Several scholars have argued that Wittgenstein held the view that the notion of number is presupposed by the notion of one-one correlation, and that therefore Hume's principle is not a sound basis for a definition of number. I offer a new interpretation of the relevant fragments on philosophy of mathematics from Wittgenstein's Nachlass, showing that if different uses of ‘presupposition’ are understood in terms of de re and de dicto knowledge, Wittgenstein's argument against the Frege-Russell definition of number turns out (...) to be valid on its own terms, even though it depends on two epistemological principles logicist philosophers of mathematics may find too ‘constructivist’. (shrink)

Wittgenstein's conception of infinity can be seen as continuing the tradition of the potential infinite that begins with Aristotle. Transfinite cardinals in set theory might seem to render the potential infinite defunct with the actual infinite now given mathematical legitimacy. But Wittgenstein's remarks on set theory argue that the philosophical notion of the actual infinite remains philosophical and is not given a mathematical status as a result of set theory. The philosophical notion of the actual infinite is not to be (...) found in the mathematics of set theory, only in a certain associated philosophy – what Wittgenstein calls a certain kind of “prose”. (shrink)

Wittgenstein's notorious sample of a ‘complete primitive language’ is often thought to be closer in kind to animal forms of communication than human language. Indeed, it has been criticised on precisely these grounds. But such debates make little sense if we take seriously Wittgenstein's idea that language is a family resemblance concept. So, rather than argue that the builders’ game ‘really is a language’, I propose to turn the debate on its head and welcome the comparison. By changing our perspective (...) in this way, I suggest that we can see that the learning of language is crucially dependent on forms of communication that are animal in nature. I then discuss how these lessons might shed light on empirical research into both the ontogenetic and phylogenetic origins of linguistic communication. (shrink)

In this paper, I examine Quine's views on the epistemology of logic. According to Quine's influential holistic account, logic is central in the “web of belief” that comprises our overall theory of the world. Because of this, revisions to logic would have devastating systematic consequences, and this explains why we are loath to make such revisions. In section1, I clarify this idea and thereby show that Quine actually takes the web of belief to have asymmetrical internal structure. This raises two (...) puzzles. First, as I show in section 2, Quine's mature thoroughly naturalized view has it that logic is simply obvious, and this is explains why we do not typically consider revising it. While Quine presents this naturalized view as a way to make good on his earlier metaphor of centrality in a web of belief, I argue that the resources of Quine's naturalized epistemology cannot adequately explain why we are reluctant to revise logic. And, Quine seems to recognize this point himself. In light of this, I explain in section 3 how Quine can allow that our overall scientific theory has systematic structure in a way that is consistent with his naturalistic strictures. Second, the asymmetrical internal structure of the web of belief seems to be inconsistent with its being a holistic web at all. I defuse this problem in section 4 by showing how Quine distinguishes between structural and confirmational considerations. I close by using this distinction to show how Quine's view can evade Michael Friedman’s criticisms, and allow for important methodological distinctions between areas of the web of belief. (shrink)

Proponents of the explanatory indispensability argument for mathematical platonism maintain that claims about mathematical entities play an essential explanatory role in some of our best scientific explanations. They infer that the existence of mathematical entities is supported by way of inference to the best explanation from empirical phenomena and therefore that there are the same sort of empirical grounds for believing in mathematical entities as there are for believing in concrete unobservables such as quarks. I object that this inference depends (...) on a false view of how abductive considerations mediate the transfer of empirical support. More specifically, I argue that even if inference to the best explanation is cogent, and claims about mathematical entities play an essential explanatory role in some of our best scientific explanations, it doesn’t follow that the empirical phenomena that license those explanations also provide empirical support for the claim that mathematical entities exist. (shrink)

This essay concerns the question of how we make genuine epistemic progress through conceptual analysis. Our way into this issue will be through consideration of the paradox of analysis. The paradox challenges us to explain how a given statement can make a substantive contribution to our knowledge, even while it purports merely to make explicit what one’s grasp of the concept under scrutiny consists in. The paradox is often treated primarily as a semantic puzzle. However, in “Sect. 1” I argue (...) that the paradox raises a more fundamental epistemic problem, and in “Sects.1 and 2” I argue that semantic proposals—even ones designed to capture the Fregean link between meaning and epistemic significance—fail to resolve that problem. Seeing our way towards a real solution to the paradox requires more than semantics; we also need to understand how the process of analysis can yield justification for accepting a candidate conceptual analysis. I present an account of this process, and explain how it resolves the paradox, in “Sect. 3”. I conclude in “Sect. 4” by considering the implications for the present account concerning the goal of conceptual analysis, and by arguing that the apparent scarcity of short and finite illuminating analyses in philosophically interesting cases provides no grounds for pessimism concerning the possibility of philosophical progress through conceptual analysis. (shrink)

This paper proposes a reading of the history of equivalence in mathematics. The paper has two main parts. The first part focuses on a relatively short historical period when the notion of equivalence is about to be decontextualized, but yet, has no commonly agreed-upon name. The method for this part is rather straightforward: following the clues left by the others for the ‘first’ modern use of equivalence. The second part focuses on a relatively long historical period when equivalence is experienced (...) in context. The method for this part is to strip the ideas from their set-theoretic formulations and methodically examine the variations in the ways equivalence appears in some prominent historical texts. The paper reveals several critical differences in the conceptions of equivalence at different points in history that are at variance with the standard account of the mathematical notion of equivalence encompassing the concepts of equivalence relation and equivalence class. (shrink)

There has emerged in recent years the recognition that the characteristically Buddhist doctrine of svasa vitti 2 (‘apperception’, as I will render it for reasons to become clear presently) was vari...

Ontological pluralism is the view that there are different ways to exist. It is a position with deep roots in the history of philosophy, and in which there has been a recent resurgence of interest. In contemporary presentations, it is stated in terms of fundamental languages: as the view that such languages contain more than one quantifier. For example, one ranging over abstract objects, and another over concrete ones. A natural worry, however, is that the languages proposed by the pluralist (...) are mere notational variants of those proposed by the monist, in which case the debate between the two positions would not seem to be substantive. Jason Turner has given an ingenious response to this worry, in terms of a principle that he calls ‘logical realism’. This paper offers a counter-response on behalf of the ‘notationalist’. I argue that, properly applied, the principle of logical realism is no threat to the claim that the languages in question are notational variants. Indeed, there seems to be every reason to think that they are. (shrink)

John MacFarlane explores how we might make sense of the idea that truth is relative. He provides new, satisfying accounts of parts of our thought and talk that have resisted traditional methods of analysis, including what we mean when we talk about what is tasty, what we know, what will happen, what might be the case, and what we ought to do.

Sets are multitudes which are also unities. It is surprising that the fact that multitudes are also unities leads to no contradictions: this is the main fact of mathematics.Kurt Gödel (Hao Wang,A Logical Journey: From Gödel to Philosophy)In what sense can something be at the same time one and many? The problem is familiar since Plato (for example,Republic524e). In recent times, Whitehead and Russell, inPrincipia Mathematica,have been struck by the difficulty of the problem: ‘If there is such an object as (...) a class, it must be in some senseoneobject, yet it is only of classes that many can be predicated. Hence, if we admit classes as objects, we must suppose that the same object can be both one and many, which seems impossible.' It is, however, in Frege's great work,The Foundations of Arithmetic(henceforth,Grundlagen),that many see the final resolution of the old question: how can something be at the same time one and many? (shrink)

Many recent writers in the philosophy of mathematics have put great weight on the relative categoricity of the traditional axiomatizations of our foundational theories of arithmetic and set theory. Another great enterprise in contemporary philosophy of mathematics has been Wright's and Hale's project of founding mathematics on abstraction principles. In earlier work, it was noted that one traditional abstraction principle, namely Hume's Principle, had a certain relative categoricity property, which here we term natural relative categoricity. In this paper, we show (...) that most other abstraction principles are not naturally relatively categorical, so that there is in fact a large amount of incompatibility between these two recent trends in contemporary philosophy of mathematics. To better understand the precise demands of relative categoricity in the context of abstraction principles, we compare and contrast these constraints to stability-like acceptability criteria on abstraction principles, the Tarski-Sher logicality requirements on abstraction principles studied by Antonelli and Fine, and supervaluational ideas coming out of Hodes' work. (shrink)

A crucial part of the contemporary interest in logicism in the philosophy of mathematics resides in its idea that arithmetical knowledge may be based on logical knowledge. Here an implementation of this idea is considered that holds that knowledge of arithmetical principles may be based on two things: (i) knowledge of logical principles and (ii) knowledge that the arithmetical principles are representable in the logical principles. The notions of representation considered here are related to theory-based and structure-based notions of representation (...) from contemporary mathematical logic. It is argued that the theory-based versions of such logicism are either too liberal (the plethora problem) or are committed to intuitively incorrect closure conditions (the consistency problem). Structure-based versions must on the other hand respond to a charge of begging the question (the circularity problem) or explain how one may have a knowledge of structure in advance of a knowledge of axioms (the signature problem). This discussion is significant because it gives us a better idea of what a notion of representation must look like if it is to aid in realizing some of the traditional epistemic aims of logicism in the philosophy of mathematics. (shrink)

By any reasonable reckoning, Gottlob Frege's ‘On Sense and Reference’ is one of the more important philosophical papers of all time. Although Frege briefly discusses the sense-reference distinction in an earlier work, it is through ‘Sense and Reference’ that most philosophers have become familiar with it. And the distinction so thoroughly permeates contemporary philosophy of language and mind that it is almost impossible to imagine these subjects without it.The distinction between the sense and the referent of a name is introduced (...) in the second paragraph of ‘Sense and Reference.’. (shrink)

Identity and distinctness facts are ones like “The Eiffel Tower is identical to the Eiffel Tower,” and “The Eiffel Tower is distinct from the Louvre.” This paper concerns one question in the metaphysics of identity: Are identity and distinctness facts metaphysically fundamental or are they nonfundamental? I provide an overview of answers to this question.

In this paper I investigate two notions of concepts that have played a dominant role in 20th century philosophy of mathematics. According to the first, concepts are definite and fixed; in contrast, according to the second notion concepts are open and subject to modifications. The motivations behind these two incompatible notions and how they can be used to account for conceptual change are presented and discussed. On the basis of historical developments in mathematics I argue that both notions of concepts (...) capture important aspects of mathematical and scientific reasoning, and, consequently, that a pluralistic approach that allows for representing both of these aspects is most useful for an adequate account of investigative practices. (shrink)

Understanding what numbers are means knowing several things. It means knowing how counting relates to numbers (called the cardinal principle or cardinality); it means knowing that each number is generated by adding one to the previous number (called the successor function or succession), and it means knowing that all and only sets whose members can be placed in one-to-one correspondence have the same number of items (called exact equality or equinumerosity). A previous study (Sarnecka & Carey, 2008) linked children's understanding (...) of cardinality to their understanding of succession for the numbers five and six. This study investigates the link between cardinality and equinumerosity for these numbers, finding that children either understand both cardinality and equinumerosity or they understand neither. This suggests that cardinality and equinumerosity (along with succession) are interrelated facets of the concepts five and six, the acquisition of which is an important conceptual achievement of early childhood. (shrink)

The existence of mereological sums can be derived from an abstraction principle in a way analogous to numbers. I draw lessons for the thesis that “composition is innocent” from neo-Fregeanism in the philosophy of mathematics.

According to the orthodox account of meaning and translation in the literature, meaning is a property of expressions of a language, and translation is a matching of synonymous expressions across languages. This linguistic account of translation gives rise to well-known skeptical conclusions about translation, objectivity, meaning and truth, but it does not conform to our best translational practices. In contrast, I argue for a textual account of meaning based on the concept of a TEXT-TYPE that does conform to our best (...) translational practices. With their semantic function in view, text-types are Archimedean points for their respective disciplines. The text-type of philosophy is no exception. Culture-transcendent conceptual analysis can proceed on firm footing without having to deny the reality of radical cultural and linguistic difference by treating components of text-types as the concepts to be analyzed. Analyses of central philosophical concepts are provided as a means of adjudicating philosophical controversy. (shrink)

Among Kant's innovations in the understanding of logic (‘general logic’) were his claims that logic had no content of its own, but was the form of the thought of any possible content, and that the unit of meaning, the truth-bearer, judgement, was essentially apperceptive. Judging was implicitly the consciousness of judging. This was for Kant a logical truth. This article traces the influence of the latter claim on Fichte, and, for most of the discussion, on Hegel. The aim is to (...) understand the relations among self-consciousness, reason and freedom in the idealist tradition. (shrink)

Ontology is the philosophical discipline which aims to understand how things in the world are divided into categories and how these categories are related together. This is exactly what information scientists aim for in creating structured, automated representations, called 'ontologies,' for managing information in fields such as science, government, industry, and healthcare. Currently, these systems are designed in a variety of different ways, so they cannot share data with one another. They are often idiosyncratically structured, accessible only to those who (...) created them, and unable to serve as inputs for automated reasoning. This volume shows, in a nontechnical way and using examples from medicine and biology, how the rigorous application of theories and insights from philosophical ontology can improve the ontologies upon which information management depends. (shrink)

It is argued that the finitist interpretation of wittgenstein fails to take seriously his claim that philosophy is a descriptive activity. Wittgenstein's concentration on relatively simple mathematical examples is not to be explained in terms of finitism, But rather in terms of the fact that with them the central philosophical task of a clear 'ubersicht' of its subject matter is more tractable than with more complex mathematics. Other aspects of wittgenstein's philosophy of mathematics are touched on: his view that mathematical (...) propositions are 'grammatical propositions' (and so, Strictly speaking, Not genuine propositions at all) and his view that in mathematics 'everything is algorithm, Nothing meaning'. His views on consistency and his anti-Foundationalism are linked with this central thesis. (shrink)

Electronic computers form an integral part of modern mathematical practice. Several high-profile results have been proven with techniques where computer calculations form an essential part of the proof. In the traditional philosophical literature, such proofs have been taken to constitute a posteriori knowledge. However, this traditional stance has recently been challenged by Mark McEvoy, who claims that computer calculations can constitute a priori mathematical proofs, even in cases where the calculations made by the computer are too numerous to be surveyed (...) by human agents. In this article we point out the deficits of the traditional literature that has called for McEvoy’s correction. We also explain why McEvoy’s defence of mathematical apriorism fails and we discuss how the debate over the epistemological status of computer-assisted mathematics contains several unfortunate conceptual reductions. (shrink)

According to the classical account, propositions are sui generis, abstract, intrinsically-representational entities and our cognitive attitudes, and the token states within us that realize those attitudes, represent as they do in virtue of their propositional objects. In light of a desire to explain how it could be that propositions represent, much of the recent literature on propositions has pressured various aspects of this account. In place of the classical account, revisionists have aimed to understand propositions in terms of more familiar (...) entities such as facts, types of mental or linguistic acts, and even properties. But we think that the metaphysical story about propositions is much simpler than either the classical theorist or the revisionist would have you believe. In what follows, we argue that a proper understanding of the nature of our cognitive relations to propositions shows that the question of whether propositions themselves represent is, at best, a distraction. We will argue that once this distraction is removed, the possibility of a very pleasing, minimalist story of propositions emerges; a story that appeals only to assumptions that are shared by all theorists in the relevant debate. (shrink)

Convinced that logic has a history and that its history always manages to surprise the philosophers, Claude Imbert has devoted much of her work to the study of the Stoic school and of the late-nineteenth-century German logician Gottlob Frege. In the fifth chapter of her book Pour une histoire de la logique, she examines the trajectory of Frege's awareness of what his new logic entails, in particular the way it subverts the project of Kant.

The idea that the function of language is its use is commonly ascribed to the Later Wittgenstein. In this paper, I argue that there is textual evidence already coming from the early manuscripts proving that Wittgenstein's philosophical development is culminating in the idea of function as use around 1929–30. I interpret a passage from Ms‐107 in order to show that Wittgenstein's practical turn has sources in different stages of his philosophical development, each of which is dominated by different ideas: the (...) idea of a picture theory; the idea of a phenomenological language and the idea of function as use. (shrink)

Extract from Hofstadter's revew in Bulletin of American Mathematical Society : http://www.ams.org/journals/bull/1980-02-02/S0273-0979-1980-14752-7/S0273-0979-1980-14752-7.pdf -/- "Aaron Sloman is a man who is convinced that most philosophers and many other students of mind are in dire need of being convinced that there has been a revolution in that field happening right under their noses, and that they had better quickly inform themselves. The revolution is called "Artificial Intelligence" (Al)-and Sloman attempts to impart to others the "enlighten- ment" which he clearly regrets not having (...) experienced earlier himself. Being somewhat of a convert, Sloman is a zealous campaigner for his point of view. Now a Reader in Cognitive Science at Sussex, he began his academic career in more orthodox philosophy and, by exposure to linguistics and AI, came to feel that all approaches to mind which ignore AI are missing the boat. I agree with him, and I am glad that he has written this provocative book. The tone of Sloman's book can be gotten across by this quotation (p. 5): "I am prepared to go so far as to say that within a few years, if there remain any philosophers who are not familiar with some of the main developments in artificial intelligence, it will be fair to accuse them of professional incom- petence, and that to teach courses in philosophy of mind, epistemology, aesthetics, philosophy of science, philosophy of language, ethics, metaphysics, and other main areas of philosophy, without discussing the relevant aspects of artificial intelligence will be as irresponsible as giving a degree course in physics which includes no quantum theory." -/- (The author now regrets the extreme polemical tone of the book.). (shrink)

Number terms and quantifiers share a range of linguistic (syntactic, semantic, and pragmatic) properties. On the basis of these similarities, one might expect these 2 classes of linguistic expression to pose similar problems to children acquiring language. We report here the results of an experiment that explicitly compared the acquisition of numerical expressions (two, four) and quantificational (some, all) expressions in younger and older 3-year-olds. Each group showed adult-like preferences for “exact” interpretations when evaluating number terms; however they did not (...) use the corresponding upper bounded interpretation when evaluating the quantifier some. Apparently, children follow different procedures for learning and evaluating numerals and quantifiers. These findings have implications for theories of number representation in child and adult grammars. (shrink)

Hilbert’s choice operators τ and ε, when added to intuitionistic logic, strengthen it. In the presence of certain extensionality axioms they produce classical logic, while in the presence of weaker decidability conditions for terms they produce various superintuitionistic intermediate logics. In this thesis, I argue that there are important philosophical lessons to be learned from these results. To make the case, I begin with a historical discussion situating the development of Hilbert’s operators in relation to his evolving program in the (...) foundations of mathematics and in relation to philosophical motivations leading to the development of intuitionistic logic. This sets the stage for a brief description of the relevant part of Dummett’s program to recast debates in metaphysics, and in particular disputes about realism and anti-realism, as closely intertwined with issues in philosophical logic, with the acceptance of classical logic for a domain reflecting a commitment to realism for that domain. Then I review extant results about what is provable and what is not when one adds epsilon to intuitionistic logic, largely due to Bell and DeVidi, and I give several new proofs of intermediate logics from intuitionistic logic+ε without identity. With all this in hand, I turn to a discussion of the philosophical significance of choice operators. Among the conclusions I defend are that these results provide a finer-grained basis for Dummett’s contention that commitment to classically valid but intuitionistically invalid principles reflect metaphysical commitments by showing those principles to be derivable from certain existence assumptions; that Dummett’s framework is improved by these results as they show that questions of realism and anti-realism are not an “all or nothing” matter, but that there are plausibly metaphysical stances between the poles of anti-realism and realism, because different sorts of ontological assumptions yield intermediate rather than classical logic; and that these intermediate positions between classical and intuitionistic logic link up in interesting ways with our intuitions about issues of objectivity and reality, and do so usefully by linking to questions around intriguing everyday concepts such as “is smart,” which I suggest involve a number of distinct dimensions which might themselves be objective, but because of their multivalent structure are themselves intermediate between being objective and not. Finally, I discuss the implications of these results for ongoing debates about the status of arbitrary and ideal objects in the foundations of logic, showing among other things that much of the discussion is flawed because it does not recognize the degree to which the claims being made depend on the presumption that one is working with a very strong logic. (shrink)

Neste ensaio, discutem-se cinco questões acerca da existência: 1. É a existência representável em termos de quantificação? 2. É a existência um predicado" real", de primeira ordem? 3. É existir o mesmo que ser? 4. Existe tudo? 5. Qual é a forma lógica de afirmações de existência? São introduzidas e examinadas algumas das mais salientes posições acerca destas questões, em especial a concepção Frege-Russell da existência e diversas concepções recentes neo-Meinongianas. Defendemos as seguintes três teses acerca daquilo que deve ser (...) um bom predicado de existência:(a) um predicado puramente lógico e parcialmente definível em termos do quantificador existencial;(b) invariavelmente um predicado de primeira ordem (dadas certas suposições de partida acerca do universo do discurso);(c) um predicado universal, verdadeiro de tudo e falso de nada. (shrink)

Words have meaning. Sentences also have meaning, but their meaning is different in kind from any collection of the meanings of the words they contain. I discuss two puzzles related to this difference. The first is how the meanings of the parts of a sentence combine to give rise to a unified sentential meaning, as opposed to a mere collection of disparate meanings (UP1). The second is why the formal ontology of linguistic meaning changes when grammatical structure is built up (...) (UP2). For example, the meaning of a sentence is a proposition evaluable for truth and falsity. In contrast, a collection of the meanings of its parts does not constitute a proposition and is not evaluable for truth. These two puzzles are closely related, since change in formal ontology is the clearest sign of the unity of meaning. The most popular strategy for answering them is taking the meanings of the parts as abstractions from primitive sentence meanings. However, I argue that, given plausible psychological constraints, sentence meanings cannot be taken as explanatory primitives. Drawing on recent work in Generative Grammar and its philosophy, I suggest that the key to both unity questions is to distinguish strictly between lexical and grammatical meaning. The latter is irreducible and determines how lexical content is used in referential acts. I argue that these referential properties determine a formal ontology, which explains why and how formal ontology changes when grammatical structure is built up (UP2). As for UP1, I suggest that, strictly speaking, lexical meanings never combine. Instead, whenever grammar specifies a formal ontology for the lexical meanings entering a grammatical derivation, further lexical (or phrasal) meanings can only specify aspects of this recursive grammatical process. In this way, contemporary grammatical theory can be used to address old philosophical problems. (shrink)

This thesis comprises three main chapters—each comprising one relatively standalone paper. The unifying theme is fragmentalism about truth, which is the view that the predicate “true” either expresses distinct concepts or expresses distinct properties. -/- In Chapter 1, I provide a formal development of alethic pluralism. Pluralism is the view that there are distinct truth properties associated with distinct domains of subject matter, where a truth property satisfies certain truth-characterizing principles. On behalf of pluralists, I propose an account of logic (...) and semantics that shows how they can answer central conceptual and logical challenges for their view. -/- In Chapter 2, I motivate and develop a modal account of propositions on the basis of an iterative conception of propositions, where the modality is logico-mathematical. The modal account of propositions takes the conception to motivate an inherently potential hierarchy of propositions. I show that the account helps provide satisfying solutions to the intensional paradoxes of Russell-Myhill, Kaplan, and Prior. -/- In Chapter 3, I propose that “true” is polysemous. I suggest that “true” is initially polysemous between correspondence truth and disquotational truth, and further polysemous between the meanings corresponding to the subconcepts of the concept truth generated by the indefinite extensibility of that concept. I show that the proposal provides satisfying solutions to the semantic paradoxes. (shrink)