Review of W. Demopoulos (ed.), Frege's Philosophy of Mathematics, Harvard University Press, Cambridge MA., 1995, xi + 464 pp. and W. W. Tait (ed.), Early Analytic Philosophy, Frege, Russell, Wittgenstein, Essays in Honor of Leonard Linsky Chicago and La Salle Illinois, Open Court, 1997, vii + 291 pp.
Frege's Philosophy of Mathematics is a collection of eighteen essays by well-known scholars. It sets out to address three main developments in recent work on Frege's philosophy of mathematics: the emerging interest in the intellectual background of Frege's logicism; the reevaluation of the mathematical content of Frege's Basic Laws of Arithmetic; and the rediscovery of what is termed "Frege's theorem" that, in the context of second-order logic, Hume's principle (i.e., the number of Fs = the number of Gs if and only if the Fs and the Gs are in one-to-one correspondence) implies the infinity of the natural numbers. In his introduction editor Demopoulos calls the "rediscovery of Frege's theorem" a major factor underlying the current, renewed interest in Frege's philosophy of mathematics. It is in fact the central theme of the book.
All but one of the papers anthologized date from the 1980s and 1990s. A major principle governing their selection was evidence of "a sympathetic, if not uncritical, reconstruction, evaluation, or extension of one or another facet of Frege's thought". Worthwhile papers not satisfying that criterion were not included in the collection (p. x). The papers are interrelated and their authors very frequently cite and thank one another in a friendly way.
The idea for the collection originated with Michael Dummett, whom Demopoulos considers to have set an intellectual standard to which most philosophers of his generation aspire. Given this, it is worthwhile to bear in mind that, while calling Frege "the best philosopher of mathematics" in the preface to Frege: Philosophy of Mathematics, Dummett opined that the reason why Frege's work in the philosophy of mathematics has been "dismissed as a total failure" is probably that his work "does not prompt any further line of investigation in mathematical logic" and "does not even appear to promise a hopeful basis for a sustainable general philosophy of mathematics ". The "evidences of the blindness and lack of generosity which were such marked features of Frege's work after 1891 combine", wrote Dummett, "with his great blunder in falling into the contradiction to suggest that he cannot have much to teach us" (p. xi-xiii).
The book is divided into three parts. In their different ways, the articles of Part One aim to situate Frege's efforts within the context of 19th century efforts to rigorize analysis and to shield it from the deleterious effects of Kant's ideas about intuitions and the synthetic a priori status of mathematical propositions. First, Alberto Coffa aims to embed logicism in a broader movement whose enemy was Kant, whose goal was the elimination of pure intuition from scientific knowledge and whose strategy was the creation of semantics as an independent discipline. This movement included the rigorization of the calculus, Frege's and Russell's theories of arithmetic, and Poincaré's and Hilbert's geometric conventionalism, which Coffa invites readers to look at as stages in a complex process that began with Bolzano.
Bolzano, writes Coffa, was the first to see that Kant had been wrong to think that all conceptual information available in a judgment was to be used up in the grounding of analytic judgments and that one was to appeal to intuition to ground the rest. Bolzano was thus prepared to "explore the possibility that all of our pure a priori knowledge -including
synthetic a priori knowledge -could be stated and grounded on concepts alone" (pp. 34-35). He and his followers, maintains Coffa, maneuvered pure intuition out of analysis and into arithmetic where Frege's gigantic fly swatter finally came to squash it out. Poincaré and Hilbert came to take up the cause of geometry. Carnap finally saw what Bolzano and Frege almost saw, namely that logical truth is truth in virtue of logical concepts. The essay closes with the words: "And then came Quine" (p. 40).
The following paper by Paul Benacerraf should be an invitation to analytic philosophers to do some thorough soul searching to determine exactly how and why they ever came to believe the views he contests. In particular, he challenges the thesis that 20th century logicists were correct to consider Frege's logicism to be a philosophical view closely allied with empiricism. Frege's view, Benacerraf maintains, "was a much more intriguing one and in its spirit directly antithetical to the philosophical motivations of his twentieth-century 'followers'" (p. 48). For Benacerraf, Foundations is first and foremost a work of mathematics and not, as he had been taught, a work in the Kantian/empiricist tradition. For him Frege was no empiricist and establishing the analyticity of arithmetical judgments was not his way of defending empiricism against Kantian attack. If Frege was a logicist, Benacerraf concludes, then he was both the first and last one.
"Frege and the Rigorization of Analysis" by Demopoulos complements Coffa's and Benacerraf's papers. Demopoulos agrees that "when Frege's foundational interests are viewed in their mathematical context, they stand in sharp contrast with the logical empiricist's attempts to show the analyticity of arithmetic and more generally of all a priori knowledge" (p. 69), but then sets out to show that Frege's interest in rigor was closely linked to his rejection of intuition in reasoning and thus had both a philosophical and a mathematical dimension. He notes that though that his remarks are "neither novel nor contentious" this
aspect of Frege's thought has been unduly neglected.
In "Frege and Arbitrary Functions" John Burgess investigates the question as to whether Frege's notion of function could be said to involve definability or even differentiability restrictions, an issue he considers to have implications for the overall interpretation of Frege's philosophy. Burgess concludes that Frege's notion of function did not involve definability restrictions and that it cannot be decided on the basis of the available texts whether it involved differentiability restrictions. Frege, he argues, cannot be expected to have espoused broader theories about functions that only began to be developed as he abandoned his original work.
Mark Wilson's "The Royal Road of Geometry" teems with facts and reflections about philosophical traditions in19th century geometry that he tries to link to Frege's published views. While it is true that as Wilson contends much "work needs to be done on how the chains of influence might have run in this period" (p. 142 n.), Wilson's principal theses should set alarm bells ringing.
For example, the stated central purpose of the paper is to show that we will better understand the character of Frege's philosophy and the manner in which he conceived his logicism by seeing it as having grown up in the shadow of the methodological concerns arising in geometry (those of complexified projective geometry and of von Staudt's theory of "concept-objects" in particular). Citing Foundations and Frege's earlier writings, Wilson argues that, contrary to popular opinion, they "stand at a remarkable distance" from the concerns of those undertaking to rigorize analysis (ex. pp. 148, 159).
Yet surely the principal reason scholars put Frege into the movement to arithmetize (and so to degeometrize) analysis is precisely that he so openly called for a more rigorous treatment of higher analysis. In Foundations §§1-4, for example, he specifically called for sharper definitions of the concepts of function, continuity, and limit and of infinity, negative
and irrational numbers. There he states his conviction that such a pursuit ultimately leads to analyses of the concept of number and of the simplest propositions holding of positive whole numbers (the leading idea of the Philosophy of Arithmetic by Weierstrass' student and assistant Husserl, who starting in the 1890s really did turn to innovations in geometry to find answers to questions about numbers that he could not answer by appealing to Weierstrass' teaching about the derivation of analysis from the concept of positive whole number).
But Wilson bypasses all talk of the arithmetization of analysis. And he does not explain away either Frege's declaration in §89 of Foundations that in calling the truths of geometry synthetic a priori Kant revealed their true nature, or the claim in §14 that the fact that the kind of innovations in geometry that Wilson discusses are possible shows that, unlike the fundamental propositions of the science of number, the axioms of geometry are independent of the primitive laws of logic and consequently synthetic. In §13 Frege had warned that we "should do well in general not to overestimate the extent to which arithmetic is akin to geometry".
Since both Burgess and Wilson advance arguments to counter what they perceive as being the "anti-Fregean conclusions" of Jaakko Hintikka and Gabriel Sandu in "The skeleton in Frege's cupboard: the standard versus nonstandard distinction" (The journal of philosophy 89), it is worth noting that those conclusions there are no more hostile to Frege than Dummett's conclusions cited above.
The four articles of Part Two investigate the mathematical content of the Begriffsschrift and Foundations. Three of them are by George Boolos, who first of all engages an imaginary, but very astute, Kantian interlocutor in a dialogue to show how and, how well, the arguments of the third part of the Begriffsschrift might be defended against a certain sort of
Kantian attack and to determine the extent to which Frege may have shown that there are many interesting examples of mathematical truths that can be reduced to logic.
Boolos scrutinizes the examples given in part three of the Begriffsschrift of judgments that can be proved by purely logical means, but which may at first sight appear to be possible only on the basis of some intuition. Far from dispensing with intuition, the Kantian responds, Frege is "up to his ears" in it. For it is an intuition of precisely the kind he believes that he has shown to be unnecessary that licenses the rule of substitution. Besides, though perhaps formally consistent, the system cannot be interpreted in Frege's way in the absence of some metaphysical doctrine of properties that Frege does not supply. Furthermore, if one reads the second-order quantifier EF as "There is a set F…", then the difficulty presented by Russell's paradox immediately arises if the range of "F" is taken to be all sets. In so arguing, the Kantian unearths some of the reasoning behind Frege's later appeal to the extensions that were to guarantee substitutivity in a purely logical way and thus pinpoints weaknesses in Frege's arguments that foreshadow problems that he later faced and that persist to this day.
Boolos replies that "a formalism like that of the Begriffsschrift can be used to schematize plural existential generalization, and our understanding of the plural forms involved in this type of inference can be appealed to in support of the claim that Frege's rule [of substitution] is properly regarded as a rule of logic" (p. 174), that "there is a way of interpreting the formulae of the Begriffsschrift that is faithful to the usual meanings of the logical operators and on which each comprehension axiom turns out to say something that can also be expressed by a sentence of the form 'if there is something…, then there are some things such that anything … is one of them and any one of them is something…' "(p.177). One does not need to be a Kantian, however, to be skeptical as to how effectively the Kantian's criticisms, which target the very heart of Frege's ill-fated enterprise, have actually been overcome.
Sandwiched in between Boolos's papers is Charles Parsons' 1964 article entitled "Frege's Theory of Number", which is intended to be a contribution towards attaining a clear view of what is true and what is false in Frege's account of arithmetic. Parsons concludes that while the criticisms of the thesis that arithmetic is a part of logic that have been made over the years do suffice to show that it is false, it should not be rejected in the unqualified way it appears to have been by mathematicians like Poincaré, Brouwer and Hilbert. Frege did, Parsons maintains, show that the logical notion of one-to-one correspondence plays a constitutive role in the notion of number.
In "The Consistency of Frege's Foundations of Arithmetic" Boolos pursues questions raised by Parsons. There he presents FA, a theory whose underlying logic is standard axiomatic second-logic written in classical notation and could have been presented as an extension of the system of the Begriffsschrift. "Numbers" is postulated as the sole nonlogical supposition of FA. It would express Frege's and Russell's conviction that existence and uniqueness are implicit in the use of the definite article in the form of the principle that for any concept F there is a unique extension of the concept 'equinumerous with F' in support of Hume's principle (the only place, Boolos points out, where Frege appeals to extensions in Foundations). Boolos is satisfied that "the principles Frege employs in the Foundations are consistent. Arithmetic can be developed on their basis in the elegant manner sketched there. And although Frege couldn't and we can't supply a reason for regarding Numbers… as a logical truth, Frege was better off than he has been thought to be" (p. 230).
"The Standard of Equality of Numbers" by Boolos asks whether Hume's Principle could have been used to help Dedekind find a proof from logical truths that there are infinitely many objects. He concludes that Dedekind would not have liked the suggestion and that we cannot
accept it either, that there is no reason for regarding Hume's Principle as a truth of logic and it is doubtful that it is a truth at all.
The last and longest part of the book is devoted to the Basic Laws of Arithmetic and to salvaging more of Frege's reasoning. Thus Richard Heck begins where Parsons and Boolos have left off by asking whether Frege's formal proofs of the axioms of arithmetic in Basic Laws depend only on Hume's principle. The presence of Basic Law V governing value-ranges leading Heck to reply that they do not, he then sets out to demonstrate that except for the ineliminable use of extensions in the proof of Hume's principle itself, all the uses of value-ranges in Frege's proof of the basic laws of arithmetic are easily and uniformly eliminable, that Frege resorted to value-ranges for reasons of convenience, simplicity and elegance.
Building on the conclusions of the preceding essays, in a second paper Heck further argues that the evaluation of Frege's efforts in Basic Laws must change. In particular, he undertakes to demonstrate that in Basic Laws, in addition to deriving axioms for arithmetic, in second-order logic from Hume's principle, Frege gave formal proofs of two celebrated theorems of Dedekind, that the book contains a proof, in Fregean arithmetic that Frege's own axioms for arithmetic determine a class of structures isomorphic to the natural numbers and a proof in pure second-order logic that all structures satisfying these axioms are isomorphic.
Howard Stein's article on the relation of the ancient Greek theory of ratio to Dedekind's work next sets the stage for three studies (by Peter Simons, Michael Dummett, Peter Neumann and S.A. Adeleke) of the theory of real numbers Frege was developing when calamity, in the form of Russell's paradox, struck. All three come to optimistic conclusions about philosophical insight afforded by Frege's never completed theory. Thus, in reconstructing and venturing to complete Frege's unfinished theory, Simons unearths a number of considerations to show that by affording us a second look at Frege's way of
supporting logicism, his aborted theory yields new insight into logicism and raises a number of philosophically interesting questions about it and mathematics in general. Simons also finds aspects of Frege's criticisms of Cantor, Dedekind and Weierstrass perceptive and of relevance now. He does not, though, entertain any illusions that a construction like his could prove any more tenable than the Frege's theory of natural numbers did.
The last three papers tackle questions of consistency. Terence Parsons sets out to show that the first-order portion of Frege's system is consistent and then to explore the significance of the model-construction technique sketched for Frege's claims about the arbitrariness of the identification of truth-values with courses of values. His tactic is to neutralize the effects of the abstraction technique that quickly leads to contradictions in first-order naïve set theory. Parsons underscores that to understand his proof one must keep in mind Frege's conviction that truth-values are objects and that terms that denote truth-values can occur syntactically in the same places that other names of objects can. Sentences being truth-values, they then can occur in all the places we would normally expect to find objects, such as flanking the identity sign, and this allowed Frege to use the identity sign for the material conditional. John Bell then undertakes to formulate and prove a stronger version of Parson's result for arbitrary first-order theories. He also shows that a natural attempt to further strengthen his result runs afoul of Tarski's theorem on the undefinability of truth.
In a final paper, entitled "Saving Frege from Contradiction", Boolos pursues his ideas about Hume's principle in an effort to repair damage done by Russell's paradox. Proposing a "New V" in the place of the unsatisfactory V' that Frege proposed in the appendix to Basic Laws II, Boolos concludes that the "development of arithmetic outlined in the Grundlagen can be carried out in the consistent theory obtained by adding Numbers to the system of the Begriffsschrift, as well as in the inconsistent system of the Grundgesetze" (p. 452).
As the reader can see this book is, as suits the spirit of the times, largely dedicated to saving or rehabilitating major chunks of Frege's logical project. It might have been entitled Doctoring Frege's Theories. Almost all of the saviors have jumped into Frege's reasoning after he had begun fixing his theory with extensions that masked certain shocking consequences of his reasoning up to that point, i.e. once he had used extensions to prove Hume's principle. But, it must be protested, Frege appealed to extensions then precisely because of some very disturbing substitution problems that his theory of identity and arithmetic was already producing, --substitution problems that are the very breeding ground of contradiction and inconsistency. Frege's proof of Hume's principle, as Boolos reminds more than once in "The Standard of Equality of Numbers", is derived from an inconsistent theory of concepts and objects
This being the case, Richard Heck puts his finger on the book's major weakness (pp. 286-87). It lies in the fact that, as Frege acknowledged in Foundations §66-67, left unmodified, the theory of arithmetic he had begun recommending was liable to produce nonsensical conclusions or be sterile and unproductive. In particular, as he wrote, one could never "decide by the means of our definitions whether any concept has the number Julius Caesar belonging to it, or whether that conqueror of Gaul is a number or not" (Foundations § 56). Onerous problems of the "Julius Caesar problem" kind induced Frege to introduce the extensions that led to the ill-fated Basic Law V of Basic Laws, which he said was the only answer that he had found to the question as to how we apprehend logical objects (quote p. 286).
So Heck is surely right to conclude that the questions really needing answers lie there. "We shall thus not", he writes, "fully understand Frege's philosophy until we understand the enormous significance the question how we apprehend logical objects, and the Caesar problem, had for him…" (p. 287). Demopoulos concurs when he requires that any
interpretation of Frege's presentation of the problem in connection with Hume's principle explain how the introduction of extensions overcomes the difficulty that the Julius Caesar problem posed for numbers while not itself succumbing to a similar objection (a major issue in this reviewer's book Rethinking Identity and Metaphysics). This problem, Demopoulos maintains, must be addressed in connection with the Basic Laws, where extensions are introduced in a way formally analogous to the contextual definition of number. Failure to address it, he warns, "implies not only that Frege's mature theory of number is ungrounded, but that Frege must have known this…" (p. 10). However, although Heck and Demopoulos make this point in the most unequivocal terms, their few pages of comments are, strangely, the only significant mentions of the Julius Caesar problem in the entire book. Conclusion: this excellent book badly needs a Caesarean section.
Lastly, it must be said that the book's general index contains no entries at all for many of the main subjects treated in the book (ex. inconsistency, contradiction, analyticity, rigorization, analysis, synthetic a priori, function) and incomplete listings for topics as basic to the work as Hume's principle, consistency, comprehension, real numbers. Though it lists Kant, it is blind to the many, and more frequent, mentions of the words "Kantianism" and "Kantian", or "anti-Kantian". Meanwhile, minor figures like Otto Hölder seem to enjoy complete listings.
Early Analytic Philosophy ,edited by W.W. Tait, grew out of a conference held in April1992 to honor Leonard Linsky upon the occasion of his retirement from the faculty of the Philosophy Department of the University of Chicago. The topic of the conference was the figures of early analytic philosophy of which Linsky was especially fond: Frege, Russell, Wittgenstein and Carnap. The papers by Tyler Burge, Michael Friedman, Warren Goldfarb,
Thomas Ricketts and Joan Weiner anthologized here "were based on or were the basis of their talks" (p. vi). Added to them are papers by Linsky's son Bernard and by Steve Gerrard and Erich Reck, who worked extensively with Linsky as graduate students. Conference participant Peter Hylton contributed a different paper and Tait added a paper of his own.
If this very diverse collection of essays could be said to have an underlying message it would perhaps be: When the history of early analytic philosophy is adequately written, and the chains of influence running through the period satisfactorily established, much scholarship on the period that seemed radical and iconoclastic will be shown to have been true to the facts. For, as the papers in this collection show, much that certain influential dogmatic strains of the analytic movement strove to weed out as unsavory (ex. intensionality, rationalism, idealism) actually mingled quite intimately in with what was otherwise judged to be the healthy beginnings of a new movement in philosophy. Likewise, much of what many philosophers would have liked to have been the case was not.
In the first paper, "Frege on Knowing the Third Realm", Tyler Burge suggests that Frege's puzzling failure to discuss our knowledge of the foundations of arithmetic is to be explained by his readiness to accept the traditional rationalist-Platonist account of the relation between reason and primitive truths. Particularly studied is Burge's conviction that "Frege assumes that we can know arithmetic and its foundations purely through reason and that individuals are reasonable and justified in believing basic foundational truths…" (p. 2).
Burge appeals to a wealth of texts that he marries with plausible conjectures as to Frege's motivations and intentions. However, Frege's rationalism was surely of one piece with his anti-empiricism and the rationalistic, Platonistic stands he took were surely, as discussed in Demopoulos' book, part of his contribution to slaying the dragon of the Kantian inspired empiricism of the times. Yet, Burge never alludes to any such thing. The word "empiricism"
never once appears in the paper. Are we to understand this neglect as part of his avowed "prescinding from complexities of Frege's epistemology" (p. 13)?
Next, Michael Friedman argues that virtually all of Carnap's most characteristic philosophical ideas and distinctions result from what Friedman calls the "deep and basically two-pronged" influence of Wittgenstein's Tractatus on the evolution of Carnap's ideas in the Aufbau and the Logical Syntax of Language. On the one hand, according to Friedman, Carnap adopted the Tractarian idea of the essentially non-factual character of analytic truth. It was, however, Friedman contends, in rejecting Wittgenstein's ideas about the ineffability of logic that Carnap came to develop his distinctive idea of philosophy as logical syntax of language and an exact science. He thus came to wed Wittgenstein's ideas on analytic truth and the metamathematical work of Hilbert, Tarski and Gödel to create his own distinctive conception of the philosophical enterprise.
In "Functions, Operations and Sense in Wittgenstein's Tractatus", Peter Hylton elucidates the role that the notion of operation plays in the difficult to interpret 5.2s of the Tractatus. He adduces a number of reasons why Wittgenstein's notion is to be considered distinct from that of propositional function in either Frege's or Russell's sense. By either Russell's or Frege's accounts, Hylton reasons, truth-functional compounding introduces new elements in the form of truth-functions, something that Wittgenstein needed to avoid. Russell, though, used a notion of propositional function which was quite distinct from that of an ordinary mathematical function, while Frege explicitly used a generalized and clarified version of the mathematical notion. The 5.2s were, Hylton argues, a direct attack on Russell's views.
Studying §§ 65-88 of Wittgenstein's Investigations, Warren Goldfarb applies Linsky's advice about reading that work against the backdrop of Frege, Russell and the early Wittgenstein in order to correct simplistic interpretations of Wittgenstein's criticisms of the
early analytic tradition. Goldfarb particularly looks at Wittgenstein's views on family resemblance, fixity of meaning, names and descriptions and presuppositionlessness.
Gerrard's and Reck's papers give the impression that Linsky encouraged students to be prolix (the two papers take up more than a third of the book!) and bold about advancing hypotheses that challenge received interpretations. Reck's goal is to reinterpret Frege's and Wittgenstein's writings and Frege's influence on Wittgenstein. Conscious that he is "flying in the face of conventional wisdom", Reck argues that the two philosophers agreed on a "reversal of metaphysics". By this, he means that, deeply sympathetic to Frege's context principle (which Reck believes underlies Frege's entire reconstruction of arithmetic and was meant to apply generally, even beyond mathematics), the later Wittgenstein modified it to: "Only in the practice of language can a word have meaning". In so doing, he adopted what Reck calls Frege's "contextual Platonism" in which the meaning of expressions is understood from the "top-down" (from sentences down to words, from a whole system of judgments down to words) and not "bottom-up" (from words to sentences) as in metaphysical Platonism.
In "Desire and Desirability: Bradley, Russell and Moore versus Mill", Gerrard helps redraw the familiar, simplistic picture we have inherited of how Russell and Moore cast off the shackles of 19th century idealism. That liberation process is studied in connection with their contact with J. S. Mill's ideas. The paper, though, reflects the still unfinished state of research into the transmogrification of Russell's and Moore's ideas on idealism, no true picture of which is going to result until they are correctly positioned in relation to philosophy on the continent at the time, in particular to the ideas of Brentano's school (Meinong most visibly), and to the idealism of Lotze, Cantor, Bolzano or Leibniz, to name some of the most obvious figures into whose work Russell and Moore delved during the years in question.
In his paper, Bernard Linsky asks whether Russell's axiom of reducibility was a principle of logic according to the views that Russell held when writing the first edition of Principia. Linsky examines the axiom from several angles, defining it, studying its origins, its drawbacks, its connection with Leibniz' identity of indiscernibles. Rightly considering that the intensional nature of the logic of Principia explains otherwise puzzling features of the work, Linsky concludes that Russell's disputed axiom is a metaphysical principle, that it was the ontology behind Principia that justified it in Russell's eyes. However, as Russell recounts in that work's preface, a very large part of the labor involved in writing that book had been expended on the contradictions and paradoxes infecting logic and set theory (p. vii). So those efforts to overcome those contradictions and paradoxes and to determine their causes really need to be taken into account in trying to determine what that axiom signified for him.
At first sight, Thomas Ricketts' paper called "Truth-Values and Courses-of -Value in Frege's Grundgesetze" looks as if it belonged in Demopoulos' book. His study cites many of the papers anthologized there and covers much of the same ground. However, Ricketts is uniformly critical of § 10 of Basic Laws, where Frege identifies the two truth-values with selected courses-of-values, and this is the main subject of the paper.
Moreover, Ricketts does not engage in the elaborate remodeling work that is the distinctive feature of Demopoulos' book and he does display the independence of mind characteristic of Early Analytic Philosophy. For instance, he maintains that in Foundations and later writings Frege is "antipathetic" to contextual definitions and that his context principle plays no substantive role in justifying the introduction of numbers. Ricketts also considers the absence a notion of logical consequence and a certain antipathy to the concept of truth to be deep-seated features of Frege's logic. Though displaying familiarity with
Boolos' and Parsons' ideas on Hume's Principle, Ricketts, helpfully I think, consistently refers to it as Cantor's Principle. He also argues that Frege's introduction of course-of -value names gives rise to the Julius Caesar problem within his conceptual notation if there are proper names that are not course-of-value names.
The two last papers stand in studied contrast to the ingenious, expert plastic surgery performed on Frege's theories in Demopoulos's book. Tackling such issues as abstraction, extensions, the creation of objects by the mind, psychologism, the meaning of "1", the cardinality of infinite sets, the notion of pure unit, equality and identity, equinumerosity and Hume's principle, Tait casts a cold eye on Frege's and Dummett's assessments of Cantor's and Dedekind's efforts. (Remember that in Principia Russell acknowledged that in "Arithmetic and series, our whole work is based on that of George Cantor" (p. viii) and that was in studying Cantor that Russell uncovered his paradox). Tait even goes so far as to charge that "Not only we have inherited from Frege, a poor regard for his contemporaries, but, taking the critical parts of his Grundlagen as a model, we in the Anglo-American tradition of analytic philosophy have inherited a poor vision of what philosophy is" (pp. 215-16).
Joan Weiner asks whether Frege had a philosophy of language and answers that he did, that for him language was a tool. She argues that, if we accept Dummett's understanding of meaning as having to do with features of language that are in no way exhausted by what is expressed in logical laws, then for Frege the philosophical treatment of the notion of truth as a philosophy of language is closely allied, not with meaning, but with science. Frege was interested in ordinary language and ordinary reasoning, she contends, only insofar as this could help him develop and explain a different sort of language, a language that could be used as a scientific tool. She even maintains that insisting that Frege was developing a theory of meaning requires one to "ascribe many serious and inexplicable blunders to him" (p. 252).