This chapter contains sections titled: Meaning as Use Language Games Mind and Matter Mathematics and Other Sciences Science, Myth, and Religion Seeing Aspects World Pictures The Inner and the Outer A Field of Diversity Further reading.

Responding to Russell is a constant throughout Wittgenstein's philosophizing. This Element focuses on Wittgenstein's criticisms of Russell's theories of judgment in the summer of 1913. Wittgenstein's response to these criticisms is of first-rate importance for his early philosophical development, setting the path to the conceptions of proposition and of logic in Tractatus Logico-Philosophicus. This Element also touches on further aspects of Wittgenstein's responses to Russell: the rejection of Russell's and Frege's logicisms in the Tractatus, the critique (...) of Russell's causal-behavioristic philosophy of mind in Wittgenstein's 'middle' period, the Russellian origins of notions of privacy dialectically treated in Philosophical Investigations, and the discussion of 'surveyability' of mathematical proof in Remarks on the Foundations of Mathematics, which is, again, a response to Russellian logicism. (shrink)

If we consider Wittgenstein's career as a whole, it appears that he wrote more on the philosophy of logic and mathematics than any other subject. Yet his writings on these subjects have exerted little influence. Indeed, the tide of response to Remarks on the Foundations of Mathematics, which contains the bulk of his latest views of mathematics, has been for the most part overwhelmingly negative. Given his later emphasis on the context-bound character of language, mathematics and logic--where language apparently operates (...) in an maximally precise, clear and general way--represent two of the most difficult cases for Wittgenstein to confront. My thesis aims to defend Wittgenstein from the charges of benighted arrogance traditionally levelled against him. I argue that Wittgenstein's later discussions of mathematics form a central part of a larger philosophical project, internally related to Philosophical Investigations and shaping in specific ways Wittgenstein's reaction to both scepticism and accounts of the nature of logical and mathematical truth. I see Wittgenstein criticizing the unreflective use of mathematical tools in philosophy, not offering a competing philosophy of mathematics. ;Too few of Wittgenstein's readers have been willing to offer detailed exegesis of his writing . This has made it difficult to understand how the aphoristic style of his late writing bears on the philosophical problems being discussed. On my view, the quality of Wittgenstein's writing is intrinsic to his later conception of the nature of logic, mathematics and philosophy. Line by line engagement with his texts is thus imperative in order to achieve an understanding of his philosophical objectives and criticisms. My dissertation offers as a paradigm of such reading a detailed exegesis and criticism of the opening five sections of Remarks on the Foundations of Mathematics. ;I briefly explore in some detail the origin and genesis of Wittgenstein's interest in rule-following, from its roots in the Tractatus's conception of logical syntax. This background illuminates Wittgenstein's later conception of logical truth and his criticisms of Frege's and Russell's arguments for logicism. To explore these criticisms, I focus on Wittgenstein' s discussions of the Frege-Russell definition of "Number" in "logical" terms. Placed in their appropriate philosophical and historical context, Wittgenstein's seemingly outrageous remarks about the nature of proof, mathematical logic and the foundations of mathematics do not simply betray his ignorance. Nor do they commit him to a revisionist attitude toward mathematical practice. Rather, they raise fundamental questions about the philosophical presuppositions lying behind attempts to bring particular mathematical results to bear in philosophy. (shrink)

Discussion of Wittgenstein's Tractatus is currently dominated by two opposing interpretations of the work: a metaphysical or realist reading and the 'resolute' reading of Diamond and Conant. Marie McGinn's principal aim in this book is to develop an alternative interpretative line, which rejects the idea, central to the metaphysical reading, that Wittgenstein sets out to ground the logic of our language in features of an independently constituted reality, but which allows that he aims to provide positive philosophical insights into (...) how language functions. McGinn takes as a guiding principle the idea that we should see Wittgenstein's early work as an attempt to eschew philosophical theory and to allow language itself to reveal how it functions. By this account, the aim of the work is to elucidate what language itself makes clear, namely, what is essential to its capacity to express thoughts that are true or false. However, the early Wittgenstein undertakes this descriptive project in the grip of a set of preconceptions concerning the essence of language that determine both how he conceives the problem and the approach he takes to the task of clarification. Nevertheless, the Tractatus contains philosophical insights, achieved despite his early preconceptions, that form the foundation of his later philosophy. -/- The anti-metaphysical interpretation that is presented includes a novel reading of the problematic opening sections of the Tractatus, in which the apparently metaphysical status of Wittgenstein's remarks is shown to be an illusion. The book includes a discussion of the philosophical background to the Tractatus, a comprehensive interpretation of Wittgenstein's early views of logic and language, and an interpretation of the remarks on solipsism. The final chapter is a discussion of the relation between the early and the later philosophy that articulates the fundamental shift in Wittgenstein's approach to the task of understanding how language functions and reveal the still more fundamental continuity in his conception of his philosophical task. (shrink)

Bertrand Russell's 1903 masterpiece "The Principles of Mathematics" places great emphasis on the need to separate propositions from psychological items such as thoughts. In 1919 Russell explicitly retracts this view, however, and defines propositions as "psychological occurrences". These psychological occurrences are held by Russell to be mental images. In this paper, I seek to explain this radical change of heart. I argue that Russell's re-psychologising of the proposition in 1919 can only be understood against the background of his (...) struggle with the problem of the unity of the proposition in earlier work. Once this is recognized, and the solution to the problem offered by the 1919 theory is appreciated, new light is also shed on Russell's naturalism. I go on to compare Russell's psychological "picture theory" with the vehemently anti-psychological picture theory of Wittgenstein's "Tractatus Logico-Philosophicus" and suggest that, once the background of the dispute is brought into clearer focus, Russell's position can be seen to have many advantages over its more celebrated rival. (shrink)

This chapter provides us with an appropriate way in to the logic of the Tractatus. Whitehead and Russell's Principia Mathematica was an attempt to vindicate “logicism”, the claim that truths of mathematics were disguised truths of logic. To overcome Russell's paradox, Russell had introduced the “theory of types”, stratifying sets, and with that the properties of sets. The resulting system was too weak to generate number theory without the addition of further axioms, including the “Axiom of (...) Reducibility”. This chapter examines the logical apparatus of language, “the logical constants” starting with what at one point Ludwig Wittgenstein calls his “fundamental thought” that the logical constants do not stand for anything. Wittgenstein stresses that operations are quite different from Russellian propositional functions. Wittgenstein's task is to introduce a single logical constant that is truth‐functional and in terms of which the whole of standard Fregean logic can be defined. (shrink)

Available from UMI in association with The British Library. Requires signed TDF. ;This work pursues certain themes in the rise and development of analytic philosophy, focussing in particular on the relationship between Aristotelian and Fregean logic, and the emergence and evolution of interest in questions of meaning, with reference to Frege and Wittgenstein. ;In the introductory chapter I consider the general relationship between analytic philosophy and the history of philosophy, suggesting the need for a closer examination of the 'paradigm shift' (...) from syllogistic theory to modern logic. In Part One I undertake this examination, with a chapter outlining the development, nature and purpose of Aristotle's logical theory, followed by a chapter describing Frege's achievement in the Begriffsschrift. My central concern here is with the emergence of a conception of sense--as grounded in the notion of logical equivalence. Aristotle, I argue, had no such conception; and it only emerged as Frege was led to reflect on the differences between his own logical system and traditional logic. ;In Part Two I consider the tightening of the 'bonds of sense' which Frege created, looking first at Frege's logicist project and then at Wittgenstein's Tractatus . I describe the epistemological motivation of Frege's work and the way that this led him to distinguish between Sinn and Bedeutung, and I argue that there is a fundamental tension in his mature thought between a semantic and an epistemic conception of sense. This tension was ironed out, however, in the early work of Wittgenstein, who rigorously thought through the implications of the semantic conception. I trace the evolution of his early ideas, in the framework provided by Frege and Russell, and explain how his saying/showing distinction shaped both his account of logic and his picture theory of language. ;In Part Three I discuss Wittgenstein's later philosophy, showing how it developed in response to problems he later perceived in both his own earlier work and the writings of Frege. I describe the shift from simples to samples, and explain his conception of meaning as use and his remarks on rule-following . In the final chapter, I consider the implications of our discussion of sense for the issues as to the nature of logic and the role of philosophical analysis. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:_Russell_ journal (home office): E:CPBRRUSSJOURTYPE2502\REVIEWS.252 : 2006-02-27 11:52 Reviews  HISTORY OF PHILOSOPHICAL ANALYSIS C P Philosophy / Purdue U. West Lafayette,  ,  @. Scott Soames. Philosophical Analysis in the Twentieth Century, Vol. : The Dawn of Analysis; Vol. : The Age of Meaning. Princeton: Princeton U. P., . Pp. xix, ; xxii, . . (hb), . (pb) for each volume. he last twenty years have seen (...) an explosion in books and papers on RusTsell ’s philosophy and its contemporary significance. There is good reason to think that this will continue as the contents of the Collected Papers are digested by Russell scholars and as more specialists contribute to the history of analytic philosophy more generally. Given all this good news, it is disconcerting to find a -page discussion of Russell, in a well-reviewed book by a first-rate philosopher, repeating many of the errors and misconceptions about Russell that scholars have worked so hard against. Soames’ discussion of Russell in the volumes under review is in fact so distressing that it alone compromises the book as a suitable introduction to the history of analytic philosophy. After briefly reviewing the outline of the two volumes, I discuss the errors concerning Russell, and conclude by drawing some lessons for Russell scholarship. Soames’ focus is on what he takes to be the most important and influential work of analytic philosophers, beginning with Moore’s Principia Ethica and ending in  with Kripke’s Naming and Necessity lectures. Kripke, in fact, marks the culmination of one of the two great achievements of analytic philos- _Russell_ journal (home office): E:CPBRRUSSJOURTYPE2502\REVIEWS.252 : 2006-02-27 11:52  Reviews ophy that Soames sees in this period: … the two most important achievements that have emerged from the analytic tradition in this period are (i) the recognition that philosophical speculation must be grounded in pre-philosophical thought, and (ii) the success achieved in understanding, and separating one from another, the fundamental methodological notions of logical consequence, logical truth, necessary truth, and apriori truth. (: xi) Moore is credited with the methodological innovation required by (i), as once we accept that what we think we know prior to philosophical reflection is a constraint on our epistemology, Moore’s response to scepticism, which Soames endorses, inevitably follows. But even Moore, and nearly every other figure that Soames discusses, is guilty of confusing necessity, analyticity and apriority. Soames endorses Kripke’s basic point that necessity is a metaphysical concept that can come apart from the epistemic notion of apriority and the semantic category of analytic propositions. Volume  repeats this charge several times, using it to undermine Moore’s views on ethics in Part One, Russell’s conception of analysis in Part Two, and logical positivism in Part Three. It is noteworthy that Soames takes Ayer’s Language, Truth and Logic as representative of logical positivism, ignoring contemporary scholarship on the Vienna Circle just as much as he ignores Russell scholarship. Part Four reconstructs Wittgenstein’s views in the Tractatus and Volume  ends with a discussion of Quine’s “Two Dogmas of Empiricism”. The second volume begins with a part on Wittgenstein’s Philosophical Investigations. This paves the way for Soames’ discussion of ordinary language philosophy in Parts Two and Three, which Soames sees as closely tied to the later Wittgenstein. Ryle’s Concept of Mind, Strawson’s early views on truth, Hare’s theory of goodness as well as Malcolm’s paradigm-case argument and Austin’s Sense and Sensibilia are investigated in these chapters. Part Four presents Grice’s theory of conversational implicature as the final nail in the coffin of ordinary language philosophy. In Part Five, Soames returns to Quine, this time discussing the ambitious arguments of Word and Object and the more general project of naturalized epistemology. Part Six articulates Davidson’s program for constructing a theory of meaning for natural languages along the lines of a Tarskistyle theory of truth, and Volume  ends with an extended discussion of the promise and limitations of Kripke’s Naming and Necessity. The material on Russell is entirely confined to four chapters in Volume : Chapter : “Logical Form, Grammatical Form, and the Theory of Descriptions ”, Chapter : “Logic and Mathematics: The Logicist Reduction”, Chapter : “Logical Constructions and... (shrink)

Bertrand Russell's work on the foundations of mathematics apparently played a decisive role in persuading Wittgenstein to abandon his aeronautical engineering studies in favor of philosophy. Wittgenstein's influence on Russell turned out to be profound as well: two years after they first met, Wittgenstein had delivered an objection to Russell's theory of judgment that was so devastating that it led to the abandonment of a major philosophical project of Russell's, leaving him reportedly “paralysed”. Two fundamental elements are (...) pivotal to the logicism of Russell's Principles of Mathematics: his conception of a proposition, and his commitment to the doctrine of the unrestricted variable. In addition to the ontological problems facing his theory of propositions, Russell also became convinced that his ontology of propositions contained fundamental logical problems, as he encountered recurring contradictions in his attempts to formalize a propositional logic that would comport with his philosophical theory. (shrink)

Ce texte discute certaines conclusions d’un article récent de F. Mülhölzer et vise à montrer que le logicisme russellien a les moyens de résister à la critique que Wittgenstein lui adresse dans la partie III des Remarques sur les fondements desmathématiques.This paper discusses some conclusions of a recent article from F.Mülhölzer. It aims at showing that Russell’s logicism has the means to overcome the criticisms Wittgenstein expounded in Remarks on the Foundations of Mathematics, part III.

For several terms at Cambridge in 1939, Ludwig Wittgenstein lectured on the philosophical foundations of mathematics. A lecture class taught by Wittgenstein, however, hardly resembled a lecture. He sat on a chair in the middle of the room, with some of the class sitting in chairs, some on the floor. He never used notes. He paused frequently, sometimes for several minutes, while he puzzled out a problem. He often asked his listeners questions and reacted to their replies. Many meetings were (...) largely conversation. These lectures were attended by, among others, D. A. T. Gasking, J. N. Findlay, Stephen Toulmin, Alan Turing, G. H. von Wright, R. G. Bosanquet, Norman Malcolm, Rush Rhees, and Yorick Smythies. Notes taken by these last four are the basis for the thirty-one lectures in this book. The lectures covered such topics as the nature of mathematics, the distinctions between mathematical and everyday languages, the truth of mathematical propositions, consistency and contradiction in formal systems, the logicism of Frege and Russell, Platonism, identity, negation, and necessary truth. The mathematical examples used are nearly always elementary. (shrink)

Several scholars have argued that Wittgenstein held the view that the notion of number is presupposed by the notion of one-one correlation, and that therefore Hume's principle is not a sound basis for a definition of number. I offer a new interpretation of the relevant fragments on philosophy of mathematics from Wittgenstein's Nachlass, showing that if different uses of ‘presupposition’ are understood in terms of de re and de dicto knowledge, Wittgenstein's argument against the Frege-Russell definition of number turns out (...) to be valid on its own terms, even though it depends on two epistemological principles logicist philosophers of mathematics may find too ‘constructivist’. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:September 27, 2008 (1:09 pm) G:\WPData\TYPE2801\russell 28,1 048RED.wpd russell: the Journal of Bertrand Russell Studies n.s. 28 (summer 2008): 69–93 The Bertrand Russell Research Centre, McMaster U. issn 0036-01631; online 1913-8032 eviews WITTGENSTEIN, CRITIC OF RUSSELL Russell Wahl English and Philosophy / Idaho State U Pocatello, id 83209, usa [email protected] Jérôme Sackur. Formes et faits: Analyse et théorie de la connaissance dans l’atomisme logique. Paris: Vrin, 2005. Pp. 313. (...) ¤31.00. isbn 978-2-7116-1722-7. Sackur joins several other recent French authors in writing penetrating analyses of Russell’s and Wittgenstein’s work. Forms and Factsz is an explication of the Tractatus, conducted by means of a contrast between Russell’s work and Wittgenstein’s. Sackur examines the interplay between Russell’s and Wittgenstein ’s versions of logical atomism, particularly, as Sackur puts it, between Russell’s original thought on the one hand, which provides the material for Wittgenstein’s position in the Tractatus, and Wittgenstein’s critique and development on the other, which, he says, provides a unity to Russell’s position (p. 12). While recognizing the inXuence of Russell on Wittgenstein’s project, Sackur in the end sees Wittgenstein as undermining and ultimately rejecting Russell’s logicism. He does not just repeat old claims about Wittgenstein, Russell and logical form. His analysis is very subtle and, perhaps unusually for a work on Wittgenstein which defends Wittgenstein over Russell, Formes et faitsz displays a surprisingly detailed knowledge of Russell’s work, before, during and after the period in question, 1913–19. Anyone who is interested in Russell’s and Wittgenstein ’s work during this period, and in particular Wittgenstein’s criticisms of Russell, will beneWt greatly from this book. There are detailed, well informed discussions of important points of disagreement, particularly Russell and Whitehead ’s account of succession and the ancestor relation, Wittgenstein’s remarks on operators, in particular the Nz operator, various remarks about the theory of types, and Wittgenstein’s now well known criticism of Russell’s theory of judgment. The subtitle of the work, “Analysis and the theory of knowledge in logical atomism”, reXects the points where Sackur sees the greatest disagreement between Russell and Wittgenstein. In the Wrst place, he sees them diTering in their attitude toward analysis, with Russell oTering translations of sentences into a September 27, 2008 (1:09 pm) G:\WPData\TYPE2801\russell 28,1 048RED.wpd 70 Reviews 1 For the Wrst point, see (among others) P.yM.yS. Hacker, Wittgenstein’s Place in Twentieth Century Philosophy (Oxford: Blackwell, 1996), pp. 69–70. For the second, see (again among others), David Pears, The False Prison (Oxford: Clarendon P., 1987), i: 63, where Pears says that Russell’s atomism was founded on the doctrine of “forced acquaintance”. 2 P.yJ. Hager, Continuity and Change in the Development of Russell’s Philosophy (Dordrecht: Klewer, 1994). perspicuous language and Wittgenstein oTering explications, or elucidations, of senses already in the sentences. Secondly, they diTer on the role that epistemology should play in logical atomism, with Wittgenstein criticizing the role of acquaintance in the development of logic. While both of these points have been made before,1 Sackur’s discussion is more thorough than previous studies and leads to some new interpretations of passages in the Tractatus. Sackur’s discussion of analysis is important to his argument. He is interested in a tension between what he sees as two aims of analysis in Wittgenstein’s work, one found in Tractatusz 3.2–3.3 and the other in Tractatusz 4.21–4.2211. He sees analysis in the Wrst group as revealing or elucidating the sense a proposition expresses, a sense understood and already there in the ordinary unanalyzed proposition, and analysis in the second group, driven by the theses of necessity and logical independence, as focusing on truth-functional analysis in terms of elementary propositions that consist of names referring to simple objects. He argues that these two aims don’t automatically go together and in fact appear to be at cross purposes. In the course of the Wrst chapter, the major distinction Sackur makes is between analysis... (shrink)

It is widely accepted that Russell wrongly took Wittgenstein to be concerned with the conditions required for an ideal language in his _Tractatus Logico-Philosophicus_. Given Russell’s relatively extensive communications with Wittgenstein, this misunderstanding is puzzling. I argue that Russell’s mistake rests on two prior assumptions for which he had some justification. First, communications with Wittgenstein were plausibly interpreted by Russell as confirming, rather than refuting, the belief that Wittgenstein shared with him the view that psychology, epistemology, and logic are interdependent. (...) Second, results from these areas in turn led Russell to the view that ordinary language is irredeemably vague and, as such, in need of replacement with an ideal language. In truth, however, Wittgenstein severed psychology and epistemology from his work and saw vagueness as a surface phenomenon only. (shrink)

The extra propositions which Wittgenstein added to Ramsey's copy of\nthe 'Tractatus' during their discussions in 1923 provide evidence,\nWrigley argues, that Wittgenstein's view of mathematics was quite\ndifferent from logicism. Contrary to this, Frascolla tries to prove\nthat the label 'no-classes logicism' tallies with the 'Tractarian'\nview of arithmetic.

The virtues of this book are many. It covers a large amount of often-neglected material introduced by Russell in his Principles of Mathematics and elsewhere, and by Russell and Whitehead in Princip...

In the Foundations of Mathematics, Ramsey attempted to amend Principia Mathematica’s logicism to meet serious objections raised against it. While Ramsey’s paper is well known, some questions concerning Ramsey’s motivations to write it and its reception still remain. This paper considers these questions afresh. First, an account is provided for why Ramsey decided to work on his paper instead of simply accepting Wittgenstein’s account of mathematics as presented in the Tractatus. Secondly, evidence is given supporting that Wittgenstein was (...) not moved by Ramsey’s objection against the Tractarian account of arithmetic, and a suggestion is made to explain why Wittgenstein reconsidered Ramsey’s account in the early thirties on several occasions. Finally, a reading is formulated to understand the basis on which Wittgenstein argues against Ramsey’s definition of identity in his 1927 letter to Ramsey. (shrink)

Wittgenstein gave a clearly erroneous refutation of Russell’s logicist project. The errors were ably pointed out by Mark Steiner. Nevertheless, I was motivated by Wittgenstein and Steiner to consider various ideas about the natural numbers. I ask which notations for natural numbers are ‘buck-stoppers’. For us it is the decimal notation and the corresponding verbal system. Based on the idea that a proper notation should be ‘structurally revelatory’, I draw various conclusions about our own concept of the natural numbers.

It is suggested that wittgenstein's picture theory of meaning is, In part a synthesis and resolution of the early metaphysics of moore and the theory of judgment held by russell about 1910. Moore's theory of the objective existence of concepts and their propositional role is considered. Russell's unsuccessful attempt at the problem of the false proposition is discussed. The ptm offers a more successful solution, Through the concept of logical form, Akin to the russellian concept of order. But this (...) solution still fails because it does not hold for ordinary language. (shrink)

Wittgenstein once wrote that 'The philosopher strives to find the liberating word, that is, the word that finally permits us to grasp what up until now has intangibly weighed down our consciousness'. Would Wittgenstein have been willing to describe the Tractatus as an attempt to find 'the liberating word'? This is the basic contention of this strikingly innovative study of the Tractatus. Matthew Ostrow argues that, far from seeking to offer a new theory in logic in the tradition (...) of Frege and Russell, Wittgenstein from the very beginning viewed all such endeavors as the ensnarement of thought. Providing a lucid and systematic analysis of the Tractatus, he argues that Wittgenstein's ultimate aim is to put an end to philosophy itself. He also highlights the intrinsic obstacles to any kind of interpretation that claims to summarize Wittgenstein's thought. (shrink)

Bertrand Russell was one of the best-known proponents of logicism: the theory that mathematics reduces to, or is an extension of, logic. Russell argued for this thesis in his 1903 The Principles of Mathematics and attempted to demonstrate it formally in Principia Mathematica (PM 1910–1913; with A. N. Whitehead). Russell later described his work as a further “regressive” step in understanding the foundations of mathematics made possible by the late 19th century “arithmetization” of mathematics and Frege’s logical definitions of (...) arithmetical concepts. The logical system of PM sought to improve on earlier attempts by solving the contradictions found in, e.g., Frege’s system, by employing a theory of types. In this article, I also consider and critically evaluate the most common objections to Russell’s logicism, including the claim that it is undermined by Gödel’s incompleteness results, and Putnam's charge of “if-thenism”. I suggest that if we are willing to accept a slightly revisionist account of what counts as a mathematical truth, these criticisms do not obviously refute Russell’s claim to have established that mathematical truths generally are a species of logical truth. (shrink)

Scott Soames’s two volume work Philosophical Analysis in the Twentieth Century1 won the American 2003 Award for Best Professional/Scholarly Book in Philosophy. It has been said to be ‘a marvellous introduction to analytic philosophy’, to deliver much ‘solid information on this dense and difficult subject’, and it has been predicted to become the standard history of twentieth-century analytic philosophy.2 Professor Soames writes clearly and candidly. At the beginning of each volume he delineates his objectives and leitmotivs. He is concerned with (...) the development of analytic philosophy from 1900 to 1975. He aims ‘to explain what the most important analytic philosophers thought and why they thought it’ (I, xi). His method is ‘to provide clear, focused and intense critical examinations of some of the most important and representative works of each major philosopher discussed. ... to provide enough detail to allow one to understand and properly evaluate the main philosophical developments of the period’ (I, xvii). A book with such laudable objectives, which holds out such high promises, and which is predicted to become the standard history of modern analytic philosophy merits careful study and considered judgement. The questions that I shall pose are dictated by the author’s aims and methods. (i) Does Soames provide an illuminating overview of analytic philosophy from 1900 to 1975? (ii) Does he correctly explain what the most important analytic philosophers thought and why they thought it? (iii) Does he select ‘some of the most important and representative works of each major philosopher discussed’? (iv) Does he properly evaluate the main developments of the period? II The broad picture Soames paints is as follows. Analytic philosophy commenced with Moore’s defence of common sense, and was continued by Russell, whose theory of descriptions, conception of analysis, logicism and logical atomism are recounted. He was followed by Wittgenstein, who argued in the Tractatus that philosophical problems arise solely from misunderstandings of language and defended the view that all necessary truths are a priori, analytic, and hence true in virtue of the meanings of words.. (shrink)

Many philosophers have puzzled over the nature of the logically simple objects, the “substance” of the world, in Wittgenstein’s Tractatus-logico-philosophicus (TLP). Such questions are misplaced because TLP is committed to the view that talk of such metaphysically problematical entities is part of the “ladder” that must be “thrown away” after one has climbed it. Further, TLP’s demotion of its logically simple objects to mere logical subjects requires an increased emphasis on the relations between these alleged objects. TLP’s account of (...) its logically simple objects is an application of Russell’s relational view of mathematical objects from his Principles of Mathematics applied to “reality”. Carnap develops an analogous relational or “structural” view of reality in The Logical Structure of the World. Despite important differences between them, these three philosophers can profitably be seen as replacing the traditional emphasis on substances with a relational model of reality. (shrink)

This paper investigates Wittgenstein's "notorious" criticism of Russell's theory of judgment. Instead of advancing a further new interpretation of it, though, I analyze and discuss some of the most promising readings of the Russell/Wittgenstein dispute put forward in the secondary literature; I aim to show that, despite their alleged reciprocal opposition, they cohere with each other because they are, at bottom, different ways of highlighting the same question. I then connect Wittgenstein's criticism of Russell to the account of the (...) nature of the proposition presented in the Tractatus, arguing that the latter can be seen as a natural development of the former. Finally, I make a quick connection between Wittgenstein's criticism of Russell and the conception of the world as the totality of facts, as presented in the Tractatus. (shrink)

It is proved that the basic framework of the premises and reasoning of Wittgenstein's “Tractatus Logico-philosophicus” corresponds quite well to the transcendental method (as formulated by H. Cohen). Whereas Kant’s philosophy proceeds from the fact of existence of mathematics and mathematised natural science and investigates their conditions of possibility, Wittgenstein proceeds from the fact that propositions of language describe reality and reveals the conditions of possibility of such descriptions. Kant, answering the question about the conditions of possibility of the (...) named sciences, comes to the idea of the transcendental subject and the distinction between the world of phenomena and the thing in itself. Wittgenstein's investigation of the conditions of possibility that the world is described by propositions leads to the assertion that both the world and language are together in logical space. The latter constitutes the a priori and transcendental condition of the possibility that language “reaches out” to reality. For both the theories - in the “Critique of Pure Reason” and in the “Tractatus Logico-Philosophicus” - the idea of the boundary is important. In the “Critique of Pure Reason” it is the boundary of possible experience and cognition, while in the “Tractatus Logico-Philosophicus” it is the boundary of what can be thought and expressed by meaningful propositions. Related to the different definitions of the boundary is the difference in the treatment of mathematised natural science. For “The Critique of Pure Reason” was created in the era of unconditional acceptance of Newtonian mechanics. And the “Tractatus Logico-Philosophicus” was created at the time of the crisis of the Newtonian paradigm and its replacement by other notions of time and space. However, the idea of boundary, which is present in both doctrines, determines closeness in the attitude towards metaphysics between the author of “The Critique of Pure Reason” and the author of “The Tractatus Logico-Philosophicus”. The study also shows that Wittgenstein did not follow logicism in his philosophy of mathematics. For him, both mathematical objects and propositions of logic are constructions. The conviction about the constructive character of mathematical and logical objects shows an affinity with the Kantian tradition in the philosophy of mathematics. (shrink)

It is fairly well known that Wittgenstein's criticisms of Russell's multiple‐relation theory of judgement had a devastating effect on the latter's philosophical enterprise. The exact nature of those criticisms however, and the explanation for the severity of their consequences, has been a source of confusion and disagreement amongst both Russell and Wittgenstein scholars. In this paper, I offer an interpretation of those criticisms which shows them to be consonant with Wittgenstein's general critique of Russell's conception of logic and (...) which serves to elucidate some of the notoriously enigmatic passages of the Tractatus. In particular, I seek to show the continuity of Wittgenstein's criticisms of the theory of judgement with his remarks on Russell's paradox and the theory of types. In addition, I place these issues in the context of Russell's own philosophical ambitions in order to reveal the deep divisions between the two over the nature of logical form and the analysis of propositional content. (shrink)

This book explores an important central thread that unifies Russell's thoughts on logic in two works previously considered at odds with each other, the Principles of Mathematics and the later Principia Mathematica. This thread is Russell's doctrine that logic is an absolutely general science and that any calculus for it must embrace wholly unrestricted variables. The heart of Landini's book is a careful analysis of Russell's largely unpublished "substitutional" theory. On Landini's showing, the substitutional theory reveals the (...) unity of Russell's philosophy of logic and offers new avenues for a genuine solution of the paradoxes plaguing Logicism. (shrink)

When in May 1930, the Council of Trinity College, Cambridge, had to decide whether to renew Wittgenstein's research grant, it turned to Bertrand Russell for an assessment of the work Wittgenstein had been doing over the past year. His verdict: "The theories contained in this new work . . . are novel, very original and indubitably important. Whether they are true, I do not know. As a logician who likes simplicity, I should like to think that they are not, but (...) from what I have read of them I am quite sure that he ought to have an opportunity to work them out, since, when completed, they may easily prove to constitute a whole new philosophy." "[ Philosophical Remarks ] contains the seeds of Wittgenstein's later philosophy of mind and of mathematics. Principally, he here discusses the role of indispensable in language, criticizing Russell's The Analysis of Mind . He modifies the Tractatus 's picture theory of meaning by stressing that the connection between the proposition and reality is not found in the picture itself. He analyzes generality in and out of mathematics, and the notions of proof and experiment. He formulates a pain/private-language argument and discusses both behaviorism and the verifiability principle. The work is difficult but important, and it belongs in every philosophy collection."--Robert Hoffman, Philosophy "Any serious student of Wittgenstein's work will want to study his Philosophical Remarks as a transitional book between his two great masterpieces. The Remarks is thus indispensible for anyone who seeks a complete understanding of Wittgenstein's philosophy."--Leonard Linsky, American Philosophical Association. (shrink)

Wittgenstein's remarks in his Tractatus on mathematics are quite obscure. Benacerraf and Putnam wrote, "In his Tractatus Loqico-Philosophicus, Wittgenstein maintained, following Russell and Frege, that mathematics was reducible to logic." On the other hand, Max Black claims, "Wittgenstein does not regard mathematics as reducible to logic, in the manner of Whitehead and Russell." I offer a detailed commentary upon Wittgenstein's remarks, concluding that his views most likely do not follow those of Frege and Russell. I reject a criticism (...) of Wittgenstein presented by Black but find severe shortcomings in the view I take Wittgenstein to be presenting. (shrink)

Russell's philosophy is rightly described as a programme of reduction of mathematics to logic. Now the theory of geometry developed in 1903 does not fit this picture well, since it is deeply rooted in the purely synthetic projective approach, which conflicts with all the endeavours to reduce geometry to analytical geometry. The first goal of this paper is to present an overview of this conception. The second aim is more far-reaching. The fact that such a theory of geometry was (...) sustained by Russell compels us to question the meaning of logicism: how is it possible to reconcile Russell's global reductionist standpoint with his local defence of the specificities of geometry? * This paper was first presented at the conference ‘Qu'est ce que la géométrie aux époques modernes et contemporaines?’, organized by the Universität Köln and the Archives Poincaré. I would like to thank Philippe Nabonnand for having enlightened me about the issues relative to projective geometry. I would like also to thank Nicholas Griffin, Brice Halimi, Bernard Linsky, Marco Panza, Ivahn Smadja for their helpful discussions. Many thanks also to the two anonymous referees for their useful suggestions. CiteULike Connotea Del.icio.us What's this? (shrink)

In 1913 Wittgenstein raised an objection to Russell’s multiple relation theory of judgment that eventually led Russell to abandon his theory. As he put it in the Tractatus, the objection was that “the correct explanation of the form of the proposition, ‘A makes the judgement p’, must show that it is impossible for a judgement to be a piece of nonsense. (Russell’s theory does not satisfy this requirement,” (5.5422). This objection has been widely interpreted to concern type restrictions on (...) the constituents of judgment. I argue that this interpretation is mistaken and that Wittgenstein’s objection is in fact a form of the problem of the unity of the proposition. (shrink)

Certain advocates of the so-called “neo-logicist” movement in the philosophy of mathematics identify themselves as “neo-Fregeans” (e.g., Hale and Wright), presenting an updated and revised version of Frege’s form of logicism. Russell’s form of logicism is scarcely discussed in this literature and, when it is, often dismissed as not really logicism at all (in light of its assumption of axioms of infinity, reducibility and so on). In this paper I have three aims: firstly, to identify more clearly (...) the primary meta-ontological and methodological differences between Russell’s logicism and the more recent forms; secondly, to argue that Russell’s form of logicism offers more elegant and satisfactory solutions to a variety of problems that continue to plague the neo-logicist movement (the bad company objection, the embarrassment of richness objection, worries about a bloated ontology, etc.); thirdly, to argue that neo-Russellian forms of logicism remain viable positions for current philosophers of mathematics. (shrink)

A detailed discussion of later Wittgenstein's take on the logicism of Frege and Russell. The focus is on Wittgenstein's views in his 1939 Lectures on the Foundations of Mathematics.

The supposition that Wittgenstein's Tractatus advances a certain metaphysics has given rise to a controversy over the ontological status of his Tractarian objects. It has been debated, for instance, whether these objects consist only of particulars or of both particulars and universals; whether they are physical, phenomenal, or phenomenological entities; and whether they correspond to Russell's objects of acquaintance or Kant's phenomena and substance. In this essay, I endorse Ishiguro's view that these objects, being formal concepts, are ontologically (...) neutral and thus are not identifiable with any ontological kind of entities. I elaborate on the coherence of this view with the propositional dependence of the meaning of Tractarian names. After showing why some arguments for ascribing a Russellian theory of meaning to these names do not work, I demonstrate why Ishiguro's account of Tractarian objects and names provides a better explanation of the unalterability of these objects. (shrink)

Influenced by the logical positivism of Moritz, and by Russell and Ramsey, Maslow's interpretation is that Wittgenstein's basic philosophy is a kind of Kantian phenomenology. In this, the first critical study of the Tractatus by an American philosopher, Maslow examines Wittgenstein's solipsism and mysticism, neglected areas of his philosophy.

We discuss a previously unnoticed resemblance between the theory of relations and predicates in The Philosophy of Logical Atomism [TPLA] by Russell and the theory of objects and names in the Tractatus Logico-Philosophicus [TLP] by Wittgenstein. Points of likeness are detected on three levels: ontology, syntax, and semantics. This analogy explains the prima facie similarities between the informal presentation of the theory of types in TPLA and the sections of the TLP devoted to this same topic. Eventually, we draw (...) some consequences concerning both sides of the analogy: for what concerns Russell, the contextual pertinence of this surprising fragment of Tractarian metaphysics and semantics is questioned on several grounds; about Wittgenstein, the interpreters who do not identify Tractarian objects with particulars are in a better position to make sense of the analogy here discovered. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Ideas 63.2 (2002) 345-363 [Access article in PDF] Tractatus Logico-Philosophicus: A "Poem" by Ludwig Wittgenstein David Rozema In the Fall term of 1911 the 22-year-old Ludwig Wittgenstein presented himself to the Cambridge philosopher of mathematics, Bertrand Russell, as a prospective student of philosophy. Wittgenstein had left off his studies as a promising young aeronautical engineer because, in the course of his engineering studies, (...) he had become obsessed with problems in the foundations of mathematics, the very problems Russell himself had tried to solve in his Principia Mathematica, a work he had completed just the year before. Wittgenstein, it seems, did not consider Russell's work to be the end of the matter, and so he had planned a book of his own on the subject. Although this work never materialized, Wittgenstein's interest in philosophy grew under his studies with Russell. And although Wittgenstein's early work in the philosophy of mathematics and logic were the initial impetus for his interest in philosophy, the events of his life soon led him back to the concerns which have always been central to the spirit of philosophy: How ought a man to live? What constitutes the good for man? What sort of life is a worthy or a beautiful one?It is likely that Wittgenstein's concern with these questions was heightened by what he saw in Russell, who followed a notoriously loose way of life outside of his academic studies. 1 If Wittgenstein ever wondered whether or not there was any connection between Russell's philosophical commitments and Russell's way of life, he never directly confronted Russell with the question. Russell was, after all, his benefactor, his mentor, and the source for his initial interest in philosophical problems. 2 Nevertheless, I maintain, Wittgenstein did confront Russell indirectly: he confronted the part of himself that was like Russell and, by attempting [End Page 345] to describe it in writing—the Tractatus Logico-Philosophicus—he discovered its barrenness. Insofar as such an act of writing is elucidatory of a person's soul in what and in how it pictures, it can legitimately be called a kind of poetry, for it bears the essential feature of all poetics: it serves to shape, through catharsis, the passions.Thus, though Wittgenstein neither considered the Tractatus to be a poem nor intentionally wrote it as a poem, it turned out to be a kind of poem in the sense that its form fits its content, both of which, in turn, accurately picture a specific form of life, and the passional result of this poetic construction is (and was, for Wittgenstein) a certain appropriate disdain for the form of life it pictures. This is what I suggest with regard to the Tractatus: it is most helpful and is best read as a poem. I will also try to show that this is how Wittgenstein himself came to think it should be read, which in turn helps to explain why he initially allowed and later continued to allow its publication.What does it mean to read the Tractatus as a poem picturing the form of life implicated by a philosophy like that of Russell's? In this poem Wittgenstein takes the fundamental principles of Russell's philosophy—that all words are signs that stand for objects or kinds of objects and that all meaningful propositions represent a possible arrangement of these objects—and draws out the full-blown implications of these principles. In the last section of the poem Wittgenstein shows the practical consequences of a life lived on these principles, concluding: "Whereof one cannot speak, thereof one must be silent." 3 The two-edged nature of this remark carries the force appropriate to the last line of a poem, for it is both descriptive and imagistic. It describes Wittgenstein's own thought when he realizes that, because of his Russellian philosophical commitments, he has nothing to say with regard to the good, the truth, and the beauty of life; and it is an image of the necessity that a... (shrink)

In A History of Philosophy in the Twentieth Century , Christian Delacampagne reviews the discipline's divergent and dramatic course and shows that its greatest figures, even the most unworldly among them, were deeply affected by events of their time. From Ludwig Wittgenstein, whose famous Tractatus was actually composed in the trenches during World War I, to Edmund Husserl and Martin Heidegger -- one who found himself barred from public life with Hitler's coming to power, the other a member of (...) the Nazi party who later refused to repudiate German war crimes. From Bertrand Russell, whose lifelong pacifism led him to turn from logic and mathematics to social and moral questions, and Jean-Paul Sartre, who made philosophy an occasion for direct and personal political engagement, to Rudolf Carnap, a committed socialist, and Karl Popper, a resolute opponent of Communism. From the Vienna Circle and the Frankfurt School to the contemporary work of philosophers as variously minded as Jacques Derrida, Jürgen Habermas, and Hilary Putnam. The thinking of these philosophers, and scores of others, cannot be understood without being placed in the context of the times in which they lived. (shrink)

In a review of Russell’s Principles from 1904, Edwin B. Wilson pays great attention to Peano’s work and that of his collaborators. His purpose was to make this work known in the USA where it “unfortunately is very little known and still less appreciated”. Wilson expands Russell’s well-known acknowledgement of Peano’s influence on his own development, seeing in Peano’s logic more than a new “mathematical tool”, describing Peano as a kind of proto-logicist, and defending him from Poincaré’s criticisms. Especially in (...) geometry, he vindicates several priority issues for Peano’s school with respect to Hilbert in the philosophy of the axiomatic method. (shrink)

Wittgenstein's 'middle period' is often seen as a transitional phase connecting his better-known early and later philosophies. The fifteen essays in this volume focus both on the distinctive character of his teaching and writing in the 1930s, and on its pivotal importance for an understanding of his philosophy as a whole. They offer wide-ranging perspectives on the central issue of how best to identify changes and continuities in his philosophy during those years, as well as on particular topics in the (...) philosophy of mind, religion, ethics, aesthetics, and the philosophy of mathematics. The volume will be valuable for all who are interested in this formative period of Wittgenstein's development. (shrink)

This book offers a detailed account and discussion of Ludwig Wittgenstein's philosophy of mathematics. In Part I, the stage is set with a brief presentation of Frege's logicist attempt to provide arithmetic with a foundation and Wittgenstein's criticisms of it, followed by sketches of Wittgenstein's early views of mathematics, in the Tractatus and in the early 1930s. Then, Wittgenstein's mature philosophy of mathematics is carefully presented and examined. Schroeder explains that it is based on two key ideas: the calculus (...) view and the grammar view. On the one hand, mathematics is seen as a human activity -- calculation -- rather than a theory. On the other hand, the results of mathematical calculations serve as grammatical norms. The following chapters explore the tension between those two key ideas and suggest a way in which it can be resolved. Finally, there are chapters analysing and defending Wittgenstein's provocative views on Hilbert's Formalism and the quest for consistency proofs and on Gödel's incompleteness theorems. (shrink)

In his highly perceptive, if underappreciated introduction to Wittgenstein’s Tractatus, Russell identifies a “lacuna” within Wittgenstein’s theory of number, relating specifically to the topic of transfinite number. The goal of this paper is two-fold. The first is to show that Russell’s concerns cannot be dismissed on the grounds that they are external to the Tractarian project, deriving, perhaps, from logicist ambitions harbored by Russell but not shared by Wittgenstein. The extensibility of Wittgenstein’s theory of number to the case of (...) transfinite cardinalities is, I shall argue, a desideratum generated by concerns intrinsic, and internal to Wittgenstein’s logical and semantic framework. Second, I aim to show that Wittgenstein’s theory of number as espoused in the Tractatus is consistent with Russell’s assessment, in that Wittgenstein meant to leave open the possibility that transfinite numbers could be generated within his system, but did not show explicitly how to construct them. To that end, I show how one could construct a transfinite number line using ingredients inherent in Wittgenstein’s system, and in accordance with his more general theories of number and of operations. (shrink)

I argue that three main interpretations of the aim of Russell’s early logicism in The Principles of Mathematics (1903) are mistaken, and propose a new interpretation. According to this new interpretation, the aim of Russell’s logicism is to show, in opposition to Kant, that mathematical propositions have a certain sort of complete generality which entails that their truth is independent of space and time. I argue that on this interpretation two often-heard objections to Russell’s logicism, deriving from (...) Gödel’s incompleteness theorem and from the non-logical character of some of the axioms of Principia Mathematica respectively, can be seen to be inconclusive. I then proceed to identify two challenges that Russell’s logicism, as presently construed, faces, but argue that these challenges do not appear unanswerable. (shrink)

Logicism is the view that mathematical truths are logical truths. But a logical truth is commonly thought to be one with a universally valid form. The form of “7 > 5” would appear to be the same as “4 > 6”. Yet one is a mathematical truth, and the other not a truth at all. To preserve logicism, we must maintain that the two either are different subforms of the same generic form, or that their forms are not (...) at all what they appear. The historical record shows that Russell pursued both these options, but that the struggle with the logical paradoxes pushed him away from the first kind of response and toward the second. An object cannot itself have a kind of inner logical complexity that makes a proposition have a different logical form merely in virtue of being about it, nor can their representatives in logical forms be single things different for different forms, at least not without postulating too many such objects and thereby creating Cantorian diagonal paradoxes. There are only apparent objects which are actually fragments of logical forms, different in different cases. (shrink)

What is at stake philosophically for Russell in espousing logicism? I argue that Russell's aims are chiefly epistemological and mathematical in nature. Russell develops logicism in order to give an account of the nature of mathematics and of mathematical knowledge that is compatible with what he takes to be the uncontroversial status of this science as true, certain and exact. I argue for this view against the view of Peter Hylton, according to which Russell uses logicism (...) to defend the unconditional truth of mathematics against various Idealist positions that treat mathematics as true only partially or only relative to a particular point of view. (shrink)

In this collection of recent and unpublished essays, leading analytic philosopher Scott Soames traces milestones in his field from its beginnings in Britain and Germany in the late nineteenth and early twentieth century, through its subsequent growth in the United States, up to its present as the world's most vigorous philosophical tradition. The central essay chronicles how analytic philosophy developed in the United States out of American pragmatism, the impact of European visitors and immigrants, the midcentury transformation of the Harvard (...) philosophy department, and the rapid spread of the analytic approach that followed. Another essay explains the methodology guiding analytic philosophy, from the logicism of Frege and Russell through Wittgenstein's linguistic turn and Carnap's vision of replacing metaphysics with philosophy of science. Further essays review advances in logic and the philosophy of mathematics that laid the foundation for a rigorous, scientific study of language, meaning, and information. Other essays discuss W.V.O. Quine, David K. Lewis, Saul Kripke, the Frege-Russell analysis of quantification, Russell's attempt to eliminate sets with his "no class theory," and the Quine-Carnap dispute over meaning and ontology. The collection then turns to topics at the frontier of philosophy of language. The final essays, combining philosophy of language and law, advance a sophisticated originalist theory of interpretation and apply it to U.S. constitutional rulings about due process. (shrink)

For different reasons, Husserl's original, thought-provoking ideas on the philosophy of logic and mathematics have been ignored, misunderstood, even despised, by analytic philosophers and phenomenologists alike, who have been content to barricade themselves behind walls of ideological prejudices. Yet, for several decades, Husserl was almost continuously in close professional and personal contact with those who created, reshaped and revolutionized 20th century philosophy of mathematics, logic, science and language in both the analytic and phenomenological schools, people whom those other makers of (...) 20th century philosophy, Russell, Frege, Wittgenstein and their followers, rarely, if ever, met. Independently of them, Husserl offered alternatives to the well-trodden paths of logicism, nominalism, formalism and intuitionism. He presented a well-articulated, thoroughly argued case for logic as an objective science, but was not philosophically naive to the point of not seeing the role of subjectivity in shaping the sense of the reality facing objective science. Given the preeminent role that philosophy of logic and philosophy of mathematics have played in transforming the way philosophy has been done since Husserl's time, and given the depth of his insights and his obvious expertise in those fields, his ideas need to be integrated into present-day, mainstream philosophy. Here, philosopher Claire Ortiz Hill and mathematician-philosopher Jairo da Silva offer a wealth of interesting insights intended to subvert the many mistaken idees recues about the development of Husserl's thought and reestablish broken ties between it and philosophy now. (shrink)

This article aims first at showing that Russell's general doctrine according to which all mathematics is deducible ‘by logical principles from logical principles’ does not require a preliminary reduction of all mathematics to arithmetic. In the Principles, mechanics (part VII), geometry (part VI), analysis (part IV–V) and magnitude theory (part III) are to be all directly derived from the theory of relations, without being first reduced to arithmetic (part II). The epistemological importance of this point cannot be overestimated: (...) class='Hi'>Russell's logicism does not only contain the claim that mathematics is no more than logic, it also contains the claim that the differences between the various mathematical sciences can be logically justified—and thus, that, contrary to the arithmetization stance, analysis, geometry and mechanics are not merely outgrowths of arithmetic. The second aim of this article is to set out the neglected Russellian theory of quantity. The topic is obviously linked with the first, since the mere existence of a doctrine of magnitude, in a work dated from 1903, is a sign of a distrust vis-à-vis the arithmetization programme. After having shown that, despite the works of Cantor, Dedekind and Weierstrass, many mathematicians at the end of the 19th Century elaborated various axiomatic theories of the magnitude, I will try to define the peculiarity of the Russellian approach. I will lay stress on the continuity of the logicist's thought on this point: Whitehead, in the Principia, deepens and generalizes the first Russellian 1903 theory. (shrink)

This book offers a detailed account and discussion of Ludwig Wittgenstein's philosophy of mathematics. In Part I, the stage is set with a brief presentation of Frege's logicist attempt to provide arithmetic with a foundation and Wittgenstein's criticisms of it, followed by sketches of Wittgenstein's early views of mathematics, in the Tractatus and in the early 1930s. Then (in Part II), Wittgenstein's mature philosophy of mathematics (1937-44) is carefully presented and examined. Schroeder explains that it is based on two (...) key ideas: the calculus view and the grammar view. On the one hand, mathematics is seen as a human activity - calculation - rather than a theory. On the other hand, the results of mathematical calculations serve as grammatical norms. The following chapters (on mathematics as grammar; rule-following; conventionalism; the empirical basis of mathematics; the role of proof) explore the tension between those two key ideas and suggest a way in which it can be resolved. Finally, there are chapters analysing and defending Wittgenstein's provocative views on Hilbert's Formalism and the quest for consistency proofs and on Gödel's incompleteness theorems. (shrink)