Dummett’s notion of indefinite extensibility is influential but obscure. The notion figures centrally in an alternative Dummettian argument for intuitionistic logic and anti-realism, distinct from his more famous, meaning-theoretic arguments to the same effect. Drawing on ideas from Dummett, a precise analysis of indefinite extensibility is proposed. This analysis is used to reconstruct the poorly understood alternative argument. The plausibility of the resulting argument is assessed.

According to the iterative conception of set, each set is a collection of sets formed prior to it. The notion of priority here plays an essential role in explanations of why contradiction-inducing sets, such as the Russell set, do not exist. Consequently, these explanations are successful only to the extent that a satisfactory priority relation is made out. I argue that attempts to do this have fallen short: understanding priority in a straightforwardly constructivist sense threatens the coherence of the empty (...) set and raises serious epistemological concerns; but the leading realist interpretations---ontological and modal interpretations of priority---are deeply problematic as well. I conclude that the purported explanatory virtues of the iterative conception are, at present, unfounded. (shrink)

There are multiple formal characterizations of the natural numbers available. Despite being inter-derivable, they plausibly codify different possible applications of the naturals – doing basic arithmetic, counting, and ordering – as well as different philosophical conceptions of those numbers: structuralist, cardinal, and ordinal. Some influential philosophers of mathematics have argued for a non-egalitarian attitude according to which one of those characterizations is ‘more basic’ or ‘more fundamental’ than the others. This paper addresses two related issues. First, we review some of (...) these non-egalitarian arguments, lay out a laundry list of different, legitimate, notions of relative priority, and suggest that these arguments plausibly employ different such notions. Secondly, we argue that given a metaphysical-cum-epistemological gloss suggested by Frege's foundationalist epistemology, the ordinals are plausibly more basic than the cardinals. This is just one orientation to relative priority one could take, however. Ultimately, we subscribe to an egalitarian attitude towards these formal characterizations: they are, in some sense, equally ‘legitimate’. (shrink)

This dissertation is centered around a set of apparently conflicting intuitions that we may have about mathematics. On the one hand, we are inclined to believe that the theorems of mathematics are true. Since many of these theorems are existence assertions, it seems that if we accept them as true, we also commit ourselves to the existence of mathematical objects. On the other hand, mathematical objects are usually thought of as abstract objects that are non-spatiotemporal and causally inert. This makes (...) it difficult to understand how we can have knowledge of them and how they can have any relevance for our mathematical theories. I begin by characterizing a realist position in the philosophy of mathematics and discussing two of the most influential arguments for that kind of view. Next, after highlighting some of the difficulties that realism faces, I look at a few alternative approaches that attempt to account for our mathematical practice without making the assumption that there exist abstract mathematical entities. More specifically, I examine the fictionalist views developed by Hartry Field, Mark Balaguer, and Stephen Yablo, respectively. A common feature of these views is that they accept that mathematics interpreted at face value is committed to the existence of abstract objects. In order to avoid this commitment, they claim that mathematics, when taken at face value, is false. I argue that the fictionalist idea of mathematics as consisting of falsehoods is counter-intuitive and that we should aim for an account that can accommodate both the intuition that mathematics is true and the intuition that the causal inertness of abstract mathematical objects makes them irrelevant to mathematical practice and mathematical knowledge. The solution that I propose is based on Rudolf Carnap's distinction between an internal and an external perspective on existence. I argue that the most reasonable interpretation of the notions of mathematical truth and existence is that they are internal to mathematics and, hence, that mathematical truth cannot be used to draw the conclusion that mathematical objects exist in an external/ontological sense. (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)

I offer an analysis of the sentence "the concept horse is a concept". It will be argued that the grammatical subject of this sentence, "the concept horse", indeed refers to a concept, and not to an object, as Frege once held. The argument is based on a criterion of proper-namehood according to which an expression is a proper name if it is so rendered in Frege's ideography. The predicate "is a concept", on the other hand, should not be thought of (...) as referring to a function. It will be argued that the analysis of sentences of the form "C is a concept" requires the introduction of a new form of statement. Such statements are not to be thought of as having function--argument form, but rather the structure subject--copula--predicate. (shrink)

Dedekind's major work on the foundations of arithmetic employs several techniques that have left him open to charges of psychologism, and through this, to worries about the objectivity of the natural-number concept he defines. While I accept that Dedekind takes the foundation for arithmetic to lie in certain mental powers, I will also argue that, given an appropriate philosophical background, this need not make numbers into subjective mental objects. Even though Dedekind himself did not provide that background, one can nevertheless (...) be found in the work of the neo-Kantian Ernst Cassirer on mathematical concept formation. (shrink)

The notion of potential infinity dominated in mathematical thinking about infinity from Aristotle until Cantor. The coherence and philosophical importance of the notion are defended. Particular attention is paid to the question of whether potential infinity is compatible with classical logic or requires a weaker logic, perhaps intuitionistic.

Logical analysis is in Frege primarily not an analysis of a concept but of its sense. Five Fregean philosophical principles are presented as constituting a framework for a theory of logical or conceptual analysis, which I call analytical explication. These principles, scattered and sometime latent in his writings are operative in Frege's critique of other views and in his constructive development of his own view. The proposed conception of analytical explication is partially rooted in Frege's notion of analytical definition. It (...) may also be the basis of what is required of a reduction of one domain to another, if it is to have the philosophical significance many reductions allegedly have. (shrink)

The categorial concepts of substance (thing) and substance (stuff) are described, and the conceptual relationships between things and their constitutive stuff delineated. The relationship between substance concepts, expressed by other count-nouns, and natural kind concepts is examined. Artefacts and their parts are argued to be substances, whereas parts of organisms are not. The confusions of seventeenth- and eighteenth-century philosophers who invoked the concept of substance are adumbrated.

This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. David Elohim examines the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable (...) propositions, and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter \textbf{2} argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal and hyperintensional cognitivism and modal and hyperintensional expressivism. Elohim develops a novel, topic-sensitive truthmaker semantics for dynamic epistemic logic, and develops a novel, dynamic two-dimensional semantics grounded in two-dimensional hyperintensional Turing machines. Chapter \textbf{3} provides an abstraction principle for two-dimensional (hyper-)intensions. Chapter \textbf{4} advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter \textbf{5} applies the fixed points of the modal $\mu$-calculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal $\mu$-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal $\mu$-calculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter \textbf{6} advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter \textbf{7} provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters \textbf{2} and \textbf{4} is availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. -/- Chapters \textbf{8-12} provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter \textbf{8} examines the interaction between Elohim's hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of $\Omega$-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing Thesis and hyperintensional epistemic set theories in the literature. Chapter \textbf{9} examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. Elohim countenances a hyperintensional semantics for novel epistemic abstractionist modalities. Elohim suggests, too, that higher observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for two-dimensional hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter \textbf{10} examines the philosophical significance of hyperintensional $\Omega$-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter \textbf{11} provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter \textbf{12} avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic two-dimensional semantics in order to account for the interaction between interpretational and objective modalities and the truthmakers thereof. This yields the first hyperintensional category theory in the literature. Elohim invents a new mathematical trick in which first-order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals in the category of Set in category theory. Chapter \textbf{13} examines modal responses to the alethic paradoxes. Elohim provides a counter-example to epistemic closure for logical deduction. Chapter \textbf{14} examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory. (shrink)

Neologicists have sought to ground mathematical knowledge in abstraction. One especially obstinate problem for this account is the bad company problem. The leading neologicist strategy for resolving this problem is to attempt to sift the good abstraction principles from the bad. This response faces a dilemma: the system of ‘good’ abstraction principles either falls foul of the Scylla of inconsistency or the Charybdis of being unable to recover a modest portion of Zermelo–Fraenkel set theory with its intended generality. This article (...) argues that the bad company problem is due to the ‘static’ character of abstraction on the neologicist's account and develops a ‘dynamic’ account of abstraction that avoids both horns of the dilemma. 1 Introduction2ion Reconstructed3 From Bad to Worse4 Abstraction Reconceived5 Bad Company Made GoodA AppendixA.1SemanticsA.2LogicA.3ConsistencyA.4Strength. (shrink)

Is it possible to characterize the sortal essence of Fs for a sortal concept F solely in terms of a criterion of identity C for F? That is, can the question ‘What sort of thing are Fs?’ be answered by saying that Fs are essentially those things whose identity can be assessed in terms of C? This paper presents a case study supporting a negative answer to these questions by critically examining the neo-Fregean suggestion that cardinal numbers can be fully (...) characterized as those things whose identity can be assessed in terms of one-one correspondence between concepts. (shrink)

This paper contains a close analysis of Frege's proofs of the axioms of arithmetic §§70-83 of Die Grundlagen, with special attention to the proof of the existence of successors in §§82-83. Reluctantly and hesitantly, we come to the conclusion that Frege was at least somewhat confused in those two sections and that he cannot be said to have outlined, or even to have intended, any correct proof there. The proof he sketches is in many ways similar to that given in (...) Grundgesetze der Arithmetik, but fidelity to what Frege wrote in Die Grundlagen and in Grundgesetze requires us to reject the charitable suggestion that it was this (beautiful) proof that he had in mind in §§82-83. (shrink)

This thesis offers a naturalist revision of Alain Badiou’s philosophy. This goal is pursued through an encounter of Badiou’s mathematical ontology and theory of truth with contemporary trends in philosophy of mathematics and philosophy of science. I take issue with Badiou’s inability to elucidate the link between the empirical and the ontological, and his residual reliance on a Heideggerian project of fundamental ontology, which undermines his own immanentist principles. I will argue for both a bottom-up naturalisation of Badiou’s philosophical approach (...) to mathematics and a top-down naturalisation. Articulating my particular understanding of what realism and naturalism should commit us to, I propose a creative fusion of Badiou’s attention to metamathematical results with a structural-informational metaphysics, proposing a ‘matherialism’ uniting the more daring speculative insights of the former with the naturalist and empiricist commitments motivating the latter. (shrink)

This paper argues that Frege is not the metaphysical platonist about mathematics that he is standardly taken to be. It is shown that Frege’s project has two distinct stages: the identification of what is true of our ordinary notions, and then the provision of a systematic account that shares the identified features. Neither of these stages involves much metaphysics. The paper criticizes in detail Dummett’s interpretation of §§55-61 of Grundlagen. These sections fall under the heading ‘Every number is a self-subsistent (...) object’ and are described by Dummett as containing the worst arguments put forward by Frege. It is argued that essentially all of Dummett’s interpretive points are mistaken. Finally, I show that Frege’s claims about the independence of mathematics from humans and their activities does not commit him to any particularly metaphysical position either. (shrink)

In this paper, I discuss the prospects of nominalistic scientific realism and show that it fails on many counts. In section 2, I discuss what is required for NSR to get off the ground. In section 3, I question the idea that theories have well-defined nominalistic content and the idea that causal activity is a necessary condition for commitment to the reality of an entity. In section 4, I challenge the notion of nominalistic adequacy of theories.

O presente manual tem como intenção constituir um guia para uma disciplina introdutória de filosofia da linguagem. Foi elaborado a partir da leccionação da disciplina de Filosofia da Linguagem I na Faculdade de Letras da Universidade do Porto desde 2001. A disciplina de Filosofia da Linguagem I ocupa um semestre lectivo e proporciona aos estudantes o primeiro contacto sistemático com a área da filosofia da linguagem. Pretende-se que este manual ofereça aos estudantes os instrumentos necessários não apenas para acompanhar uma (...) iniciação ao campo da filosofia da linguagem mas também para o estudo e a investigação autónomos posteriores. Para isso é percorrido um trajecto que conduz das intuições pré-teóricas acerca de linguagem, de que todos dispomos, até um conhecimento disciplinar específico, histórico e temático, da filosofia da linguagem. Em termos práticos, são considerados como precedentes da disciplina de Filosofia da Linguagem I – mesmo se, pelo existem actualmente precedências em sentido estrito – as disciplinas de Lógica I e II e de Filosofia do Conhecimento I e II. Os programas dessas disciplinas nos últimos anos estiveram presentes por trás da elaboração do manual. (shrink)

This paper explores the relationship between Hume's Prinicple and Basic Law V, investigating the question whether we really do need to suppose that, already in Die Grundlagen, Frege intended that HP should be justified by its derivation from Law V.

Are Fregean thoughts compositionally complex and composed of senses? We argue that, in Begriffsschrift, Frege took 'conceptual contents' to be unstructured, but that he quickly moved away from this position, holding just two years later that conceptual contents divide of themselves into 'function' and 'argument'. This second position is shown to be unstable, however, by Frege's famous substitution puzzle. For Frege, the crucial question the puzzle raises is why "The Morning Star is a planet" and "The Evening Star is a (...) planet" have different contents, but his second position predicts that they should have the same content. Frege's response to this antinomy is of course to distinguish sense from reference, but what has not previously been noticed is that this response also requires thoughts to be compositionally complex, composed of senses. That, however, raises the question just how thoughts are composed from senses. We reconstruct a Fregean answer, one that turns on an insistence that this question must be understood as semantic rather than metaphysical. It is not a question about the intrinsic nature of residents of the third realm but a question about how thoughts are expressed by sentences. (shrink)

This paper puts forward and defends an account of mathematical truth, and in particular an account of the truth of mathematical axioms. The proposal attempts to be completely nonrevisionist. In this connection, it seeks to satisfy simultaneously both horns of Benacerrafs work on informal rigour. Kreisel defends the view that axioms are arrived at by a rigorous examination of our informal notions, as opposed to being stipulated or arrived at by trial and error. This view is then supplemented by a (...) Fregean account of the objectivity and our knowledge of abstract objects. It is then argued that the resulting view faces no insurmountable metaphysical or epistemic obstacles. (shrink)

I consider different versions of a structuralist view of mathematical objects, according to which characteristic mathematical objects have no more of a 'nature' than is given by the basic relations of a structure in which they reside. My own version of such a view is non-eliminative in the sense that it does not lead to a programme for eliminating reference to mathematical objects. I reply to criticisms of non-eliminative structuralism recently advanced by Keränen and Hellman. In replying to the former, (...) I rely on a distinction between 'basic' and 'constructed' structures. A conclusion is that ideas from the metaphysical tradition can be misleading when applied to the objects of modern mathematics. (shrink)

I consider a grand, yet neglected proposal put forward by Shoemaker—a functionalist theory of all properties. I argue that two possible ways of developing this proposal meet with substantial objections. However, if we are prepared to endorse an ontology of tropes, one of these functionalist analyses can be developed into an original and informative theory of properties.

In his groundbreaking Grundlagen, Frege (1884) pointed out that number words like ‘four’ occur in ordinary language in two quite different ways and that this gives rise to a philosophical puzzle. On the one hand ‘four’ occurs as an adjective, which is to say that it occurs grammatically in sentences in a position that is commonly occupied by adjectives. Frege’s example was (1) Jupiter has four moons, where the occurrence of ‘four’ seems to be just like that of ‘green’ in (...) (2) Jupiter has green moons. On the other hand, ‘four’ occurs as a singular term, which is to say that it occurs in a position that is commonly occupied by paradigmatic cases of singular terms, like proper names: (3) The number of moons of Jupiter is four. Here ‘four’ seems to be just like ‘Wagner’ in (4) The composer of Tannhäuser is Wagner, and both of these statements seem to be identity statements, ones with which we claim that what two singular terms stand for is identical. But that number words can occur both as singular terms and as adjectives is puzzling. Usually adjectives cannot occur in a position occupied by a singular term, and the other way round, without resulting in ungrammaticality and nonsense. To give just one example, it would be ungrammatical to replace ‘four’ with ‘the number of moons of Jupiter’ in (1): (5) *Jupiter has the number of moons of Jupiter moons. This ungrammaticality results even though ‘four’ and ‘the number of moons of Jupiter’ are both apparently singular terms standing for the same object in (3). So, how can it be that number words can occur both as singular terms and as adjectives, while other adjectives cannot occur as singular terms, and other singular terms cannot occur as adjectives? (shrink)

Discussions of the applicability of mathematics in the natural sciences have been flawed by failure to realize that there are multiple senses in which mathematics can be ‘applied’ and, correspondingly, multiple problems that stem from the applicability of mathematics. I discuss semantic, metaphysical, descriptive, and and epistemological problems of mathematical applicability, dwelling on Frege's contribution to the solution of the first two types. As for the remaining problems, I discuss the contributions of Hartry Field and Eugene Wigner. Finally, I argue (...) that there are epistemological problems concerning the applicability of mathematics that nobody in the philosophical community has yet confronted, though the problems are well known to physicists. (shrink)

Russell had two theories of definite descriptions: one for singular descriptions, another for plural descriptions. We chart its development, in which ‘On Denoting’ plays a part but not the part one might expect, before explaining why it eventually fails. We go on to consider many-valued functions, since they too bring in plural terms—terms such as ‘4’ or the descriptive ‘the inhabitants of London’ which, like plain plural descriptions, stand for more than one thing. Logicians need to take plural reference seriously (...) if only because mathematicians take many-valued functions seriously. We assess the objection (by Russell, Frege and others) that many-valued functions are illegitimate because the corresponding terms are ambiguous. We also assess the various methods proposed for getting rid of them. Finding the objection ill-founded and the methods ineffective, we introduce a logical framework that admits plural reference, and use it to answer some earlier questions and to raise some more. (shrink)

Neo-Fregeanism contends that knowledge of arithmetic may be acquired by second-order logical reflection upon Hume's principle. Heck argues that Hume's principle doesn't inform ordinary arithmetical reasoning and so knowledge derived from it cannot be genuinely arithmetical. To suppose otherwise, Heck claims, is to fail to comprehend the magnitude of Cantor's conceptual contribution to mathematics. Heck recommends that finite Hume's principle be employed instead to generate arithmetical knowledge. But a better understanding of Cantor's contribution is achieved if it is supposed that (...) Hume's principle really does inform arithmetical practice. More generally, Heck's arguments misconceive the epistemological character of neo-Fregeanism. (shrink)

In this paper, we investigate (1) what can be salvaged from the original project of "logicism" and (2) what is the best that can be done if we lower our sights a bit. Logicism is the view that "mathematics is reducible to logic alone", and there are a variety of reasons why it was a non-starter. We consider the various ways of weakening this claim so as to produce a "neologicism". Three ways are discussed: (1) expand the conception of logic (...) used in the reduction, (2) allow the addition of analytic-sounding principles to logic so that the reduction is not to "logic alone" but to logic and truths knowable a priori, and (3) revise the conception of "reducible". We show how the current versions of neologicism fit into this classification scheme, and then focus on a kind of neologicism which we take to have the most potential for achieving the epistemological goals of the original logicist project. We argue that that the "weaker" the form of neologicism, the more likely it is to be a new form of logicism, and show how our preferred system, though mathematically weak, is metaphysically and epistemogically strong, and can "reduce" arbitrary mathematical theories to logic and analytic truths, if given a legitimate new sense of "reduction". (shrink)

On the neo-Fregean approach to the foundations of mathematics, elementary arithmetic is analytic in the sense that the addition of a principle wliich may be held to IMJ explanatory of the concept of cardinal number to a suitable second-order logical basis suffices for the derivation of its basic laws. This principle, now commonly called Hume's principle, is an example of a Fregean abstraction principle. In this paper, I assume the correctness of the neo-Fregean position on elementary aritlunetic and seek to (...) explain one way in which it may be extended to encompass the theory of real numbers, introducing the reals, by means of suitable further abstraction principles, as ratios of quantities. (shrink)

A common view is that natural language treats numbers as abstract objects, with expressions like the number of planets, eight, as well as the number eight acting as referential terms referring to numbers. In this paper I will argue that this view about reference to numbers in natural language is fundamentally mistaken. A more thorough look at natural language reveals a very different view of the ontological status of natural numbers. On this view, numbers are not primarily treated abstract objects, (...) but rather 'aspects' of pluralities of ordinary objects, namely number tropes, a view that in fact appears to have been the Aristotelian view of numbers. Natural language moreover provides support for another view of the ontological status of numbers, on which natural numbers do not act as entities, but rather have the status of plural properties, the meaning of numerals when acting like adjectives. This view matches contemporary approaches in the philosophy of mathematics of what Dummett called the Adjectival Strategy, the view on which number terms in arithmetical sentences are not terms referring to numbers, but rather make contributions to generalizations about ordinary (and possible) objects. It is only with complex expressions somewhat at the periphery of language such as the number eight that reference to pure numbers is permitted. (shrink)

A core commitment of Bob Hale and Crispin Wright’s neologicism is their invocation of Frege’s Constraint—roughly, the requirement that the core empirical applications for a class of numbers be “built directly into” their formal characterization. According to these neologicists, if legitimate, Frege’s Constraint adjudicates in favor of their preferred foundation—Hume’s Principle—and against alternatives, such as the Dedekind–Peano axioms. In this paper, we consider a recent argument for legitimating Frege’s Constraint due to Hale, according to which the primary empirical application of (...) the naturals is transitive counting, or answering ‘how many’-questions using numerals. We make two claims regarding Hale’s argument. First, it fails to legitimate Frege’s Constraint in virtue of resting on unsupported and highly contentious assumptions. Secondly, even if sound, Hale’s argument would vindicate a version of Frege’s Constraint which fails to adjudicate in favor of Hume’s Principle over alternative characterizations of the naturals. (shrink)

The way scientific discovery has been conceptualized has changed drastically in the last few decades: its relation to logic, inference, methods, and evolution has been deeply reloaded. The ‘philosophical matrix’ moulded by logical empiricism and analytical tradition has been challenged by the ‘friends of discovery’, who opened up the way to a rational investigation of discovery. This has produced not only new theories of discovery, but also new ways of practicing it in a rational and more systematic way. Ampliative rules, (...) methods, heuristic procedures and even a logic of discovery have been investigated, extracted, reconstructed and refined. The outcome is a ‘scientific discovery revolution’: not only a new way of looking at discovery, but also a construction of tools that can guide us to discover something new. This is a very important contribution of philosophy of science to science, as it puts the former in a position not only to interpret what scientists do, but also to provide and improve tools that they can employ in their activity. (shrink)

Scientific models are important, if not the sole, units of science. This thesis addresses the following question: in virtue of what do scientific models represent their target systems? In Part i I motivate the question, and lay out some important desiderata that any successful answer must meet. This provides a novel conceptual framework in which to think about the question of scientific representation. I then argue against Callender and Cohen’s attempt to diffuse the question. In Part ii I investigate the (...) ideas that scientific models are ‘similar’, or structurally morphic, to their target systems. I argue that these approaches are misguided, and that at best these relationships concern the accuracy of a pre-existing representational relationship. I also pay particular attention to the sense in which target systems can be appropriately taken to exhibit a ‘structure’, and van Fraassen’s recent argument concerning the pragmatic equivalence between representing phenomena and data. My next target is the idea that models should not be seen as objects in their own right, but rather what look like descriptions of them are actually direct descriptions of target systems, albeit not ones that should be understood literally. I argue that these approaches fail to do justice to the practice of scientific modelling. Finally I turn to the idea that how models represent is grounded, in some sense, in their inferential capacity. I compare this approach to anti-representationalism in the philosophy of language and argue that analogous issues arise in the context of scientific representation. Part iii contains my positive proposal. I provide an account of scientific representation based on Goodman and Elgin’s notion of representation-as. The result is a highly conventional account which is the appropriate level of generality to capture all of its instances, whilst remaining informative about the notion. I illustrate it with reference to the Phillips-Newlyn machine, models of proteins, and the Lotka-Volterra model of predator-prey systems. These examples demonstrate how the account must be understood, and how it sheds light on our understanding of how models are used. I finally demonstrate how the account meets the desiderata laid out at the beginning of the thesis, and outline its implications for further questions from the philosophy of science; not limited to issues surrounding the applicability of mathematics, idealisation, and what it takes for a model to be ‘true’. (shrink)

Adherents of Ockham’s fundamental razor contend that considerations of ontological parsimony pertain primarily to fundamental objects. Derivative objects, on the other hand, are thought to be quite unobjectionable. One way to understand the fundamental vs. derivative distinction is in terms of the Aristotelian distinction between ontologically independent and dependent objects. In this paper I will defend the thesis that every natural number greater than 0 is an ontologically dependent object thereby exempting the natural numbers from Ockham’s fundamental razor.

Frege’s argument against the ancient Greek conception of numbers as 'multitudes of units’ has been hailed as one of the most successful in his "Grundlagen". The aim of this paper is to show that despite Frege’s best efforts, the Euclidean conception remains a viable alternative to the Fregean conception of numbers by arguing that neither a dilemma argument Frege brings against the Euclidean conception nor a possible argument against it based on the truth of what is known as "Hume’s Principle" (...) succeeds. (shrink)

Social choice and welfare economics are subjects at the frontier of many disciplines. Even if economics played the major role in their development, sociology, psychology and, principally, political science, mathematics and philosophy have been central for the manifold inventiveness of the employed methods and for the diversity of the studied topics. This phenomenon can be compared with game theory, a subject which has, of course, many connections with social choice and welfare. This fact is reflected by the disciplinary origins of (...) the contributors to the subject and, as an anecdote, by the disciplinary origins of the board of editors of this journal. Philosophers are expected to contribute mainly to the study of social justice and related ethical questions. But there is a tradition among logicians for studying voting theory. A famous example is C. L. Dodgson (Lewis Carroll), even though the complete works of Dodgson on voting occupy only a few pages. A major recent example is Michael Dummett. Michael Dummett is famous among social choice theorists for his joint paper with Robin Farquharson published in Econometrica in 1961. Later he wrote two important books on voting (Dummett (1984, 1997); for an overview see Salles (2006)). But it must be outlined that Michael Dummett is also, and above all, one of the greatest contemporary philosophers whose work on the German logician Frege, on intuitionism, realism, anti-realism, justificationism has been central for the development of analytical philosophy in the second part of the last century and in this century (an example is the Symposium in a recent issue of Mind (see Peacocke (2005) and Dummett (2005)). Sir Michael Dummett is Wykeham Professor of Logic emeritus at Oxford University. His interview was conducted at New College, Oxford in September 2004. (shrink)

Crispin Wright and Bob Hale have defended the strategy of defining the natural numbers contextually against the objection which led Frege himself to reject it, namely the so-called ‘Julius Caesar problem’. To do this they have formulated principles (called sortal inclusion principles) designed to ensure that numbers are distinct from any objects, such as persons, a proper grasp of which could not be afforded by the contextual definition. We discuss whether either Hale or Wright has provided independent motivation for a (...) defensible version of the sortal inclusion principle and whether they have succeeded in showing that numbers are just what the contextual definition says they are. (shrink)

Frege's development of the theory of arithmetic in his Grundgesetze der Arithmetik has long been ignored, since the formal theory of the Grundgesetze is inconsistent. His derivations of the axioms of arithmetic from what is known as Hume's Principle do not, however, depend upon that axiom of the system--Axiom V--which is responsible for the inconsistency. On the contrary, Frege's proofs constitute a derivation of axioms for arithmetic from Hume's Principle, in (axiomatic) second-order logic. Moreover, though Frege does prove each of (...) the now standard Dedekind-Peano axioms, his proofs are devoted primarily to the derivation of his own axioms for arithmetic, which are somewhat different (though of course equivalent). These axioms, which may be yet more intuitive than the Dedekind-Peano axioms, may be taken to be "The Basic Laws of Cardinal Number", as Frege understood them. Though the axioms of arithmetic have been known to be derivable from Hume's Principle for about ten years now, it has not been widely recognized that Frege himself showed them so to be; nor has it been known that Frege made use of any axiomatization for arithmetic whatsoever. Grundgesetze is thus a work of much greater significance than has often been thought. First, Frege's use of the inconsistent Axiom V may invalidate certain of his claims regarding the philosophical significance of his work (viz., the establishment of Logicism), but it should not be allowed to obscure his mathematical accomplishments and his contribution to our understanding of arithmetic. Second, Frege's knowledge that arithmetic is derivable from Hume's Principle raises important sorts of questions about his philosophy of arithmetic. For example, "Why did Frege not simply abandon Axiom V and take Hume's Principle as an axiom?". (shrink)

A feature of Frege's philosophy of arithmetic that has elicited a great deal of attention in the recent secondary literature is his contention that numbers are ‘self‐subsistent’ objects. The considerable interest in this thesis among the contemporary philosophy of mathematics community stands in marked contrast to Kreisel's folk‐lore observation that the central problem in the philosophy of mathematics is not the existence of mathematical objects, but the objectivity of mathematics. Although Frege was undoubtedly concerned with both questions, a goal of (...) the present paper is to argue that his success in securing the objectivity of arithmetic depends on a less contentious commitment to numbers as objects than either he or his critics have supposed. As such, this paper is an articulation and defense of both Frege's analysis of arithmetic and Kreisel's observation. (shrink)

How do scientific models represent in a way that enables us to discover new truths about reality and draw inferences about it? Contemporary accounts of scientific discovery answer this question by focusing on the cognitive mechanisms involved in the generation of new ideas and concepts in terms of a special sort of reasoning—or model-based reasoning—involving imagery. Alternatively, I argue that answering this question requires that we recognise the crucial role of the propositional imagination in the construction and development of models (...) for the purpose of generating hypotheses that are plausible candidates for truth. I propose simple fictionalism as a new account of models as Waltonian games of make-believe and suggest that models can lead to genuine scientific discovery when they are used as representations that denote real world phenomena and generate two main kinds of theoretical hypotheses, model-world comparisons and direct attributions. (shrink)

Recent discussions on Fregean and neo-Fregean foundations for arithmetic and real analysis pay much attention to what is called either ‘Application Constraint’ ($AC$) or ‘Frege Constraint’ ($FC$), the requirement that a mathematical theory be so outlined that it immediately allows explaining for its applicability. We distinguish between two constraints, which we, respectively, denote by the latter of these two names, by showing how$AC$generalizes Frege’s views while$FC$comes closer to his original conceptions. Different authors diverge on the interpretation of$FC$and on whether it (...) applies to definitions of both natural and real numbers. Our aim is to trace the origins of$FC$and to explore how different understandings of it can be faithful to Frege’s views about such definitions and to his foundational program. After rehearsing the essential elements of the relevant debate (§1), we appropriately distinguish$AC$from$FC$(§2). We discuss six rationales which may motivate the adoption of different instances of$AC$and$FC$(§3). We turn to the possible interpretations of$FC$(§4), and advance a Semantic$FC$(§4.1), arguing that while it suits Frege’s definition of natural numbers (4.1.1), it cannot reasonably be imposed on definitions of real numbers (§4.1.2), for reasons only partly similar to those offered by Crispin Wright (§4.1.3). We then rehearse a recent exchange between Bob Hale and Vadim Batitzky to shed light on Frege’s conception of real numbers and magnitudes (§4.2). We argue that an Architectonic version of$FC$is indeed faithful to Frege’s definition of real numbers, and compatible with his views on natural ones. Finally, we consider how attributing different instances of$FC$to Frege and appreciating the role of the Architectonic$FC$can provide a more perspicuous understanding of his foundational program, by questioning common pictures of his logicism (§5). (shrink)

This comment focuses on Chapter 9 of The Boundary Stones of Thought and the argument, due to William Tait, that Ian Rumfitt there sustains for the indeterminacy of set. I argue that Michael Dummett’s argument, based on the notion of indefinite extensibility and set aside by Rumfitt, provides a more powerful basis for the same conclusion. In addition, I outline two difficulties for the way Rumfitt attempts to save classical logic from acknowledged failures of the principle of bivalence, one specifically (...) for his treatment of the set-theoretic case, the other of more general bearing but especially germane to the case of vagueness. (shrink)

In §57 of the Foundations of Arithmetic, Frege famously turns to natural language to support his claim that numbers are ‘self-subsistent objects’:I have already drawn attention above to the fact th...

The paper proves a conjecture of Solomon Feferman concerning the indefiniteness of the continuum hypothesis relative to a semi-intuitionistic set theory.

In his treatise Die Grundlagen der Arithmetik, Gottlob Frege tries to find a definition of number. First he rejects the idea that number could be a property of external objects. Then he comes with a suggestion that a numerical statement expresses a property of a concept, namely it indicates how many objects fall under the concept. Subsequently Frege rejects, or at least essentially modifies, also this definition, because in his view that a number cannot be a property – it should (...) be an object. In this article, I try to show that Frege’s first definition of number seems to be, despite his own opinion, much more promising than he supposed. I also argue that Frege’s argumentation against the empirical character of number is by no means convincing. (shrink)

In the aftermath of a normalized Foucaultian world with an all encompassing web of biopower, one remaining hope is to cultivate nimbleness. Nimbleness is an embodied aesthetic sensitivity to the material presence. Cultivating nimbleness is a particular style of cultivation; it is to willfully gather together one’s self in the wake of a formative force far richer than the derivative web of living power relationships of human embeddness within a horizon of social, economical, political and historical subjectivating power relations; which (...) are chronicled and labeled by Michel Foucault as the normalizing practices of biopower. In other words to have freedom, one must start by rejecting the categories and labels normally internalized in order to relearn to learn from the material presence. Such a style of cultivation is a means of resisting normalizing power relations which co-opt cultivating practices to engross their own dominance which has had the by-product of an impotence to negate the gross material injustices present. This normalizing style of cultivation is a prevalent, corrupted, semblance which denies the importance of beauty for that of efficiency, rejects non-human purposiveness, and limits its measure of ethics to short term economical pragmatism. The thesis acknowledges that something is awry with the world and that giving care to beauty might help. The aim is to examine the event of Beauty as depicted by the philosopher Immanuel Kant and to apply this characterization to elective architectural spaces such that it may motivate individuals to cultivate their own nimbleness in relation to a formative force of nature. However given the revealed need for sensitivity to the particular material presence, the thesis can not be a rule book or catalog for beautiful design. Rather it is a rehabilitation for architects who are already heterospatially curious, with the desired outcome of architects cultivating their own nimbleness to reflectively judge as a ground up, multi-node, rhizomatic means of resistance to normalizing power practices as manifest in bad architecture. (shrink)

The Frege-Russell view is that existence is a second-order property rather than a property of individuals. One of the most compelling arguments for this view is based on the premise that there is an especially close connection between existence and number. The most promising version of this argument is by C.J.F Williams (1981, 1992). In what follows, I argue that this argument fails. I then defend an account according to which both predications of number and existence attribute properties to individuals.