a collection of articles by philosophers of mathematics on themes associated with the work of Paul Benacceraf.

Hilary Putnam first published the consistency objection against Ludwig Wittgenstein’s account of mathematics in 1979. In 1983, Putnam and Benacerraf raised this objection against all conventionalist accounts of mathematics. I discuss the 1979 version and the scenario argument, which supports the key premise of the objection. The wide applicability of this objection is not apparent; I thus raise it against an imaginary axiomatic theory T similar to Peano arithmetic in all relevant aspects. I argue that a conventionalist can explain the (...) consistency of T and suggest that an analogous explanation can be provided for the consistency of Peano arithmetic. (shrink)

I attempt to accommodate the phenomenon of vagueness with classical logic and bivalence. I hold that for any vague predicate there is a sharp cut-off between the things that satisfy it and the things that do not; I claim that this is due to the greater naturalness of one of the candidate meanings of that predicate. I extend the thought to the problem of the many and Benacerraf cases. I go on to explore the idea that it is ontically indeterminate (...) what the most natural meanings are, and hence ontically indeterminate where the sharp cut-off in a sorites series is. (shrink)

When I first began to take an interest in the debate over nominalism in philosophy of mathematics, some twenty-odd years ago, the issue had already been under discussion for about a half-century. The terms of the debate had been set: W. V. Quine and others had given “abstract,” “nominalism,” “ontology,” and “Platonism” their modern meanings. Nelson Goodman had launched the project of the nominalistic reconstruction of science, or of the mathematics used in science, in which Quine for a time had (...) joined him before turning against him. William Alston, Rudolf Carnap, and Michael Dummett had raised doubts about what the point of Goodman’s exercise could be, and though they had unfortunately been largely ignored, Quine’s contention that the exercise cannot be successfully completed had gained wide publicity as the so-called “indispensability” argument against nominalism. By contrast, two subtle discussions of Paul Benacerraf had been appropriated by nominalists and turned into the so-called “multiple reductions” and “epistemological” arguments for nominalism. (shrink)

One textbook may introduce the real numbers in Cantor’s way, and another in Dedekind’s, and the mathematical community as a whole will be completely indifferent to the choice between the two. This sort of phenomenon was famously called to the attention of philosophers by Paul Benacerraf. It will be argued that structuralism in philosophy of mathematics is a mistake, a generalization of Benacerraf’s observation in the wrong direction, resulting from philosophers’ preoccupation with ontology.

The paper is devoted to analysis of P. Benacerraf’s argument against set-theoretic reductionist realism which is a fragment of a broader argument, know as the “identification problem”. The analyzed fragment of P. Benacerraf’s argument concerns the possibility of reducing of mathematical notions to set-theoretic notions. The paper presents a reconstruction of P. Benacerraf’s original argumentation, its analysis and also several possible objections proposed by P. Benacerraf himself about 30 years later after the original publication. Namely, he claimed (1) that a (...) set-theoretic definition of natural numbers in G. Frege’s fashion can serve as a proper and unique set-theoretic definition, (2) that his argument doesn’t undermine eliminative reductionsts’ position, (3) that even if there are no argument possible in favor of some particular set-theoretic definition of natural numbers one may take set-theoretic realism for granted. An analysis of the mentioned possible objections shows their dependence on a number of additional premises. The paper demonstrates that P. Benacerraf’s objections on his own argument against set-theoretic realism either have a pragmatic character themselves or essentially rely on additional arguments that are justified pragmatically or require additional argumentation. For example, his possible objections require that set theory is considered as the only true foundational theory in mathematics, and that it has several important pragmatic virtues, like convenience of use to formalize other mathematical theories. In some cases, P. Benacerraf’s objections on their own, or the indicated additional principles may well be called into question, which demonstrates the insufficiency of P. Benacerraf’s objections against his original argument. Without the mentioned pragmatic arguments P. Benacerraf’s objections become a kind of belief in mysticism. Accordingly, his doubts about his own argument against set-theoretical realism seem insufficient to reject the problem of identification and save the position of set-theoretical realism from collapse. (shrink)

The argument is a dialectical one. It is not a direct proof that the mind is something more than a machine, but a schema of disproof for any particular version of mechanism that may be put forward. If the mechanist maintains any specific thesis, I show that [146] a contradiction ensues. But only if. It depends on the mechanist making the first move and putting forward his claim for inspection. I do not think Benacerraf has quite taken the point. He (...) criticizes me both for "failing to notice" that my ability to show that the Gödel sentence of a formal system is true "depends very much on how he is given. (shrink)

The twentieth century has witnessed an unprecedented 'crisis in the foundations of mathematics', featuring a world-famous paradox, a challenge to 'classical' mathematics from a world-famous mathematician, a new foundational school, and the profound incompleteness results of Kurt Gödel. In the same period, the cross-fertilization of mathematics and philosophy resulted in a new sort of 'mathematical philosophy', associated most notably with Bertrand Russell, W. V. Quine, and Gödel himself, and which remains at the focus of Anglo-Saxon philosophical discussion. The present collection (...) brings together in a convenient form the seminal articles in the philosophy of mathematics by these and other major thinkers. It is a substantially revised version of the edition first published in 1964 and includes a revised bibliography. The volume will be welcomed as a major work of reference at this level in the field. (shrink)

The twentieth century has witnessed an unprecedented 'crisis in the foundations of mathematics', featuring a world-famous paradox (Russell's Paradox), a challenge to 'classical' mathematics from a world-famous mathematician (the 'mathematical intuitionism' of Brouwer), a new foundational school (Hilbert's Formalism), and the profound incompleteness results of Kurt Gödel. In the same period, the cross-fertilization of mathematics and philosophy resulted in a new sort of 'mathematical philosophy', associated most notably (but in different ways) with Bertrand Russell, W. V. Quine, and Gödel himself, (...) and which remains at the focus of Anglo-Saxon philosophical discussion. The present collection brings together in a convenient form the seminal articles in the philosophy of mathematics by these and other major thinkers. It is a substantially revised version of the edition first published in 1964 and includes a revised bibliography. The volume will be welcomed as a major work of reference at this level in the field. (shrink)

The present collection brings together in a convenient form the seminal articles in the philosophy of mathematics by these and other major thinkers.

What Numbers Could Not Be’) that an adequate account of the numbers and our arithmetic practice must satisfy not only the conditions usually recognized to be necessary: (a) identify some w-sequence as the numbers, and (b) correctly characterize the cardinality relation that relates a set to a member of that sequence as its cardinal number—it must also satisfy a third condition: the ‘<’ of the sequence must be recursive. This paper argues that adding this further condition was a mistake—any w-sequence (...) would do, no matter how undecidable its ‘<’ relation. (shrink)

The incredible achievements of modern scientific theories lead most of us to embrace scientific realism: the view that our best theories offer us at least roughly accurate descriptions of otherwise inaccessible parts of the world like genes, atoms, and the big bang. In Exceeding Our Grasp, Stanford argues that careful attention to the history of scientific investigation invites a challenge to this view that is not well represented in contemporary debates about the nature of the scientific enterprise. The historical record (...) of scientific inquiry, Stanford suggests, is characterized by what he calls the problem of unconceived alternatives. Past scientists have routinely failed even to conceive of alternatives to their own theories and lines of theoretical investigation, alternatives that were both well-confirmed by the evidence available at the time and sufficiently serious as to be ultimately accepted by later scientific communities. Stanford supports this claim with a detailed investigation of the mid-to-late 19th century theories of inheritance and generation proposed in turn by Charles Darwin, Francis Galton, and August Weismann. He goes on to argue that this historical pattern strongly suggests that there are equally well-confirmed and scientifically serious alternatives to our own best theories that remain currently unconceived. Moreover, this challenge is more serious than those rooted in either the so-called pessimistic induction or the underdetermination of theories by evidence, in part because existing realist responses to these latter challenges offer no relief from the problem of unconceived alternatives itself. Stanford concludes by investigating what positive account of the spectacularly successful edifice of modern theoretical science remains open to us if we accept that our best scientific theories are powerful conceptual tools for accomplishing our practical goals, but abandon the view that the descriptions of the world around us that they offer are therefore even probably or approximately true. (shrink)

The place of the emotions among the passions -- The analytic of the emotions I -- The analytic of the emotions II -- The dialectic of the emotions -- Pride, arrogance, and humility -- Shame, embarrassment, and guilt -- Envy -- Jealousy -- Anger -- Love -- Friendship -- Sympathy and empathy.

Some scientific categories seem to correspond to genuine features of the world and are indispensable for successful science in some domain; in short, they are natural kinds. This book gives a general account of what it is to be a natural kind and puts the account to work illuminating numerous specific examples.

By the time of his death in 2006, Sir Peter Strawson was regarded as one of the world's most distinguished philosophers. Unavailable for many years,_ Scepticism and Naturalism_ is a profound reflection on two classic philosophical problems by a philosopher at the pinnacle of his career. Based on his acclaimed Woodbridge lectures delivered at Columbia University in 1983, Strawson begins with a discussion of scepticism, which he defines as questioning the adequacy of our grounds for holding various beliefs. He then (...) draws deftly on Hume and Wittgenstein to argue that we must distinguish between 'hard', scientific naturalism; or 'soft', humanistic naturalism. In the remaining chapters the author takes up several issues in which sceptical doubts play an important role, in particular the nature of transcendental arguments and including the objectivity of moral philosophy, the mental and the physical, and the existence of abstract entities. _Scepticism and Naturalism_ is essential reading for those seeking an introduction to the work of one of the twentieth century’s most important and original philosophers. This reissue includes a substantial new foreword by Quassim Cassam and a fascinating intellectual autobiography by Strawson, which together form an excellent introduction to his life and work. (shrink)

The concept of normal is so familiar that it can be hard to imagine contemporary life without it. Yet the term entered everyday speech only in the mid-twentieth century. Before that, it was solely a scientific term used primarily in medicine to refer to a general state of health and the orderly function of organs. But beginning in the middle of the twentieth century, normal broke out of scientific usage, becoming less precise and coming to mean a balanced condition to (...) be maintained and an ideal to be achieved. In Normality, Peter Cryle and Elizabeth Stephens offer an intellectual and cultural history of what it means to be normal. They explore the history of how communities settle on any one definition of the norm, along the way analyzing a fascinating series of case studies in fields as remote as anatomy, statistics, criminal anthropology, sociology, and eugenics. Cryle and Stephens argue that since the idea of normality is so central to contemporary disability, gender, race, and sexuality studies, scholars in these fields must first have a better understanding of the context for normality. This pioneering book moves beyond binaries to explore for the first time what it does—and doesn’t—mean to be normal. (shrink)

By the time of his death in 2006, Sir Peter Strawson was regarded as one of the world's most distinguished philosophers. Unavailable for many years,_ Scepticism and Naturalism_ is a profound reflection on two classic philosophical problems by a philosopher at the pinnacle of his career. Based on his acclaimed Woodbridge lectures delivered at Columbia University in 1983, Strawson begins with a discussion of scepticism, which he defines as questioning the adequacy of our grounds for holding various beliefs. He then (...) draws deftly on Hume and Wittgenstein to argue that we must distinguish between 'hard', scientific naturalism; or 'soft', humanistic naturalism. In the remaining chapters the author takes up several issues in which sceptical doubts play an important role, in particular the nature of transcendental arguments and including the objectivity of moral philosophy, the mental and the physical, and the existence of abstract entities. _Scepticism and Naturalism_ is essential reading for those seeking an introduction to the work of one of the twentieth century’s most important and original philosophers. This reissue includes a substantial new foreword by Quassim Cassam and a fascinating intellectual autobiography by Strawson, which together form an excellent introduction to his life and work. (shrink)