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 Two yet unpublished essays by Benacerraf have been included in the volume: an early version of “Mathematical Truth” from 1968 and an essay on “What Numbers Could Not Be” from the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)

Tim Crane characterizes the cognitive role of singular thought via singular mental files: the application of such files to more than one object is senseless. As many do, he thus stresses the contrast between ‘singular’ and ‘general’. I give a counterexample, plurally-directed singular thought, and I offer alternative characterizations of singular thought—better described as ‘objects-directed thought’—initially in terms of the defeasibility of the descriptions associated with one's thinking of an object, and then more broadly in terms of whether descriptions of (...) the object or description-independent epistemic routes to the object are primarily operative in an agent's thinking. (shrink)

If we must take mathematical statements to be true, must we also believe in the existence of abstract eternal invisible mathematical objects accessible only by the power of pure thought? Jody Azzouni says no, and he claims that the way to escape such commitments is to accept true statements which are about objects that don't exist in any sense at all. Azzouni illustrates what the metaphysical landscape looks like once we avoid a militant Realism which forces our (...) commitment to anything that our theories quantify. Escaping metaphysical straitjackets, while retaining the insight that some truths are about objects that do exist, Azzouni says that we can sort scientifically-given objects into two categories: ones which exist, and to which we forge instrumental access in order to learn their properties, and ones which do not, that is, which are made up in exactly the same sense that fictional objects are. He offers as a case study a small portion of Newtonian physics, and one result of his classification of its ontological commitments, is that it does not commit us to absolute space and time. (shrink)

Many philosophers hold that physical laws have a unique modal status known as nomic necessity which is weaker than metaphysical necessity. This orthodox view has come into question in the past few decades. In particular, the metaphysical view known as essentialism has provided an argument that the laws of nature are necessary in the strongest possible sense. It seems obvious to many that at least some essentialist arguments in favor of the necessity of scientific claims are going to be sound. (...) For example, the view that claims like "water is H2O" are necessary has itself 2 become an orthodox view. However, the question of whether laws, like the law of conservation of energy, or the law of gravity, are necessary is far more contentious. Philosophers divide roughly into two camps, law necessitarians1 who hold that the laws are necessary in the strongest sense and contingency theorists who hold that they are at least in some sense contingent. One argument for the necessitarian position is via an essentialist theory of the transworld identity of properties. In this paper I defend such a theory of the identity of properties and its necessitarian consequences from one major criticism. To focus the paper, I center the discussion on a single critic, E. J. Lowe. In his book, The Four Category Ontology, he offers a criticism of the essentialist argument for necessitarianism via an analogy with other forms of transworld identity and intuitions about the contingency of the physical constants2. I undermine the usefulness of Lowe's analogy by examining the purposes of attributions of properties. I also show that the essentialist's position can allow it to accommodate the intuitions of contingency in a way that fits best with the purpose behind property attributions. (shrink)

Jody Azzouni argues that we involuntarily experience certain physical items, certain products of human actions, and certain human actions themselves as having meaning-properties. We understand these items as possessing meaning or as having truth values. For example, a sign on a door reading "Drinks Inside" strikes native English speakers as referring to liquids in the room behind the door. The sign has a truth value--if no drinks are found in the room, the sign is misleading. Someone pointing in (...) a direction has the same effect: we experience her gesture as significant. Azzouni does not suggest that we don't recognize the expectations or intentions of speakers ; we do recognize that the person pointing in a certain direction intends for us to understand her gesture's significance. Nevertheless, Azzouni asserts that we experience that gesture as having significance independent of her intentions. The gesture is meaningful on its own. The same is true of language, both spoken and written. We experience the meanings of language artifacts as independent of their makers' intentions in the same way that we experience an object's shape as a property independent of the object's color. There is a distinctive phenomenology to the experience of understanding language, and Semantic Perception shows how this phenomenology can be brought to bear as evidence for and against competing theories of language. (shrink)

Most philosophers of mathematics try to show either that the sort of knowledge mathematicians have is similar to the sort of knowledge specialists in the empirical sciences have or that the kind of knowledge mathematicians have, although apparently about objects such as numbers, sets, and so on, isn't really about those sorts of things as well. Jody Azzouni argues that mathematical knowledge really is a special kind of knowledge with its own special means of gathering evidence. He analyses (...) the linguistic pitfalls and misperceptions philosophers in this field are often prone to, and explores the misapplications of epistemic principles from the empirical sciences to the exact sciences. What emerges is a picture of mathematics both sensitive to mathematical practice, and to the ontological and epistemological issues that concern philosophers. (shrink)

When ordinary people - mathematicians among them - take something to follow from something else, they are exposing the backbone of our self-ascribed ability to reason. Jody Azzouni investigates the connection between that ordinary notion of consequence and the formal analogues invented by logicians. One claim of the book is that, despite our apparent intuitive grasp of consequence, we do not introspect rules by which we reason, nor do we grasp the scope and range of the domain, as (...) it were, of our reasoning. This point is illustrated with a close analysis of a paradigmatic case of ordinary reasoning: mathematical proof. (shrink)

All sides in the recent debates over the Quine‐Putnam Indispensability thesis presuppose Quine's criterion for determining what a discourse is ontologically committed to. I subject the criterion to scrutiny, especially in regard to the available competitor‐criteria, asking what means of evaluation there are for comparing alternative criteria against each other. Finding none, the paper concludes that ontological questions, in a certain sense, are philosophically indeterminate.

I revisit my earlier arguments for the (trivial) inconsistency of natural languages, and take up the objection that no such argument can be established on the basis of surface usage. I respond with the evidential centrality of surface usage: the ways it can and can't be undercut by linguistic science. Then some important ramifications of having an inconsistent natural language are explored: (1) the temptation to engage in illegitimate reductio reasoning, (2) the breakdown of the knowledge idiom (because its facticity (...) isn't comfortable with every sentence being true and false). Restoring the utility of the knowledge idiom motivates regimentation - the consistentist reinterpretation of natural language inferences (as they occur in real time). It's then sketched how to give the semantics of a natural language despite its inconsistency: Necessary and sufficient truth conditions are dropped in favor of necessary truth conditions and sufficient truth conditions. (shrink)

A normative constraint on theories about objects which we take to be real is explored: such theories are required to track the properties of the objects which they are theories of. Epistemic views in which observation (and generalizations of it) play a central role, and holist views which see epistemic virtues as applicable only to whole theories, are contrasted in the light of this constraint. It's argued that global-style epistemic virtues can't meet the constraint, although (certain) epistemic views within which (...) observation (and, more generally, thick epistemic access) are centralized do meet it. (shrink)

In this book Jody Azzouni challenges existing epistemological conventions about knowledge: what it means to know something, who or what is seen as knowing, and how we talk about it. He argues that the classic restrictive conditions philosophers routinely place on knowers only hold in special cases, and suggests that knowledge can be equally attributed to children, sophisticated animals, unsophisticated animals, and machinery or devices. Through this perspective and a close examination of its relation to linguistics and psychology, (...) Azzouni freshly approaches longstanding epistemological puzzles including the dogmatism paradox, Gettier puzzles, Agrippa's trilemma, and the surprise-exam paradox. (shrink)

Philosophers are very fond of making non-factualist claims—claims to the effect that there is no fact of the matter as to whether something is the case. But can these claims be coherently stated in the context of classical logic? Some care is needed here, we argue, otherwise one ends up denying a tautology or embracing a contradiction. In the end, we think there are only two strategies available to someone who wants to be a non-factualist about something, and remain within (...) the province of classical logic. But one of these strategies is rather controversial, and the other requires substantially more work than is often supposed. Being a non-factualist is no easy business, and it may not be the most philosophically perspicuous way to go. (shrink)

Our experience of objects is very rich. We perceive objects as possessing individuation conditions. This, however, is a projection of our senses and thinking. Azzouni shows the resulting austere metaphysics tames many ancient philosophical problems about constitution, as well as contemporary puzzles about reductionism.

Ordinary language and scientific language enable us to speak about, in a singular way, what we recognize not to exist: fictions, the contents of our hallucinations, abstract objects, and various idealized but nonexistent objects that our scientific theories are often couched in terms of. Indeed, references to such nonexistent items-especially in the case of the application of mathematics to the sciences-are indispensable. We cannot avoid talking about such things. Scientific and ordinary languages thus enable us to say things about Pegasus (...) or about hallucinated objects that are true, such as "Pegasus was believed by the ancient Greeks to be a flying horse," or "That elf I'm now hallucinating over there is wearing blue shoes." Standard contemporary metaphysical views and semantic analyses of singular idioms on offer in contemporary philosophy of language have not successfully accommodated these routine practices of saying true and false things about the nonexistent while simultaneously honoring the insight that such things do not exist in any way at all. That is, philosophers often feel driven to claim that such objects do exist, or they claim that all our talk isn't genuine truth-apt talk, but only pretence. This book reconfigures metaphysics in radical ways that allow the accommodation of our ordinary ways of speaking of what does not exist while retaining the absolutely crucial presupposition that such objects exist in no way at all, have no properties, and so are not the truth-makers for the truths and falsities that are about them. (shrink)

A distinction between the epistemic practices in mathematics and in the empirical sciences is rehearsed to motivate the epistemic role puzzle. This is distinguished both from Benacerraf's 1973 epistemic puzzle and from sceptical arguments against our knowledge of an external world. The stipulationist position is described, a position which can address this puzzle. Methods of avoiding the stipulationist position by using pure logic to provide knowledge of mathematical abstracta are discussed and criticized.

An argument for Trivialism, the view that natural languages are logically inconsistent, is provided that does not rely on contentious empirical assumptions about natural language terms such as “and” or “or.” Further, the view is defended against an important objection recently mounted against it by Thomas Hofweber.

An extensive exploration of the special properties of ‘exist’ is here undertaken. Two of several results are: Denying that `exist’ has associated with it a set of necessary and sufficient conditions has seemed to a number of philosophers to imply metaphysical nihilism . This is because it has seemed that without such conditions the target domain of `existence’ is arbitrarily open. I show this is wrong. Second, my analysis sheds light on the puzzling question of what we are asking when (...) we ask of something, `Does it exist?’ and mean that question in an ontically relevant way. (shrink)

The form of nominalism known as 'mathematical fictionalism' is examined and found wanting, mainly on grounds that go back to an early antinominalist work of Rudolf Carnap that has unfortunately not been paid sufficient attention by more recent writers.

May the theory of radical interpretation developed by Donald Davidson on the basis of Quine's arguments for the indeterminacy of translation help fix the meaning of the logical constants? In particular, may the theory exclude ways of conferring meaning on the constants which, although developed within the Davidsonian framework, would lead to unexpected results? Could an interpreter fix the meaning of the constants in a non classical way, although still in accordance with the guiding principles of the interpretative strategy? Or, (...) on the contrary, does the theory incorporate constraints on interpretation which are stronger than those imposed by the possibility of non classical, or deviant logics? I examine the particular case of negation within the framework of the natural deduction system developed by Gentzen and Prawitz, and conclude that Davidson's theory, understood as an empirical theory of truth, leaves the problem of knowing which meaning should be assigned to the constant for negation without a satisfactory solution. Although interpetation has to be carried out according to the principle of charity, disagreements may come up regarding which meaning should be assigned to "not" in a properly regimented natural language. (shrink)

This monograph presents Azzouni’s new approach to the rule-following paradox. His solution leaves intact an isolated individual’s capacity to follow rules, and it simultaneously avoids replacing the truth conditions for meaning-talk with mere assertability conditions for that talk. Kripke’s influential version of Wittgenstein’s rule-following paradox—and Wittgenstein’s views more generally—on the contrary, make rule-following practices and assertions about those practices subject to community norms without which they lose their cogency. Azzouni summarizes and develops Kripke’s original version of Wittgenstein’s rule-following (...) paradox to make salient the linchpin assumptions of the paradox. By doing so, Azzouni reveals how compelling Kripke’s earlier work on the paradox was. Objections raised over the years by Fodor, Forbes Ginsborg, Goldfarb, Tait, Wright, and many others, are all shown to fail. No straight solution can be made to work. Azzouni illustrates this in detail by showing that a popular family of straight solutions due to Lewis and refined by Williams, “reference magnetism,” fail as well. And yet an overlooked sceptical solution is still available in logical space. Azzouni describes a series of “disposition-meaning” private languages that he shows can be successfully used by a population of speakers to communicate with one another despite their ideolectical character. The same sorts of languages enable solitary “Robinson Crusoes” to survive and flourish in their island habitats. These languages—sufficiently refined—have the same properties normal human languages have; and this is the key to solving the rule-following paradox without sacrificing the individual’s authority over her self-imposed rules or her ability to follow those rules. Azzouni concludes this unusual monograph by uncovering a striking resemblance between the rule-following paradox and Hume’s problem of induction: he shows the rule-following paradox to be a corollary of Hume’s problem that arises when the problem of induction is applied to an individual’s own abilities to follow rules. (shrink)

_Knowledge and Reference in Empirical Science_ is a fascinating study of the bounds between science and language: in what sense, and of what, does science provide knowledge? Is science an instrument only distantly related to what's real? Can the language of science be used to adequately describe the truth? In this book, Jodi Azziouni investigates the technology of science - the actual forging and exploiting of causal links, between ourselves and what we endeavor to know and understand.

Kant discovered a philosophical problem with mathematical proof. Despite being a priori , its methodology involves more than analytic truth. But what else is involved? This problem is widely taken to have been solved by Frege’s extension of logic beyond its restricted (and largely Aristotelian) form. Nevertheless, a successor problem remains: both traditional and contemporary (classical) mathematical proofs, although conforming to the norms of contemporary (classical) logic, never were, and still aren’t, executed by mathematicians in a way that transparently reveals (...) why these proofs—written in the vernacular to this very day—succeed in conforming to those norms. (shrink)

There are several epistemic distinctions among truths that I have argued for in this paper. First, there are those truths which holdof every rationally accessible conceptual scheme (class A truths). Second, there are those truths which holdin every rationally accessible conceptual scheme (class B truths). And finally, there are those truths whose truthvalue status isindependent of the empirical sciences (class C truths). The last category broadly includes statementsabout systems and the statements they contain, as well as statements true by virtue (...) of the rules of language itself.At the risk of anachronism, I'll describe the positions of Carnap (1956); Quine (1980); Grice et al. (1956); the various Putnam's and myself in terms of the above distinctions: both Carnap and Quine (pretty much) think there are no class A or class B truths. Both Putnam (1975) and Putnam (1983c) think there are class A and class B truths, and that these classes overlap. I deny there are class B truths but affirm the existence of class A truths (although I haven't given explicit examples of the latter here). Finally, everyone here but Quine (1980) thinks there are class C truths (of one sort or another). Putnam (1975) attempts to show that certain class C truths are simultaneously class A and class B truths. Grice et al. (1956) take pains to distinguish the claim that there are class C truths from the claim that there are class A truths, and claim (against Quine, 1980) that no argument showing there are no class A truths shows there are no class C truths.On my interpretation of Quine (1980) he thinks that the nonexistence of class A truths shows there are no class C truths-given the extra bit of argument that a notion of ‘true by convention’ or ‘true by virtue of meaning’ without epistemic content, is a distinction without significance. But that issue, which is the one Grice et al. (1956) are concerned with, has not been the focus in this paper-and so in a sense I have shifted the terms of the original debate.Here I have been primarily concerned to distinguish epistemic notions and sort out how and in what ways they relate to each other. A primary tool in this exercise has been the explicit recognition that formal models of truth make universality an unlikely property of our conceptual schemes. If I have not convinced anyone that the epistemic notions sort out the way I think they do, I hope at least that some burden-shifting has occurred: that philosophers do not either take it for granted that certain notionsmust be expressible in any conceptual scheme or treat the fact that conceptual schemes must be (in some sense) limited as of little (philosophical) moment.On the other hand, if I am right about the epistemology, it follows that previous attempts to mark out the necessary structures in rationally accessible conceptual schemes via a priori truths is hopeless. What I think must replace their role, what I call globally incorrigible sets of sentences, is a topic for another time. (shrink)

Mistaken reasons for thinking diagrammatic proofs aren't rigorous are explored. The main result is that a confusion between the contents of a proof procedure (what's expressed by the referential elements in a proof procedure) and the unarticulated mathematical aspects of a proof procedure (how that proof procedure is enabled) gives the impression that diagrammatic proofs are less rigorous than language proofs. An additional (and independent) factor is treating the impossibility of naturally generalizing a diagrammatic proof procedure as an indication of (...) lack of rigor. (shrink)

Selon le holisme sémantique, les propriétés sémantiques sont par nature anatomiques, c’est-à-dire intrinsèquement collectives. Selon le holisme de l’interprétation, la signification ou le contenu sont attribués collectivement. La thèse de constitution holistique de la signification peut être raisonnablement défendue à l’aide d’une contrainte de molécularité, qui introduit une hiérarchie dans la complexité logique des phrases, et un ordre ou une articulation dans le langage, en faisant droit à une condition finitaire d’anatomicité.According to semantic holism, semantic properties are anatomic by nature, (...) i.e. intrinsically collective. According to interpretation holism, meaning, or content, is collectively attributed. The thesis that meaning is holistically constituted may be defended provided we take into account a molecularity requirement. The requirement’s rôle is to introduce order or articulation in the langage, via a hierarchy of logical complexity, and by paying due attention to a finitary condition of anatomicity. (shrink)

I dub a certain central tradition in philosophy of language (and mind) the de re tradition. Compelling thought experiments show that in certain common cases the truth conditions for thoughts and public-language expressions categorically turn on external objects referred to, rather than on linguistic meanings and/or belief assumptions. However, de re phenomena in language and thought occur even when the objects in question don't exist. Call these empty de re phenomena. Empty de re thought with respect to numeration is explored (...) in this paper, and such thought with respect to hallucinations is touched on. (shrink)

Bradley Armour-Garb and James A. Woodbridge, in Pretense and Pathology, make an ambitious and far-ranging case that philosophical fictionalism (particularly the pretence variety that they favour) illuminates several long-standing philosophical puzzles posed by words in ordinary language, such as ‘exist’, ‘true’ and ‘means that’, as well as the more technical, ‘refers to’, ‘proposition’ and ‘satisfies’. Along the way, Armour-Garb and Woodbridge discuss topics in the philosophy of language, philosophical logic, ontology, epistemology and more. An important aspect of their project is (...) its nominalist aspiration: by showing (if they do) that pretence is involved when we use the foregoing terms, they hope to show that it’s false that these uses induce ontological commitments (to, e.g., propositions or properties). In this respect, their pretence approach is yet another attempt to undercut the family of indispensability arguments (pioneered by W.V.O. Quine) that prima facie force ontological commitments by means of indispensable language usage. (shrink)

The ramifications are explored of taking physical theories to commit their advocates only to ‘physically real’ entities, where ‘physically real’ means ‘causally efficacious’ (e.g., actual particles moving through space, such as dust motes), the ‘physically significant’ (e.g., centers of mass), and the merely mathematical—despite the fact that, in ordinary physical theory, all three sorts of posits are quantified over. It's argued that when such theories are regimented, existential quantification, even when interpreted ‘objectually’ (that is, in terms of satisfaction via variables, (...) rather than by substitution-instances) need not imply any ontological commitments. (shrink)

A new approach to truth is offered which dispenses with the truth predicate, and replaces it with a special kind of quantifier which simultaneously binds variables in sentential and nominal positions. The resulting theory of truth for a (first-order) language is shown to be able to handle blind truth ascriptions, and is shown to be compatible with a characterization of the semantic and syntactic principles governing that language. Comparisons with other approaches to truth are drawn. An axiomatization of AU-quantifiers and (...) a model theory for them is given, and an appendix contains a completeness proof. (shrink)

Ask a philosopher what a proof is, and you’re likely to get an answer hii empaszng one or another regimentationl of that notion in terms of a finite sequence of formalized statements, each of which is either an axiom or is derived from an axiom by certain inference rules. (Wecan call this the formal conception of proof) Ask a mathematician what a proof is, and you will rbbl poay get a different-looking answer. Instead of stressing a partic- l uar regimented (...) notion of proof, the answer the mathematician will give ilikl.. (shrink)

First, I discuss the older “theory-centered” and the more recent semantic conception of scientific theories. I argue that these two perspectives are nothing more than terminological variants of one another. I then offer a new theory-centered view of scientific theories. I argue that this new view captures the insights had by each of these earlier views, that it’s closer to how scientists think about their own theories, and that it better accommodates the phenomenon of inconsistent scientific theories.

Starting-point epistemology (SPE) is a new position that, coupled with agent-centered rationality, is the key to resolving philosophical scepticism. SPE acknowledges that metacognitively-sophisticated agents know that they know things and know (something) about the methods by which this happens. Agent-centered rationality implies that a metacognitively-sophisticated agent should only desert a knowledge claim because of a challenge they recognize to be fatal to that claim. Scepticism is metacognitive pathology: Except in those rare cases when an individual is cognitively damaged, sceptical arguments (...) should fail. This book studies the various ways the cognitively healthy can protect themselves from prematurely distrusting what they take themselves to know. A sceptical position results from an agent's failure to correctly monitor their own processes of knowledge-gathering. (shrink)

Benjamin Callard has recently suggested that causation between Platonic objects—standardly understood as atemporal and non-spatial—and spatio-temporal objects is not ‘a priori’ unintelligible. He considers the reasons some have given for its purported unintelligibility: apparent impossibility of energy transference, absence of physical contact, etc. He suggests that these considerations fail to rule out a priori Platonic-object causation. However, he has overlooked one important issue. Platonic objects must causally affect different objects differently, and different Platonic objects must causally affect the same objects (...) differently. How are Platonic objects—ones outside space and time—supposed to do that? CiteULike Connotea Del.icio.us What's this? (shrink)

This chapter contains sections titled: Realism vs. Anti‐realism in the Semantics of Mathematical Language Anti‐realism about the Empirical Realm and, in Particular, about the Past Historical Significance and Historical Insignificance Generality and Holistic Explanations The Objectivity of Historiography Conclusion Bibliography Further Reading.

Recent discussions of Geach sentences by Braun and Salmon are reprised. It is shown that the intractability of providing semantics for Geach sentences (using standard logical tools) is due to the assumption that quantifiers are ontologically committing. Representing the content of these statements is easy using neutral quantifiers. An important concern is consistent identity conditions for nonreferring terms. It may be thought that Meinongian-object approaches handle this better than Azzouni's no-objects-in-any-sense-at-all approach. This is shown to be false. How our (...) truth-inducing practices determine truth values for empty-term statements is indicated—in particular, it is argued that our identification and distinguishing practices with “fictional entities” are parasitic on our practices with entities we take to be real. Examples from Austin—white dots on the horizon and specks—are among the test cases for the views developed here. (shrink)