This is a preprint version of the paper that appeared in definitive form in The Monist, July 1994, vol. 77, no. 3, pp. 345-357 and in Husserl or Frege? by Claire Ortiz Hill and Guillermo Rosado Haddock. The published version should be consulted for all citations.
One hundred years ago Gottlob Frege published a damaging, abusive review (Frege 1894)1 of Edmund Husserl's Philosophy of Arithmetic (Husserl 1891).2 Although rather a lot has now been written about Frege's review and the role it might have played in the development of Husserl's thought,3 much still remains to be rectified regarding Frege's assessment of the book and the credence his review has been accorded. Philosophers have generally been all too willing to trust Frege's judgement, and so all too ready to dismiss Husserl's book as irredeemably muddled.
Up until now it has been easy to underrate the Philosophy of Arithmetic because, although philosophers have long been familiar with the tenor of Frege's review, Husserl's book is only now beginning to receive the attention it deserves,4 and numerous misconceptions still abound concerning it. Those who have examined the issues closely, however, have often risen to its defense.5 Some have drawn attention to the fact that Frege attacked views Husserl had not espoused.6
Moreover, Frege himself actually seemed to indicate something of the kind when he wrote at the end of his review that he had taken it to be his task to exhibit the damage caused by the irruption of psychology into logic "in the proper light," but conceded that the errors he believed he "was compelled to point out are to be laid to the charge, not so much of the author, as of a widespread philosophical disease" (Frege 1894, 209).
Frege's statement raises an interesting question as to the extent to which he might actually have been condemning someone else's abuses. Here I will argue that a close look at his criticisms of Husserl's views on abstraction, extensionality, and psychologism yields important clues as to the answer to that question. I will endeavor to show the extent to which Frege used his review of the Philosophy of Arithmetic as a forum for attacking Georg Cantor's theory of number. By so doing I hope to help put Frege's objections "in the proper light," and so undo some of the damage done to Husserl's book.
In his review, Frege repeatedly criticized Husserl's use of a theory of number abstraction in the Philosophy of Arithmetic. It is worthwhile to examine Frege's objections closely because Husserl's use of a theory of abstraction cannot just be dismissed off hand. Husserl was in quite respectable company there. Richard Dedekind, Georg Cantor, Giuseppe Peano and Bertrand Russell all had theories of abstraction. In fact, Frege himself is often said to have espoused theory of abstraction.7 So it is very important to understand what exactly Frege and Husserl each meant when they used the word.
Husserl clearly describes the theory of abstraction he had come to advocate in Chapter 4 of the Philosophy of Arithmetic where he defines number abstraction as a procedure by which, while actually engaged in the counting process, the counter "abstracts from" the particular properties of the individual members of the multiplicity, or set of items being counted, the particular way in which they are given, and any relations obtaining among them, only regarding the set as being composed of distinct featureless items to be counted (pp. 85-88). Anticipating the objection that in so doing the items themselves and any relations obtaining between them would naturally disappear (p. 84), Husserl adds that abstracting from the particular properties of the items to be counted merely means not directing one's attention toward them while actually counting. ''That absolutely does not have the effect of making these contents and the relations obtaining among them disappear" (p. 85). He ends his chapter by condemning Aristotle's, John Locke's and J. S. Mill's theories of abstraction (pp. 91-92).
This being Husserl's view, Frege was perfectly correct to write that Husserl would have one "abstract from the peculiar constitution of the individual contents that make up the multiplicity and retain each one only in so far as it is a something or a one". Frege was also correct in writing that Husserl's "process of abstracting the number goes hand in hand with a process of emptying all content" (Frege 1894, 196).
And by so characterizing Husserl's theory of abstraction, Frege rather neatly put Husserl into the category of thinkers to which he belonged when writing the Philosophy of Arithmetic, i.e., into the school of Karl Weierstrass and Georg Cantor, two men whose ideas Frege also vehemently opposed.8 Lecture notes which have survived from Husserl's student years indicate that Husserl had been pointed in this direction by his teacher Weierstrass9 who exercised a profound influence on him.10
Georg Cantor, also a Weierstrass student, was Husserl's close friend, colleague and mentor at the University of Halle from 1886 to 1900.11 And Husserl's description of the abstraction process in the Philosophy of Arithmetic is actually quite similar to descriptions of the same process Cantor made during those years. A look at Husserl's personal copies of Cantor's Contributions to the Theory of the Transfinite (1886-90) in fact shows that Husserl marked and underlined precisely those passages (and almost exclusively those passages) in which Cantor defined the abstraction process. For instance, Husserl marked and underlined the passage in which Cantor wrote:
By the power of cardinal number of a set M . . . I understand the general concept or species concept (universal) which one obtains when one abstracts from the set both the nature of its elements as well as all relationships which the elements have either between themselves or to other things, in particular to the order that may prevail between the elements, and only reflects on that which is common with all sets which are equivalent with M.12
But, unwilling just to condemn Husserl for holding views he really did hold in the Philosophy of Arithmetic, Frege quite unfairly went on to charge in his review that Husserl's procedure would "cleanse things of their peculiarities... in the wash-tub of the mind" where things "assume a quite peculiar pliancy...." There, Frege charges, "we can easily change objects by directing our attention towards them or away from them.... We attend less to a property, and it disappears. By thus making one characteristic mark after another disappear, we obtain more and more abstract concepts" (Frege 1894, 197). Then Frege went on to provide this caricature of the procedure Husserl had advocated:
Suppose, e.g., that there are a black and 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 --but they continue presumably to be separate. We have thus perhaps obtained from each of them a general concept of cat. By continued application of this procedure, each object is transformed into a more and more bloodless phantom. (Frege 1894, 197-98).
In Foundations of Arithmetic (§ 34), Frege had explicitly stated that in the abstraction process the things themselves do not lose any of their characteristics. One might disregard the properties which distinguish a white cat and a black cat, but the white cat would still remain white, and the black one black whether or not one thought about their colors, or made any inference from their difference in this regard. The cats would not become colorless. They would remain precisely as before.
However, ten years later, he apparently could no longer imagine that abstraction did not actually change objects because in his review of the Philosophy of Arithmetic he maintained that it must surely be assumed 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).
Nonetheless, in spite of the force of the charges he directed against Husserl in his review, Frege was honest enough to admit that Husserl himself did not hold that the mind creates new objects or changes old ones, and actually acknowledged that Husserl "disputes this in the most vehement terms (p. 139)." (Frege 1894, 205)
When, however, Frege goes on to charge that Husserl had taken "the road of magic rather than of science" (Frege, 1894, 205), we have a good initial clue as to whom else Frege wishes to attack, for he had accused Cantor of the very same thing in reviewing the Contributions to the Theory of the Transfinite. There Frege complained about Cantor's use of the verb 'abstract' which Frege branded "a psychological expression and, as such, to be avoided in mathematics" (Frege 1984,180; 181).
A posthumously published draft of this review (Frege 1979, 68-71) is more revealing and considerably less tame than the one actually published. There Frege likens mathematicians who like Cantor talk of abstraction to "negroes from the heart of Africa." For these mathematicians, Frege contends, words like "abstraction" are supposed to have "the kind of magical effects" that enable them to abstract from any properties of things which bother them (p. 69). In the spirit of the cat example in his review of Husserl, Frege complains that mathematicians like Cantor find a whole host of things in mice which are unworthy to form a part of the number. "Nothing simpler", Frege writes:
one abstracts from the whole lot. Indeed when you get down to it 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, even from those in virtue of which we call them animals, three-dimensional beings.... (p. 70)
In this review, Frege even alludes to someone who he says he suspects is one of Cantor's pupils and who when asked what general concept he arrives at when given a pencil exerts himself "to the utmost in abstracting from the nature of the pencil and the order in which its elements are given," to answer 'the cardinal number one' (p. 71).
2. Misery Loves Company
After summarizing the fundamental ideas of the Philosophy of Arithmetic, Frege characterizes Husserl's endeavor as "an attempt to justify a naïve conception of number". Frege calls naïve "any view on which a statement of number is not a statement about a concept or about the extension of a concept." "For," he contends, "when one first reflects on number, one is led by a certain necessity to such a conception" (Frege 1894, 197). "If the author had used the word 'extension of a concept' in the same sense as I," Frege declares, "we should hardly differ in opinion about the sense of a number statement." (Frege 1894, 201-02)
In the Philosophy of Arithmetic, however, Frege complained, multiplicities, sets, are more indeterminate and more general than numbers. Husserl would first analyze the concept of multiplicity and use it in determining definite numbers and the generic concept of number presupposing them. He would go from the general to the particular and back again (Frege 1894, 195). Husserl had even argued that the relationship between the number and the generic concept of what is numbered was in certain respects the opposite of what Frege supposed (Husserl 1891, 186).
Frege's criticism can only be fairly assessed in connection with his own struggles with extensionality, because on several occasions he confessed that he had never become completely reconciled to using extensions, and he expressed profound reservations about them throughout his career. As it happens the 1894 review of Husserl's book actually contains some of the most forceful statements Frege ever made in their favor.
Frege first appealed to extensions in his 1884 Foundations of Arithmetic (§ 68) in an attempt to eliminate certain undesirable consequences (§§ 66-67, 107) of the theory of number he advocated there. And he devoted several sections of that work to discussing the pros and cons of introducing them (§§ 68-73). In one of its concluding sections he wrote of extensions that: "This way of getting over the difficulty cannot be expected to meet with universal approval, and many will prefer other methods of removing the doubt in question. I attach no decisive importance even to bringing in the extensions of concepts at all. (§ 107)
Although Frege had ended the Foundations of Arithmetic claiming that he did not attach any decisive importance to bringing in extensions, he did not propose any alternative. And when Georg Cantor wrote in an 1885 review of the book that it was unfortunate that Frege had taken extensions of concepts as the foundation of the number concept,13 Frege immediately rose to defend himself against Cantor's charge that it was "a reversal of the proper order when one undertakes to base the latter concepts on the concept of the extension of a concept" because the extension of a concept was generally something quantitatively completely undetermined, and that for such quantitative determination, the concept of number would have to have been given from somewhere else (Frege 1984, 122). So the Frege and Husserl exchange on the relationship between numbers and extensions of concepts described in the first two paragraphs of this section was actually an echo of the earlier exchange Frege and Cantor had had on the same subject.
In the Philosophy of Arithmetic, Husserl surely further incurred Frege's wrath by writing that all Frege's definitions become true and correct propositions when extensions of concepts are substituted for concepts, but then they are absolutely self-evident and without value. The results of Frege's endeavors, Husserl wrote, were such as to make one wonder how anyone could believe them to be true other than temporarily (Husserl 1891,134). Husserl in fact advanced several specific arguments against Frege's theory of identity, substitutivity and extensionality, and even pointed to the problem regarding the equivalence of the identity of equivalent propositional functions which years later Frege would blame for the collapse of his logical edifice.15
Frege explained in the preface to the 1893 Basic Laws of Arithmeticl6 that he was trying to prove the theory of number he had expressed in Foundations of Arithmetic. The extensions he had introduced, he now explained to his readers had made for "far greater flexibility" and "also have a great fundamental importance" (p. ix). "In fact", he writes, "I even define number itself as the extension of a concept...." (p. x). As far as he could see, he wrote, his basic law about extensions (Basic Law V or Principle V) was the only place in which a dispute could arise. This would be the place where the decision would have to be made (p. vii).
Then, a year later Frege came out fighting for extensions in his review of Husserl. In seeming reply to Cantor's charge regarding the indeterminacy of extensions, Frege defiantly wrote in his review: "A concept under which only one object falls has a determinate extension, as does a concept under which no object falls, or a concept under which infinitely many objects fall" (Frege 1894, 202).
Nonetheless, in spite of the sureness regarding extensions Frege expressed in that review, upon receiving the news of the paradox of the class of all classes that are not members of themselves in a letter from Russell in 1902, Frege immediately wrote back that it was the basic law about extensions that was at fault, and that its collapse seemed to undermine the foundations of arithmetic he had proposed for arithmetic.17 ''I have never disguised from myself," Frege wrote of his law in his 1903 appendix to Basic Laws of Arithmetic II,
its lack of the self-evidence that belongs to the other axioms.... I should have gladly dispensed with this foundation if I had known of any substitute for it. And even now I do not see how arithmetic can be scientifically established ... unless we are permitted ... to pass from a concept to its extension ... Solatium miseris socios habuisse malorum.... Everybody who in his proofs has made use of extensions of concepts, classes, sets, is in the same position as I.18
So, he concluded there, the way he had introduced extensions was not legitimate (p. 216), and the interpretation he had so far put on the words 'extension of a concept' needed to be corrected. Then he set out to track down the origin of the contradiction (p. 217). There was nothing, he decided, to stop him from transforming an equality holding between two concepts into an equality of extensions in conformity with the first part of his law, but from the fact that concepts are equal in extension we cannot infer that whatever falls under one falls under the other. The extension may fall under only one of the two concepts whose extension it is. This can in no way be avoided and so the second part of the law fails. This, however, really abolishes the extension of the concept, he concluded (p. 214 note f; pp. 218-23).
Frege never believed that his law about extensions recovered from the shock it had sustained from Russell's paradox (Frege 1979, 182), and came to rue having used the expression 'extension of a concept' which he finally concluded 'leads us into a thicket of contradictions" (Frege 1980, 55).19 By criticizing Frege's use of extensions, Husserl and Cantor had obviously struck a sensitive chord and so provoked an angry reaction by Frege.
3. Confusing Logic, Psychologism and Empiricism
In his destructive review of Philosophy of Arithmetic, Frege further charged that Husserl's theory of number was conceived in the sin of confusing logic and psychologism. Frege's principal charges turn on Husserl's use of the word 'Vorstellung', or 'presentation',20 which Frege had decided to use only to designate the subjective, psychological phenomena (Frege 1884, §27 n.; Frege 1979, 72-76) which in "On Sense and Reference" he explicitly identified with the traditional empirical sources of knowledge, i.e. sense impressions, direct experiences, internal images arising from memories of sense impressions, and any object in so far as it was sensibly perceptible or spatial.21 For Frege psychologism and empiricism were closely linked. Arithmetic, however, was a branch of logic for which no ground of proof need be drawn from experience or from intuition (ex. Frege 1984, 180; Frege 1884, v, §§ 8,14).
As far as I know no one has ever used the word 'Vorstellung' in Frege's special way, and he was most certainly correct to say that Husserl did not. In the Philosophy of Arithmetic, Husserl incurred Frege's wrath by using the word to designate both what Frege had decided was subjective and what Frege thought of as objective. In so doing, Frege complained, Husserl had transposed everything into the subjective mode (Frege 1894, 198). He had turned concepts, objects, and meanings of words (Frege 1894, 197) and numbers into "Vorstellungen, the results of mental processes or activities…." (Frege 1894, 207) And, Frege further charges that since in Husserl's book "all things are Vorstellungen we can easily change objects by directing our attention towards them or away from them" (Frege 1894, 197). We can make properties disappear one after another, and actually essentially change objects. (Frege 1894, 197-98)
Husserl's use of the word 'Vorstellung' in the Philosophy of Arithmetic is ambiguous. A close look at the context, however, usually makes the intended reference clear, and several passages from the Philosophy of Arithmetic clearly indicate that Husserl was not guilty of confusing logic and psychology in the sense of obliterating the differences between mental processes and objects.22 For example, the following passages from the book clearly show that Husserl did not change meanings into mental entities:
Suppose we have a Vorstellung of a set of objects A, B, C, D. According to the order of succession in which the whole is formed finally, only D is given as a sensory Vorstellung. The remaining contents are, however, merely given as imaginary Vorstellungen altered in time and content. If we go in the opposite direction, from D to A, the phenomenon is different. The logical meaning nullifies all these differences.... While we make the Vorstellung of the set, we pay no attention to the fact that, in the grouping process, changes have occurred in the content. It is our intention to hold them together and unify them and so the logical content of the Vorstellung is not, for example, some D, then C just before, B even earlier, until we get to the most radically changed, A, but rather: (A, B, C, D). (Husserl 1891, 28-29)
This passage shows that Husserl did not reduce everything to subjectivity. He considered the theory that mental acts can engender relations to be untenable. In Philosophy of Arithmetic he wrote:
Our mental activity does not make relations; they are simply there, and when interest is directed toward them they are noticed just like any other content. Genuinely creative acts that would produce any new content . . . are absurd from the psychological point of view ... the act can in no way generate its content.... (Husserl 1891, 42)
Husserl even plainly wrote: "Isn't it obvious that a 'number' and the 'presentation of a number' are not the same thing?" (Husserl 1891, 30)
Once again, Cantor's presence is felt when Frege writes that according to Husserl "numbers are supposed to be Vorstellungen, the results of mental processes or activities," and charges Husserl with actually removing elements. Frege had also charged Cantor with "creating" numbers (Frege 1884, §96 note), and with engaging in the "psychological and hence empirical" activity of psychically removing elements.23 Moreover, when criticizing Husserl for writing that when the number of items to be counted is beyond our capacity for presentation we are then to "idealize" our capacity for presentation (Frege 1894, 207, re. Husserl 1891, 251), Frege was surely also thinking of Cantor's statement in the Mannigfaltigkeitslehre that whenever "one comes to no greater number, one imagines a new one".24
The conflict between what is mental and the ideal realm of pure logic is plainly manifest and unresolved in Cantor's writings25 where subjective and objective are mixed together in a puzzling way that cries out for clarification. In the same passage of the Mannigfaltigkeitslehre in which Cantor explicitly rejects the belief that "the source of knowledge and certainty is located in the senses or in the so-called form of pure intuition of the world of Vorstellungen," he goes to write that "certain knowledge . . . can only be obtained through concepts an ideas (Ideen), which are at best only stimulated by outer experience, but which are principally formed through inner induction, like something which, so to speak, already lay within us and is only awakened and brought to consciousness."26 In Contributions to the Theory of the Transfinite he would write that "the act of abstraction with respect to nature and order … effects or rather awakens in my intellect the concept 'five' ,27 and that the cardinal number belonging to a set was "an abstract image in our intellect."28 Frege was perfectly justified in qualifying such appeals to "inner intuition" as "rather mysterious" (Frege 1884, §86). Cantor's philosophy was mystical to say the least.29
Though Husserl often used similar language, a close examination of the context shows that his statements are almost always qualified and explicated,30 which makes them philosophically more sophisticated than Cantor's words are.31 In particular, if in his famous review Frege seems to be attacking a more extreme form of psychologism than Husserl ever espoused, this is because the psychologism Husserl actually tried out in the Philosophy of Arithmetic was a variation on a theme by the realist and empiricist Franz Brentano.32 By obliterating the differences between Husserl's empiricistic use of the word 'Vorstellung' and Cantor's more Platonic ideas, Frege rather unfairly glossed over the differences between Husserl's and Cantor's ideas in this regard, making Husserl's ideas appear less reasoned than they actually were. In many respects the Philosophy of Arithmetic can be read as a first attempt on Husserl's part to, as he termed it, "banish the metaphysical fog and mysticism from mathematical investigations like Cantor's,"33 and to redress some of the very weaknesses in accounts of the relationship between the subjectivity of knowing and the objectivity of mathematics34 which Frege found so irritating.
Husserl was finally "deeply dissatisfied" with the analyses of the Philosophy of Arithmetic.35 He later wrote of being "tormented" by "the incredibly strange worlds . . . of pure logic and the world of act-consciousness" while struggling to outline the logic of mathematical thought and calculation.36 The idea of set, he recalled having reasoned, was supposed to arise out of the collective combination, and since everything which could be grasped intuitively was either physical or psychical, the collective had to be psychical. So it had to arise through psychological reflection in Brentano's sense, through reflection upon the act of collecting. But Husserl was finally compelled to conclude that Brentano's theories of presentation "could not help,"37 and that there was "an essential, quite unbridgeable difference between the sciences of the ideal and the sciences of the real," the correct assessment of which presupposed "the complete abandonment of the empiricistic theory of abstraction, whose present dominance renders all logical matters unintelligible."38
In the 1893 Basic Laws of Arithmetic, Frege accorded extensions "great fundamental importance" (p. x). A year later, he published an angry attack on the ideas of two Halle mathematicians who had criticized his use of extensions. His famous review of Husserl was partly a veiled assault on Cantor's ideas as reflected in the first book of a younger critic. Frege actually directly incorporated into his review of Husserl several specific criticisms he had already made of Cantor's work. The review was also partly a reflection of certain personal psychological problems Frege had regarding extensions and of his inability to accept criticism gracefully.
By drawing attention to these facts and to the relationship between Cantor's and Husserl's ideas, I have tried to contribute to putting Frege's attack on Husserl "in the proper light" by providing some insight into some of the issues underling criticisms which Frege himself suggested were not purely aimed at Husserl's book. I have tried to undermine the popular idea that Frege's review of the Philosophy of Arithmetic is a straightforward, objective assessment of Husserl's book, and to give some specific reasons for thinking that the uncritical reading of Frege's review has unfairly distorted philosophers' perception of a work they do not know very well.
To put Frege's objections "in the proper light," the Philosophy of Arithmetic also needs to be evaluated in connection with kindred attempts, "to provide a more detailled analysis of the concepts of arithmetic and a deeper foundation for its theorems."39 For instance, Husserl was not the only mathematician to try to marry Brentano's ideas on presentation and Cantor's theory of arithmetic. The logical empiricist Bertrand Russell did likewise.40 "In Arithmetic ... our whole work is based on that of Georg Cantor" Russell wrote in the preface to Principia Mathematica.41 Russell even for a time confused his own ideas on Vorstellungen with Frege's ideas, and those of Brentano's student Alexius Meinong.42
1 G. Frege, "Rezension von E. Husserl: Philosophie der Arithmetik", Zeitschrift für Philosophie und philosophische Kritik 103, 1894, pp.313-32. In the text I cite as (Frege 1894) the translation in Frege's Collected Papers on Mathematics, Logic und Philosophy, Blackwell, Oxford, 1984, pp.195-209. Other articles in the Collected Papers are cited in the text as (Frege 1984).
2 E. Husserl, Philosophie der Arithmetik, Pfeffer, Halle, 1891, cited in the text as (Husserl 1891).
3 For example: D. Føllesdal's, "Husserl and Frege: A Contribution to Elucidating the Origins of Phenomenological Philosophy," Mind, Meaning and Mathematics, L. Haaparanta (ed.), Kluwer, Dordrecht, 1994, a translation of his1958 Norwegian Master's thesis; J.N. Mohanty, Husserl and Frege, Indiana University Press, Bloomington IN, 1982; D. W. Smith and R. McIntyre, Husserl on Intentionality, Reidel, Dordrecht, 1982; C. O. Hill, Word and Object in Husserl, Frege, and Russell, Ohio University Press, Athens OH, 1991; M. Dummett, Frege, Philosophy of Mathematics, Harvard University Press, Cambridge MA, 1991, Chapter.12.
4 For example in J. P. Miller, Numbers in Presence and Absence, M. Nijhoff, The Hague, 1982; D. Willard, Logic and the Objectivity of Knowledge, Ohio University Press, Athens OH., 1984; D. Bell, Husserl, Routledge, London, 1990.
5 M. Farber, The Foundations of Phenomenology, Harvard University Press, Cambridge, MA, 1943, pp. 16-17, 25-60; J. N. Findlay, "Translator's Introduction" to Husserl's Logical Investigations", Routledge and Kegan Paul, London, 1970, p.1; A. Church, "Review of M. Farber's The Foundations of Phenomenology," Journal of Symbolic Logic 9, 1944, p. 64; G. E. Rosado Haddock, "Remarks on Sense and Reference in Frege and Husserl", Chapter 2 of this book; Bell pp.67, 69, 77.
6 Hill, Word and Object in Husserl, Frege and Russell, Chapters 2 & 5; Bell, pp. 79-81; Willard, pp. 43, 62-63, 69-70, 118-21; Mohanty, p.26.
7 R. Dedekind, "The Nature and Meaning of Numbers" § VI, Essays on the Theory of Numbers, Dover, New York, 1963 (1887); G. Cantor, Gesammelte Abhandlungen, E. Zermelo (ed.), Springer, New York, 1932, pp.379, 387, 411-12; G. Frege, Foundations of Arithmetic, Oxford, Blackwell, 1986 (1884), §§ 29-44, cited in the text as (Frege 1884); B. Russell, Principles of Mathematics, Norton, London, 1903, pp. ll5, 166, 219, 285, 305, 314-15, 497, 519. See also J. Dauben, Georg Cantor, His Mathematics and Philosophy of the Infinite, Princeton University Press, Princeton NJ, 1979, pp. 151, 170-71, 176-77, 220-28; M. Hallett, Cantorian Set Theory and Limitation of Size, Clarendon, Oxford, 1984, pp. 54-85, 119-64; I. Grattan-Guinness, "Georg Cantor's Influence on Bertrand Russell," History and Philosophy of Logic 1, l980, 68-71; C. Thiel, "Gottlob Frege: Die Abstraktion," Studies on Frege, Vol. l, M. Schirn (ed.), Frommann-Holzboog, Stuttgart,1976, pp. 243-64; Dummett, pp.50-52, 82-85, 45,167-68.
8 Frege criticizes Cantor and Weierstrass in Grundgesetze der Arithmetik, Vol. 2, Olms, Hildesheim, 1966, §§ 68-86 and 143-55; Also see Frege's "Draft Towards a Review of Cantor's Gesammelte Abhandlungen zur Lehre vom Transfiniten," in Frege's Posthumous Writings, Blackwell, Oxford, 1980, pp. 68-71, cited in the text as (Frege 1980); "Review of Georg Cantor, Zur Lehre vom Transfiniten: Gesammelte Abhandlungen," in Frege's Collected Papers on Mathematics, Logic and Philosophy, pp. 178-81; Dummett, Chapter 21 and p. 243 where Dummett notes that in criticizing Weierstrass, Frege descended "rapidly into the grossest abuse"; Dauben, pp. 220-28.
9 L. Eley, "Einleitung des Herausgebers", Philosophie der Arithmetik mit ergänzenden Texten, Husserliana Vol. XII, M. Nijhoff, The Hague, 1970; Miller, pp. 1-6, 19.
10 K. Schuhmann, Husserl-Chronik, M. Nijhoff, The Hague, 1977, pp. 6-9; Miller, pp. 2-10; M. Kusch, Language as Calculus vs. Language as Universal Medium, Kluwer, Dordrecht, 1989, pp. 14-15.
11 Schuhmann, pp. 19, 22; A. Fraenkel, "Georg Cantor", Jahresbericht der deutschen Mathematiker Vereinigung 39, pp. 221, 253 note, 257 (abridged in Cantor's Gesammelte Abhandlungen); E. Husserl, Introduction to the Logical Investigations, A Draft of a Preface to the Logical Investigations 1913, M. Nijhoff, The Hague, 1975, p. 37 and notes; Eley, pp. XXIII-XXV; R. Schmit, Husserls Philosophie der Mathematik, Bouvier, Bonn, 1981, pp. 44, 58; M. Husserl, "Skizze eines Lebensbildes von E. Husserl" Husserl Studies 5, 1988, p. 114; W. Illemann, Husserls vorphänomenologische Philosophie, Hirzel, Leipzig, 1932, p. 50; I must thank Ivor Grattan-Guinness for bringing to my attention the letters published in Georg Cantor Briefe, H. Meschkowski and W. Nilson (eds.), Springer, New York, 1991, pp. 321, 373-74, 379-80, 423-24, and W. Purkert and H. Ilgauds, Georg Cantor 1845-1918, Birkhäuser, Basel, 1991, pp. 206-07.
12 Cantor, p. 387; Dauben, p. 221; Grattan-Guinness, pp. 68-69. I personally examined Husserl's copies of Cantor's works at the Husserl Archives in Leuven in June 1993.
13 Cantor, "Rezension von Freges Grundlagen", Gesammelte Abhandlungen, pp. 440-41.
14 G. Frege, "Erwiderung auf Cantors Rezension der Grundlagen der Arithmetik", Deutsche Literaturzeitung 6, 1885, Nr. 28, Sp. 1030; Frege, Collected Papers on Mathematics, Logic and Philosophy, p.122.
15 I discuss Husserl's arguments at length in Word and Object in Husserl, Frege and Russell, Chapter 4, and "Husserl and Frege on Substitutivity", Chapter 1 of this book.
16 G. Frege, Basic Laws of Arithmetic, University of California Press, Berkeley CA., 1964 (1893), pp. vii-x.
17 G. Frege, Philosophical and Mathematical Correspondence, Blackwell, Oxford, 1979, pp. 131-32, cited in the text as (Frege 1979).
18 G. Frege, Translations from the Philosophical Writings, Blackwell, Oxford, 3rd ed., 1980, p. 214.
19 I document this more fully in Chapter 6 of Word and Object in Husserl, Frege and Russell and in "Frege's Letters," From Dedekind to Gödel, Essays on the Development of the Foundations of Mathematics, J. Hintikka (ed.), Kluwer, Dordrecht, 1995, pp. 97-118. Dummett discusses problems with extensionality at length in Frege, Philosophy of Mathematics cited above.
20 Following Russell's translation in "On the Nature of Acquaintance", Logic and Knowledge, Allen and Unwin, London, 1956, pp. 169-70. Frege's translators have often misleadingly opted for "idea".
21 Frege, Translations from the Philosophical Writings, pp. 59-64. I discuss this at length in Word and Object in Husserl, Frege and Russell, Chapters 5 and 7.
22 See Miller, pp. 7-8, 21-23.
23 Frege, Collected Papers on Mathematics, Logic and Philosophy, pp. 180-81; Dauben, pp. 220-28, 239.
24 Cantor, p. 195; Dauben, p. 206.
25 Hallett, pp. 16-18, 34-35, 121, 128-33, 146-58; Dauben, p. 132.
26 Cantor, p. 207 note 6; Hallett, p. 15.
27 Cantor, p. 418 note 1; Hallett p. 128.
28 Cantor, p. 416; Hallett p. 128.
29 Dauben, Chapter 6, and pp. 236-39; Hallett, pp. 9-11, 35-36.
30 Hill, Word and Object in Husserl, Frege and Russell, p. 14.
31 Dauben, pp. 6, 49, 120, 121, 127, 147, 150, 154, 159; Hallett, pp. xi, 67, 49.
32 H. Spiegelberg, The Context of the Phenomenological Movement, M. Nijhoff, The Hague, 1981, pp. 7-63; Bell, pp. 3-28.
33 Husserl, Logical Investigations, p. 242
34 Ibid. p. 42
35 Husserl, Introduction to the Logical Investigations, pp. 34-35.
36 Edmund Husserl, "Persönliche Aufzeichnungen," Philosophy and Phenomenological Research 16 (1956), p. 294; Miller, Chapters 1, 3 and 5; Kusch, pp. 12-55.
37 Husserl, Introduction to the Logical Investigations, pp. 34-35.
38 Husserl, Logical Investigations, p. 185.
39 G. Frege, "Begriffsschrift", From Frege to Gödel, J. van Heijenoort (ed.), Harvard University Press, Cambridge MA,1967, p. 8.
40 B. Russell, My Philosophical Development, Allen & Unwin, London, 1985 (1959), p. 100; Russell links his views on presentation with those of Brentano's school most explicitly in the articles he published in Mind 8, 13-16 (1899-1907) on Brentano's student A. Meinong, reprinted in Essays on Analysis, Braziller, New York, 1973. He discusses his views on presentation at length in "The Nature of Acquaintance", Logic and Knowledge, pp. 127-74; "Knowledge by Acquaintance and Knowledge by Description", Chapter 10 of Mysticism and Logic, Allen and Unwin, London, 1986 (1917); Chapter 5 of Problems of Philosophy, Oxford University Press, Oxford, 1967 (l912). See also A. Garciadiego, Bertrand Russell and the Origins of the Set-theoretic Paradoxes, Birkhäuser, Basel, 1992; Grattan-Guinness, "Georg Cantor's Influence on Bertrand Russell" as cited in note 7 above.
41 B. Russell and A. N. Whitehead, Principia Mathematica to *56, Cambridge University Press, Cambridge, 1964 (l92, 2nd ed. rev.), p. viii.
42 Hill, Word and Object in Husserl, Frege and Russell, pp. 61-66, 134-35.