Charles Parsons’ book “Mathematical Thought and Its Objects” of 2008 (Cambridge University Press, New York) is critically discussed by concentrating on one of Parsons’ main themes: the role of intuition in our understanding of arithmetic (“intuition” in the specific sense of Kant and Hilbert). Parsons argues for a version of structuralism which is restricted by the condition that some paradigmatic structure should be presented that makes clear the actual existence of structures of the necessary sort. Parsons’ paradigmatic structure is the (...) so-called ‘intuitive model’ of arithmetic realized by Hilbert’s strings of strokes. This paper argues that Hilbert’s strings, considered as given in intuition, cannot play the role Parsons assigns to them: the criteria of identity of these strings do not have the sharpness that Parsons wants to see in them, and Parsons inadvertently projects abstract structures into his ‘intuitive model’. This diagnosis is exemplified with respect to (a) Parsons’ distinction between addition and multiplication on the one hand and exponentiation on the other and (b) his analysis of arithmetical knowledge in simple cases like “7 + 5 = 12”. All in all, it is claimed that Parsons book contains many important insights with respect to, for example, different versions structuralism, the notion of “natural number” and its uniqueness, induction, predicativity and other things, for which he is rightly famous, but that his way of drawing on the notion of intuition leaves too many questions unanswered. (shrink)

I argue that a practice can only be taken to be one of apparent rule following if it contains a practice of policing moves within the practice. So the existence of an apparently rule-governed practice entails the existence of, what I call, a policing practice. I then argue that this entailment cannot be reconciled with a non-factualist construal of the policing practice. Thus non-factualism about the policing practice is false. Factualism about the policing practice entails realism about rules. So I (...) conclude that we ought to be realists about rules. Finally I distinguish a position which I call ultra-realism about rules and note that this too is a casualty of the view developed here. (shrink)

RésuméPaul Boghossian soutient contre Wittgenstein que le normativisme au sujet de la logique et des mathématiques est incompatible avec le fait de tenir les énoncés logiques et mathématiques pour vrais et que le normativisme entraîne une régression indue. Je soutiens, pour ma part, que le normativisme n'entraîne pas une telle régression, parce que les normes peuvent être implicites et que le normativisme peut bien être «factualiste» si l'on rejette ce que Rockney Jacobsen appelle le «cognitivisme sémantique». Je tiens en outre (...) que la position de Boghossian repose sur la façon trompeuse dont il découpe le terrain conceptuel. (shrink)

This paper outlines a solution to what can be called “the problem of domination by reason”, “conceptual domination” or “clarificatorory injustice”, connected with how a philosopher may appear to be in a position to legitimately coerce, by means of arguments, an interlocutor who shares with her a concept or a conceptual system to accept a philosophical characterization of a concept or whatever the concept concerns. The proposed solution is based on a particular interpretation of what Wittgenstein means by agreement in (...) his later philosophy, when he says that philosophy only states what anyone grants to it. Wittgenstein’s view and the proposed solution are characterized by their continued recognition of the value of logic and reason, truth and knowledge, as opposed to attempting to solve the problem by embracing relativism and questioning the value of the logic, reason, truth and knowledge. Relevant kind of disagreements licence no relativistic conclusions, because problems relating to them can be solved without going this far. Keywords: domination, relativism, Wittgenstein, method, agreement. (shrink)

A set of hypotheses is formulated for a connectionist approach to cognitive modeling. These hypotheses are shown to be incompatible with the hypotheses underlying traditional cognitive models. The connectionist models considered are massively parallel numerical computational systems that are a kind of continuous dynamical system. The numerical variables in the system correspond semantically to fine-grained features below the level of the concepts consciously used to describe the task domain. The level of analysis is intermediate between those of symbolic cognitive models (...) and neural models. The explanations of behavior provided are like those traditional in the physical sciences, unlike the explanations provided by symbolic models.Higher-level analyses of these connectionist models reveal subtle relations to symbolic models. Parallel connectionist memory and linguistic processes are hypothesized to give rise to processes that are describable at a higher level as sequential rule application. At the lower level, computation has the character of massively parallel satisfaction of soft numerical constraints; at the higher level, this can lead to competence characterizable by hard rules. Performance will typically deviate from this competence since behavior is achieved not by interpreting hard rules but by satisfying soft constraints. The result is a picture in which traditional and connectionist theoretical constructs collaborate intimately to provide an understanding of cognition. (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)

In §8 of Remarks on the Foundations of Mathematics (RFM), Appendix 3 Wittgenstein imagines what conclusions would have to be drawn if the Gödel formula P or ¬P would be derivable in PM. In this case, he says, one has to conclude that the interpretation of P as “P is unprovable” must be given up. This “notorious paragraph” has heated up a debate on whether the point Wittgenstein has to make is one of “great philosophical interest” revealing “remarkable insight” in (...) Gödel’s proof, as Floyd and Putnam suggest (Floyd (2000), Floyd (2001)), or whether this remark reveals Wittgenstein’s misunderstanding of Gödel’s proof as Rodych and Steiner argued for recently (Rodych (1999, 2002, 2003), Steiner (2001)). In the following the arguments of both interpretations will be sketched and some deficiencies will be identified. Afterwards a detailed reconstruction of Wittgenstein’s argument will be offered. It will be seen that Wittgenstein’s argumentation is meant to be a rejection of Gödel’s proof but that it cannot satisfy this pretension. (shrink)

This volume examines the limitations of mathematical logic and proposes a new approach to logic intended to overcome them. To this end, the book compares mathematical logic with earlier views of logic, both in the ancient and in the modern age, including those of Plato, Aristotle, Bacon, Descartes, Leibniz, and Kant. From the comparison it is apparent that a basic limitation of mathematical logic is that it narrows down the scope of logic confining it to the study of deduction, without (...) providing tools for discovering anything new. As a result, mathematical logic has had little impact on scientific practice. Therefore, this volume proposes a view of logic according to which logic is intended, first of all, to provide rules of discovery, that is, non-deductive rules for finding hypotheses to solve problems. This is essential if logic is to play any relevant role in mathematics, science and even philosophy. To comply with this view of logic, this volume formulates several rules of discovery, such as induction, analogy, generalization, specialization, metaphor, metonymy, definition, and diagrams. A logic based on such rules is basically a logic of discovery, and involves a new view of the relation of logic to evolution, language, reason, method and knowledge, particularly mathematical knowledge. It also involves a new view of the relation of philosophy to knowledge. This book puts forward such new views, trying to open again many doors that the founding fathers of mathematical logic had closed historically. (shrink)

In this excellent book Sebastien Gandon focuses mainly on Russell's two major texts, Principa Mathematica and Principle of Mathematics, meticulously unpicking the details of these texts and bringing a new interpretation of both the mathematical and the philosophical content. Winner of The Bertrand Russell Society Book Award 2013.

This book brings together philosophers, mathematicians and logicians to penetrate important problems in the philosophy and foundations of mathematics. In philosophy, one has been concerned with the opposition between constructivism and classical mathematics and the different ontological and epistemological views that are reflected in this opposition. The dominant foundational framework for current mathematics is classical logic and set theory with the axiom of choice. This framework is, however, laden with philosophical difficulties. One important alternative foundational programme that is actively pursued (...) today is predicativistic constructivism based on Martin-Löf type theory. Associated philosophical foundations are meaning theories in the tradition of Wittgenstein, Dummett, Prawitz and Martin-Löf. What is the relation between proof-theoretical semantics in the tradition of Gentzen, Prawitz, and Martin-Löf and Wittgensteinian or other accounts of meaning-as-use? What can proof-theoretical analyses tell us about the scope and limits of constructive and predicative mathematics? (shrink)

What is the source of logical and mathematical truth? This book revitalizes conventionalism as an answer to this question. Conventionalism takes logical and mathematical truth to have their source in linguistic conventions. This was an extremely popular view in the early 20th century, but it was never worked out in detail and is now almost universally rejected in mainstream philosophical circles. Shadows of Syntax is the first book-length treatment and defense of a combined conventionalist theory of logic and mathematics. It (...) argues that our conventions, in the form of syntactic rules of language use, are perfectly suited to explain the truth, necessity, and a priority of logical and mathematical claims, as well as our logical and mathematical knowledge. (shrink)

This thesis proposes a perspective on language and its development by starting from two approaches. The first is the ecological-enactive approach to cognition. In opposition to the widespread idea that cognition is information-processing in the brain, the ecological-enactive approach explains human cognition in relational terms, as skilful interactions with a sociomaterial environment shaped by practices. The second is the metalinguistic approach to language, which holds that reflexive or metalinguistic language use – talking about talking – is crucial for understanding language (...) and its development. In particular, I defend two theses: 1. A child’s initial communicative behaviour can be explained in terms of attentional actions: social actions that function by directing someone else’s attention. 2. In order for the child’s communicative behaviour to be sensitive to key properties of language, such as semantic content and normativity, she needs to learn metalinguistic skills. The development of this ecological-enactive perspective on language serves two functions. First, the ecological-enactive approach started by considering basic behaviour, such as locomotion and grasping. An approach in the cognitive sciences, however, should be able to account for the full gamut of human cognition. If the perspective developed in this thesis is viable, this is a contribution to extending the ecological-enactive approach to typically human forms of cognition. Second, this perspective throws new light on philosophical problems concerning language. In the different chapters, I deal with questions concerning the nature of linguistic knowledge, explanations of communicative behaviour, and the origins of semantic content and linguistic normativity. (shrink)

Arguing that our attachment to Aristotelian modes of discourse makes a revision of their conceptual foundations long overdue, the author proposes the consideration of unacknowledged factors that play a central role in argument itself. These are in particular the subjective imprint and the dynamics of argumentation. Their inclusion in a four-dimensional framework and the focus on thesis validity allow for a more realistic view of our discourse practice. Exhaustive analyses of fascinating historical and contemporary arguments are provided. These range from (...) Columbus’s advocacy of the Western Passage to India, over the trial of King Louis XVI during the French Revolution, to today’s highly charged controversies surrounding euthanasia and embryo research. Excavating foundational issues such as the purpose of argument itself and the contested role of argument as a generator of knowledge, the book culminates in a discussion of the relationship between rationality and reasonableness and criticizes the restrictions of ‘rational’ argument relying on fixed logical, economic or cultural criteria that in reality are mutable. Here, a true, open argument requires the infusion of Paul Lorenzen’s principle of ‘transsubjectivity’, which recognizes but transcends the partiality of the individual and which can be seen in the pragmatic and expanding consensus that humanity can control itself to safeguard the future of a fragile, damaged world. (shrink)

The conceptual framework of behaviorism is reconstructed in a logical scheme rather than along chronological lines. The resulting reconstruction is faithful to the history of behaviorism and yet meets the contemporary challenges arising from cognitive science, psycholinguistics, and philosophy. In this reconstruction, the fundamental premise is that psychology is to be a natural science, and the major corollaries are that psychology is to be objective and empirical. To a great extent, the reconstruction of behaviorism is an elaboration of behaviorist views (...) of what it is for a science to be objective and empirical. The reconstruction examines and evaluates behaviorist positions on observation and the rejection of introspection, the behavioral data language, theory construction, stimulus-response psychology, the organization of behavior, complex processes, agency, and the interpretation of mentalistic language. The resulting reconstruction shows behaviorism to be a pragmatic psychological version of positivism based on a behavioral epistemology. (shrink)

* This paper was originally given as one of the Doris Lee Lectures on February 20th 1975, at the University of London Institute of Education.

This is a short, and therefore necessarily very incomplete discussion of one of the great questions of modern philosophy. I return to a station at which an interpretative train of thought of mine came to a halt in a paper written almost 20 years ago, about Wittgenstein and Chomsky,[1] hoping to advance a little bit further down the track. The rule-following passages in the Investigations and Remarks on the Foundations of Mathematics in fact raise a number of distinct issues about (...) rules, meaning, objectivity, and reasons, whose conflation is encouraged by the standard caption, "the Rule-following Considerations".[2] So, let me begin by explaining my focus here. (shrink)

Methodological reductionists practice ‘wannabe reductionism’. They claim that one should pursue reductionism, but never propose how. I integrate two strains in prior work to do so. Three kinds of activities are pursued as “reductionist”. “Successional reduction” and inter-level mechanistic explanation are legitimate and powerful strategies. Eliminativism is generally ill-conceived. Specific problem-solving heuristics for constructing inter-level mechanistic explanations show why and when they can provide powerful and fruitful tools and insights, but sometimes lead to erroneous results. I show how traditional metaphysical (...) approaches fail to engage how science is done. The methods used do so, and support a pragmatic and non-eliminativist realism. (shrink)

A number of prominent philosophers advance the following ideas: (1) Meaning is use. (2) Meaning is an intrinsically normative notion. Call (1) the use thesis, hereafter UT, and (2) the normativity thesis, hereafter NT. They come together in the view that for a linguistic expression to have meaning is for there to be certain proprieties governing its employment.1 These ideas are often associated with a third.

The relation between Wittgenstein's philosophy of mathematics and mathematical Intuitionism has raised a considerable debate. My attempt is to analyse if there is a commitment in Wittgenstein to themes characteristic of the intuitionist movement in Mathematics and if that commitment is one important strain that runs through his Remarks on the foundations of mathematics. The intuitionistic themes to analyse in his philosophy of mathematics are: firstly, his attacks on the unrestricted use of the Law of Excluded Middle; secondly, his distrust (...) of non-constructive proofs; and thirdly, his impatience with the idea that mathematics stands in need of a foundation. These elements are Fogelin's starting point for the systematic reconstruction of Wittgenstein's conception of mathematics. (shrink)

This is a critical discussion of Part III of Peter Railton's recent book Facts, Values, and Norms: Essays Toward a Morality of Consequence (Cambridge University Press, 2003).

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)

In “Truth by Convention” W.V. Quine gave an influential argument against logical conventionalism. Even today his argument is often taken to decisively refute logical conventionalism. Here I break Quine’s arguments into two— the super-task argument and the regress argument—and argue that while these arguments together refute implausible explicit versions of conventionalism, they cannot be successfully mounted against a more plausible implicit version of conventionalism. Unlike some of his modern followers, Quine himself recognized this, but argued that implicit conventionalism was explanatorily (...) idle. Against this I show that pace Quine’s claim that implicit conventionalism has no content beyond the claim that logic is firmly accepted, implicit rules of inference can be used to distinguish the firmly accepted from the conventional. As part of my case, I argue that positing syntactic rules of inference as part of our linguistic competence follows from the same methodology that leads contemporary linguists and cognitive scientists to posit rules of phonology, morphology, and grammar. The upshot of my discussion is a diagnosis of the fallacy in Quine’s master critique of logical conventionalism and a re-opening of possibilities for an attractive conventionalist theory of logic. (shrink)

I propose to sketch my views on several aspects of the philosophy of mathematics that I take to be especially relevant to philosophy as a whole. The relevance of my discussion would, I think, become more evident, if the reader keeps in mind the function of (the philosophy of) mathematics in philosophy in providing us with more transparent aspects of general issues. I shall consider: (1) three familiar examples; (2) logic and our conceptual frame; (3) communal agreement and objective certainty; (...) (4) the transcommunal universality of mathematics; (5) the big jump to the potential infinite; (6) the reconciliation of local creation with the hypothesis of discovery; (7) Platonism as realism plus conceptualism; (8) foundational studies and mathematical practice; and (9) the decomposition of philosophical disagreements. The views of Gödel and Wittgenstein are emphasized in order to add specificity to the discussions. (shrink)

This article questions social constructionists' claims to introduce Wittgenstein's philosophy to psychology. The philosophical fiction of a neonate Crusoe is introduced to cast doubt on the interpretations and use of the private language argument to support a new psychology developed by the constructionists. It is argued that a neonate Crusoe's viability in philosophy and apparent absence in psychology offends against the integrity of the philosophical contribution Wittgenstein might make to psychology. The consequences of accepting Crusoe's viability are explored as they (...) appear in both philosophy and psychology. (shrink)

Deleuze's theory of time set out in Difference and Repetition is a complex structure of three different syntheses of time – the passive synthesis of the living present, the passive synthesis of the pure past and the static synthesis of the future. This article focuses on Deleuze's third synthesis of time, which seems to be the most obscure part of his tripartite theory, as Deleuze mixes different theoretical concepts drawn from philosophy, Greek drama theory and mathematics. Of central importance is (...) the notion of the cut, which is constitutive of the third synthesis of time defined as an a priori ordered temporal series separated unequally into a before and an after. This article argues that Deleuze develops his ordinal definition of time with recourse to Kant's definition of time as pure and empty form, Hölderlin's notion of ‘caesura’ drawn from his ‘Remarks on Oedipus’ (1803) and Dedekind's method of cuts as developed in his pioneering essay ‘Continuity and Irrational Numbers’ (1872). Deleuze then ties together the conceptions of the Kantian empty form of time and the Nietzschean eternal return, both of which are essentially related to a fractured I or dissolved self. This article aims to assemble the different heterogeneous elements that Deleuze picks up on and to show how the third synthesis of time emerges from this differential multiplicity. (shrink)

There is definitely a family resemblance between what contemporary contextualism maintains in philosophy of language and some of the claims about meaning put forward by the later Wittgenstein. Yet the main contextualist thesis, namely that linguistic meaning undermines truth-conditions, was not defended by Wittgenstein. If a claim in this regard can be retrieved in Wittgenstein despite his manifest antitheoretical attitude, it is instead that truth-conditions trivially supervene on linguistic meaning. There is, however, another Wittgensteinian claim that truly has a contextualist (...) flavour, namely that linguistic meaning is itself wide-contextual. To be sure, this claim does not lead to the eliminativist/intentionalist conception of linguistic meaning that radical contextualists have recently developed. Rather, it goes together with a robust conception of linguistic meaning as intrinsically normative. Yet it may explain why Wittgenstein is taken to be a forerunner of contemporary contextualism. (shrink)

According to Barry Stroud, Wittgenstein thought that language is social only in this minimal way: we cannot make sense of the idea of someone having a language unless we can describe her as using signs in conformity with the linguistic practices of some community. Since a solitary person could meet this condition, Stroud concludes that, for Wittgenstein, solitary languages are possible. I argue that Wittgenstein in fact thought that language is social in a much more robust way. Solitary languages are (...) not possible because we cannot make sense of the idea of someone having a language unless we can think of her as actively participating in the linguistic practices that fix the standards governing the applications of her words. (shrink)

According to the dominant view, the later Wittgenstein identified the meaning of an expression with its use in the language and vehemently rejected any kind of mentalism or intentionalism about linguistic meaning. I argue that the dominant view is wrong. The textual evidence, which has either been misunderstood or overlooked, indicates that at least since the Blue Book Wittgenstein thought speakers' intentions determine the contents of linguistic utterances. His remarks on use are only intended to emphasize the heterogeneity of natural (...) language. Taking into account remarks written after he finished the Investigations, I show how Wittgenstein anticipated the basic tenets of Gricean intention-based semantics. These are, in particular, the distinction between ‘natural’ and ‘non-natural’ meaning and the distinction between what a speaker means by an utterance and what the expression uttered means in the speaker’s natural language. Importantly, Wittgenstein also believed that only the meaning of the speaker determined the content of ambiguous expressions, such as ‘bank’, on a particular occasion of utterance. (shrink)

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

The core entities of computer science include formal languages, spec-ifications, models, programs, implementations, semantic theories, type inference systems, abstract and physical machines. While there are conceptual questions concerning their nature, and in particular ontological ones (Turner 2018), our main focus here will be on the relationships between them. These relationships have an extensional aspect that articulates the propositional connection between the two entities, and an intentional one that fixes the direction of governance. An analysis of these two aspects will drive (...) our investigation; an investigation that will touch upon some of the central concerns of the philosophy of computer science (Turner 2017). (shrink)

Throughout his whole work, Wittgenstein seizes on a distinction between logical and physical possibility, and impossibility. Despite this continuity and although, Wittgenstein brings in this distinction in various contexts and from different vantage points, he often solely brushes over it without elaborating in detail. In the so-called Big Typescript, however, he dedicates himself not only to the distinction between logical and physical possibility but also to the distinction between logical possibility and impossibility in particular investigations. In the course of these (...) investigations, another aspect arises and is tossed and turned repeatedly by Wittgenstein – namely, the place of “imaginability” in these considerations. On the basis of three focussed chapters in the Big Typescript, I argue that “imaginability” as an utterance of the form “being able to imagine ‘what it would be like’” can be allocated the place of a criterion for logical possibility. To this end, I will first outline the chapters 96., 27. and 26. in one section each. Although in these chapters, Wittgenstein only indicates rather than claiming explicitly “imaginability” to be a criterion for logical possibility, I will discuss in the last section how this conclusion can be drawn by combining the results of the previous sections. (shrink)

A set of hypotheses is formulated for a connectionist approach to cognitive modeling. These hypotheses are shown to be incompatible with the hypotheses underlying traditional cognitive models. The connectionist models considered are massively parallel numerical computational systems that are a kind of continuous dynamical system. The numerical variables in the system correspond semantically to fine-grained features below the level of the concepts consciously used to describe the task domain. The level of analysis is intermediate between those of symbolic cognitive models (...) and neural models. The explanations of behavior provided are like those traditional in the physical sciences, unlike the explanations provided by symbolic models. Higher-level analyses of these connectionist models reveal subtle relations to symbolic models. Parallel connectionist memory and linguistic processes are hypothesized to give rise to processes that are describable at a higher level as sequential rule application. At the lower level, computation has the character of massively parallel satisfaction of soft numerical constraints; at the higher level, this can lead to competence characterizable by hard rules. Performance will typically deviate from this competence since behavior is achieved not by interpreting hard rules but by satisfying soft constraints. The result is a picture in which traditional and connectionist theoretical constructs collaborate intimately to provide an understanding of cognition. (shrink)

Since the rise of modern natural science there has been deep tension between the conceptual and the natural. Wittgenstein's discussion of how we learn a sensation-language contains important resources that can help us relieve this tension. The key here, I propose, is to focus our attention on animal nature, conceived as partially re-enchanted. To see how nature, so conceived, helps us relieve the tension in question, it is crucial to gain a firm and detailed appreciation of how the primitive-instinctive, a (...) central part of animal nature, actually serves the conceptual. I offer such an appreciation by closely examining §244 of the Philosophical Investigations and Peter Winch's discussion of it. The general aim is to bring out a certain kind of Wittgensteinian “naturalism”, a “naturalism” that is fully alive to the rootedness of conceptuality in nature. A concomitant aim is to illustrate the truth of Wittgenstein's saying that in philosophy one often has to pay close attention to details. (shrink)

In this paper, I take three snapshots of Wittgenstein's philosophical work in order to jot a few notes on the issue of the continuity in his philosophy. I use Wittgenstein's distinction between what can be 'said' and what can only be “shown” in order to highlight Wittgenstein's continual insistence that our basic relation with reality is seamless. I propose that Wittgenstein holds, throughout his philosophical career, that our thinking does not stop short of the world. In brief, I suggest that (...) Wittgenstein would note that our natural history is largely what the mediaevals would call second nature. (shrink)