This is a preprint version of the paper that appeared in definitive form in Mind, Meaning and Mathematics, Leila Haaparanta (ed.), Dordrecht: Kluwer, 1994 and in 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.
In the critical discussion of Gottlob Frege's logic in Edmund Husserl's Philosophy of Arithmetic,1 Husserl outlines his objections to the use Frege makes of Leibniz's principle of the substitutivity of identicals in the Foundations of Arithmetic.2 In the 1903 appendix to Basic Laws II,3 Frege linked these same criticisms with Russell's paradox when, without mentioning Husserl's name, he traced the source of the paradox to points Husserl had made in the Philosophy of Arithmetic. For many philosophical, linguistic and historical reasons4 these two facts have gone virtually uncommented. In the belief that Husserl's discussion of identity and substitutivity in Frege's theory of number may actually he able to shed light on some dark areas surrounding the significance of Russell's paradox for logic and epistemology I propose here to examine Husserl's criticisms and systematically tie his arguments in with observations made by Bertrand Russell and others who have studied Frege's work.
First, however, I must preface my discussion with a short historical digression aimed at showing how Husserl fit into Frege's intellectual world and his competency to deal with Frege's ideas. This is necessary because Husserl is not generally thought of as having been someone who could have understood Frege's work in 1891. Louis Couturat5 Alonzo Church6 and Dallas Willard7 are among the very few people who seem to have noticed that Husserl wrote anything worthwhile or insightful at all about Frege's logic in the Philosophy of Arithmetic. Husserl is most often wrongly thought of as having been a kind of intellectual infant when he wrote it,8 and for a long time it was thought that his intellectual awakening only began in 1894 with Frege's bitter review9 of the book.10
The many people who still underrate Husserl's ability in 1891 to publish an insightful work concerning the philosophy of arithmetic are not, however, in possession of the facts for during the years in which his philosophical ideas were developing, Husserl actually had the unusual privilege of directly participating in the early development of the very mathematical, logical and philosophical ideas that would go on to determine the course of philosophical thought in English-speaking countries in the twentieth century. He was, in fact, directly and intimately involved in the earliest discussions of such pivotal issues in twentieth century logic, mathematics and analytic philosophy as number theory, the continuum problem, set theory, the axiomatic foundations of geometry, Russell's paradox, infinity, function theory, intentionality, intensionality, analyticity, identity, sense and reference, and completeness, all of which are philosophical issues which still, a hundred years later, present thorny problems for philosophers, filling the pages of the journals and books they read. Husserl's ideas now need to he knit back into the intellectual context that produced them.
1. Weierstrass, Brentano, Stumpf and Cantor
It was Karl Weierstrass's courses on the theory of functions that, in the late 1870s and early 1880s, first awakened Edmund Husserl's interest in seeking radical foundations for mathematics. Husserl was impressed by his teacher's emphasis on clarity and logical stringency.11 He was receptive to Weierstrass's efforts to further the work begun by Bernard
Bolzano to instill rigor in mathematical analysis12 and to transform the "mixture of reason and irrational instincts" it then was into a purely rational discipline. Weierstrass exercised a deep influence on Husserl and in 1883 Husserl became his assistant. It was from Weierstrass, Husserl once said, that he acquired the ethos of his scientific striving.13
After serving as Weierstrass's assistant for a year, Husserl traveled to Austria to study under Franz Brentano. Like Weierstrass, Brentano was working on Bolzano's ideas,14 and under Brentano, Husserl studied Bolzano's writings and the Paradoxes of the Infinite in particular.15 Brentano was then engaged in reforming logic16 and was vigorously trying to revise old traditions, paying particular attention to matters of linguistic expression.17 He was influenced by British empiricism18 and Michael Dummett, for one, considers him to have been, "roughly comparable to Russell and Moore" in England.19 Russell himself actually explicitly acknowledged the kinship between his own ideas and those of Brentano, and there was enough superficial kinship between Russell's views on reference and those of Brentano's school for Russell to have at one point confused his ideas and theirs.20
Brentano sent Husserl to Halle to prepare his Habilitationsschrift on number theory under the direction of Carl Stumpf,21 a man to whom Frege had appealed in 1882 for help in making his Begriffsschrift known. In his reply to Frege's request, Stumpf had mentioned how pleased he was that Frege was working on logical problems because it was an area where there was a great need for cooperation between mathematicians and philosophers. He agreed with Frege that arithmetical and algebraic judgments were analytic and expressed his own interest in working on that problem. He also suggested that Frege's ideas might be more favorably received if he first explained them in ordinary language. Frege appears to have taken Stumpf's advice by expressing his ideas in prose in the 1884 Foundations of Arithmetic.22 Husserl began studying Frege's Foundations as Stumpf's colleague in the late 1880s, and he used it extensively as he worked on the Philosophy of Arithmetic.23
In Halle, Husserl befriended another man who, like him, had been profoundly influenced by Karl Weierstrass. This was the creator of the theory of sets, Georg Cantor.24 Cantor too was carrying on the work Bolzano had begun,25 and enough kinship is apparent between Husserl's and Cantor's work to have prompted scholars to speak of the influence Husserl may have had on Cantor's work,26 and of Cantor's influence on Husserl's work 27 Enough of a kinship exists between Frege's and Cantor's work to have prompted Michael Dummett to speak of Georg Cantor as "the mathematician whose pioneering work was closest to that of Frege ..."28 and as one "who ought, of all philosophers and mathematicians, to have been the most sympathetic" to Frege's work.29
Russell thought his own debt to Cantor was evident. In Russell's opinion Cantor had "conquered for the intellect a new and vast province which had been given over to Chaos and Night".30 "In arithmetic and theory of series, our whole work is based on that of Georg Cantor", he wrote in the preface to Principia Mathematica.31 And it was while studying Cantor's work that Russell found the paradoxes to which Frege's logic leads.32 In The Principles of Mathematics, Russell pays homage to Weierstrass for the happy changes he, Dedekind, Cantor and their followers had wrought in mathematics by adding "quite immeasurably to theoretical correctness" and thereby remedying "a diminution of logical precision and a loss in subtlety of distinction".33 "No mathematical subject", he wrote there, "has made, in recent years, greater advances than the theory of Arithmetic. The movement in favour of correctness in deduction, inaugurated by Weierstrass, has been brilliantly continued by Dedekind, Cantor, Frege and Peano .…"34 Through the labors of Weierstrass and Cantor, the fundamental problem of infinity and continuity had undergone a complete transformation, Russell considered.35
Husserl had actually rather fortuitously found himself in the right place at the right time, and by the time he published the Philosophy of Arithmetic, he had long been involved in philosophical investigations into the principles of mathematics and in the work to obtain greater precision in mathematics that ultimately made extensive formalization of mathematics possible and led to comprehensive formal systems like that of Russell's Principia Mathematica.36
In contrast, Frege remained aloof and apparently loath to undertake even the short train journey that would have taken him to Göttingen, Leipzig, or Halle and the likes of Cantor, Zermelo or Hilbert, or a bit farther to Berlin, Paris, Austria or Cambridge where he could have met with Weierstrass, Brentano, Peano, or Russell.37 And Frege actually devoted several sections of the second volume of the Basic Laws of Arithmetic to refuting Cantor's and Weierstrass's views.38
Russell didn't learn of Weierstrass's and Cantor's work until the mid-1890s and he first came into contact with Frege's work several years later than Husserl, Hilbert, Cantor or Brentano's circle did. Most of these people already knew Frege's work in the 1880's. Russell was too young and too faraway actually to interact with the imposing figures whom Husserl regularly frequented over long periods.39 So more than Frege, Russell or Wittgenstein, Husserl was actually present and witnessed the very earliest stages of twentieth century Anglo-American philosophy, and the Philosophy of Arithmetic was written under the influence of the same mathematicians and philosophers that ultimately played such a key role in determining the course of philosophy in English speaking countries.
2. Husserl's Encounter with Frege's Foundations of Arithmetic
Husserl first obtained a copy of Frege's Foundations of Arithmetic in the late 1880s.40 Though, he did not use Frege's book at all in his Habilitationsschrift called On the Concept of Number 41 that he defended before Cantor and Stumpf at the University of Halle in 1887,42 he thoroughly studied Frege's book in the Philosophy of Arithmetic published four years later. There he cites Frege more often than any other author. In a letter Frege himself once acknowledged Husserl's interest in his Foundations, noting that Husserl's study was perhaps the most thorough one that had been done up to that time. Husserl replied to Frege saying how much stimulation he had derived from Frege's work and acknowledged having "derived constant pleasure from the originality of mind, clarity... and honesty" of Frege's investigations which, he wrote, "nowhere stretch a point or hold back a doubt, to which all vagueness in thought and word is alien, and which everywhere try to penetrate to the ultimate foundations." While writing the Philosophy of Arithmetic, no other book, Husserl claimed, had provided him with nearly as much enjoyment as Frege's remarkable work had.43
Much of what Husserl had written about Frege's ideas in the Philosophy of Arithmetic was, though, critical and in the Logical Investigations Husserl would make a point of retracting certain of the objections he had voiced concerning Frege's views on analyticity and his opposition to psychologism there. A close look at Husserl's statement of retraction, however, shows that he only retracted three pages of his criticisms of Frege's logic (not eight as a typographical error in the English edition suggests), leaving most of his basic criticisms of Frege's logical project intact.44 For instance, Husserl never retracted his statements that theories of number like Frege's are unjustified and scientifically useless, that all Frege's definitions become true and correct propositions when one substitutes extensions of concepts for the concepts, but that then they are absolutely self-evident and without value, and that the results of Frege's endeavors are such as to make one wonder how anyone could believe they were true other than temporarily.45
3. Substitutivity in Frege's Foundations of Arithmetic
"Now, it is actually the case that in universal substitutability all the laws of identity are contained", wrote Frege in § 65 of Foundations. And in the brief summary of his views Frege offers in the last pages of that book, he repeats this conviction that: "... a certain condition has to be satisfied, namely that it must be possible in every judgement to substitute without loss of truth the right-hand side of our putative identity for the left side. Now at the outset, and until we bring in further definitions, we do not know of any other assertion concerning either side of such an identity except the one, that they are identical. We had only to show, therefore, that the substitution is possible in an identity" (§ 107). It is evident from this that substitutivity was destined to play a very central role in Frege's theories, so it is very important to examine the arguments of Foundations so as to understand exactly how substitutivity operated in Frege's philosophy of arithmetic.
Frege's Foundations of Arithmetic is divided into five parts, the first three of which are largely devoted to the refutation of views of number which Frege opposes. In part four he outlines his own theory and in part five he summarizes the results of his work. Frege begins outlining his own theories by affirming that numbers are independent objects (§ 55) which figure as such in identity statements like '1 + 1 = 2'. Though in everyday discourse numbers are often used as adjectives rather than as nouns, in arithmetic, he argues, their independent status is apparent at every turn and any appearance to the contrary "can always be got around", for example by rewriting the statement 'Jupiter has four moons' as 'the number of Jupiter's moons is four'. In the new version, Frege argues, the word 'is' is not the copula, but the 'is' of identity and means "is identical with" or "is the same as". "So", he concludes, "... what we have is an identity, stating that the expression 'the number of Jupiter's moons' signifies the same object as the word 'four"'. Using the same reasoning he concludes that Columbus is identical with the discoverer of America for "it is the same man that we call Columbus and the discoverer of America" (§ 57). (Note that Frege here, as always, quite perspicuously distinguishes between words and objects. In his identity statements he is asserting the sameness of one object as given in two different ways by different linguistic expressions.)
Now that Frege is satisfied that he has established numbers as independent objects and acquired a set of meaningful statements in which a number is recognized as the same again, he turns to the question of establishing a criterion for deciding in all cases whether b is the same as a. For him this means defining the sense of the statement: 'the number which belongs to the concept F is the same as that which belongs to the concept G'. This, he believes, will provide a general criterion for the identity of numbers (§ 63).
Not wanting to introduce a special definition of identity for this, but wishing rather "to use the concept of identity, taken as already known, as a means for arriving at that which is to be regarded as being identical", Frege explicitly adopts Leibniz's principle that "things are the same as each other, of which one can be substituted for the other without loss of truth" (§ 65). However, even as he is writing Leibniz's formula right into the foundations of his logic, Frege modifies Leibniz's dictum in a way which, as I hope to show, has presented thorny problems for those who have tried to further Frege's insights and answer some of the really hard questions his logic raises. Although, as Husserl would point out in the Philosophy of Arithmetic, Leibniz's law defines identity, complete coincidence, Frege, here as elsewhere, 46 explicitly maintains that for him "whether we use 'the same' as Leibniz does, or 'equal' is not of any importance. 'The same' may indeed he thought to refer to complete agreement in all respects, 'equal' only to agreement in this respect or that." (§ 65)47
Frege believed that by rewriting the sentences of ordinary language, these differences between equality and identity could be made to vanish. So here he recommends rewriting the sentence 'the segments are equal in length' as 'the length of the segments are equal or the same' and 'the surfaces are identical in color' as 'the color of the surfaces is identical'. Since he believed all the laws of identity were contained in universal substitutivity, to justify his definition he believed he only needed "to show that it is possible, if line a is parallel to line b, to substitute 'the direction of line b' everywhere for 'the direction of line a'. This task is made simpler", he notes, "by the fact that we are being taken initially to know of nothing that can be asserted about the direction of a line except the one thing, that it coincides with the direction of some other line. We should thus have to show only that substitution was possible in an identity of this one type, or in judgement-contents containing such identities as constituent elements." (§ 65)
In these examples he has transformed statements about objects which are equal under a certain description into statements expressing complete identity. By erasing the difference between identity and equality he in fact is arguing that being the same in any one way is equivalent to being the same in all ways. While conceivably one could use this principle to stipulate substitution conditions for symbols, very few objects could satisfy its conditions and, outside of strictly mathematical contexts where differences between equality and identity seem not to apply in the same way as they do elsewhere, many of the inferences that could be made by appealing to such a principle would lead to evidently false and absurd conclusions.
Frege himself acknowledged that left unmodified this procedure was liable to produce nonsensical conclusions, or be sterile and unproductive. For him, the source of the nonsense lay in the fact that, as he himself points out, his definition provides no way of deciding whether, for example, England is or is not the same as the direction of the Earth's axis. Though he is certain that no one would be inclined to confuse England with the direction of the Earth's axis this, he acknowledges, would not be owing to his definition which, he notes, "says nothing as to whether the proposition 'the direction of a is identical with q' should be affirmed or denied except for the one case where q is given in the form of 'the direction of b'" (§ 66).
As it stood, the definition was unproductive, according to him, because were we "to adopt this way out, we should have to be presupposing that an object can only be given in one single way .… All identities would then amount simply to this, that whatever is given to us in the same way is to be reckoned as the same .… We could not, in fact, draw from it any conclusion which was not the same as one of our premisses." Surely, he concludes, identities play such an important role in so many fields "because we are able to recognize something as the same again even although it is given in a different way." (§ 67; also § 107).
Seeing that he could not by these methods alone "obtain any concept of direction with sharp limits to its application, nor therefore, for the same reasons, any satisfactory concept of Number either", Frege felt obliged to introduce extensions to guarantee that "if line a is parallel to line b, then the extension of the concept 'line parallel to line a' is identical with the extension the concept 'line parallel to line b' and conversely, if the extensions of the two concepts just named are identical, then a is parallel to b." (§ 67; also § 107)
While Frege maintained in Foundations that he attached "no decisive importance even to bringing in the extensions of concepts at all" (§ 107), by the time he wrote Basic Laws he felt obliged to accord them a fundamental role. There he would argue that the generality of an identity can always be transformed into an identity of courses-of-values and conversely, an identity of courses-of-values may always he transformed into the generality of an identity. By this he meant that if it is true that (x)(x) = (x), then those two functions have the same extension and that, vice versa, functions having the same extension are identical (Basic Laws §§ 9 & 21). "This possibility" he wrote then, "must he regarded as a law of logic, a law that is invariably employed, even if tacitly, whenever discourse is carried on about extension of concepts. The whole Leibniz-Boole calculus of logic rests upon it. One might perhaps regard this transformation as unimportant or even as dispensable. As against this, I recall the fact that in my Grundlagen der Arithmetik I defined a Number as the extension of a concept ....48
4. Husserl's Criticisms
Husserl had the following remarks to make about Frege's theory of number as described above.49 In his first objection to it, he appeals to common linguistic usage which distinguishes between the equality and the identity of two objects. Leibniz's definition, he points out, defines identity, not equality, so that as long as the least difference remains there will be propositions for which the elements in question will not be interchangeable salva veritate (p. 104). Here Husserl is appealing to the ordinary, non-mathematical, use of the words 'equality' and 'identity'. For example, we commonly say that the United States of America was dedicated to the proposition that all men are created equal with respect to their legal rights, but I believe that no one has ever said, nor would be so foolish as to say, that all men are created identical. (It should he noted that Husserl's remarks never concern the identity or equality of signs, but only the equality or identity of objects and the properties that might be predicated of them.)
According to dictionaries, two things are identical when they are the same in every way. They are equal when they are the same under a specific description, as given in a particular way. The difference between equality and identity would then be the difference between sharing any given property or properties, or having all properties in common. Husserl's point is that if x were to have even one property that y does not have, then though they may be equal in one or in many respects, they are not identical and there will be statements in which substitution will fail, and so affect the truth-value of statements made referring to them, or the outcome of one's inquiries regarding them.
In another argument, Husserl alludes to the problems that arise when one begins examining the grounds for determining the equality of two objects (pp. 108-09). One can declare two simple, unanalyzed objects equal without much further ado, he notes. But there is a certain ambiguity in ordinary language with regard to complex objects. If two objects are the same, then it follows that they must have all their properties in common. But the inverse does not seem to hold. Sometimes two objects have their properties in common and we still do not say that they are the same.
At first sight, Husserl's point may seem illogical for it seems that he is saying that x and y could be different without there being any discernible difference between them. Before condemning his analysis outright, however, it should he noted that, tangling with problems surrounding extensionality, identity and classes, Bertrand Russell was moved to make the same observations. Writing in Introduction to Mathematical Philosophy on classes and problems connected with Leihniz's law of the identity of indiscernibles he argued that it was just "as it were, an accident, a fact about ... this higgedly-piggedly job-lot of a world in which chance has imprisoned us" that no two particulars were precisely the same and he hypothesized that "there might quite well, as a matter of abstract logical possibility, be two things which had exactly the same predicates."50 He also wrote in Principia Mathematica that: "It is plain that if x and y are identical, and jx is true, then jy is true ... the statement must hold for any function. But we cannot say conversely: 'If, with all values of j, jx implies jy, then x and y are identical'; ... we cannot without the help of an axiom be sure that x and y are identical if they have the same predicates. Leibniz's 'identity of indiscernibles' supplied this axiom.''51
We may in fact, Husserl continues his argument, declare the same objects to be equal in one case and different in another (pp. 108-09). He offered the following example to illustrate his point: two straight lines may in one case be said to be equal because they have the same length, but might otherwise be deemed unequal because to be equal two segments of a line must be parallel and have the same direction. Husserl tries to overcome the ambiguity involved by concluding that two objects are to be considered equal if they share the specific properties which constitute the main focus of interest of the investigation and these properties are the same.
An example will help illustrate Husserl's point. A few years ago a Jerusalem courtroom found a retired Ohio autoworker named John Demjanjuk guilty of being Ivan the Terrible, the murderer of hundreds of thousands of Jews. Throughout his fourteen month trial, Demjanjuk had insisted that he was a victim of mistaken identity. For the Jerusalem courtroom that condemned him to death the only thing that mattered involved determining whether or not he was the same man who had operated gas chambers at Treblinka during World War II, any of the innumerably many other things that could be truthfully predicated of him were beside the point. Their reasoning was of the form: F(x), and if F(y) then x ºy. Killing hundreds of thousands of Jews was true of Ivan the Terrible and if the same were true of John Demjanjuk, then he would be Ivan the Terrible --and liable to hanging.
Numerous other examples can be found to illustrate the differences between equality and identity. For instance, is a person in an irreversible coma following an accident who is entirely dependent on machines to sustain her bodily functions identical to the person she was before the accident took place? Think of the innumerable things that could have been predicated of her before which are no longer true, and the truly macabre propositions that could result from substitution rules which do not take sufficient account of the difference between equality and identity. Certainly her family would never have entertained the thought of depriving her of the minimum means necessary to support her life before she was in the coma. And don't differences between equality and identity figure in many other dilemmas actually faced in medical practice today. Surely, some of the very real moral issues involved in abortion rights turn on whether a fetus is in all essential respects the same as the person that will develop from it if the pregnancy is not terminated. Such contexts make it hard to dismiss differences between equality and identity as being merely linguistic or psychological.
In another argument Husserl sides with those who hold that characteristics, properties, attributes and concepts are not liable to the same identity conditions as objects are, so that talk of them cannot for him be reducible to talk of the objects they are about (pp. 134-35). He argues that Leibniz's definition turns the real state of affairs upside down (pp. 104-05). Assuming that, against all odds, one manages to find objects satisfying the conditions it sets down, then by what right can one replace one with the other in certain true propositions or all? The only precise answer, he replies, would he the identity or equality of the referents. However, here Husserl comes up against the same difficulty that Frege himself would encounter when confronted with Russell's paradox more than a decade later (see §5 of this text). Though characteristics which are the same form propositions which are the same, Husserl wrote, having equivalent propositions does not mean that the characteristics figuring in those propositions are the same. In other words, though two propositions may be formally equivalent by virtue of the fact that what is asserted of the reference in them is the same, from the fact that two propositions have the same reference it cannot be concluded that what has been asserted of their referent is the same. In Introduction to Mathematical Philosophy, Russell would give a reason for the problem Husserl perceived: "For many purposes, a class and a defining characteristic of it are practically interchangeable. The vital difference between the two consists in the fact that there is only one class having a given set of members, whereas there are always many different characteristics by which a given class may be defined."52
Husserl is making several interrelated, but different points which shed light on Frege's and Russell's difficulties. Quine's proposition that 'all bachelors are unmarried men'53 provides a happy, relatively unproblematic example of equal properties coinciding in extension and so illustrates Husserl's point that if F = G, then if (x) F(x) then G(x). If being married to Jackie Kennedy is the same thing as being her husband, then anyone who is married to Jackie Kennedy is her husband. However, and this is Husserl's second point, were Jackie Kennedy's husband found out to be Marilyn Monroe's lover, it would not then follow that being Jackie Kennedy's husband and being Marilyn Monroe's lover are the same. The statement 'The husband of Jackie Kennedy was the lover of Marilyn Monroe' could be determined to be true if both the descriptions figuring in it were found to be true of the same man, John Kennedy.
However, trivial, self-evident statements resulting from substitution like 'John Kennedy was the husband of Jackie Kennedy' and 'John Kennedy is John Kennedy', though true, are not its equivalent. Likewise, if it could he determined that a certain retired Ohio autoworker and a certain sadistic concentration camp guard were the same person, this would not then mean that everything that is true of sadistic concentration camp guards is true of retired Ohio autoworkers. In the latter case it could be argued that the matter has been complicated by changes that have taken place over the course of time. That is not, however, the case in the example preceding it since a person can be married and have an extra-marital affair at one and the same time.
Husserl's objection lies in the fact that predicates which are true of the same objects are not always themselves interchangeable salva veritate. Russell made the same point in 1918 to his audience at Gordon Square. The two propositional functions 'x is a man' and 'x is a featherless biped' are formally equivalent, he notes. When one is true, so is the other, and vice versa. However, he points out that there are a certain number of things that you can say about a propositional function which will not be true if you substitute another formally equivalent propositional function in its place. "For instance," he writes, "the propositional function 'x is a man' is one which has to do with the concept of humanity. That will not be true of 'x is a featherless biped'. Or if you say 'so-and-so asserts that such and such is a man' the propositional function 'x is a man' comes in there, but 'x is a featherless biped' does not."54 In an age of organ transplants, the point made by appealing to the old example of the statements 'creature with a heart' and 'creature with kidneys' coinciding in extension, but not in intension is less abstract than it may have been earlier in the century. Although almost anyone would concede that everyone who has a beating heart has at least one kidney and vice versa, who would think of undergoing a heart or kidney transplant or operation with a surgeon who believed that having a heart and having a kidney were the same thing? We cannot conclude from (x) F(x) = G (x) that F º G.
Husserl further notes that arguing that two objects are equal because they are interchangeable obviously misses the point (p. 104). In a case like John Demjanjuk's, this requirement would in fact have quite disturbing consequences. For instance, appealing to it one might reason that if John Demjanjuk could take Ivan the Terrible's place at the gallows, then John Demjanjuk and Ivan the Terrible were the same man. Obviously if, as Demjanjuk claimed, he was not Ivan the Terrible, but rather a victim of mistaken identity, such a conclusion would constitute a very grave miscarriage of justice. It would also be quite unthinkable to write off the differences between being the retired Ohio autoworker and the sadistic concentration camp guard as being merely linguistic or psychological.
There is, in fact, no way of determining whether two things are interchangeable which does not presume knowledge of their identity, Husserl concluded (p. 105). If substitutivity is to serve as the criterion of identity, then establishing the identity of two things implies that one has already established their interchangeability, but this would imply undertaking innumerably many acts in which what is predicated of one object is established as being the same as what is predicated of the other object and establishing this would require once again establishing that the same things can be predicated of each one of these pairs and so on. Michael Dummett makes the same point when he writes that:
the truth of an identity statement cannot he established by an appeal to Leihniz's law since, apart from the impossibility of running through the totality of first level concepts, there will often be no other way of ascertaining that a particular predicate which is true of the bearer of one name is also true of the bearer of the other except by establishing that the two names have the same bearer. There is, for example, no way of showing that the predicate 'is visible shortly before sunrise', which is plainly true of the Morning Star, is also true of the Evening Star, which does not depend upon showing that the Morning Star and the Evening Star are one and the same celestial body.55
Though Husserl provided the above criticisms of Frege's use of Leihniz's principle of substitutivity of identicals, it was not Husserl, but rather Frege himself who established the link between some of these very problems and the significance of Russell's paradox for the logical theories propounded in Foundations and Basic Laws. This is what I hope to show now.
5. Some of the Broader Philosophical Issues Involved
Knowing the central role Frege accorded to fixing the sense of an identity and of the link he made between substitutivity and identity in the Foundations, 56 and knowing that Frege ultimately concluded that Russell's paradox meant that there were irremediable flaws in the foundations proposed for arithmetic in Foundations and Basic Laws, it is important to take a close look at the connections between Frege's views on substitutivity and his reasons for despair regarding the tenability of his logical theories. The philosophical questions involving substitutivity and Russell's paradox are surely in this way tied into fundamental matters of vital concern to many philosophers in this century. In this section I want to look at what Frege himself thought was the fatal flaw in his reasoning and at how Russell would tackle the same problems. I examine these issues from another angle in Chapter 5, a sister chapter to this one.
In writing the Basic Laws of Arithmetic, Frege set out to actually demonstrate the theory of number he had advanced in Foundations.57 Once Russell informed him of the famous paradox, Frege immediately traced the source of the problem to Basic Law V (or Principle V) as promulgated in Basic Laws. 58 Basic Law V was Frege's axiom of extensionality which codified, or rather mandated, the views regarding identity and substitutivity Frege believed his system required. Right from the beginning, the discovery of the paradox indicated to Frege that Basic Law V was false. Although transforming of the generality of an identity into an identity of ranges of values was allowable, he concluded, the converse is not always permissible.59
Frege initially thought a solution to the problems raised by Russell's paradox might be found, 60 and he finally proposed one which involved a modification of the problematic law.61 By mid-1906, however, he had apparently decided that all efforts to repair his logical edifice were destined to failure.62
Frege was also very specific about precisely what it is about Basic Law V that leads to the paradoxes. In the several texts in which Frege pinpoints what he believed was the source of the difficulties, he consistently cites Basic Law V's transformation of concepts into objects for extensional treatment as being at fault.63 In §§ 146-47 of the 1903 Basic Laws II, he is quite specific about the nature of the procedure he had come to advocate:
If a (first-level) function (of one argument) and another function are such as always to have the same graph for the same argument, then we may say instead that the graph of the first is the same as that of the second. We are then recognizing something common to the two functions ... We must regard it as a fundamental law of logic that we are justified in thus recognizing something common, and that accordingly we may transform an equality holding generally into an equation (identity).64
An article on the logical paradoxes of set theory he was working on in 1906 gives further insight into his reasoning:
Let the letters 'j' and 'y' stand in for concept-words (nomina appellativa). Then we designate subordination in sentences of the form 'If something is a j, then it is a y'. In sentences of the form 'If something is a j, then it is a y and if something is a y then it is a j' we designate mutual subordination, a second level relation which has strong affinities with the first level relation of equality (identity) ... And this compels us ineluctably to transform a sentence in which mutual subordination is asserted of concepts into a sentence expressing an equality.... Admittedly, to construe mutual subordination simply as equality is forbidden …. Only in the case of objects can there be a question of equality (identity). And so the said transformation can only occur by concepts being correlated with objects in such a way that concepts which are mutually subordinate are correlated with the same object.65
He gives as an example the sentence: 'Every square root of 1 is a binomial coefficient of the exponent -1 and every binomial coefficient of the exponent -1 is a square root of 1.' According to his theory this sentence is to be rewritten as 'The extension of the concept square root of 1 is equal to (coincides with) the extension of the concept binomial coefficient of the exponent -1.' The words 'the extension of the concept square root of 1' are now to be regarded as a proper name as, Frege claims, is indicated by the definite article. Such a transformation acknowledges that there is one and only one object designated by the proper name. Frege explains that:
By permitting the transformation, you concede that such proper names have meanings. But by what right does such a transformation take place, in which concepts correspond to extensions of concepts, mutual subordination to equality? An actual proof can scarcely be furnished. We will have to assume an unprovable law here. Of course it isn't as self-evident as one would wish for a law of logic. And if it was possible for there to be doubts previously, these doubts have been reinforced by the shock the law has sustained from Russell's paradox.66
In 1912 he would write of how he had originally silenced his doubts and, in order to obtain objects out of concepts, had decided to admit the passage from concepts to their extensions, which until he died he maintained leads to Russell's paradox.67 An article and a letter Frege wrote just prior to his death eighteen years later both clearly show that for the rest of his life he remained firm in his conviction that this flaw in his system was the precise cause of Russell's paradox and so undermined his whole life's work. He wrote:
One feature of language that threatens to undermine the reliability of thinking is its tendency to form proper names to which no objects correspond…. A particularly noteworthy example of this is the formation of a proper name after the pattern of 'the extension of the concept a' ... Because of the definite article, this expression appears
to designate an object; but there is no object for which this phrase could be a linguistically appropriate designation. From this has arisen the paradoxes of set theory which have dealt the deathblow to set theory itself.68
This procedure, Frege concluded, leads "into a thicket of contradictions".... "Confusion is bound to arise if a concept word, as a result of its transformation into a proper name, comes to be in a place for which it is unsuited." People should be warned against changing a concept into an object.69
Bertrand Russell also eventually established a connection between certain puzzles involving descriptions containing the definite article and the contradictions, or paradoxes, connected with set theory.70 So as he struggled to find solutions to puzzles and paradoxes connected with Frege's logic Russell was actually faced with resolving the problem Frege described above. Russell would even go so far as to write in his 1919 Introduction to Mathematical Philosophy that he considered analyses of the word 'the' to be of so very great importance to a correct understanding of descriptions and classes that he would give the doctrine of the word if he were dead from the waist down and not merely in prison (where he was at the time).71 No account of Frege's problems with substitutivity is complete, then, without a look at Russell's attempts to solve them.
"The whole theory of definition, of identity, of classes, of symbolism is wrapped up in the theory of denoting", Russell declared in his 1903 Principles of Mathematics.72 So, it is perhaps not surprising to find him recounting that during 1903 and 1904 his work was almost entirely devoted to denoting problems which he thought were probably relevant to his problems with the contradictions of set theory. His 1905 theory of denoting proved to him that they were and it represented his first major breakthrough in finding a solution to the paradoxes.73
One of the most significant problems associated with descriptions containing the definite article dealt with by Russell's new theory of denoting is directly tied in with Leibniz's principle of the substitutivity of identicals. If identity can only hold between x and y if they are different symbols for the same object it would not then seem to have much importance, Russell noted. However, he observed, identity statements containing descriptive phrases of the form 'the so-and-so' constitute an exception and lead to a puzzle he explained in these words:
If a is identical with b, whatever is true of the one is true of the other, and either may be substituted for the other in any proposition without altering the truth or falsehood of that proposition. Now George IV wished to know whether Scott was the author of Waverley; and in fact Scott was the author of Waverley. Hence we may substitute Scott for the author of 'Waverley', and thereby prove that George IV wished to know whether Scott was Scott. Yet an interest in the law of identity can hardly be attributed to the first gentleman of Europe.74
In Frege's logic descriptions like 'the discoverer of America' or 'the extension of the word 'star'' were treated as objects. His Basic Law V would have made such descriptions subject to the same formal rules of identity as those governing objects, and so amenable to extensional treatment and substitution. Frege ultimately felt obliged to condemn this operation as illicit and judged his reliance upon it to be the fatal flaw in his logic and the source of the paradoxes associated with set theory.
Like Frege, Russell considered statements equating a term standing for an object with a description to be identities, and identities to be statements which equated objects. Russell had originally believed that all expressions denote directly, "that, if a word means something, there must he something that it means".75 But his struggle to solve the paradoxes forced him to come to terms with serious logical problems which seemed to him to be unavoidable when definite descriptions are regarded as standing for genuine constituents of the propositions in which they figure. For instance, if the expression 'the author of Waverley' really does denote some object c, Russell finally reasoned, the proposition 'Scott is the author of Waverley' would be of the form 'Scott is c'. But if c denoted any one other than Scott, this proposition would be false; while if c denoted Scott, the resulting proposition would be 'Scott is Scott', which is self-evident and trivial and plainly different from 'Scott is the author of Waverley' which may be true or false. So, Russell concluded, c does not stand simply for Scott, nor for anything else76 because
No one outside a logic-book ever wishes to say that 'x is x', and yet assertions of identity are often made in such forms as 'Scott was the author of Waverley'.… The meaning of such propositions cannot be stated without the notion of identity, although they are not simply statements that Scott is identical with another term, the author of Waverley …. The shortest statement of 'Scott is the author of Waverley' seems to be 'Scott wrote Waverley; and it is always true of y that if y wrote Waverley, y is identical with Scott'.77
Analyzed in this way, the description may be substituted for y in any propositional function fy and a significant proposition will be the result. These reflections on substitutivity and definite descriptions led Russell to the conclusion that all phrases (other than propositions) containing the word 'the' (in the singular) are incomplete symbols which have a meaning in use but when taken out of context do not actually denote anything at all.78
Russell also came to consider classes to be incomplete symbols in the same sense descriptions are, and so this new way of analyzing away incomplete symbols represented for him a major breakthrough in resolving the contradictions apt to result when descriptive phrases are wrongly assumed to stand for an entity. In 1918 he told his listeners at Gordon Square:
you find that all the formal properties that you desire of classes, all their formal uses in mathematics, can be obtained without supposing for a moment that there are such things as classes, without supposing, that is to say, that a proposition in which symbolically a class occurs, does in fact contain a constituent corresponding to that symbol, and when rightly analysed that symbol will disappear, in the same sort of way as descriptions disappear when the propositions are rightly analysed in which they occur.79
Russell considered this theory of definite descriptions to have been his most valuable contribution to philosophy and frequently spoke in enthusiastic terms of its role in resolving his logical problems, and the paradoxes in particular.80 He once claimed that his success in his 1905 article "On Denoting" was the source of all his subsequent progress. As a consequence of his new theory of denoting, he said, he found at last that substitution would work, and all went swimmingly.81 This new theory, he claimed, afforded a "clean shaven picture of reality" and "swept away a host of otherwise insoluble problems". It did not, however, sweep away all the problems and in spite of Russell's enthusiastic appraisals and the acclaim it has received, even Russell recognized that additional measures were needed82 to guarantee that the unwanted whiskers wouldn't come back. So Russell marshalled the theory of types and the axiom of reducibility into his barbershop to try to finish the job.
However, even as he expounded the theory of types, Russell realized that it only solved some of "the paradoxes for the sake of which" it was "invented" and something more would be necessary to solve the other contradictions.83 In 1917 he would go so far as to concede that "... the theory of types emphatically does not belong to the finished and certain part of our subject: much of the theory is still inchoate, confused and obscure."84
From the axiom of reducibility all the usual properties of identity and classes would follow. Two formally equivalent functions would determine the same class and, conversely, two functions which determine the same class would he formally equivalent. Without it or some equivalent axiom "many of the proofs of Principia become fallacious" Russell believed and "we should be compelled to regard identity as indefinable." 85
Frege had introduced classes into his logical system to uphold the theory of substitutivity and identity his work on the foundations of arithmetic called for. Russell's paradox finally convinced him that this was an ill-fated move and he gave up. Russell devised a way of analyzing away classes and descriptions and proposed the very problematic axiom of reducibility to mandate the properties of identity, classes and substitutivity the logic of Principia seemed to need. Neither he nor Frege ever felt they had ever solved the substitutivity problem.
6. Concluding Remarks
The chief objective of this paper has been to show Husserl's ability to evaluate Frege's logical work and pinpoint genuine problems in his reasoning. I have also tried to show that, independently of whether or not one is persuaded of the gravity of the problems with extensionality discussed here, Frege himself ultimately concluded that such problems were serious enough to sink his logic, and that in trying to free Frege's work from paradox, Russell felt obliged to come to terms with these same problems. A closer look at the Husserl-Frege relationship, in fact, reveals that the two men clashed swords on many more matters than have yet been discussed in the literature and that their ideas overlapped on many issues which still figure importantly in philosophical discussions today. Of course, the arguments of this paper but raise deeper questions regarding Russell's theory of definite descriptions and how Husserl's philosophy might deal with the questions raised. Naturally these are matters calling for in depth study extending well beyond the limits of this paper and I have tried to begin to answer them elsewhere.86 I would, however, like to make one more remark concerning them.
In his now classic article on Russell's mathematical logic87 Kurt Gödel complained about Principia Mathematica's lack of formal precision in the foundations and the fact that in it Russell omits certain syntactical considerations even in cases where they are necessary for the cogency of the proofs. In particular, he cites Russell's treatment of incomplete symbols, which he complains are introduced, not by explicit definitions, but by rules describing how to translate sentences containing them into sentences not containing them. The matter is especially doubtful, he notes, for the rule of substitution and it is chiefly this rule which would have to be proved.88
As an example of Russell's analysis of basic logical concepts, Gödel examines Russell's treatment of the definite article 'the' in connection with problems about what descriptive phrases signify (in deference to Frege, Gödel uses signify and signification in the place of Russell's denote and denotation). Gödel agrees with Russell that the apparently obvious answer that the description 'the author of Waverley' signifies Walter Scott leads to unexpected difficulties. For, Gödel reasons, if one admits an axiom of extensionality according to which the signification of a composite expression containing constituents which have themselves a signification depends only on the signification of these constituents, and not on the way in which this signification is expressed, then it follows that the statements 'Scott is the author of Waverley' and 'Scott is Scott' have the same signification. Of Russell's technique for solving this puzzle, Gödel writes that he could not help but feel that the problem raised by Frege's puzzling conclusion had only been evaded by Russell's theory of descriptions and that there was something behind it which is not yet completely understood.89
In a later paper Gödel wrote of the significance of Russell's paradox for set theory that "it might seem at first that the set-theoretical paradoxes would doom to failure such an undertaking, but closer examination shows that they cause no trouble at all. They are a very serious problem, not for mathematics, however, but rather for logic and epistemology."90 Because of the connection both Frege and Russell saw between the paradoxes of set theory and the fact that descriptions are not as immediately translatable into the extensional language their theories required, I am inclined to think that Gödel's intriguing statements regarding Russell's theory of definite descriptions and epistemological and logical questions raised by Russell's paradox might be linked, and that with the theory of definite descriptions Russell artfully swept under the carpet a whole host of deep logical and epistemological problems the set-theoretical paradoxes raise for philosophy. I also believe that further work on Husserl's logic will show that he quite lucidly addressed these very issues. I myself pursue these questions further in my Rethinking Identity and Metaphysics, On the Foundations of Analytic Philosophy91 and in Chapter 5 of this book.
Notes
1 E. Husserl, Philosophie der Arithmetik, Pfeffer, Halle, 1891.
2 G. Frege, Foundations of Arithmetic, Blackwell, Oxford, 1986 (1884).
3 G. Frege, The Basic Laws of Arithmetic, University of California Press, Berkeley, 1964 (1893), pp. 127-43.
4 I go into these in detail in my Word and Object in Husserl, Frege and Russell, Ohio University Press, Athens, 1991.
5 G. Frege, Philosophical and Mathematical Correspondence, Blackwell, Oxford, 1980, p. 7 (Couturat's July 1, 1899 letter to Frege).
6 A. Church, "Review of M. Farber, The Foundations of Phenomenology" Journal of Symbolic Logic 9, 1944, pp. 63-65.
7 D. Willard, Logic and the Objectivity of Knowledge, A Study in Husserl's Early Philosophy, University of Ohio Press, Athens OH, 1984. Two comments on Husserl's psychological objections (as opposed to his logical objections) to Frege's views on identity appear in H. Ishiguro's, Leibniz's Philosophy of Logic and Language, Cambridge University Press, Cambridge UK, 1990, p. 209 n. 43; and in D. Wiggins', Sameness and Substance, Blackwell, Oxford, 1980, pp. 20-21, n. 7.
8 Willard, pp. xii-xiv comments.
9 G. Frege, "Review of Dr. E. Husserl's Philosophy of Arithmetic", Mind 81, no. 323, July 1972, pp. 321-37. Translation of "Rezension von Dr E. G. Husserl: Philosophie der Arithmetik", Zeitschrift für Philosophie und philosophische Kritik 103, 1894, pp. 313-32.
10 M. Dummett, Frege, Philosophy of Language, Duckworth, London, 1981 (2nd ed. rev.), p. xlii; M. Dummett, The Interpretation of Frege's Philosophy, Harvard University Press, Cambridge MA, 1981, p. 56. Frege, Philosophical and Mathematical Correspondence, pp. 60-61; Comments Willard, pp. xii-xiii, p. 118.
1l K. Schuhmann, Husserl-Chronik, M. Nijhoff, The Hague, 1977, pp. 6-11 on Weierstrass and Husserl.
12 M. Kline, Mathematical Thought from Ancient to Modern Times, Vol. 3, Oxford University Press, New York, 1972, pp. 950-56, 960-66, 972 on Weierstrass and Bolzano
13 Schuhmann, pp. 7, 11; Willard, pp. 3-4, 21-22; A. Osborn, The Philosophy of E. Husserl in its Development to his First Conception of Phenomenology in the Logical Investigations, International Press, New York, 1934, p. 12.
14 L. McAlister, The Philosophy of Franz Brentano, Duckworth, London, 1976, p. 49.
15 Osborn, p.18.
16 McAlister, pp. 45, 53.
17 Osborn, p. 21.
18 Ibid. p. 17.
19 Dummett, Interpretation of Frege's Philosophy, pp. 72-73, 496-97; Dummett, Frege, Philosophy of Language, p. 683.
20 B. Russell, My Philosophical Development, Allen and Unwin, London, 1975 (1959), p. 100. See my discussions of this in my Word and Object in Husserl, Frege and Russell, Chapters 5 (§ 1) and 7.
21 Osborn, p. 29; Willard, pp. 32-34.
22 Frege, Philosophical and Mathematical Correspondence, pp. 171-72; Frege's letter to Marty pp. 99-102 may have actually been addressed to Stumpf; see p. 99.
23 Schuhmann, p. 18; Frege, Philosophical and Mathematical Correspondence, p. 64.
24 A. Fraenkel, "Georg Cantor", Jahresbericht der deutschen Mathematiker Vereinigung 39, 1930, pp. 221, 253n., 257.
25 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. Comparing the English edition with other editions one discovers the typographical error.
26 J. Cavaillès, Philosophie Mathématique, Hermann, Paris, 1962, p. 182, in reference to the Fraenkel biography cited in note 24.
27 R. Schmit, Husserls Philosophie der Mathematik, Bouvier, Bonn, 1981, pp. 40-48; pp. 58-62; L. Eley, "Einleitung des Herausgebers", Philosophie der Arithmetik mit ergänzenden Texten, Vol. XII, M. Nijhoff, The Hague, 1970, pp. XXIII-XXV.
28 Dummett, Frege, Philosophy of Language, p. 630.
29 Dummett, The Interpretation of Frege's Philosophy, p. 21.
30 B. Russell, "The Study of Mathematics", Mysticism and Logic, Allen & Unwin, London, 1963 (1917), pp. 66; and B. Russell, Principles of Mathematics, Norton, New York, 1903, p. xviii.
31 B. Russell and A. N. Whitehead, Principia Mathematica to *56, Cambridge University Press, Cambridge UK, 1964 (1927, 2nd ed. rev.), p. viii.
32 Russell, Principles of Mathematics, §§ 100, 344, 500; Frege, Philosophical and Mathematical Correspondence, pp. 133, 147.
33 Russell, Principles of Mathematics, § 149.
34 Ibid. § 107.
35 Ibid. § 249.
36 Schuhmann, p. 13
37 Frege, Philosophical and Mathematical Correspondence, pp. 6-7, 52, 170; G. Frege, "Gottlob Frege: Briefe an Ludwig Wittgenstein", Grazer philosophische Studien 33/34, 1989, pp. 11, 14.
38 G. Frege, Grundgesetze der Arithmetik, Olms, Hildesheim, 1966, §§ 68-85 of the second volume for Cantor and pp. 148-155 for Weierstrass. Basic Laws II is not available in English. §§ 56-67, 86-137, 139-44 are translated in G. Frege, Translations from the
Philosophical Writings, Blackwell, Oxford, 1980, thus omitting Frege's criticisms of Cantor and Weierstrass.
39 B. Russell, "My Mental Development", The Philosophy of Bertrand Russell, P. Schilpp (ed.), Northwestern University Press, Evanston, 1944 (there have been several editions of this work published by different publishers); C. Kilmister, Russell, The Harvester Press, Brighton, 1984, pp. 5-6, 54-66; I. Grattan-Guinness, Dear Russell-Dear Jourdain, Duckworth, London, 1977, pp. 132, 143-44.
40 Schuhmann, p. 18.
41 E. Husserl, Über den Begriff der Zahl, Psychologische Analyse, Heynemansche Buchdruckerei, Halle, 1887. In English in Husserl Shorter Works, P. McCormick and F. Elliston (eds.), Notre Dame University Press, Notre Dame, 1981, pp. 92-119.
42 Schuhmann, p. 19.
43 Frege, Philosophical and Mathematical Correspondence, pp. 63-65.
44 E. Husserl, The Logical Investigations, Routledge and Kegan Paul, London, 1970, p. 179 n.
45 Husserl, Philosophie der Arithmetik, pp. 104-05, 134.
46 For example Frege, Translations from the Philosophical Writings, pp. 22-23, 56, 120-21, 141n., 146n., 159-61, 210; Frege, Philosophical and Mathematical Correspondence, p. 141; G. Frege, Posthumous Writings, Blackwell, Oxford, 1979, pp. 120-21, 182; Frege, "Review of Dr. E. Husserl's Philosophy of Arithmetic", pp. 327, 331.
47 I have had to alter Austin's translation of § 65 to make it agree with other Frege texts and common usage according to which 'equal' means 'agreement in this respect or that' and 'identical', or 'the same' refer to complete agreement in all respects.
48 Frege, Basic Laws of Arithmetic, § 9, see also p. 6 and Frege, Translations from the Philosophical Writings, pp. 159-61, 214; Frege, Philosophical and Mathematical Correspondence, pp. 140-41, 191.
49 I cite the page numbers to the original 1891 edition of Husserl's Philosophie der Arithmetik within the text.
50 B. Russell, Introduction to Mathematical Philosophy, Allen and Unwin, London, 1919, p. 192.
51 Russell, Principia Mathematica, p. 57.
52 Russell, Introduction to Mathematical Philosophy, p. 13.
53 W. Quine, "Two Dogmas of Empiricism", From a Logical Point of View, Harper and Row, New York, 1961 (1953), pp. 27-32.
54 B. Russell, "The Philosophy of Logical Atomism", Logic and Knowledge, Allen and Unwin, London, 1956, pp. 265-66.
55 Dummett, Frege, Philosophy of Language, pp. 544-45.
56 Frege, Foundations of Arithmetic, p. x, §§ 62, 65, 107.
57 Frege, Basic Laws of Arithmetic, pp. 5, 7, 129, and §§ 38-47.
58 Ibid., p. 127; Frege, Philosophical and Mathematical Correspondence, pp. 130-32.
59 Frege, Basic Laws of Arithmetic, p. 132; Frege, Philosophical and Mathematical Correspondence, pp. 73, 132.
60 Ibid.
61 Frege, Basic Laws of Arithmetic, pp. 139-43.
62 Dummett, The Interpretation of Frege's Philosophy, pp. 21-27; Frege, Posthumous Writings, p. 176.
63 Frege, Philosophical and Mathematical Correspondence, pp. 54-56, 191; Frege, Posthumous Writings, pp. 181-82, 269-70.
64 Frege, Translations from the Philosophical Writings, pp. 159-60.
65 Frege, Posthumous Writings, pp. 181-82.
66 Ibid.
67 Frege, Philosophical and Mathematical Correspondence, p. 191.
68 Frege, Posthumous Writings, p. 269.
69 Frege, Philosophical and Mathematical Correspondence, p. 55.
70 Russell makes this connection in his My Philosophical Development, p. 60; "The Philosophy of Logical Atomism", Logic and Knowledge, pp. 262-66; "My Mental Development", pp. 13-14; and his Essays in Analysis, Allen & Unwin, London, 1973, p. 165.
71 Russell, Introduction to Mathematical Philosophy, p. 167.
72 Russell, Principles of Mathematics, p. 56.
73 Grattan-Guinness, pp. 79-80, 94; Russell, My Philosophical Development, p. 60.
74 Russell, "On Denoting", Logic and Knowledge, pp. 47-48.
75 Russell, My Philosophical Development, pp. 62-63
76 Russell and Whitehead, Principia Mathematica, p. 67; Russell, "The Philosophy of Logical Atomism", Logic and Knowledge, pp. 245-48.
77 Russell, "On Denoting", Logic and Knowledge, p. 55.
78 Russell and Whitehead, Principia Mathematica, pp. 67-68.
79 Russell, "The Philosophy of Logical Atomism", Logic and Knowledge, p. 266. Also Russell and Whitehead, Principia Mathematica, chapter III and the texts cited in note 70.
80 Kilmister, pp. 102, 108, 123, 138; Grattan-Guinness, pp. 70, 94 and the texts cited in note 70.
81Grattan-Guinness, pp. 78-79
82 Russell, My Philosophical Development, pp. 60-65.
83 Russell, "Mathematical Logic as Based on the Theory of Types", Logic and Knowledge, pp. 79-82.
84 Russell, Introduction to Mathematical Philosophy, p. 13
85 Russell and Whitehead, Principia Mathematica, pp. 55-59, 75-77, 166-72.
86 My Word and Object in Husserl, Frege and Russell.
87 K. Gödel, "Russell's Mathematical Logic", Collected Works, Vol. 2, Oxford University Press, New York, 1990, pp. l l9-41.
88 Ibid., p. 120.
89 Ibid., pp. 121-23.
90 Ibid., p. 258.
91 C. Hill, Rethinking Identity and Metaphysics, On the Foundations of Analytic Philosophy, Yale University Press, New Haven, 1997.