a consistent, complete deductive system (Husserl 1970, 441).
This solution was in keeping with Cantor's conviction that mathematical concepts need only be both non-self-contradictory and stand in systematically determined relations established through definition from the previously formed, proven concepts one already has on hand and that mathematicians are only obliged to provide definitions of the new numbers determinate in this way and, if need be, to establish this relationship to the older numbers (Cantor 1883,182). But Husserl 's Mannigfaltigkeiten were Riemannian more than they were Cantorian.
By showing how Husserl's and Cantor's ideas overlapped and crisscrossed during Husserl's time in Halle, I have tried to shed light on a complex period in the development of Husserl's thought. Four stages in that interaction are discernible from this study.
Initially, Husserl's ideas about mathematics and philosophy, sets, abstraction, and the arithmetization of analysis fit in with Cantor's ideas. That may be attributable to the influence of Weierstrass, to Cantor's position on Husserl's Habilitationskommittee, and to the fact that many of Cantor's theories might seem amenable to clarification through Brentano's teachings. For example: Brentano's collective unification might be what Cantor meant by the "special relationship" binding elements of a set; the objects of thought and intuition of Cantor's sets might be Brentano's intentional objects; Cantor's technique of extracting numbers from reality through abstraction might be a psychological process; and with Brentano's positivism Husserl might have been able to "banish all metaphysical fog and all mysticism" from mathematical investigations into numbers and manifolds like Cantor's (Husserl 1900-01, 242).
There was then a second stage during which Cantor's work must have been instrumental in unseating Husserl from his earliest convictions by raising hard questions about imaginary numbers, sets, consciousness and pure logic, idealism, etc., something not so astonishing when one considers that Husserl suffered the direct, early impact of theories about transfinite sets that played a role in rattling the logical assumptions of Bertrand Russell, Gottlob Frege, Richard Dedekind and many others who worked on the foundations of arithmetic.
In a third stage Husserl drew near to some ideas Cantor was rather alone in espousing (metaphysical idealism and the renunciation of psychologism, empiricism, and naturalism) and turned away from other of his key ideas (his theory of sets, the arithmetization of analysis). Husserl then felt obliged to set out on his own, turning to Lotze's more sophisticated ideas about Platonic Ideas and to Bolzano's more sophisticated work on pure logic for guidance.
A fourth stage would consist of the assimilation of certain of Cantor's ideas into Husserl's phenomenology and extends far beyond the compass of this study. Here it would be a matter of studying the relationship between Cantor's theories and, for example, Husserl's Mannigfaltigkeitslehre, his theories about eidetic intuition, the phenomenological reductions, noemata, horizons, infinity, whole and part, formal logic, and of how Husserl's philosophy of logic and mathematics relates to Frege's, Russell's or Gödel's ideas, in order to arrive at a fairer assessment of Husserl's place in the history of philosophy, logic, set theory and the foundations of mathematics than has been possible, dreamt of, or even thought desirable up until now.
* I wish to thank Miss Emiko Ima and Mr. William Gallagher for their help. Emi is a particularly diligent worker. I have occasion to thank Mr. Ivor Grattan-Guinness in the notes, but that does not by any means account for all the friendly assistance he has provided. So I
wish to thank him again here.
1 Exceptions: Fraenkel 1930, pp. 221, 253 n., 257; Schmit 1981, pp. 24, 40-48, 70-72, 77, 94, 124, 131-37; Picker 1962, pp. 266-73, 290, 302, 309-11, 328-29, 345-46; Rosado Haddock 1973, pp. 140-43; Hill 1991, pp. 2, 17, 20, 91; Grattan-Guinness, 1980, pp. 81-82, n. 56; Cavaillès 1962, p. 180; Illemann 1932, p. 50.
2 Primary sources are: M. Husserl 1988, p. 114 and letters of reference Cantor wrote for Husserl published in Cantor 1991, pp. 321, 373-74, 379-80, 423-24, and W. Purkert and H. Ilgauds 1991, pp. 206-07, which I must thank Ivor Grattan-Guinness for bringing to my attention; See also L. Eley's, "Editor's Introduction" to Husserl 1970, pp. XXIII-XXVIII; Husserl 1913, p. 37 and notes; Schuhmann 1977, pp. 19, 22.
3 I must thank Ivor Grattan-Guinness for sending this to me which he obtained from the Mittag-Leffler Institute in Sweden. The text is translated in Hallett 1984, pp. 6-7.
4 See Jourdain 1908-14; Fraenkel 1930, pp. 193, 194, 199, 200, 208, 232-33, 236, 247, 251; Russell 1903, pp. 115, 166, 219, 285, 305, 314-15, 497, 519; Dauben 1979, pp. 151, 170-71, 176-77, 220-28, Hallett 1984, pp. 51-85, ll9-64; Grattan-Guinness 1980, pp. 68-71.
5 Becker 1930, pp. 40-41; Schuhmann 1977, pp. 6-9, 11, 345. See also Picker 1962, pp. 266-70, 289-91, 302; Schmit 1981, pp. 24, 34-36; Osborn 1934, pp. 10-17, 37; Mahnke 1966, pp. 75-76. Miller 1982, pp. 1-13. Willard 1980, pp. 46-64, and 1984, pp. 3, 21, 110, 130; Kusch 1989, pp. 12-15.
6 Personally examined at the Husserl Archives and Library in Leuven, Belgium.
7 At this time in his career Husserl used the terms 'set', 'multiplicity', 'totality', 'manifold', etc. interchangeably. See Husserl 1891, p. 8 and note and Husserl 1887, p. 96.
8 It is worthwhile to note here that Cantor maintained that the transfinite numbers were "in a certain sense new irrationals", that both the transfinite numbers and the finite irrational numbers were definite, delineated forms of or modifications of the actual infinite (Cantor 1887/8, pp. 395-96; Cantor 1991, p.182; Dauben 1979, p.128).
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