A novel theoretical framework for an embodied, non-representational approach to language that extends and deepens enactive theory, bridging the gap between sensorimotor skills and language. -/- Linguistic Bodies offers a fully embodied and fully social treatment of human language without positing mental representations. The authors present the first coherent, overarching theory that connects dynamical explanations of action and perception with language. Arguing from the assumption of a deep continuity between life and mind, they show that this continuity extends to language. (...) Expanding and deepening enactive theory, they offer a constitutive account of language and the co-emergent phenomena of personhood, reflexivity, social normativity, and ideality. Language, they argue, is not something we add to a range of existing cognitive capacities but a new way of being embodied. Each of us is a linguistic body in a community of other linguistic bodies. The book describes three distinct yet entangled kinds of human embodiment, organic, sensorimotor, and intersubjective; it traces the emergence of linguistic sensitivities and introduces the novel concept of linguistic bodies; and it explores the implications of living as linguistic bodies in perpetual becoming, applying the concept of linguistic bodies to questions of language acquisition, parenting, autism, grammar, symbol, narrative, and gesture, and to such ethical concerns as microaggression, institutional speech, and pedagogy. (shrink)

There is an urgent need in philosophy of mathematics for new approaches which pay closer attention to mathematical practice. This book will blaze the trail: it offers philosophical analyses of important characteristics of contemporary mathematics and of many aspects of mathematical activity which escape purely formal logical treatment.

From Brouwer To Hilbert: The Debate on the Foundations of Mathematics in the 1920s offers the first comprehensive introduction to the most exciting period in the foundation of mathematics in the twentieth century. The 1920s witnessed the seminal foundational work of Hilbert and Bernays in proof theory, Brouwer's refinement of intuitionistic mathematics, and Weyl's predicativist approach to the foundations of analysis. This impressive collection makes available the first English translations of twenty-five central articles by these important contributors and many others. (...) The articles have been translated for the first time from Dutch, French, and German, and the volume is divided into four sections devoted to (1) Brouwer, (2) Weyl, (3) Bernays and Hilbert, and (4) the emergence of intuitionistic logic. Each section opens with an introduction which provides the necessary historical and technical context for understanding the articles. Although most contemporary work in this field takes its start from the groundbreaking contributions of these major figures, a good, scholarly introduction to the area was not available until now. Unique and accessible, From Brouwer To Hilbert will serve as an ideal text for undergraduate and graduate courses in the philosophy of mathematics, and will also be an invaluable resource for philosophers, mathematicians, and interested non-specialists. (shrink)

The period from 1900 to 1935 was particularly fruitful and important for the development of logic and logical metatheory. This survey is organized along eight "itineraries" concentrating on historically and conceptually linked strands in this development. Itinerary I deals with the evolution of conceptions of axiomatics. Itinerary II centers on the logical work of Bertrand Russell. Itinerary III presents the development of set theory from Zermelo onward. Itinerary IV discusses the contributions of the algebra of logic tradition, in particular, Löwenheim (...) and Skolem. Itinerary V surveys the work in logic connected to the Hilbert school, and itinerary V deals specifically with consistency proofs and metamathematics, including the incompleteness theorems. Itinerary VII traces the development of intuitionistic and many-valued logics. Itinerary VIII surveys the development of semantical notions from the early work on axiomatics up to Tarski's work on truth. (shrink)

Social factors have so far been neglected in embodied theories of perception despite the wealth of phenomenological insights and empirical evidence indicating their importance. I examine evidence from developmental psychology and neuroscience and attempt an initial classification according to whether social factors play a contextual, enabling, or constitutive role in the ability to perceive objects in a detached manner, i.e. beyond their immediate instrumental use. While evidence of cross-cultural variations in perceptual styles and the influence of social cues on visual (...) attention could not be said to play more than a contextual role, other factors such as the intricate developmental links between dyadic and triadic interactions in infancy, as well as episodes of peer-learning in children, play enabling roles. A common element in these factors is the presence and resolution of interpersonal conflict. Detached object perception could not develop without these social factors. I argue, in addition, that social skills such as managing partial social acts which are addressed to and completed by others, linguistic mediation, makebelieve play, and the ability to control perspectival switches are constitutive -- i.e. are of the essence -- of the ability to see objects as present with a detached attitude. I discuss the prospects of incorporating such social elements into dynamical interpretations of the sensorimotor approach through the enactive notion of participatory sense-making. (shrink)

Open peer commentary on the article “Perception-Action Mutuality Obviates Mental Construction” by Martin Flament Fultot, Lin Nie & Claudia Carello. Upshot: I contrast enactivist and ecological perspectives on some of the themes raised by the authors. I discuss some of their worries about the notion of sense-making and other epistemological aspects of enactivism.

The temporality of intentions and actions in situations of social interaction can sometimes be paradoxical. I argue that in these situations it may sometimes be possible to conceive of individual acts that can, in a strong sense, be intended retroactively. This could happen when the relational patterns in social interaction literally alter the virtual structure of a participant's past corporeal intentions resulting in an odd experience of having intended something all along without knowing it. I propose that this possibility should (...) be interpreted as more than just a narrowly epistemic phenomenon. Examining this claim involves clarifying the enactive perspective on intentionality, which I do here. The enactive approach rejects the model of a causal relation between intention and action for one of an intrinsic qualitative relation between the two as facets of sense-making. I develop this idea and compare it with Merleau-Ponty's Fundierung model of the mutual relation between corporeal and reflexive intentionality to show that co-regulated moves/affections during social interaction may modulate both arcs of this relation, creating the possibility of a re-signification that alters not the actuality but the virtual tendencies that preceded the social act. (shrink)

Cantor’s theory of cardinal numbers offers a way to generalize arithmetic from finite sets to infinite sets using the notion of one-to-one association between two sets. As is well known, all countable infinite sets have the same ‘size’ in this account, namely that of the cardinality of the natural numbers. However, throughout the history of reflections on infinity another powerful intuition has played a major role: if a collectionAis properly included in a collectionBthen the ‘size’ ofAshould be less than the (...) ‘size’ ofB(part–whole principle). This second intuition was not developed mathematically in a satisfactory way until quite recently. In this article I begin by reviewing the contributions of some thinkers who argued in favor of the assignment of different sizes to infinite collections of natural numbers (Thabit ibn Qurra, Grosseteste, Maignan, Bolzano). Then, I review some recent mathematical developments that generalize the part–whole principle to infinite sets in a coherent fashion (Katz, Benci, Di Nasso, Forti). Finally, I show how these new developments are important for a proper evaluation of a number of positions in philosophy of mathematics which argue either for the inevitability of the Cantorian notion of infinite number (Gödel) or for the rational nature of the Cantorian generalization as opposed to that, based on the part–whole principle, envisaged by Bolzano (Kitcher). (shrink)

An Introduction to Proof Theory provides an accessible introduction to the theory of proofs, with details of proofs worked out and examples and exercises to aid the reader's understanding. It also serves as a companion to reading the original pathbreaking articles by Gerhard Gentzen. The first half covers topics in structural proof theory, including the Gödel-Gentzen translation of classical into intuitionistic logic, natural deduction and the normalization theorems, the sequent calculus, including cut-elimination and mid-sequent theorems, and various applications of these (...) results. The second half examines ordinal proof theory, specifically Gentzen's consistency proof for first-order Peano Arithmetic. The theory of ordinal notations and other elements of ordinal theory are developed from scratch, and no knowledge of set theory is presumed. The proof methods needed to establish proof-theoretic results, especially proof by induction, are introduced in stages throughout the text. Mancosu, Galvan, and Zach's introduction will provide a solid foundation for those looking to understand this central area of mathematical logic and the philosophy of mathematics. (shrink)

How accurate is the picture of the human mind that has emerged from studies in neuroscience, psychology, and cognitive science? Anybody with an interest in how minds work - how we learn about the world and how we remember people and events - may feel dissatisfied with the answers contemporary science has to offer. Sensorimotor Life draws on current theoretical developments in the enactive approach to life and mind. It examines and expands the premises of the sciences of the human (...) mind, while developing an alternative picture closer to people's daily experiences. Enactive ideas are applied and extended, providing a theoretically rich, naturalistic account of meaning and agency. The book includes a dynamical systems description of different types of sensorimotor regularities or sensorimotor contingencies; a dynamical interpretation of Piaget's theory of equilibration to ground the concept of sensorimotor mastery; and a theory of agency as organized networks of sensorimotor schemes, as well as its implications for embodied subjectivity. Written for students and researchers of cognitive science, the authors offer a fuller view of the mind, a view better attuned to the experiences of people who live, work, love, struggle, and age, thrown into a world of meaningful relations they help create. Additionally, the book is of interest to neuroscientists, psychiatrists, and philosophers of science. (shrink)

Summary This article presents three hitherto unpublished letters by Frank Plumpton Ramsey on the foundations of mathematics with commentary. One of the letters was sent to Abraham Fraenkel and the other two letters to Heinrich Behmann. The transcription of the letters is preceded by an account that details the extent of Ramsey's known contacts with mathematical logicians on the Continent.

In the last two decades there has been renewed interest in visualization in logic and mathematics. Visualization is usually understood in different ways but for the purposes of this article I will take a rather broad conception of visualization to include both visualization by means of mental images as well as visualizations by means of computer generated images or images drawn on paper, e.g. diagrams etc. These different types of visualization can differ substantially but I am interested in offering a (...) characterization of visualization that is as broad as possible. The article describes and explains (1) the way in which visual thinking fell into disrepute, (2) the renaissance of visual thinking in mathematics over recent decades, (3) the ways in which visual thinking has been rehabilitated in epistemology of mathematics and logic. (shrink)

At the same time, the book is a contribution to recent philosophical debates, in particular on the prospects for a successful nominalist reconstruction of .

In previous work by Baaz and Iemhoff, a Gentzen calculus for intuitionistic logic with existence predicate is presented that satisfies partial cut elimination and Craig's interpolation property; it is also conjectured that interpolation fails for the implication-free fragment. In this paper an equivalent calculus is introduced that satisfies full cut elimination and allows a direct proof of interpolation via Maehara's lemma. In this way, it is possible to obtain much simpler interpolants and to better understand and overcome the failure of (...) interpolation for the implication-free fragment. (shrink)

It is reported that in reply to John Wisdom’s request in 1944 to provide a dictionary entry describing his philosophy, Wittgenstein wrote only one sentence: “He has concerned himself principally with questions about the foundations of mathematics”. However, an understanding of his philosophy of mathematics has long been a desideratum. This was the case, in particular, for the period stretching from the Tractatus Logico-Philosophicus to the so-called transitional phase. Marion’s book represents a giant leap forward in this direction. In the (...) preface, Marion provides a diagnosis for why it has taken such a long time to obtain an accurate picture of Wittgenstein’s ideas in the foundations of mathematics. When the Remarks on the Foundations of Mathematics came out in 1956 the reception by specialists, such as Kreisel, was negative. This led many Wittgenstein scholars to set aside the issues in the foundations of mathematics and go on with the business of the day, focusing on Wittgenstein’s philosophy of language and psychology. However, these early negative appraisals were severely limited by two facts. First of all, the nature of the Remarks was a hindrance to an understanding of what Wittgenstein was up to. The book was edited by Wittgenstein’s literary executors by collating together passages from several manuscripts dating from 1937 to 1944, cutting, however, many passages in between remarks. Second, many of the original manuscripts and typescripts of the transitional phase were not available and thus it was virtually impossible to obtain a balanced picture of Wittgenstein’s development. Finally, a stark contraposition between the early and late Wittgenstein did not encourage people to work seriously on the development of Wittgenstein’s position from the Tractatus through the transitional period to the later writings. (shrink)

The paper deals with the art of cross-examination or probe (peirastiké) wich is, according to Aristotle, a part of dialectic and aims at revealing falle pretensions to knowledge. A detailed discussion of the relevant passages from the Sophistical Refutations shows that "peirastic" argumenta were not easily distinguishablc forro the arguments used by the sophists against the scientists in order to undermine their reputation. It is also shows that the difficulty of telling dialectic from sophistry is a theme Aristotle inherits frorn (...) Plato's Sophist. (shrink)

In Sophistical Refutations 8 Aristotle claims that every sophistical refutation depends on a false belief which is implicitly held by the victim of the fallacy and can normally be elicited from him as an explicit additional premiss. In this case the fallacious argument will be turned into a valid one, albeit with a false premiss. The paper discusses the nature of the FVP and tries to discover how it works when it tacitly causes the false appearance of a fallacious argument.

We present a sequent calculus for extensional mereology. It extends the classical first-order sequent calculus with identity by rules of inference corresponding to well-known mereological axioms. Structural rules, including cut, are admissible.

The paper examines Posterior Analytics II 11, 94a20-36 and makes three points. (1) The confusing formula ‘given what things, is it necessary for this to be’ [τίνων ὄντων ἀνάγκη τοῦτ᾿ εἶναι] at a21-22 introduces material cause, not syllogistic necessity. (2) When biological material necessitation is the only causal factor, Aristotle is reluctant to formalize it in syllogistic terms, and this helps to explain why, in II 11, he turns to geometry in order to illustrate a kind of material cause that (...) can be expressed as the middle term of an explanatory syllogism. (3) If geometrical proof is viewed as a complex construction built on simpler constructions, it can in effect be described as a case of purely material constitution. (shrink)

Tarski, Carnap and Quine spent the academic year 1940?1941 together at Harvard. In their autobiographies, both Carnap and Quine highlight the importance of the conversations that took place among them during the year. These conversations centred around semantical issues related to the analytic/synthetic distinction and on the project of a finitist/nominalist construction of mathematics and science. Carnap's Nachlaß in Pittsburgh contains a set of detailed notes, amounting to more than 80 typescripted pages, taken by Carnap while these discussions were taking (...) place. In my article, I present a survey of these notes with special emphasis on Tarski's rejection of the analytic/synthetic distinction, the passage from typed languages to first-order languages, Tarski's finitism/nominalism, and the construction of a finitist language for mathematics and science. (shrink)

We will discuss a mathematical proof found in Wittgenstein’s Nachlass, a constructive version of Euler’s proof of the infinity of prime numbers. Although it does not amount to much, this proof allows us to see that Wittgenstein had at least some mathematical skills. At the very last, the proof shows that Wittgenstein was concerned with mathematical practice and it also gives further evidence in support of the claim that, after all, he held a constructivist stance, at least during the transitional (...) period of his thought (1929-33). (shrink)