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)

This volume is a collection of essays in appreciation, analysis and honor of Paul Ziff, one of the leading American philosophers of the post-World War II period. The essays address questions that loomed large in Ziff's own work. Essays by Zeno Vendler, Jay Rosenberg, and Tom Patton address topics in philosophy of language: understanding, misunderstanding, rules, regularities, and proper names. Michael Resnik examines the nature of numbers, Rita Nolan addresses `mutant predicates', and Peter Alexander discusses microscopes and corpuscles. Douglas (...) C. Long ruminates on Ziff's claim that machines can neither think nor feel. The essays of Dale Jamieson, Bill E. Lawson, Douglas Dempster, and Joseph Ullian address various questions in aesthetics: aesthetic appreciation and morality, expression, the scope of appreciation, and the aesthetics of sport. In the spirit of Ziff, Douglas Stalker criticizes some of the `mush' that looms large in our intellectual lives. The volume begins with a reminiscence by Paul Benacerraf, and ends with selections from an unpublished volume of plays by Paul Ziff. The volume should appeal to anyone whose work has been influenced by Ziff, or is interested in central philosophical problems concerning language, mind, and art. (shrink)

This volume features essays about and by Paul Benacerraf, whose ideas have circulated in the philosophical community since the early nineteen sixties, shaping key areas in the philosophy of mathematics, the philosophy of language, the philosophy of logic, and epistemology. The book started as a worskhop held in Paris at the Collège de France in May 2012 with the participation of Paul Benacerraf. The introduction addresses the methodological point of the legitimate use of so-called “Princess Margaret (...) Premises” in drawing philosophical conclusions from Gödel’s first incompleteness theorem. The book is then divided into three sections. The first is devoted to an assessment of the improved version of the original dilemma of “Mathematical Truth” due to Hartry Field: the challenge to the platonist is now to explain the reliability of our mathematical beliefs given the very subject matter of mathematics, either pure or applied. The second addresses the issue of the ontological status of numbers: Frege’s logicism, fictionalism, structuralism, and Bourbaki’s theory of structures are called up for an appraisal of Benacerraf’s negative conclusions of “What Numbers Could Not Be.” The third is devoted to supertasks and bears witness to the unique standing of Benacerraf’s first publication: “Tasks, Super-Tasks, and Modern Eleatics” in debates on Zeno’s paradox and associated paradoxes, infinitary mathematics, and constructivism and finitism in the philosophy of mathematics. Two yet unpublished essays by Benacerraf have been included in the volume: an early version of “Mathematical Truth” from 1968 and an essay on “What Numbers Could Not Be” from the mid 1970’s. A complete chronological bibliography of Benacerraf’s work to 2016 is provided.Essays by Jody Azzouni, Paul Benacerraf, Justin Clarke-Doane, Sébastien Gandon, Brice Halimi, Jon Pérez Laraudogoitia, Mary Leng, Antonio Leòn-Sànchez and Ana Leòn-Mejía, Marco Panza, Fabrice Pataut, Philippe de Rouilhan, Andrea Sereni, and Stewart Shapiro. (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)

Benacerraf criticizes Lucas’ argument against Mechanism because, in his opinion, it depends too much on how the system we are talking about is presented and because the argument put in form of challenge reduces itself to a contest of wits between Lucas and the mechanists. In Benacerraf opinion, Lucas should clarify the sense of utilised notions and the argument would have to be reconstructed as formally as possible, in order to determine the involved philosophical premises. Moreover Benacerraf (...) maintains that, instead of abandoning the idea that human mind is a machine, we could assume that minds are machines for which it is not possible to prove the consistency or that they are inadequate for arithmetic; moreover minds could be machines whose characteristics we are not able to specify. However, Lucas answers that the requirement of reconstructing his argument in a formal way misunderstands his project: his argument is not a direct proof but a dialectical argument, a schema of disproof for any particular version of mechanist argument, and so the attempt to reconstruct it as a rigorous proof is a distortion of the original argument, that is essentially dialectical. What about the hypothesis suggested by Benacerraf, Lucas disputes that we are able to manage arithmetic and we don’t seem as inconsistent as an inconsistent system is, because we are selective while an inconsistent system is not; at the other hand, the idea that we are machine but we don’t know anything about what kind of machine we are evacuates Mechanism of all content. (shrink)

In "Mathematical Truth", Paul Benacerraf articulated an epistemological problem for mathematical realism. His formulation of the problem relied on a causal theory of knowledge which is now widely rejected. But it is generally agreed that Benacerraf was onto a genuine problem for mathematical realism nevertheless. Hartry Field describes it as the problem of explaining the reliability of our mathematical beliefs, realistically construed. In this paper, I argue that the Benacerraf Problem cannot be made out. There simply (...) is no intelligible problem that satisfies all of the constraints which have been placed on the Benacerraf Problem. The point generalizes to all arguments with the structure of the Benacerraf Problem aimed at realism about a domain meeting certain conditions. Such arguments include so-called "Evolutionary Debunking Arguments" aimed at moral realism. I conclude with some suggestions about the relationship between the Benacerraf Problem and the Gettier Problem. (shrink)

Paul Benacerraf’s argument that mathematical realism is apparently incompatible with mathematical knowledge has been widely thought to also show that a priori knowledge in general is problematic. Although many philosophers have rejected Benacerraf’s argument because it assumes a causal theory of knowledge, some maintain that Benacerraf nevertheless put his finger on a genuine problem, even though he didn’t state the problem in its most challenging form. After diagnosing what went wrong with Benacerraf’s argument, I argue (...) that a new, more challenging, version of Benacerraf’s problem can be constructed. The new version—what I call the Defeater Version—of Benacerraf’s problem makes use of a no-defeater condition on knowledge and justification. I conclude by arguing that the best way to avoid the problem is to construct a theory of how a priori judgments reliably track the facts. I also suggest four different kinds of theories worth pursuing. (shrink)

Hartry Field’s epistemological challenge to the mathematical platonist is often cast as an improvement on Paul Benacerraf’s original epistemological challenge. I disagree. While Field’s challenge is more difficult for the platonist to address than Benacerraf’s, I argue that this is because Field’s version is a special case of what I call the ‘sociological challenge’. The sociological challenge applies equally to platonists and fictionalists, and addressing it requires a serious examination of mathematical practice. I argue that the non-sociological (...) part of Field’s challenge amounts to a minor reformulation of Benacerraf’s original challenge. So, I contend, Field’s challenge is not an improvement on Benacerraf’s. What is new to Field’s challenge is as much a problem for the fictionalist as it is for the platonist. (shrink)

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

I discuss Steinhart’s argument against Benacerraf’s famous multiple-reductions argument to the effect that numbers cannot be sets. Steinhart offers a mathematical argument according to which there is only one series of sets to which the natural numbers can be reduced, and thus attacks Benacerraf’s assumption that there are multiple reductions of numbers to sets. I will argue that Steinhart’s argument is problematic and should not be accepted.

A natural realist approach to the philosophy of arithmetic is defended by way of considering and arguing against contemporary attempts to solve Paul Benacerraf's dilemma . The first horn of the dilemma concerns the existence of abstract mathematical objects, which seems necessitated by a desire for a unified semantics. Benacerraf adopts an extensional semantics whereby the reference of terms for natural numbers must be abstract objects. The second horn concerns a desirable causal constraint on knowledge, according to (...) which "for X to know that S is true requires some causal relation to obtain between X and the referents of the names, predicates, and quantifiers of S". Within the philosophical tradition, Benacerraf's dilemma crystallizes the tension between realists and empiricists . It is shown that natural realism and naturalism meet. ;Both horns of the dilemma are amended.objects are conceived along conceptualist lines such that their existence is not mind-independent. The causal constraint is refined by drawing upon the work of Mark Steiner . Natural realism is the notion that the truth-values of arithmetical statements are recognition-transcendent. Natural realism explains why one and only one arithmetic is applicable. It is in some sense discovered, which is captured by the idea that arithmetical statements have truth-values even if one is not able to adduce what they are. ;It is argued that with proper emendation, like that of Philip Kitcher , empiricism as envisioned by J. S. Mill is defensible. Furthermore, the epistemology of arithmetical knowledge is divided into two tiers. The first principles of arithmetic are acquired by causal interaction with physical objects. As Kitcher has shown, empiricism functions at the first-tier. The a priori is revised such that arithmetical knowledge generated from the first-tier is considered non-empirical. ;It is argued that pragmatists' indispensability argument, once qualified, allows that arithmetic's applicability is one reason to consider natural realism for that domain . Furthermore, this thesis provides a case where aspects of Hilary Putnam's early and later views are rendered consistent. By way of utilizing both Putnam's writing in favour of realism , and those that break with them , it is suggested that he should not have abandoned his earlier view. In a nutshell, the realization of knowledge can depend upon agents' values, methods, and so on, without forcing the abandonment of realism . Putnam's early view sets the criteria for when one can be a realist about a given domain. (shrink)

Evolutionary debunking arguments purport to show that robust moral realism, the metaethical view that there are non-natural and mind-independent moral properties and facts that we can know about, is incompatible with evolutionary explanations of morality. One of the most prominent evolutionary debunking arguments is advanced by Sharon Street, who argues that if moral realism were true, then objective moral knowledge is unlikely because realist moral properties are evolutionary irrelevant and moral beliefs about those properties would not be selected for. However, (...) no evolutionary, causal explanation plays an essential role in reaching the argument’s epistemological conclusion. Street’s argument depends on the Benacerraf-Field challenge, which is the challenge to explain the reliability of our moral beliefs about causally inert moral properties. The Benacerraf-Field challenge relies on metaphysically necessary facts about realist moral properties, rather than on contingent Darwinian facts about the origin of our moral beliefs. Attempting to include an essential causal empirical premise yet avoiding recourse to the Benacerraf-Field problem yields an argument that is either self-defeating or of limited scope. Ultimately, evolutionary, causal explanations of our moral beliefs and their consequences do not present the strongest case against robust moral realism. Rather, the question is whether knowledge of casually-inert, mind-intendent properties is plausible at all. (shrink)

Paul Benacerraf’s wide reputation as a philosopher of mathematics rests on a surprisingly small output of articles. The complete bibliography included in this volume shows only 11 articles, together with his doctoral thesis on logicism and the two versions of the introduction to the excellent anthology in the philosophy of mathematics that he edited with Hilary Putnam. Of these, two articles stand out, “What Numbers Could Not Be,” published in 1965, and “Mathematical Truth,” published in 1973. Most of (...) the articles in this volume comment on one or both of these two famous essays. (shrink)

Does Wittgenstein have a coherent philosophy of mathematics? Here, I will be concerned with showing that the answer is positive. However, given that his life-long philosophical perspective on mathematics tends to be misleading, I focus on the specific problem posed by Paul Benacerraf in ‘Mathematical Truth’, that is: the puzzle about how to reconcile the metaphysics with the epistemology for mathematics. My aim is to show that there is an adequate anti-platonistic solution to that puzzle in the mature (...) writings of the Austrian philosopher. (shrink)

Abstract Robert Hanna has recently advanced a theory of non-conceptual content, the central claim of which is that "it is perfectly possible for there to be directly referential intuitions without concepts". Hanna bases this claim in Kant's account of intuition in the Critique of Pure Reason, and so extends his Kantian non-conceptualism beyond the epistemology of empirical knowledge into the realm of mathematics. Thus, Hanna has proposed a Kantian non-conceptualist solution to a well-known dilemma set out by Paul (...) class='Hi'>Benacerraf in his 1973 paper, "Mathematical Truth". I argue that Hanna is right about Kant's non-conceptualism, but mistaken in its application to Benacerraf's Dilemma. (shrink)

Hewitt's Conceptual Integrated Science is the most widely used textbook in Integrated Science courses. This course covers chemistry, physics, biology, earth science, and astronomy and is mostly taken by Elementary-Education Majors, i.e. future grade-school teachers who are required to take a survey-of-science course.

Johannes Kepler (1571-1630) spielte als genialer Entdecker von Naturgesetzen eine zentrale Rolle in der frühen Naturphilosophie Schellings und Hegels; die Romantik feierte ihn als Prototypen des Genies schlechthin. Um 1840 setzt sich Schelling in einem veränderten Kontext für die erste Gesamtausgabe der Werke Keplers ein: Die Naturphilosophie wird nun vom Empirismus und Induktivismus scharf kritisiert. Neu entdeckte Dokumente belegen, wie man dennoch auf Kepler zurückgreifen konnte; gezeigt wird, dass sich idealistische und nach-idealistische Philosophieauffassungen also nicht ausschließen, so...

In some previous work, I tried to give a concept-based account of the nature of our entitlement to certain very basic inferences (see the papers in Part III of Boghossian 2008b). In this previous work, I took it for granted, along with many other philosophers, that we understood well enough what it is for a person to infer. In this paper, I turn to thinking about the nature of inference itself. This topic is of great interest in its own right (...) and surprisingly understudied by philosophers. A correct understanding of inference promises to shed light on a number of important topics. In particular, it threatens to undermine the sort of concept-based story about entitlement to which I had previously been attracted. (shrink)