This book offers an historically-informed critical assessment of Dummett's account of abstract objects, examining in detail some of the Fregean presuppositions whilst also engaging with recent work on the problem of abstract entities.

In this paper, I develop a theory on which each of a thing’s abundant properties is immanent in that thing. On the version of the theory I will propose, universals are abundant, each instantiated universal is immanent, and each uninstantiated universal is such that it could have been instantiated, in which case it would have been immanent. After setting out the theory, I will defend it from David Lewis’s argument that such a combination of immanence and abundance is absurd. I (...) will then advocate the theory on the grounds that it accomplishes all of Lewis’s “new work” while providing a gain in parsimony and a new account of fine-grained content. I will close with a discussion of how the theory also affords a new reply to two objections to uninstantiated universals: Armstrong’s charge that they are inconsistent with naturalism, and a Benacerraf-Field-style objection about epistemic access. (shrink)

S. Shapiro, The Oxford handbook of philosophy of mathematics and logic Oxford and New York: Oxford University Press, 2005. xv + 833 pp. £52.00. ISBN 0-19-514...

Causation appears to present us with an interpretative difficulty. While arguably a redundant relation given fundamental physics, it is nevertheless apparently pragmatically indispensable. This chapter revisits certain arguments made previously by the author for these claims with the benefit of hindsight, starting with the role of causal models in the human sciences, and attempting to explain why it is not possible to straightforwardly ground such models in fundamental physics. This suggests that further constraints, going beyond physics, are needed to legitimate (...) such models. These supplementary constraints could be reified, but that would seem to conflict with the completeness of physics. A response is to emphasize the practical role of causal talk, and the author suggests that a fictionalist approach might be worth exploring. After clarifying fictionalism as a general approach, they carry out in some detail the project of clarifying what a fictionalist attitude to causation would involve. (shrink)

Intuition is sometimes derided as an abstruse or esoteric phenomenon akin to crystal-ball gazing. Such derision appears to be fuelled primarily by the suggestion, evidently endorsed by traditional rationalists such as Plato and Descartes, that intuition is a kind of direct, immediate apprehension akin to perception. This paper suggests that although the perceptual analogy has often been dismissed as encouraging a theoretically useless metaphor, a quasi-perceptualist view of intuition may enable rationalists to begin to meet the challenge of supplying a (...) theoretically satisfying treatment of their favoured epistemic source. It is argued, first, that intuitions and perceptual experiences are at a certain level of abstraction the same type of mental state, presentations, which are distinct from beliefs, hunches, inclinations, attractions, and seemings. The notion of a presentation is given a positive explication, which identifies its characteristic features, accounts for several of its substantive psychological roles, and systematically locates it in a threefold division among types of contentful states. Subsequently, it is argued that presentations, intuitive no less than sensory, are by their nature poised to play a distinctive epistemic role. Specifically, in the case of intuition, we encounter an intellectual state that is so structured as to provide justification without requiring justification in turn—something which may, thus, be thought of as ‘given’. (shrink)

There is no doughnut without a hole, the saying goes. And that’s true. If you think you can come up with an exception, it simply wouldn’t be a doughnut. Holeless doughnuts are like extensionless color, or durationless sound—nonsense. Does it follow, then, that when we buy a doughnut we really purchase two sorts of thing—the edible stuff plus the little chunk of void in the middle? Surely we cannot just take the doughnut and leave the hole at the grocery store, (...) as we cannot just eat the doughnut and save the hole for later. But then, again, surely when we eat a doughnut we do not also eat the hole. Or do we? (shrink)

If mathematics is regarded as a science, then the philosophy of mathematics can be regarded as a branch of the philosophy of science, next to disciplines such as the philosophy of physics and the philosophy of biology. However, because of its subject matter, the philosophy of mathematics occupies a special place in the philosophy of science. Whereas the natural sciences investigate entities that are located in space and time, it is not at all obvious that this is also the case (...) with respect to the objects that are studied in mathematics. In addition to that, the methods of investigation of mathematics differ markedly from the methods of investigation in the natural sciences. Whereas the latter acquire general knowledge using inductive methods, mathematical knowledge appears to be acquired in a different way, namely, by deduction from basic principles. The status of mathematical knowledge also appears to differ from the status of knowledge in the natural sciences. The theories of the natural sciences appear to be less certain and more open to revision than mathematical theories. For these reasons mathematics poses problems of a quite distinctive kind for philosophy. Therefore philosophers have accorded special attention to ontological and epistemological questions concerning mathematics. (shrink)

Lynne Baker’s case for the incompatibility of naturalism with the first-person perspective raises a range of questions about the relationship between naturalism and the various properties involved in first-person perspectives. After arguing that non-qualitative properties—most notably, haecceities like being Lynne Baker—are ineliminably tied to first-person perspectives, this paper considers whether naturalism and non-qualitative properties are, in fact, compatible. In doing so, the discussion focus on Shamik Dasupgta’s argument against individuals and, in turn, non-qualitative properties. Several strategies for undermining Dasgupta's argument (...) are considered, drawing on de re laws and haecceitistic possibilities. Finally, an analogy is drawn between naturalism and platonism regarding mathematical entities and naturalism's parallel commitment to individuals. I conclude that naturalists are obliged to posit non-qualitative properties. (shrink)

In this paper, I consider an argument for the claim that any satisfactory epistemology of mathematics will violate core tenets of naturalism, i.e. that mathematics cannot be naturalized. I find little reason for optimism that the argument can be effectively answered.

This article attempts to motivate a new approach to anti-realism (or nominalism) in the philosophy of mathematics. I will explore the strongest challenges to anti-realism, based on sympathetic interpretations of our intuitions that appear to support realism. I will argue that the current anti-realistic philosophies have not yet met these challenges, and that is why they cannot convince realists. Then, I will introduce a research project for a new, truly naturalistic, and completely scientific approach to philosophy of mathematics. It belongs (...) to anti-realism, but can meet those challenges and can perhaps convince some realists, at least those who are also naturalists. (shrink)

The indispensability argument for abstract mathematical entities has been an important issue in the philosophy of mathematics. The argument relies on several assumptions. Some objections have been made against these assumptions, but there are several serious defects in these objections. Ameliorating these defects leads to a new anti-realistic philosophy of mathematics, mainly: first, in mathematical applications, what really exist and can be used as tools are not abstract mathematical entities, but our inner representations that we create in imagining abstract mathematical (...) entities; second, the thoughts that we create in imagining infinite mathematical entities are bounded by external conditions. (shrink)

Correspondence theories are frequently either too vaguely expressed – “true statements correspond to the way things are in the world,” or implausible – “true statements mirror raw, mind-independent reality.” I address this problem by developing features and roles that ought to characterize what I call ldquo;modest” correspondence theories. Of special importance is the role of correspondence in directing our responses to cases of suspected non-factuality; lack of straightforward correspondence shows the need for, and guides us in our choice of, various (...) kinds of reconstrual projects. This, I argue, is in contrast to the approaches suggested by deflationism and coherence, and thus modest correspondence theories are appropriately distinct from rivals. (shrink)

I argue that certain species of belief, such as mathematical, logical, and normative beliefs, are insulated from a form of Harman-style debunking argument whereas moral beliefs, the primary target of such arguments, are not. Harman-style arguments have been misunderstood as attempts to directly undermine our moral beliefs. They are rather best given as burden-shifting arguments, concluding that we need additional reasons to maintain our moral beliefs. If we understand them this way, then we can see why moral beliefs are vulnerable (...) to such arguments while mathematical, logical, and normative beliefs are not—the very construction of Harman-style skeptical arguments requires the truth of significant fragments of our mathematical, logical, and normative beliefs, but requires no such thing of our moral beliefs. Given this property, Harman-style skeptical arguments against logical, mathematical, and normative beliefs are self-effacing; doubting these beliefs on the basis of such arguments results in the loss of our reasons for doubt. But we can cleanly doubt the truth of morality. (shrink)

Nominalism faces the task of explaining away the ontological commitments of applied mathematical statements. This paper reviews an argument from the philosophy of logic that focuses on this task and which has been used as an objection to certain specific formulations of nominalism. The argument as it is developed in this paper aims to show that nominalism in general does not have the epistemological advantages its defendants claim it has. I distinguish between two strategies that are available to the nominalist: (...) The Evaluation Programme, which tries to preserve the common truth‐values of mathematical statements even if there are no mathematical objects, and Fictionalism, which denies that mathematical sentences have significant truth‐values. It is argued that the tenability of both strategies depends on the nominalist’s ability to account for the notion of consequence. This is a problem because the usual meta‐logical explications of consequence do themselves quantify over mathematical entities. While nominalists of both varieties may try to appeal to a primitive notion of consequence, or, alternatively, to primitive notions of logical or structural possibilities, such measures are objectionable. Even if we are equipped with a notion of either consequence or possibility that is primitive in the relevant sense, it will not be strong enough to account for the consequence relation required in classical mathematics. These examinations are also useful in assessing the possible counter‐intuitive appeal of the argument from the philosophy of logic. (shrink)

The purpose of the present paper is to challenge some received assumptions about the logical analysis of modal English, and to show that these assumptions are crucial to certain debates in current philosophy of language. Specifically, I will argue that the standard analysis in terms of quantified modal logic mistakenly fudges important grammatical distinctions, and that the validity of Kripke's modal argument against description theories of proper names crucially depends on ensuing equivocations.

Theodore Sider’s recent book, “Writing the Book of the World”, employs a primitive notion of metaphysical structure in order to make sense of substantive metaphysics. But Sider and others who employ metaphysical primitives face serious epistemological challenges. In the first section I develop a specific form of this challenge for Sider’s own proposed epistemology for structure; the second section develops a general reliability challenge for Sider’s theory; and the third and final section argues for the rejection of Siderean structure in (...) the course of answering a transcendental argument against such rejection. (shrink)

This paper formulates a general epistemological argument against what I call non-causal realism, generalizing domain specific arguments by Benacerraf, Field, and others. First I lay out the background to the argument, making a number of distinctions that are sometimes missed in discussions of epistemological arguments against realism. Then I define the target of the argument—non-causal realism—and argue that any non-causal realist theory, no matter the subject matter, cannot be given a reasonable epistemology and so should be rejected. Finally I discuss (...) and respond to several possible responses to the argument. In addition to clearing up and avoiding numerous misunderstandings of arguments of this kind that are quite common in the literature, this paper aims to present and endorse a rigorous and fully general epistemological argument against realism. (shrink)

The issues I explore in this paper are best introduced by the table with which it begins. The left-hand entry in each row gives expression to a kind objectivity; the right-hand entry affirms the existence of a special kind of object. When philosophers believe in any of the entities on the right, it is typically because they think them necessary to ground the facts on the left. By the same token, when philosophers deny any of the facts on the left, (...) it is often because they cannot bring themselves to believe in the associated kind of object on the right. My project is to explore the extent to which it is possible to have the objectivity without the objects—for example, absolute motion without substantival space, objective predication without universals, objective synonymy without propositions, and objective modality without possible worlds. Each of these combinations would have been congenial to A.N. Prior. (shrink)

This paper gives a semantics for schematic logic, proving soundness and completeness. The argument for soundness is carried out in ontologically innocent fashion, relying only on the existence of formulae which are actually written down in the course of a derivation in the logic. This makes the logic available to a nominalist, even a nominalist who does not wish to rely on modal notions, and who accepts the possibility that the universe may in fact be finite.

This paper contains a defense against anti-realism in mathematics in the light both of incompleteness and of the fact that mathematics is a ‘cultural artifact.’. Anti-realism (here) is the view that theorems, say, of aritltmetic cannot be taken at face value to express true propositions about the system of numbers but must be reconstrued to be about somctliiiig else or about nothing at all. A ‘bite-the-bullet’ aspect of the defease is that, adopting new axioms, liitherto independent, is not. a matter (...) of recognizing trutlis wliich had previoasly been unrecognized, but of extending the domain of what is true. (shrink)

A strong form of representationalism says that every conscious property of every mental state can be identified with some part of the state’s representational properties. A weaker representationalism says that some conscious property of some mental state can be identified with some part of the state’s representational properties. David Papineau has recently argued that all such theories are incorrect since they construe consciousness as consisting in “relations to propositions or other abstract objects outside space and time”, whereas consciousness is “concrete” (...) and “here and now”. Papineau defends instead a kind of “qualia theory” according to which all conscious properties are intrinsic non-relational properties of subjects. He argues that this theory bypasses the difficulties he identifies for representationalism. Similar worries about representationalism, and similar ideas to the effect that some qualia theory, adverbial theory, or sense-datum theory fares better with respect to these worries are relatively wide-spread. I argue that Papineau’s theory does not bypass the difficulties he identifies for representationalism. In fact, Papineau’s theory arguably has no advantage at all over representationalism with regard to these issues. The features that concern Papineau about representationalist views do not derive—or do not derive solely—from the representationalism of these views. They derive from a common-factorism of these views. And this common-factorism is embraced by Papineau as well as by most theories of consciousness and perception. (shrink)

A strong form of representationalism says that every conscious property of every mental state can be identified with some part of the state’s representational properties. A weaker representationalism says that some conscious property of some mental state can be identified with some part of the state’s representational properties. David Papineau has recently argued that all such theories are incorrect since they construe consciousness as consisting in “relations to propositions or other abstract objects outside space and time”, whereas consciousness is “concrete” (...) and “here and now”. Papineau defends instead a kind of “qualia theory” according to which all conscious properties are intrinsic non-relational properties of subjects. He argues that this theory bypasses the difficulties he identifies for representationalism. Similar worries about representationalism, and similar ideas to the effect that some qualia theory, adverbial theory, or sense-datum theory fares better with respect to these worries are relatively wide-spread. I argue that Papineau’s theory does not bypass the difficulties he identifies for representationalism. In fact, Papineau’s theory arguably has no advantage at all over representationalism with regard to these issues. The features that concern Papineau about representationalist views do not derive—or do not derive solely—from the representationalism of these views. They derive from a common-factorism of these views. And this common-factorism is embraced by Papineau as well as by most theories of consciousness and perception. (shrink)

Thin Objects: Anionist Account, by LinneboØystein. Oxford: Oxford University Press, 2018. Pp. xviii + 238.

The use of tensed language and the metaphor of set ‘formation’ found in informal descriptions of the iterative conception of set are seldom taken at all seriously. Both are eliminated in the nonmodal stage theories that formalise this account. To avoid the paradoxes, such accounts deny the Maximality thesis, the compelling thesis that any sets can form a set. This paper seeks to save the Maximality thesis by taking the tense more seriously than has been customary (although not literally). A (...) modal stage theory, MST, is developed in a bimodal language, governed by a tenselike logic. Such a language permits a very natural axiomatisation of the iterative conception, which upholds the Maximality thesis. It is argued that the modal approach is consonant with mathematical practice and a plausible metaphysics of sets and shown that MST interprets a natural extension of Zermelo set theory less the axiom of Infinity and, when extended with a further axiom concerning the extent of the hierarchy, interprets Zermelo–Fraenkel set theory. (shrink)

The purpose of this paper is to apply Crispin Wright’s criteria and various axes of objectivity to mathematics. I test the criteria and the objectivity of mathematics against each other. Along the way, various issues concerning general logic and epistemology are encountered.

In everyday speech we seem to refer to such things as abstract objects, moral properties, or propositional attitudes that have been the target of metaphysical and/or epistemological objections. Many philosophers, while endorsing scepticism about some of these entities, have not wished to charge ordinary speakers with fundamental error, or recommend that the discourse be revised or eliminated. To this end a number of non-revisionary antirealist strategies have been employed, including expressivism, reductionism and hermeneutic fictionalism. But each of these theories faces (...) forceful objections. In particular, we argue, proponents of these strategies face a dilemma: either concedes that their theory is revisionary, or adopt an implausible account of speaker-meaning whereby the content of certain types of utterance is opaque to their speakers. In this paper we introduce a new type of antirealist strategy, which is thoroughly non-revisionary, and leaves speaker-meaning transparent to speakers. We draw on work on pragmatics in the philosophy of language to develop a theory we call ‘pragmatic antirealism’. The pragmatic antirealist holds that while the sentences of the discourses in question have metaphysically contentious truth conditions, ordinary utterances of them are pragmatically modified in context in such a way that speakers do not incur commitment to those truth conditions. After setting out the theory, we show how it might be developed for both mathematical and ethical discourse, before responding to some likely objections. (shrink)

Mereological nihilism is the view that all concrete objects are simple. Existence monism is the view that the only concrete object is one big simple: the world. I will argue that nihilism culminates in monism. The nihilist demands the simplest sufficient ontology, and the monist delivers it.

The paper argues that no second order theory is ontologically commited to anything beyond what its individual variables range over.

It is often claimed that nominalistic programmes to reconstruct mathematics fail, since they will at some point involve the notion of logical consequence which is unavailable to the nominalist. In this paper we use an idea of Goodman and Quine to develop a nominalistically acceptable explication of logical consequence.

Las teorías que usamos para representar el mundo pueden ser extremadamente complejas. Abordan temas tales como electrones, campos cuánticos, estrellas de neutrones, materia oscura, redes neuronales, mercados económicos, la atmósfera y muchas otras entidades que suponemos existen en el universo. Al formular nuestras teorías, recurrimos a lenguajes exactos que nos permiten minimizar la vaguedad y expresarnos lo más precisa y cuantitativamente posible. Recurrimos a la matemática. Cuando formulamos nuestras teorías fácticas en lenguaje matemático, estas se refieren no solamente a objetos (...) que nosotros interpretamos como materiales, tales como partículas o personas, sino también a entidades más raras de un mundo abstracto: conjuntos, números, funciones, espacios algebraicos, variedades, topologías, y otras entidades similares. Estos objetos no son materiales en el sentido en que nosotros decimos que una manzana es material. Ellos no existen en el espacio-tiempo, no interactúan, no cambian o evolucionan. Sin embargo, allí están, profundamente arraigados en nuestras teorías más apreciadas acerca del mundo. (shrink)

C. S. Jenkins has recently proposed an account of arithmetical knowledge designed to be realist, empiricist, and apriorist: realist in that what’s the case in arithmetic doesn’t rely on us being any particular way; empiricist in that arithmetic knowledge crucially depends on the senses; and apriorist in that it accommodates the time-honored judgment that there is something special about arithmetical knowledge, something we have historically labeled with ‘a priori’. I’m here concerned with the prospects for extending Jenkins’s account beyond arithmetic—in (...) particular, to set theory. After setting out the central elements of Jenkins’s account and entertaining challenges to extending it to set theory, I conclude that a satisfactory such extension is unlikely. (shrink)

Peano's axiomatizations of geometry are abstract and non-intuitive in character, whereas Peano stresses his appeal to concrete spatial intuition in the choice of the axioms. This poses the problem of understanding the interrelationship between abstraction and intuition in his geometrical works. In this article I argue that axiomatization is, for Peano, a methodology to restructure geometry and isolate its organizing principles. The restructuring produces a more abstract presentation of geometry, which does not contradict its intuitive content but only puts it (...) into a particular form. (shrink)

Since the linguistic turn, many have taken semantics to guide ontology. Here, I argue that semantics can, at best, serve as a partial guide to ontological commitment. If semantics were to be our guide, semantic data and semantic treatments would need to be taken seriously. Through an examination of plurals and their treatments, I argue that there can be multiple, equally semantically adequate, treatments of a natural language theory. Further, such treatments can attribute different ontological commitments to a theory. Given (...) this, I argue that semantics can fail to deliver determinate ontological commitments and determinate answers to ontological questions, more generally. (shrink)

Syntactic Reductionism, as understood here, is the view that the ‘logical forms’ of sentences in which reference to abstract objects appears to be made are misleading so that, on analysis, we can see that no expressions which even purport to refer to abstract objects are present in such sentences. After exploring the motivation for such a view, and arguing that no previous argument against it succeeds, sentences involving generalized quantifiers, such as ‘most’, are examined. It is then argued, on this (...) basis, that Syntactic Reductionism is untenable. (shrink)

This is a survey of the concept of continuity. Efforts to explicate continuity have produced a plurality of philosophical conceptions of continuity that have provably distinct expressions within contemporary mathematics. I claim that there is a divide between the conceptions that treat the whole continuum as prior to its parts, and those conceptions that treat the parts of the continuum as prior to the whole. Along this divide, a tension emerges between those conceptions that favor philosophical idealizations of continuity and (...) those that are more readily available for mathematico-scientific use. (shrink)

Forthcoming in Philosophical Compass. I explain why plural quantifiers and predicates have been thought to be philosophically significant.

For Quine, the ontological commitments of a discourse are what fall under its (objectual) quantifiers. The recent literature, however, is beginning to move away from this picture. There are direct challenges to Quine's criterion, and there are also attempts to provide alternatives. Azzouni suggests that the ontological commitments of a discourse should be determined by an existence predicate instead. The availability of this alternative forces an adjudication between Qune's criterion and the predicate approach to ontological commitment. I argue that to (...) adjudicate between these criteria for ontological commitment, we need first to adjudicate between criteria for what exists. (shrink)

Yet, he also says that it is philosophically indeterminate which criterion for what exists is correct. Nominalism is the view that certain objects ( i.e ., abstract objects) do not exist, and not the view that it is philosophically indeterminate whether or not they do. I resolve the dilemma that Azzouni's claims pose: Azzouni is a non-factualist about what exists, but he is a factualist about which criterion for what exists our community of speakers has adopted. It is in the (...) latter sense only that Azzouni can call himself a nominalist. My thanks to Jody Azzouni and to an anonymous referee for helpful suggestions. CiteULike Connotea Del.icio.us What's this? (shrink)

In this essay, the tension that Benacerraf identifies for theories of mathematical truth is used as the vehicle for arguing against a particular desideratum for semantic theories. More specifically, I place in question the desideratum that a semantic theory, provided for some area of discourse, should run in parallel with the semantic theory holding for the rest of the language. The importance of this desideratum is also made clear by means of tracing out the subtle implications of its rejection.

The platonism/nominalism debate in the philosophy of mathematics concerns the question whether numbers and other mathematical objects exist. Platonists believe the answer to be in the positive, nominalists in the negative. According to non-factualists, the question is ‘moot’, in the sense that it lacks a correct answer. Elaborating on ideas from Stephen Yablo, this article articulates a non-factualist position in the philosophy of mathematics and shows how the case for non-factualism entails that standard arguments for rival positions fail. In particular, (...) showing how and why non-factualists reject nominalism illuminates the originality and interest of their position. (shrink)

Conflicting accounts of the role of mathematics in our physical theories can be traced to two principles. Mathematics appears to be both (1) theoretically indispensable, as we have no acceptable non-mathematical versions of our theories, and (2) metaphysically dispensable, as mathematical entities, if they existed, would lack a relevant causal role in the physical world. I offer a new account of a role for mathematics in the physical sciences that emphasizes the epistemic benefits of having mathematics around when we do (...) science. This account successfully reconciles theoretical indispensability and metaphysical dispensability and has important consequences for both advocates and critics of indispensability arguments for platonism about mathematics. (shrink)

As our data will show, negative existential sentences containing socalled empty names evoke the same strong semantic intuitions in ordinary speakers and philosophers alike.Santa Claus does not exist.Superman does not exist.Clark Kent does not exist.Uttering the sentences in (1) seems to say something truth-evaluable, to say something true, and to say something different for each sentence. A semantic theory ought to explain these semantic intuitions.The intuitions elicited by (1) are in apparent conflict with the Millian view of proper names. According (...) to Millianism, the meaning (or 'semantic value') of a proper name is just its referent. But empty names, such as 'Santa Claus' and 'Superman', appear to lack a .. (shrink)

In “Mathematical Truth,” Paul Benacerraf raises an epistemic challenge for mathematical platonists. In this paper, I examine the assumptions that motivate Benacerraf’s original challenge, and use them to construct a new causal challenge for the epistemology of mathematics. This new challenge, which I call ‘Gödel’s Gap’, appeals to intuitive insights into mathematical knowledge. Though it is a causal challenge, it does not rely on any obviously objectionable constraints on knowledge. As a result, it is more compelling than the original challenge. (...) It is also more general; the challenge applies equally to platonistic and non-platonistic accounts of mathematical truth. And it can be generalized beyond the case of mathematical knowledge to pose a challenge for, e.g., moral knowledge. (shrink)

This article discusses the argument we cannot have knowledge of abstract entities because they are not part of the causal order. The claim of this article is that the argument fails because of equivocation. Assume that the “causal order” is concerned with contingent facts involving time and space. Even if the existence of abstract entities is not contingent and does not involve time or space it does not follow that no truths about abstract entities are contingent or involve time or (...) space. I argue that it is the latter which is required to obtain the desired conclusion. (shrink)