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

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.

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)

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

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 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)

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)

Now in a third edition, Introduction to Philosophy: Classical and Contemporary Readings is a highly acclaimed, topically organized collection that covers five major areas of philosophy--theory of knowledge, philosophy of religion, philosophy of mind, freedom and determinism, and moral philosophy. Editor Louis P. Pojman enhances the text's topical organization by arranging the selections into a pro/con format to help students better understand opposing arguments. He also includes accessible introductions to each chapter, subsection, and individual reading, a unique feature for an (...) anthology of this depth. While the book focuses on a compelling sampling of classical material--including selections from Plato, Aristotle, Descartes, Locke, Berkeley, Hume, and Kant--it also incorporates some of philosophy's best twentieth-century and contemporary work, featuring articles by Bertrand Russell, Richard Taylor, John Searle, Thomas Nagel, and others. This third edition contains an expanded glossary, more extensive section introductions, and twelve new selections: Karl Popper: "Epistemology without a Knowing Subject" Richard Rorty: "Dismantling Truth: Solidarity Versus Objectivity" Daniel Dennett: "Postmodernism and Truth" Bruce Russell: "The Problem of Evil: Too Much Suffering" David Chalmers: "Against Materialism: Can Consciousness Be Reductively Explained?" Baron Paul Henri d'Holbach: "A Defense of Determinism" Michael Levin: "A Compatibilist Defense of Moral Responsibility" Plato: Socratic Morality: "Crito" Herodotus: "Custom Is King" J. L. Mackie: "The Subjectivity of Values" Louis P. Pojman: "A Critique of Mackie's Theory of Moral Subjectivism" Thomas Nagel: "Moral Luck". (shrink)

In ”Skepticism,” Peter Klein distinguishes between the “Academic Skeptic” who proposes that we cannot have knowledge of a certain set of propositions and the “Pyrrhonian Skeptic” who refrains from opining about whether we can have knowledge. Klein argues that Academic Skepticism is plausibly supported by a “Closure Principle‐style” argument based on the claim that if x entails y and S has justification for x, then S has justification for y. He turns to contextualism to see if it can contribute to (...) the discussion between one who claims that we can have knowledge about some epistemically interesting class of propositions and the Academic Skeptic. He outlines the background of Pyrrhonian Skepticism, pointing out that the Pyrrhonist withholds assent concerning our knowledge‐bearing status because reason cannot provide an adequate basis for assent. He assesses three possible patterns of reasoning, and concludes that the Pyrrhonist view, that reason cannot resolve matters concerning the nonevident, is vindicated. (shrink)

First published in 1987. Routledge is an imprint of Taylor & Francis, an informa company.

First published in 1987. Routledge is an imprint of Taylor & Francis, an informa company.

This paper advances the research on personalization of political communication by investigating whether this process of focusing on politicians instead of political issues plays a role on Twitter. Results of a content analysis of 5,530 tweets posted in the run-up to the German federal election provide evidence that Twitter communication refers more often to politicians than to issues. However, tweets containing personal characteristics about political leaders play only a marginal role. When distinguishing among different groups of actors on Twitter (journalists, (...) politicians, citizens), we find that citizens focus more on candidates than do journalists or politicians. Investigating the impact of a televised debate on Twitter communication, we observe that this person-centered event puts the focus on individual politicians instead of issues. (shrink)

It has been over two decades since Miranda Fricker labeled epistemic injustice, in which an agent is wronged in their capacity as a knower. The philosophical literature has proliferated with variants and related concepts. By considering cases in popular music, we argue that it is worth distinguishing a parallel phenomenon of art-interpretive injustice, in which an agent is wronged in their creative capacity as a possible artist. In section 1, we consider the prosecutorial use of rap lyrics in court as (...) a central case of this injustice. In section 2, we distinguish art-interpretive injustice from other categories already discussed in recent literature. In section 3, we discuss the relationship between genre discourse and identity prejudice. The case for recognizing the category of art-interpretive injustice is that it allows one to recognize a class of harms as being importantly related in ways that one would otherwise overlook. (shrink)