This volume aims to establish the starting point for the development, evaluation and appraisal of the phenomenology of mathematics.

When it comes to Wittgenstein's philosophy of mathematics, even sympathetic admirers are cowed into submission by the many criticisms of influential authors in that field. They say something to the effect that Wittgenstein does not know enough about or have enough respect for mathematics, to take him as a serious philosopher of mathematics. They claim to catch Wittgenstein pooh-poohing the modern set-theoretic extensional conception of a real number. This article, however, will show that Wittgenstein's criticism is well grounded. A real (...) number, as an 'extension', is a homeless fiction; 'homeless' in that it neither is supported by anything nor supports anything. The picture of a real number as an 'extension' is not supported by actual practice in calculus; calculus has nothing to do with 'extensions'. The extensional, set-theoretic conception of a real number does not give a foundation for real analysis, either. The so-called complete theory of real numbers, which is essentially an extensional approach, does not define (in any sense of the word) the set of real numbers so as to justify their completeness, despite the common belief to the contrary. The only correct foundation of real analysis consists in its being 'existential axiomatics'. And in real analysis, as existential axiomatics, a point on the real line need not be an 'extension'. (shrink)

A dialetheia is a sentence, A, such that both it and its negation, ¬A, are true (we shall talk of sentences throughout this entry; but one could run the definition in terms of propositions, statements, or whatever one takes as her favourite truth-bearer: this would make little difference in the context). Assuming the fairly uncontroversial view that falsity just is the truth of negation, it can equally be claimed that a dialetheia is a sentence which is both true and false.

The Knowability Paradox purports to show that the controversial but not patently absurd hypothesis that all truths are knowable entails the implausible conclusion that all truths are known. The notoriety of this argument owes to the negative light it appears to cast on the view that there can be no verification-transcendent truths. We argue that it is overly simplistic to formalize the views of contemporary verificationists like Dummett, Prawitz or Martin-Löf using the sort of propositional modal operators which are employed (...) in the original derivation of the Paradox. Instead we propose that the central tenet of verificationism is most accurately formulated as follows: if φ is true, then there exists a proof of φ Building on the work of Artemov (Bull Symb Log 7(1): 1-36, 2001), a system of explicit modal logic with proof quantifiers is introduced to reason about such statements. When the original reasoning of the Paradox is developed in this setting, we reach not a contradiction, but rather the conclusion that there must exist non-constructed proofs. This outcome is evaluated relative to the controversy between Dummett and Prawitz about proof existence and bivalence. (shrink)

Mathematical fictionalism (or as I'll call it, fictionalism) is best thought of as a reaction to mathematical platonism. Platonism is the view that (a) there exist abstract mathematical objects (i.e., nonspatiotemporal mathematical objects), and (b) our mathematical sentences and theories provide true descriptions of such objects. So, for instance, on the platonist view, the sentence ‘3 is prime’ provides a straightforward description of a certain object—namely, the number 3—in much the same way that the sentence ‘Mars is red’ provides a (...) description of Mars. But whereas Mars is a physical object, the number 3 is (according to platonism) an abstract object. And abstract objects, platonists tell us, are wholly nonphysical, nonmental, nonspatial, nontemporal, and noncausal. Thus, on this view, the number 3 exists independently of us and our thinking, but it does not exist in space or time, it is not a physical or mental object, and it does not enter into causal relations with other objects. This view has been endorsed by Plato, Frege (1884, 1893-1903, 1919), Gödel (1964), and in some of their writings, Russell (1912) and Quine (1948, 1951), not to mention numerous more recent philosophers of mathematics, e.g., Putnam (1971), Parsons (1971), Steiner (1975), Resnik (1997), Shapiro (1997), Hale (1987), Wright (1983), Katz (1998), Zalta (1988), and Colyvan (2001). (shrink)

Los estudios sociales de la ciencia y, en particular, la sociología del conocimiento científico, han criticado las filosofías de la ciencia por fundarse en epistemologías centradas en el individuo como sujeto de conocimiento, en detrimento de análisis que den cuenta de las comunidades científicas; una explicación del conocimiento científico centrada en el individuo es incapaz de dar cuenta de las tradiciones y actual estado de la ciencia. Este artículo sostiene, sin embargo, que la SSK no diluye el sujeto en la (...) comunidad científica sino que, por el contrario, permite definirlo en su interacción con otros, en la interpretación y organización del conocimiento de la naturaleza. Esto se hace recurriendo a la noción de identidad del sujeto en tanto practicante de la ciencia. (shrink)

A dialetheia is a sentence, A, such that both it and its negation, A, are true (we shall talk of sentences throughout this entry; but one could run the definition in terms of propositions, statements, or whatever one takes as her favourite truth bearer: this would make little difference in the context). Assuming the fairly uncontroversial view that falsity just is the truth of negation, it can equally be claimed that a dialetheia is a sentence which is both true and (...) false. (shrink)

In this paper, elementary but hitherto overlooked connections are established between Wittgenstein's remarks on mathematics, written during his transitional period, and free-variable finitism. After giving a brief description of theTractatus Logico-Philosophicus on quantifiers and generality, I present in the first section Wittgenstein's rejection of quantification theory and his account of general arithmetical propositions, to use modern jargon, as claims (as opposed to statements). As in Skolem's primitive recursive arithmetic and Goodstein's equational calculus, Wittgenstein represented generality by the use of free (...) variables. This has the effect that negation of unbounded universal and existential propositions cannot be expressed. This is claimed in the second section to be the basis for Wittgenstein's criticism of the universal validity of the law of excluded middle. In the last section, there is a brief discussion of Wittgenstein's remarks on real numbers. These show a preference, in line with finitism, for a recursive version of the continuum. (shrink)

This paper offers an interpretation of Poincaré's conventionalism, distinguishing it from the Duhem–Quine thesis, on the one hand, and, on the other, from the logical positivist understanding of conventionalism as a general account of necessary truth. It also confronts Poincaré's conventionalism with some counter-arguments that have been influential: Einstein's (general) relativistic argument, and the linguistic rejoinders of Quine and Davidson. In the first section, the distinct roles played by the inter-translatability of different geometries, the inaccessibility of space to direct observation, (...) and general holistic considerations are identified. Together, they form a constructive argument for conventionalism that underscores the impact of fact on convention. The second section traces Poincaré's influence on the general theory of relativity and Einstein's ensuing ambivalence toward Poincaré. Lastly, it is argued that neither Quine nor Davidson has met the conventionalist challenge. (shrink)

This paper discusses the coverage of ordinary language philosophy in Scott Soames' Philosophical Analysis in the Twentieth Century. After praising the book's virtues, I raise three points where I dissent from Soames' take on the history. First, I suggest that there is more to ordinary language philosophy than the rather implausible version of it that Soames sees to have been destroyed by Grice. Second, I argue that confusions between analyticity, necessity and priority are less important to the ordinary language period (...) than Soames takes them to be. Finally, I claim that Soames' criticisms of Ryle turn in part on attributing reductionist positions to Ryle that Ryle did not hold. (shrink)

We document the influence on programming language semantics of the Platonism/formalism divide in the philosophy of mathematics.

Philosophy strives to give us a firmer hold on our concepts. But what about their hold on us? Why place ourselves under the sway of a concept and grant it the authority to shape our thought and conduct? Another conceptualization would carry different implications. What makes one way of thinking better than another? This book develops a framework for concept appraisal. Its guiding idea is that to question the authority of concepts is to ask for reasons of a special kind: (...) reasons for concept use, which tell us which concepts to adopt, adhere to, or abandon, thereby shoring up—or undercutting—the reasons for action and belief that guide our deliberations. Traditionally, reasons for concept use have been sought either in timeless rational foundations or in concepts’ inherent virtues, such as precision and consistency. Against this, the book advances two main claims: that we find reasons for concept use in the conceptual needs we discover when we critically distance ourselves from a concept by viewing it from the autoethnographic stance; and that sometimes, concepts that conflict, or exhibit other vices such as vagueness or superficiality, are just what we need. By considering not what concepts are absolutely best, but what concepts we now need, we can reconcile ourselves to the contingency of our concepts, determine the proper place of efforts to tidy up thought, and adjudicate between competing conceptions of things—even things as contested as liberty or free will. A needs-based approach separates helpful clarification from hobbling tidy-mindedness, and authoritative definition from conceptual gerrymandering. (shrink)

This volume draws connections between Wittgenstein's philosophy and the work of Saul Kripke, especially his Naming and Necessity. Saul Kripke is regarded as one of the foremost representatives of contemporary analytic philosophy. His most important contributions include the strict distinction between metaphysical and epistemological questions, the introduction of the notions of contingent a priori truth and necessary a posteriori truth and original accounts of names, descriptions, identity, necessity and realism. The chapters in this book elucidate the relevant connections between Kripke's (...) work and Wittgenstein, specifically concerning the standard meter, contingent apriori and rule-following. The contributions shed light on how Kripke's philosophical outlook was influenced by Wittgenstein, and how mainstream analytic philosophy and Wittgensteinian philosophy can fruitfully engage with one another. Kripke and Wittgenstein will be of interest to philosophers working on Wittgenstein, Kripke, and the history of analytic philosophy. (shrink)

This chapter argues that several aspects of Bernard Williams’s style, methodology, and metaphilosophy can be read as evolving dialectically out of Wittgenstein’s own. After considering Wittgenstein as a stylistic influence on Williams, especially as regards ideals of clarity, precision, and depth, Williams’s methodological debt to Wittgenstein is examined, in particular his anthropological interest in thick concepts and their point. The chapter then turns to Williams’s explicit association, in the 1990s, with a certain form of Wittgensteinianism, which he called ‘Left Wittgensteinianism’. (...) It is shown how this is not a sudden conversion, but the direct product of Williams’s longstanding critical engagement with Wittgenstein’s methodology and metaphilosophy: Williams arrives at this position by envisaging a Wittgensteinianism that thinks in concrete sociohistorical terms, embraces genuine explanation, and relinquishes its insistence on the purity of philosophy. When properly understood, moreover, this critique turns out to be continuous with Williams’s advocacy of a conception of philosophy as a humanistic discipline. Finally, it is shown that Williams inherits from Wittgenstein a certain understanding of how philosophy can help us to live, in particular the therapeutic ambition to liberate us from distortions in our self-understanding by assembling reminders. (shrink)

Once upon a time, logical conventionalism was the most popular philosophical theory of logic. It was heavily favored by empiricists, logical positivists, and naturalists. According to logical conventionalism, linguistic conventions explain logical truth, validity, and modality. And conventions themselves are merely syntactic rules of language use, including inference rules. Logical conventionalism promised to eliminate mystery from the philosophy of logic by showing that both the metaphysics and epistemology of logic fit into a scientific picture of reality. For naturalists of all (...) stripes, this was paradise. Alas, paradise was lost. By the end of the twentieth century, logical conventionalism had been almost universally abandoned. But more recently, it has been revitalized. This chapter provides an overview of logical conventionalism and its history, clears away misunderstandings, and briefly responds to both the canonical objections to conventionalism (from Quine, Dummett, Boghossian, and many others) as well as new objections (from Field and Golan). From paradise lost, to paradise regained: logical conventionalism is back. (shrink)

This volume announces a new era in the philosophy of God. Many of its contributions work to create stronger links between the philosophy of God, on the one hand, and mathematics or metamathematics, on the other hand. It is about not only the possibilities of applying mathematics or metamathematics to questions about God, but also the reverse question: Does the philosophy of God have anything to offer mathematics or metamathematics? The remaining contributions tackle stereotypes in the philosophy of religion. The (...) volume includes 35 contributions. It is divided into nine parts: 1. Who Created the Concept of God; 2. Omniscience, Omnipotence, Timelessness and Spacelessness of God; 3. God and Perfect Goodness, Perfect Beauty, Perfect Freedom; 4. God, Fundamentality and Creation of All Else; 5. Simplicity and Ineffability of God; 6. God, Necessity and Abstract Objects; 7. God, Infinity, and Pascal's Wager; 8. God and (Meta-)Mathematics; and 9. God and Mind. (shrink)

The distinction between the a priori and the a posteriori is an old and influential one. But both the distinction itself and the crucial notion of a priori knowledge face powerful philosophical challenges. Many philosophers worry that accepting the a priori is tantamount to accepting epistemic magic. In contrast, this Element argues that the a priori can be formulated clearly, made respectable, and used to do important epistemological work. The author's conception of the a priori and its role falls short (...) of what some historical proponents of the notion may have hoped for, but it allows us to accept and use the notion without abandoning either naturalism or empiricism, broadly understood. This Element argues that we can accept and use the a priori without magic. (shrink)

How does Quine fare in the first decades of the twenty-first century? In this paper I examine a cluster of Quinean theses that, I believe, are especially fruitful in meeting some of the current challenges of epistemology and ontology. These theses offer an alternative to the traditional bifurcations of truth and knowledge into factual and conceptual-pragmatic-conventional, the traditional conception of a foundation for knowledge, and traditional realism. To make the most of Quine’s ideas, however, we have to take an active (...) stance: accept some of his ideas and reject others, sort different versions of the relevant ideas, sharpen or revise some of the ideas, connect them with new, non-Quinean ideas, and so on. As a result the paper pits Quine against Quine, in an attempt to identify those Quinean ideas that have a lasting value and sketch potential developments. (shrink)

In this work we argue that there is no strong demarcation between pure and applied mathematics. We show this first by stressing non-deductive components within pure mathematics, like axiomatization and theory-building in general. We also stress the “purer” components of applied mathematics, like the theory of the models that are concerned with practical purposes. We further show that some mathematical theories can be viewed through either a pure or applied lens. These different lenses are tied to different communities, which endorse (...) different evaluative standards for theories. We evaluate the distinction between pure and applied mathematics from a late Wittgensteinian perspective. We note that the classical exegesis of the later Wittgenstein’s philosophy of mathematics, due to Maddy, leads to a clear-cut but misguided demarcation. We then turn our attention to a more niche interpretation of Wittgenstein by Dawson, which captures aspects of the aforementioned distinction more accurately. Building on this newer, maverick interpretation of the later Wittgenstein’s philosophy of mathematics, and endorsing an extended notion of meaning as use which includes social, mundane uses, we elaborate a fuzzy, but more realistic, demarcation. This demarcation, relying on family resemblance, is based on how direct and intended technical applications are, the kind of evaluative standards featured, and the range of rhetorical purposes at stake. (shrink)

This paper addresses a problem about aesthetic normativity raised by Kant. Can aesthetic experiences be appropriate or inappropriate to their objects? And, if so, how is that possible given that, according to Kant, aesthetic experience is not objective? Kant thought the answer to the first question was yes. But his official answer to the second question, in terms of the free play of the faculties, is obscure. The paper offers a clearer answer, inspired by Kant, which invokes Wittgenstein’s notion of (...) “knowing how to go on.” Aesthetic normativity is problematic only on the assumption that claims to the normativity of one’s responses to things must be based on the recognition of objective properties of those things. However, Wittgenstein’s discussion shows that we need not accept that assumption. There can be legitimate claims to the normativity of one’s responses which do not rely on those responses’ reflecting appreciation of objective facts. (shrink)

Since the publication of the Remarks on the Foundations of Mathematics, Wittgenstein’s interpreters have endeavored to reconcile his general constructivist/anti-realist attitude towards mathematics with his confessed anti-revisionary philosophy. In this article, the author revisits the issue and presents a solution. The basic idea consists in exploring the fact that the so-called “non-constructive results” could be interpreted so that they do not appear non-constructive at all. The author substantiates this solution by showing how the translation of mathematical results, given by the (...) possibility of translation between logics, can be seen as a tool for partially implementing the solution Wittgenstein had in mind. (shrink)

In this paper, I offer a close reading of Wittgenstein's remarks on inconsistency, mostly as they appear in the Lectures on the Foundations of Mathematics. I focus especially on an objection to Wittgenstein's view given by Alan Turing, who attended the lectures, the so-called ‘falling bridges’-objection. Wittgenstein's position is that if contradictions arise in some practice of language, they are not necessarily fatal to that practice nor necessitate a revision of that practice. If we then assume that we have adopted (...) a paraconsistent logic, Wittgenstein's answer to Turing is that if we run into trouble building our bridge, it is either because we have made a calculation mistake or our calculus does not actually describe the phenomenon it is intended to model. The possibility of either kind of error is not particular to contradictions nor inconsistency, and thus contradictions do not have any special status as a thing to be avoided. (shrink)

The goal of the paper is to develop and propose a general model of the state space of AI. Given the breathtaking progress in AI research and technologies in recent years, such conceptual work is of substantial theoretical interest. The present AI hype is mainly driven by the triumph of deep learning neural networks. As the distinguishing feature of such networks is the ability to self-learn, self-learning is identified as one important dimension of the AI state space. Another dimension is (...) recognized as generalization, the possibility to go over from specific to more general types of problems. A third dimension is semantic grounding. Our overall analysis connects to a number of known foundational issues in the philosophy of mind and cognition: the blockhead objection, the Turing test, the symbol grounding problem, the Chinese room argument, and use theories of meaning. It shall finally be argued that the dimension of grounding decomposes into three sub-dimensions. And the dimension of self-learning turns out as only one of a whole range of “self-x-capacities” (based on ideas of organic computing) that span the self-x-subspace of the full AI state space. (shrink)

PLEASE NOTE: This is the corrected 2nd eBook edition, 2021. ●●●●● _Critique of Impure Reason_ has now also been published in a printed edition. To reduce the otherwise high price of this scholarly, technical book of nearly 900 pages and make it more widely available beyond university libraries to individual readers, the non-profit publisher and the author have agreed to issue the printed edition at cost. ●●●●● The printed edition was released on September 1, 2021 and is now available through (...) all booksellers, including Barnes & Noble, Amazon, and brick-and-mortar bookstores under ISBN 978-0-578-88646-6. ●●●●● In light of the length of this book, readers who would like to have a more detailed description of the book's objectives and method may find it helpful to read the detailed and clearly written Wikipedia entry about this work: From the Wikipedia search page, use the search phrase "Critique of Impure Reason". At least at the time of this writing (11/29/2021), the Wikipedia entry is well-researched and accurate. ●●●●● In addition, a "Primer on Bartlett's CRITIQUE OF IMPURE REASON" has been made available by the author. It is available under its title through PhilPapers and other philosophy online archives. ●●●●● COMMENDATIONS OF THIS WORK, from the back cover of the published edition: ●●●●● “I admire its range of philosophical vision.” – Nicholas Rescher, Distinguished University Professor of Philosophy, University of Pittsburgh, author of more than 100 books. ●●●●● “Bartlett’s _Critique of Impure Reason_ is an impressive, bold, and ambitious work. Careful scholarship is balanced by original analyses that lead the reader to recognize the limits of meaning, knowledge, and conceptual possibility. The work addresses a host of traditional philosophical problems, among them the nature of space, time, causality, consciousness, the self, other minds, ontology, free will and determinism, and others. The book culminates in a fascinating and profound new understanding of relativity physics and quantum theory.” – Gerhard Preyer, Professor of Philosophy, Goethe-University, Frankfurt am Main, Germany, author of many books including _Concepts of Meaning_, _Beyond Semantics and Pragmatics_, _Intention and Practical Thought_, and _Contextualism in Philosophy_. ●●●●● “[This work’s] goal is of a unique and difficult species: Dr. Bartlett seeks to develop a formal logical calculus on the basis of transcendental philosophical arguments; in fact, he hopes that this calculus will be the formal expression of the transcendental foundation of knowledge.... I consider Dr. Bartlett’s work soundly conceived and executed with great skill.” – C. F. von Weizsäcker, philosopher and physicist, former Director, Max-Planck-Institute, Starnberg, Germany. ●●●●● “Bartlett has written an American “Prolegomena to All Future Metaphysics.” He aims rigorously to eliminate meaningless assertions, reach bedrock, and place philosophy on a firm foundation that will enable it, like science and mathematics, to produce lasting results that generations to come can build on. This is a great book, the fruit of a lifetime of research and reflection, and it deserves serious attention.” — Martin X. Moleski, former Professor, Canisius College, Buffalo, NY, studies of scientific method, the presuppositions of thought, and the self-referential nature of epistemology. ●●●●● “Bartlett has written a book on what might be called the underpinnings of philosophy. It has fascinating depth and breadth, and is all the more striking due to its unifying perspective based on the concepts of reference and self-reference.” – Don Perlis, Professor of Computer Science, University of Maryland, author of numerous publications on self-adjusting autonomous systems and philosophical issues concerning self-reference, mind, and consciousness. ●●●●● ●●●●● The _Critique of Impure Reason: Horizons of Possibility and Meaning_ comprises a major and important contribution to philosophy. Thanks to the generosity of its publisher, this massive 885-page volume has been published as a free open access eBook (3.75MB) as well as an open access printed edition. It inaugurates a revolutionary paradigm shift in philosophical thought by providing compelling and long-sought-for solutions to a wide range of philosophical problems. In the process, the work fundamentally transforms the way in which the concepts of reference, meaning, and possibility are understood. The book includes a Foreword by the celebrated German philosopher and physicist Carl Friedrich von Weizsäcker. ●●●●● In Kant’s _Critique of Pure Reason_ we find an analysis of the preconditions of experience and of knowledge. In contrast, but yet in parallel, the new _Critique_ focuses upon the ways—unfortunately very widespread and often unselfconsciously habitual—in which many of the concepts that we employ _conflict_ with the very preconditions of meaning and of knowledge. ●●●●● This is a book about the boundaries of frameworks and about the unrecognized conceptual confusions in which we become entangled when we attempt to transgress beyond the limits of the possible and meaningful. We tend either not to recognize or not to accept that we all-too-often attempt to trespass beyond the boundaries of the frameworks that make knowledge possible and the world meaningful. ●●●●● The _Critique of Impure Reason_ proposes a bold, ground-breaking, and startling thesis: that a great many of the major philosophical problems of the past can be solved through the recognition of a viciously deceptive form of thinking to which philosophers as well as non-philosophers commonly fall victim. For the first time, the book advances and justifies the criticism that a substantial number of the questions that have occupied philosophers fall into the category of “impure reason,” violating the very conditions of their possible meaningfulness. ●●●●● The purpose of the study is twofold: first, to enable us to recognize the boundaries of what is referentially forbidden—the limits beyond which reference becomes meaningless—and second, to avoid falling victims to a certain broad class of conceptual confusions that lie at the heart of many major philosophical problems. As a consequence, the boundaries of _possible meaning_ are determined. ●●●●● Bartlett, the author or editor of more than 20 books, is responsible for identifying this widespread and delusion-inducing variety of error, _metalogical projection_. It is a previously unrecognized and insidious form of erroneous thinking that undermines its own possibility of meaning. It comes about as a result of the pervasive human compulsion to seek to transcend the limits of possible reference and meaning. ●●●●● Based on original research and rigorous analysis combined with extensive scholarship, the _Critique of Impure Reason_ develops a self-validating method that makes it possible to recognize, correct, and eliminate this major and pervasive form of fallacious thinking. In so doing, the book provides at last provable and constructive solutions to a wide range of major philosophical problems. ●●●●● CONTENTS AT A GLANCE ▪▪▪▪▪ Preface ▪▪▪▪▪ Foreword by Carl Friedrich von Weizsäcker ▪▪▪▪▪ Acknowledgments ▪▪▪▪▪ Avant-propos: A philosopher’s rallying call ▪▪▪▪▪ Introduction ▪▪▪▪▪ A note to the reader ▪▪▪▪▪ A note on conventions ▪▪▪▪▪ ▪▪▪▪▪ ▪▪▪▪▪ PART I ▪▪▪▪▪ ▪▪▪▪▪ WHY PHILOSOPHY HAS MADE NO PROGRESS AND HOW IT CAN ▪▪▪▪▪ 1 Philosophical-psychological prelude ▪▪▪▪▪ 2 Putting belief in its place: Its psychology and a needed polemic ▪▪▪▪▪ 3 Turning away from the linguistic turn: From theory of reference to metalogic of reference ▪▪▪▪▪ 4 The stepladder to maximum theoretical generality ▪▪▪▪▪ ▪▪▪▪▪ ▪▪▪▪▪ PART II ▪▪▪▪▪ THE METALOGIC OF REFERENCE ▪▪▪▪▪ A New Approach to Deductive, Transcendental Philosophy ▪▪▪▪▪ 5 Reference, identity, and identification ▪▪▪▪▪ 6 Self-referential argument and the metalogic of reference ▪▪▪▪▪ 7 Possibility theory ▪▪▪▪▪ 8 Presupposition logic, reference, and identification ▪▪▪▪▪ 9 Transcendental argumentation and the metalogic of reference ▪▪▪▪▪ 10 Framework relativity ▪▪▪▪▪ 11 The metalogic of meaning ▪▪▪▪▪ 12 The problem of putative meaning and the logic of meaninglessness ▪▪▪▪▪ 13 Projection ▪▪▪▪▪ 14 Horizons ▪▪▪▪▪ 15 De-projection ▪▪▪▪▪ 16 Self-validation ▪▪▪▪▪ 17 Rationality: Rules of admissibility ▪▪▪▪▪ ▪▪▪▪▪ ▪▪▪▪▪ PART III ▪▪▪▪▪ PHILOSOPHICAL APPLICATIONS OF THE METALOGIC OF REFERENCE ▪▪▪▪▪ Major Problems and Questions of Philosophy and the Philosophy of Science ▪▪▪▪▪ 18 Ontology and the metalogic of reference ▪▪▪▪▪ 19 Discovery or invention in general problem-solving, mathematics, and physics ▪▪▪▪▪ 20 The conceptually unreachable: “The far side” ▪▪▪▪▪ 21 The projections of the external world, things-in-themselves, other minds, realism, and idealism ▪▪▪▪▪ 22 The projections of time, space, and space-time ▪▪▪▪▪ 23 The projections of causality, determinism, and free will ▪▪▪▪▪ 24 Projections of the self and of solipsism ▪▪▪▪▪ 25 Non-relational, agentless reference and referential fields ▪▪▪▪▪ 26 Relativity physics as seen through the lens of the metalogic of reference ▪▪▪▪▪ 27 Quantum theory as seen through the lens of the metalogic of reference ▪▪▪▪▪ 28 Epistemological lessons learned from and applicable to relativity physics and quantum theory ▪▪▪▪▪ ▪▪▪▪▪ PART IV ▪▪▪▪▪ HORIZONS ▪▪▪▪▪ 29 Beyond belief ▪▪▪▪▪ 30 _Critique of Impure Reason_: Its results in retrospect ▪▪▪▪▪ ▪▪▪▪▪ SUPPLEMENT ▪▪▪▪▪ The Formal Structure of the Metalogic of Reference ▪▪▪▪▪ APPENDIX I ▪▪▪▪▪ The Concept of Horizon in the Work of Other Philosophers ▪▪▪▪▪ APPENDIX II ▪▪▪▪▪ Epistemological Intelligence ▪▪▪▪▪ References ▪▪▪▪▪ Index ▪▪▪▪▪ About the author. 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The notion of the riddle plays a pivotal role in Wittgenstein’s Tractatus . By examining the comparisons he draws between philosophical problems and riddles, this paper offers a reassessment of the aims and methods of the book. Solving an ordinary riddle does not consist in learning a new fact; what it requires is that we transform the way we use words. Similarly, Wittgenstein proposes to transform the way philosophers understand the nature of their problems. But since he holds that these (...) problems are ultimately unsolvable, rather than attempting to solve the riddles of philosophy, he aims to dissolve them. (shrink)

We can simply define Gödel 1931 Incompleteness away by redefining the meaning of the standard definition of Incompleteness: A theory T is incomplete if and only if there is some sentence φ such that (T ⊬ φ) and (T ⊬ ¬φ). This definition construes the existence of self-contradictory expressions in a formal system as proof that this formal system is incomplete because self-contradictory expressions are neither provable nor disprovable in this formal system. Since self-contradictory expressions are neither provable nor disprovable (...) only because they are self-contradictory we could define them as unsound instead of defining the formal system as incomplete. (shrink)

Our approach to the liar paradox is based on the Wittgensteinian approach to semantic and logical paradoxes. The main aim of this article is to point out that the liar sentence is only seemingly intelligible, and that it has not been given any sense. First, we will present the traditional solutions of the paradox, especially those which we call modificational. Then we will determine what the defects of these solutions are. Our main objection is that the modificational approaches assume that (...) we can express in languages certain senses which are improper. Next, we will explain why we think that the liar sentence is a mere nonsense. This sentence does not have any role in any language game – it is completely useless. We will also respond to several objections to our approach. 1. That it is not consistent with the principle of compositionality of sense. 2. According to the Quineian philosophy of logic, paradoxical sentences can be conceived as false assumptions leading to crises of logical paradigms. 3. The liar sentence seems to be, contrary to our approach, intelligible. (shrink)

In a famous passage from the Philosophical Investigations, Wittgenstein describes a pupil who has been learning to write out various sequences of numbers in response to orders such as “+1” and “+2”. He has shown himself competent for numbers up to 1000, but when we have him continue the “+2” sequence beyond 1000, he writes the numerals 1004, 1008, 1012. As Wittgenstein describes the case: We say to him, “Look what you’re doing!” — He doesn’t understand us. We say “You (...) should have added two; look how you began the series!” — He answers: “Yes! Isn’t it right? I thought that’s how I should [sollen] do it. — Or suppose he were to say, pointing to the series, “But I went on in the same way!”1The passage continues:— It would now be no use to us to say “But don’t you see…?” — and repeat for him the old explanations and examples. — In such a case, we might perhaps say: this person naturally understands our order, once given our explanations, as we would understand the order “Add 2 up to 1000, 4 up to 2000, 6 up to 3000, and so on.”. (shrink)

A. J. Ayer’s Language, Truth, and Logic had been responsible for introducing the Vienna Circle’s ideas, developed within a Germanophone framework, to an Anglophone readership. Inevitably, this migration from one context to another resulted in the alteration of some of the concepts being transmitted. Such alterations have served to facilitate a number of false impressions of Logical Empiricism from which recent scholarship still tries to recover. In this paper, I will attempt to point to the ways in which LTL has (...) helped to foster the various mistaken stereotypes about Logical Empiricism which were combined into the received view. I will begin by examining Ayer’s all too brief presentation of an Anglocentric lineage for his ideas. This lineage, as we shall see, simply omits the major 19th century Germanophone influences on the rise of analytic philosophy. The Germanophone ideas he presents are selectively introduced into an Anglophone context, and directed towards various concerns that arose within that context. I will focus on the differences between Carnap’s version of the overcoming of metaphysics, and Ayer’s reconfiguration into what he calls the elimination of metaphysics. Having discussed the above, I will very briefly outline the consequences that Ayer’s radicalisation of the Vienna Circle’s doctrines had on the subsequent Anglophone reception of Logical Empiricism. (shrink)

In recent years, philosophers have used expressions of Wittgenstein’s (e.g. “language-games,” “form of life,” and “family resemblance”) in attempts to conceive of the discipline of philosophy in a broad, open, and perhaps global way. These Wittgenstein-inspired approaches indicate an awareness of the importance of cultural and historical diversity for approaching philosophical questions. While some philosophers have taken inspiration from Wittgenstein in embracing contextualism in philosophical hermeneutics, Wittgenstein himself was more instrumental than contextual in his treatment of other philosophers; his focus (...) in his writings was on his own philosophical problems. Does this mean that Wittgensteinian philosophy is a poor resource after all for comparative, cross-cultural, or globally-engaged philosophy (i.e. if it is properly Wittgensteinian)? In this article, I examine the relevance of Wittgenstein to contextually-sensitive philosophy through studies of his conceptions of history and culture, his interest in Spengler’s philosophy of history, and recent scholarship by Hans-Johann Glock and Hans Sluga on the place of contextualism in Wittgenstein’s analysis of philosophical problems. Ultimately, this article advances the view that there are strong resources in Wittgenstein’s philosophy for those seeking a more globally-engaged approach to the field. (shrink)

The basic human ability to treat quantitative information can be divided into two parts. With proto-arithmetical ability, based on the core cognitive abilities for subitizing and estimation, numerosities can be treated in a limited and/or approximate manner. With arithmetical ability, numerosities are processed (counted, operated on) systematically in a discrete, linear, and unbounded manner. In this paper, I study the theory of enculturation as presented by Menary (2015) as a possible explanation of how we make the move from the proto-arithmetical (...) ability to arithmetic proper. I argue that enculturation based on neural reuse provides a theoretically sound and fruitful framework for explaining this development. However, I show that a comprehensive explanation must be based on valid theoretical distinctions and involve several stages in the development of arithmetical knowledge. I provide an account that meets these challenges and thus leads to a better understanding of the subject of enculturation. (shrink)

This paper portrays the later Wittgenstein’s conception of contradictions and his therapeutic approach to them. I will focus on and give relevance to the Lectures on the Foundations of Mathematics, plus the Remarks on the Foundations of Mathematics. First, I will explain why Wittgenstein’s attitude towards contradictions is rooted in: a rejection of the debate about realism and anti-realism in mathematics; and Wittgenstein’s endorsement of logical pluralism. Then, I will explain Wittgenstein’s therapeutic approach towards contradictions, and why it means that (...) a contradiction is not a problem for logic and mathematics. Rather, contradictions are problematic when we do not know what to infer from them. Once a meaning is established through a new rule of inference, the contradiction becomes a usable expression like many others in our inferential apparatus. Thus, the apparent problem is dissolved. Finally, I will take three examples of dissolved contradictions from Wittgenstein to clarify further his notion. I will conclude considering why his position on contradictions led him to clash with Alan Turing, and whether the latter was convinced by the Wittgensteinian proposal. (shrink)

Latest Sermon from the Church of Fundamentalist Naturalism by Pastor Hofstadter. Like his much more famous (or infamous for its relentless philosophical errors) work Godel, Escher, Bach, it has a superficial plausibility but if one understands that this is rampant scientism which mixes real scientific issues with philosophical ones (i.e., the only real issues are what language games we ought to play) then almost all its interest disappears. I provide a framework for analysis based in evolutionary psychology and the work (...) of Wittgenstein (since updated in my more recent writings). -/- Those wishing a comprehensive up to date framework for human behavior from the modern two systems view may consult my book ‘The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle’ 2nd ed (2019). Those interested in more of my writings may see ‘Talking Monkeys--Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet--Articles and Reviews 2006-2019 3rd ed (2019), The Logical Structure of Human Behavior (2019), and Suicidal Utopian Delusions in the 21st Century 4th ed (2019). (shrink)

This paper seeks to widen the dialogue between the “epistemology of peer disagreement” and the epistemology informed by Wittgenstein’s last notebooks, later edited as On Certainty. The paper defends the following theses: not all certainties are groundless; many of them are beliefs; and they do not have a common essence. An epistemic peer need not share all of my certainties. Which response to a disagreement over a certainty is called for, depends on the type of certainty in question. Sometimes a (...) form of relativism is the right response. Reasonable, mutually recognized peer disagreement over a certainty is possible.—The paper thus addresses both interpretative and systematic issues. It uses Wittgenstein as a resource for thinking about peer disagreement over certainties. (shrink)

Reply to eight critical reviews of my paper "Franz Brentano and Higher-Order Theories of Consciousness" in the same issue of the journal Argumentos.

The present article aims at investigating the political aspects of Wittgenstein’s later philosophy, focusing mainly on the Philosophical Investigations. This theme remains rather marginal within Wittgensteinian scholarship, facing the key challenge of the sparsity of explicit discussions of political issues in Wittgenstein’s writings. Based on the broader anthropological and synecdochic character of Wittgenstein’s later philosophy, the main objective of the article is to make explicit the implicit political import of some of the main themes of the Philosophical Investigations. This is (...) done by correlating them with certain political concepts and issues, such as reification, mystification, emancipation, self-institution and the common. In such a way, a new light is shed on Wittgenstein’s influence on and his relevance to contemporary political theory, while at the same time the first steps are taken for the enrichment of the discussion regarding personal and social autonomy and the critique of late capit... (shrink)

The aim of this paper is to provide a dynamic interpretation of Kant’s logical hylomorphism. Firstly, various types of the logical hylomorphism will be illustrated. Secondly, I propose to reevaluate Kant’s constitutivity thesis about logic. Finally, I focus on the design of logical norms as specific kinds of artefacts.

According to some scholars, such as Rodych and Steiner, Wittgenstein objects to Gödel’s undecidability proof of his formula $$G$$, arguing that given a proof of $$G$$, one could relinquish the meta-mathematical interpretation of $$G$$ instead of relinquishing the assumption that Principia Mathematica is correct. Most scholars agree that such an objection, be it Wittgenstein’s or not, rests on an inadequate understanding of Gödel’s proof. In this paper, I argue that there is a possible reading of such an objection that is, (...) in fact, reasonable and related to Gödel’s proof. (shrink)

I here present and defend what I call the Triviality Theory of Truth, to be understood in analogy with Matti Eklund’s Inconsistency Theory of Truth. A specific formulation of is defended and compared with alternatives found in the literature. A number of objections against the proposed notion of meaning-constitutivity are discussed and held inconclusive. The main focus, however, is on the problem, discussed at length by Gupta and Belnap, that speakers do not accept epistemically neutral conclusions of Curry derivations. I (...) first argue that the facts about speakers’ reactions to such Curry derivations do not constitute a problem for the Triviality Theory specifically. Rather, they follow from independent, uncontroversial facts. I then propose a solution which coheres with the theory as I understand it. Finally, I consider a normative reading of their objection and offer a response. (shrink)

In this essay, I seek to demonstrate the interplay of philosophical voices – particularly, that of a platonist voice and a community-agreement-view voice – that drives Wittgenstein’s rule-following dialectic forward; and I argue that each voice succumbs to a particular form of dialectical oscillation that renders its response to the problem of rule-following philosophically inadequate. Finally, I suggest that, by seeing and taking stock of the dilemma in which these responses to the skeptical problem are caught, we can come to (...) appreciate Wittgenstein’s own view of what might constitute a proper a response to the so-called problem of rule-following. This view can be preliminarily characterized by saying that Wittgenstein’s aim is to dissolve the temptation to philosophically rebut the skeptical challenge posed by the rule-following dialectic, an aim he achieves by revealing the semantic emptiness of the apparent sentences that raise the skeptical problem. (shrink)

A "concept" in the sense favoured by Wittgenstein is a paradigm for a transition between parts of a notational system. A concept-determining sentence such as "There is no reddish green" registers the absence of such a transition. This suggests a plausible account of what is perceived in an experiment that was first designed by Crane and Piantanida, who claim to have induced perceptions of reddish green. I shall propose a redescription of the relevant phenomena, invoking only ordinary colour concepts. This (...) redescription is not ruled out by anything the experimenters say. It accounts for certain peculiarities in both their descriptions and their subjects', and suggests that instead of discovering forbidden colours the experimenters introduced a new use of "-ish". Still, there is a point in speaking of "reddish green" in their context, which can be motivated by invoking what Wittgenstein calls a "physiognomy". (shrink)

In the scholarship on Wittgenstein’s later philosophy of mathematics, the dominant interpretation is a theoretical one that ascribes to Wittgenstein some type of ‘ism’ such as radical verificationism or anti-realism. Essentially, he is supposed to provide a positive account of our mathematical practice based on some basic assertions. However, I claim that he should not be read in terms of any ‘ism’ but instead should be read as examining philosophical pictures in the sense of unclear conceptions. The contrast here is (...) that basic assertions that frame philosophical ‘isms’ are propositional such that they are subject to normal argumentative evaluation, while pictures in Wittgenstein’s sense are non-propositional—they lack a clear truth condition. They, therefore, need clarification rather than argumentation. In this paper, I provide a detailed analysis of Wittgenstein’s treatment of philosophical pictures with special focus on his argument on contradiction. I begin by explaining the problem with this trend of theoretical interpretation, taking Steve Gerrard’s otherwise excellent interpretation as a representative example and pointing out why it is problematic. Next, I will argue that those problems do not arise if we take Wittgenstein’s task as the clarification of philosophical pictures. I do this, first, by explaining Wittgenstein’s method using his argument concerning the Augustinian Picture in Philosophical Investigations and then pointing out that the same method can be identified in the crucial arguments in his philosophy of mathematics. Finally, in order to connect my interpretation with the current scholarship, I will explain the relation of my interpretation with those of New Wittgensteinian scholars. (shrink)

Russell’s initial project in philosophy (1898) was to make mathematics rigorous reducing it to logic. Before August 1900, however, Russell’s logic was nothing but mereology. First, his acquaintance with Peano’s ideas in August 1900 led him to discard the part-whole logic and accept a kind of intensional predicate logic instead. Among other things, the predicate logic helped Russell embrace a technique of treating the paradox of infinite numbers with the help of a singular concept, which he called ‘denoting phrase’. Unfortunately, (...) a new paradox emerged soon: that of classes. The main contention of this paper is that Russell’s new conception only transferred the paradox of infinity from the realm of infinite numbers to that of class-inclusion. Russell’s long-elaborated solution to his paradox developed between 1905 and 1908 was nothing but to set aside of some of the ideas he adopted with his turn of August 1900: (i) With the Theory of Descriptions, he reintroduced the complexes we are acquainted with in logic. In this way, he partly restored the pre-August 1900 mereology of complexes and simples. (ii) The elimination of classes, with the help of the ‘substitutional theory’, and of propositions, by means of the Multiple Relation Theory of Judgment, completed this process. (shrink)