It is our contention that an ontological commitment to propositions faces a number of problems; so many, in fact, that an attitude of realism towards propositions—understood the usual “platonistic” way, as a kind of mind- and language-independent abstract entity—is ultimately untenable. The particular worries about propositions that marshal parallel problems that Paul Benacerraf has raised for mathematical platonists. At the same time, the utility of “proposition-talk”—indeed, the apparent linguistic commitment evident in our use of 'that'-clauses (in offering explanations and making (...) predictions)—is also in need of explanation. We account for this with a fictionalist analysis of our use of 'that'-clauses. Our account avoids certain problems that arise for the usual error-theoretic versions of fictionalism because we apply the notion of semantic pretense to develop an alternative, pretense-involving, non-error-theoretic, fictionalist account of proposition-talk. (shrink)

The main aim of this thesis is to defend moral realism. In chapter 1, I argue that moral realism is best understood as the view that moral sentences have truth-value, there are moral properties that make some moral sentences true, and moral properties are not reducible to non- moral properties. Realism is contrasted with non-cognitivism, error-theory and reductionism, which, in brief, deny, and, respectively. In the introductory chapter, it is also argued that there are some prima facie reasons to assume (...) that non-cognitivism and error-theory are erroneous. In chapters 2 and 3, I suggest that the two main forms of reductionism, analytic and synthetic reductionism, are mistaken. In chapter 4, I argue that the considerations in the previous chapters in relation to non-cognitivism, error-theory and reductionism provide support to moral realism. It is also suggested that these considerations make it plausible to hypothesise that moral properties depend on non- moral properties in a way I refer to as ‘the realist formula’. The realist formula confirms moral realism since it implies that moral properties are not reducible to non- moral properties. In chapters 5, 6 and 7, I argue that moral realism, much owing to the realist formula, is able to explain significant meta-ethical issues regarding moral disagreement, moral reason and moral motivation. Among other things, externalism concerning moral motivation is defended. The explanatory value of moral realism in relation to these meta-ethical issues is taken to suggest that this view is preferable to non-cognitivism, error-theory and reductionism. Some of the meta-ethical issues discussed in these chapters, particularly moral disagreement and motivation, have been thought to provide support to non-cognitivism and error-theory. I maintain that since realism, unlike reductionism, is able to counter these arguments, it justifies us in upholding the view that moral sentences have truth-value and the view that there are moral properties. In chapter 8, various objections against realism with regard to the dependence of moral properties on non- moral properties are responded to. In chapter 9, I consider an influential argument to the effect that moral properties are not involved in causal explanations. I maintain that this argument fails and that it therefore is reasonable to assume that moral properties are natural properties. However, the discussions in chapters 8 and 9 also suggest that moral realism might face problems that cannot be thoroughly discussed in this thesis. (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.

Some authors have begun to appeal directly to studies of argumentation in their analyses of mathematical practice. These include researchers from an impressively diverse range of disciplines: not only philosophy of mathematics and argumentation theory, but also psychology, education, and computer science. This introduction provides some background to their work.

This paper argues that Philip Kitcher's epistemology of mathematics, codified in his Naturalistic Constructivism, is not naturalistic on Kitcher's own conception of naturalism. Kitcher's conception of naturalism is committed to (i) explaining the correctness of belief-regulating norms and (ii) a realist notion of truth. Naturalistic Constructivism is unable to simultaneously meet both of these commitments.

Platonism in the philosophy of mathematics is the doctrine that there are mathematical objects such as numbers. John Burgess and Gideon Rosen have argued that that there is no good epistemological argument against platonism. They propose a dilemma, claiming that epistemological arguments against platonism either rely on a dubious epistemology, or resemble a dubious sceptical argument concerning perceptual knowledge. Against Burgess and Rosen, I show that an epistemological anti- platonist argument proposed by Hartry Field avoids both horns of their dilemma.

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)

According to standard mathematical platonism, mathematical entities (numbers, sets, etc.) are abstract entities. As such, they lack causal powers and spatio-temporal location. Platonists owe us an account of how we acquire knowledge of this inaccessible mathematical realm. Some recent versions of mathematical platonism postulate a plenitude of mathematical entities, and Mark Balaguer has argued that, given the existence of such a plenitude, the attainment of mathematical knowledge is rendered non-problematic. I assess his epistemology for such a profligate platonism and find (...) it unsatisfactory because it lacks an adequate semantics, in particular, an adequate account of reference. (shrink)

In her recent book, Realism in mathematics, Penelope Maddy attempts to reconcile a naturalistic epistemology with realism about set theory. The key to this reconciliation is an analogy between mathematics and the physical sciences based on the claim that we perceive the objects of set theory. In this paper I try to show that neither this claim nor the analogy can be sustained. But even if the claim that we perceive some sets is granted, I argue that Maddy's account fails (...) to explain the key issue faced by an epistemology for mathematics, namely the step from knowledge of the finite to knowledge of the infinite. (shrink)

James Ladyman has recently proposed a view according to which all that exists on the level of microphysics are structures "all the way down". By means of a comparative reading of structuralism in philosophy of mathematics as proposed by Stewart Shapiro, I shall present what I believe structures could not be. I shall argue that, if Ladyman is indeed proposing something as strong as suggested here, then he is committed to solving problems that proponents of structuralism in philosophy of mathematics (...) such as Shapiro are trying to solve. Attempting to do so, however, brings out a tacit tension in Ladyman's position. I shall argue that the upshot of this is that the ontological import that Ladyman attributes to structures is rather epistemological import properly understood. (shrink)

I argue here that a properly Platonic theory of the nature of number is still viable today. By properly Platonic, I mean one consistent with Plato's own theory, with appropriate extensions to take into account subsequent developments in mathematics. At Parmenides 143a-4a the existence of numbers is proven from our capacity to count, whereby I establish as Plato's the theory that numbers are originally ordinal, a sequence of forms differentiated by position. I defend and interpret Aristotle's report of a Platonic (...) distinction between form and mathematical numbers, arguing that mathematical numbers alone are cardinals, by reference to certain non-technical features of a set-theoretical approach and other considerations in philosophy of mathematics. Finally I respond to the objections that such a conception of number was unavailable in antiquity and that this theory is contradicted by Aristotle's report in Metaph . XIII that Platonic numbers are collections of units. I argue that Aristotle reveals his own misinterpretation of the terms in which Plato's theory was expressed. (shrink)

According to the dominant computational approach in cognitive science, cognitive agents are digital computers; according to the alternative approach, they are dynamical systems. This target article attempts to articulate and support the dynamical hypothesis. The dynamical hypothesis has two major components: the nature hypothesis (cognitive agents are dynamical systems) and the knowledge hypothesis (cognitive agents can be understood dynamically). A wide range of objections to this hypothesis can be rebutted. The conclusion is that cognitive systems may well be dynamical systems, (...) and only sustained empirical research in cognitive science will determine the extent to which that is true. (shrink)

Many philosophers think all abstract objects are causally inert. Here, focusing on novels, I argue that some abstracta are causally efficacious. First, I defend a straightforward argument for this view. Second, I outline an account of object causation—an account of how objects cause effects. This account further supports the view that some abstracta are causally efficacious.

This article elaborates the epistemic indispensability argument, which fully embraces the epistemic contribution of mathematics to science, but rejects the contention that such a contribution is a reason for granting reality to mathematicalia. Section 1 introduces the distinction between ontological and epistemic readings of the indispensability argument. Section 2 outlines some of the main flaws of the first premise of the ontological reading. Section 3 advances the epistemic indispensability argument in view of both applied and pure mathematics. And Sect. 4 (...) makes a case for the epistemic approach, which firstly calls into question the appeal to inference to the best explanation in the defense of the indispensability claim; secondly, distinguishes between mathematical and physical posits; and thirdly, argues that even though some may think that inference to the best explanation works in the postulation of physical posits, no similar considerations are available for postulating mathematicalia. (shrink)

I develop a novel argument against the claim that ordinals are sets. In contrast to Benacerraf’s antireductionist argument, I make no use of covert epistemic assumptions. Instead, my argument uses considerations of ontological dependence. I draw on the datum that sets depend immediately and asymmetrically on their elements and argue that this datum is incompatible with reductionism, given plausible assumptions about the dependence profile of ordinals. In addition, I show that a structurally similar argument can be made against the claim (...) that cardinals are sets. (shrink)

In a series of works, Jody Azzouni has defended deflationary nominalism, the view that certain sentences quantifying over mathematical objects are literally true, although such objects do not exist. One alleged attraction of this view is that it avoids various philosophical puzzles about mathematical objects. I argue that this thought is misguided. I first develop an ontologically neutral counterpart of Field’s reliability challenge and argue that deflationary nominalism offers no distinctive answer to it. I then show how this reasoning generalizes (...) to other philosophically problematic entities. The moral is that puzzle avoidance fails to motivate deflationary nominalism. (shrink)

In this paper I will take into examination the relevance of the main indispensability arguments for the comprehension of the applicability of mathematics. I will conclude not only that none of these indispensability arguments are of any help for understanding mathematical applicability, but also that these arguments rather require a preliminary analysis of the problems raised by the applicability of mathematics in order to avoid some tricky difficulties in their formulations. As a consequence, we cannot any longer consider the applicability (...) problems as subordinate to ontological ones: no ontological stance on mathematical entities can offer an easy road to the comprehension of the applicability of mathematics. (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 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)

Mathematical platonism is the view that abstract mathematical objects exist. Ontological pluralism is the view that there are many modes of existence. This paper examines the prospects for...

Philosophers of science since Nagel have been interested in the links between intertheoretic reduction and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relationship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important particular case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. In defense (...) of this claim, I offer evidence from mathematical practice, and I respond to contrary suggestions due to Steinhart, Maddy, Kitcher and Quine. I then show how, even if set-theoretic reductions are generally not explanatory, set theory can nevertheless serve as a legitimate foundation for mathematics. Finally, some implications of my thesis for philosophy of mathematics and philosophy of science are discussed. In particular, I suggest that some reductions in mathematics are probably explanatory, and I propose that differing standards of theory acceptance might account for the apparent lack of unexplanatory reductions in the empirical sciences. (shrink)

Evidential holism begins with something like the claim that “it is only jointly as a theory that scientific statements imply their observable consequences.” This is the holistic claim that Elliott Sober tells us is an “unexceptional observation”. But variations on this “unexceptional” claim feature as a premise in a series of controversial arguments for radical conclusions, such as that there is no analytic or synthetic distinction that the meaning of a sentence cannot be understood without understanding the whole language of (...) which it is a part and that all knowledge is empirical knowledge. This paper is a survey of what evidential holism is, how plausible it is, and what consequences it has. Section 1 will distinguish a range of different holistic claims, Sections 2 and 3 explore how well motivated they are and how they relate to one another, and Section 4 returns to the arguments listed above and uses the distinctions from the previous sections to identify holism's role in each case. (shrink)

In this paper a parallel is drawn between the problem of epistemic access to abstract objects in mathematics and the problem of epistemic access to idealized systems in the physical sciences. On this basis it is argued that some recent and more traditional approaches to solving these problems are problematic.

This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. I examine 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. I also develop a novel topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel dynamic two-dimensional semantics grounded in two-dimensional hyperintensional Turing machines. Chapter \textbf{3} provides an abstraction principle for epistemic (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} are 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 my 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. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest, too, that 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 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. I invent 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. I provide 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)

In many diagrams one seems to perceive necessity – one sees not only that something is so, but that it must be so. That conflicts with a certain empiricism largely taken for granted in contemporary philosophy, which believes perception is not capable of such feats. The reason for this belief is often thought well-summarized in Hume's maxim: ‘there are no necessary connections between distinct existences’. It is also thought that even if there were such necessities, perception is too passive or (...) localized a faculty to register them. We defend the perception of necessity against such Humeanism, drawing on examples from mathematics. (shrink)

Pyrrhonists provide a way of investigating the world in which conflicting views about a given topic are critically compared, assessed, and juxtaposed. Since Pyrrhonists are ultimately unable to decide between these views, they end up suspending judgment about the issues under examination. In this paper, I consider the question of whether Pyrrhonists can be realists or anti-realists about science, focusing, in particular, on contemporary philosophical discussions about it. Althoughprima faciethe answer seems to be negative, I argue that if realism and (...) anti-realism are understood as philosophical stances rather than particular doctrines—that is, if they are conceptualized in terms of a mode of engagement, a style of reasoning, and some propositional attitudes—the apparent tension between Pyrrhonism, realism, and anti-realism vanishes. The result is a first step in the direction of bringing Pyrrhonism to bear on contemporary debates in the philosophy of science. (shrink)

Nosso objetivo neste artigo é o de lançar luz sobre alguns aspectos das concepções de Gödel acerca da percepção de conceitos. Começamos investigando a natureza e o papel da analogia entre percepção sensível e percepção de conceitos. A seguir, examinamos as conexões entre percepção de conceitos, razão e compreensão, tentando mostrar que a percepção de conceitos é compreensão de conceitos. Por fim, examinamos aqueles aspectos da concepção de Gödel em que a percepção de conceitos de fato se aproxima perigosamente da (...) ideia de um "olho da mente". Contudo, argumentamos que esses aspectos problemáticos não são intrínsecos à noção de percepção de conceitos, mas sim resultam de uma má compreensão da distinção entre conceito e objeto. Our goal in this article is to illuminate some aspects of Gödel's conception of the perception of concepts. We begin by investigating the nature and function of Gödel's analogy between sense perception and perception of concepts. After that, we examine the connections between perception of concepts, reason, and understanding, trying to show that the perception of concepts consists in the understanding of concepts. Finally, we examine those aspects of Gödel's conception where the perception of concepts actually comes dangerously close to the idea of a "mind's eye". However, we argue that these problematic aspects are not intrinsic to the notion of perception of concepts, but result from a misunderstanding of the distinction between concept and object. (shrink)

Does science justify any part of mathematics and, if so, what part? These questions are related to the so-called indispensability arguments propounded, among others, by Quine and Putnam; moreover, both were led to accept significant portions of set theory on that basis. However, set theory rests on a strong form of Platonic realism which has been variously criticized as a foundation of mathematics and is at odds with scientific realism. Recent logical results show that it is possible to directly formalize (...) almost all, if not all, scientifically applicable mathematics in a formal system that is justified simply by Peano Arithmetic (via a proof-theoretical reduction). It is argued that this substantially vitiates the indispensability arguments. (shrink)

This long-awaited volume is a must-read for anyone with a serious interest in philosophy of mathematics. The book falls into two parts, with the primary focus of the first on ontology and structuralism, and the second on intuition and epistemology, though with many links between them. The style throughout involves unhurried examination from several points of view of each issue addressed, before reaching a guarded conclusion. A wealth of material is set before the reader along the way, but a reviewer (...) wishing to summarize the author's views crisply will be frustrated. The chapter-by-chapter survey below conveys at best a very incomplete and imperfect impression of the work's virtues, and even of its contents, falling short even of supplying a full menu for the banquet of food for thought that Parsons serves up to his readers. (shrink)

Maddy's (1990) arguments against Aggregate Theory were undermined by the shift in her position in 1997. The present paper considers Aggregate Theory in the light of this, and the recent search for `New Axioms for Mathematics'. If Set Theory is the part-whole theory of singletons, then identifying singletons with their single members collapses Set Theory into Aggregate Theory. But if singletons are not identical to their single members, then they are not extensional objects and so are not a basis for (...) Science. Either way, the Continuum Hypothesis has no physical interest. (shrink)

A phenomenological approach to mathematical practice is sketched out, and some problems with this sort of approach are considered. The approach outlined takes mathematical practices as its data, and seeks to provide an empirically adequate philosophy of mathematics based on observation of these practices. Some observations are presented, based on two case studies of some research into the classification of C*-algebras. It is suggested that an anti-realist account of mathematics could be developed on the basis of these and other studies, (...) locating the substance of mathematics in the various informal argument methods used by mathematicians. (shrink)

At their extremes, the modernization of ancient mathematical texts (absolute presentism) leaves nothing of the source and the refusal to modernize (absolute antiquarism) changes nothing. The extremes exist only as tendencies. This paper attempts to justify the admissibility of broad modernization of mathematical sources (presentism) in the context of a socio-cultural (non-fundamentalist) philosophy of mathematics.

The subjects of this thesis are truth, grounding and dependence. The thesis consists of an introduction and five free-standing essays. The purpose of the introduction is not merely to summarize the papers, but to provide a general background to the discussions in the essays. The introduction is divided into four chapters, each of which splits into a number of sections and/or subsections. Chapter 1. concerns the notion of ontological dependence. I start by making a distinction between two different types of (...) ontological dependence and discuss how well these notions deal with a number of philosophical issues. I then go on to consider the role that ontological dependence plays in hierarchies of natural kinds. In Chapter 2., I discuss a related notion, namely that of grounding. I sketch the theoretical framework by specifying the logical form of grounding statements and a set of structural principles that govern grounding. The chapter ends with a brief discussion on some philosophical applications of grounding. Chapter 3. deals with the notion of truthmaking and how it squares with the grounding framework developed in the previous chapter. I present the reader with the so-called Truthmaker Principle, and provide answers to a number of questions that it raises. The fourth and final chapter summarizes the five papers. The six essays are divided into three categories. Paper I deals with the notion of ontological dependence in hierarchies of natural kinds. Paper II concerns the notion of grounding and resemblance orderings among powers. Papers III, IV and V discuss various aspects of truthmaker theory. (shrink)

Some mathematical philosophers believe that we can achieve a new and better version of mathematical Platonism, by eliminating defects of original Platonism. According to Brown's version of Platonism, that here we call it “Modern Platonism”, the nature of mathematics can be formulated in these seven theses: realism, abstraction, particularity, Intuitiveness, priority, fallibility, and extensibility. This paper criticizes and evaluates the New Platonism, according to two major criteria: the social acceptability, and the methodological acceptability. The social acceptability of a theory, according (...) to my definition, is the interest and attitude of the people to that theory; and it can be measured on the frequency or percentage of interested parties. But the methodological acceptance of a theory means to match it with criteria such as consistency, simplicity and explanatory power; and its value can be assessed based on its success in solving philosophical problems related to it. According to Brown, the new Platonism, is the best philosophical theory about the nature of mathematics, both sociological and methodological. As for sociological criteria, we can be sympathetic and agree with Brown. That is, it seems that the new version of Platonism is still acceptable. But it needs to prove its methodological acceptability. Because the access problem and the certainty problem are still not resolved. (shrink)

sBuilding models as a practical aspect of ecological theory has as a principal purpose the determination of relations in formal language. In this paper, the authors provide a formalization of ecological models based on impure systems theory. Impure systems contain objects and subjects: subjects are human beings. We can distinguish a person as an observer that by definition is the subject himself and part of the system. In this case he acquires the category of object. Objects are significances, which are (...) the consequence of perceptual beliefs on the part of the subject about material or energetic objects with certain characteristics. The impure system approach is as follows: objects are perceptual significances of material or energetic objects. The set of these objects will form an impure set of the first order. The existing relations between these relative objects will be of two classes: transactions of matter and/or energy and inferential relations. Transactions can have alethic modality: necessity, possibility, impossibility and contingency. In this work we define measures which let us choose the more suitable variables to relate both the model with the ecosystem and with different models. In this way we define different comparison indexes. (shrink)

Can institutional objects be identified with physical objects that have been ascribed status functions, as advocated by John Searle in The Construction of Social Reality (1995)? The paper argues that the prospects of this identification hinge on how objects persist – i.e., whether they endure, perdure or exdure through time. This important connection between reductive identification and mode of persistence has been largely ignored in the literature on social ontology thus far.

This paper outlines a second-philosophical account of arithmetic that places it on a distinctive ground between those of logic and set theory.

It is sometimes suggested that impure sets are spatially co-located with their members (and hence are located in space). Sets, however, are in important respects like numbers. In particular, sets are connected to concepts in much the same manner as numbers are connected to concepts—in both cases, they are fundamentally abstracts of (or corresponding to) concepts. This parallel between the structure of sets and the structure of numbers suggests that the metaphysics of sets and the metaphysics of numbers should parallel (...) each other in relevant ways. This entails, in turn, that impure sets are not co-located with their members (nor are they located in space). (shrink)

Fitch's basic logic is an untyped illative combinatory logic with unrestricted principles of abstraction effecting a type collapse between properties (or concepts) and individual elements of an abstract syntax. Fitch does not work axiomatically and the abstraction operation is not a primitive feature of the inductive clauses defining the logic. Fitch's proof that basic logic has unlimited abstraction is not clear and his proof contains a number of errors that have so far gone undetected. This paper corrects these errors and (...) presents a reasonably intuitive proof that Fitch's system K supports an implicit abstraction operation. Some general remarks on the philosophical significance of basic logic, especially with respect to neo-logicism, are offered, and the paper concludes that basic logic models a highly intensional form of logicism. (shrink)

Paul Benacerraf's argument from multiple reductions consists of a general argument against realism about the natural numbers (the view that numbers are objects), and a limited argument against reductionism about them (the view that numbers are identical with prima facie distinct entities). There is a widely recognized and severe difficulty with the former argument, but no comparably recognized such difficulty with the latter. Even so, reductionism in mathematics continues to thrive. In this paper I develop a difficulty for Benacerraf's argument (...) against reductionism that is of comparable severity to the now widely recognized difficulty with his general argument against realism. Thanks to Kit Fine, Hartry Field, Jeff Sebo, Ted Sider, Stephen Schiffer, and anonymous referees at Philosophia Mathematica for helpful comments on earlier versions of this paper. Thanks to Aron Edidin for many helpful discussions of the problems that inspired it. CiteULike Connotea Del.icio.us What's this? (shrink)

This essay explores an ideal notion of form (mathematical structure) that embraces logical, phenomenological, and ontological form. Husserl envisioned a correlation among forms of expression, thought, meaning, and object—positing ideal forms on all these levels. The most puzzling formal entities Husserl discussed were those he called ‘manifolds’. These manifolds, I propose, are forms of complex states of affairs or partial possible worlds representable by forms of theories (compare structuralism). Accordingly, I sketch an intentionality-based semantics correlating these four Husserlian levels of (...) form—thereby integrating logic, phenomenology, and ontology. (shrink)

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

ParkWoosuk. _Philosophy’s Loss of Logic to Mathematics: An Inadequately Understood Take-Over _. Studies in Applied Philosophy, Epistemology, and Rational Ethics; 43. Springer, 2018. ISBN: 978-3-319-95146-1 ; 978-3-030-06984-1 978-3-319-95147-8. Pp. xii + 230. doi: 10.1007/978-3-319-95147-8.

Do numbers exist? Carnap (1956 [1950]) famously argues that this question can be understood in an “internal” and in an “external” sense, and calls “external” questions “non-cognitive”. Carnap also says that external questions are raised “only by philosophers” (p. 207), which means that, in his view, philosophers raise ”non-cognitive” questions. However, it is not clear how the internal/external distinction and Carnap’s related views about philosophy should be understood. This paper provides a new interpretation. I draw attention to Carnap’s distinction between (...) purely external statements, which are independent from all frameworks, and pragmatic external statements, which concern which framework one should adopt, and argue that the latter express noncognitive mental states. Specifically, I argue that “frameworks” are systems of rules for the assessment of statements, which are utterances of ordinary language sentences. Pragmatic external statements express noncognitive dispositions to follow only certain rules of assessment. For instance, “numbers exist” understood as a pragmatic external statement expresses a noncognitive disposition to use only rules of assessment according to which numbers do exist. Carnap can thereby be understood as proposing a distinctive form of noncognitivism about ontology that is in some respects analogous to norm-expressivism in metaethics. (shrink)

In this paper six of the most important issues in the philosophy of logic are examined from a standpoint that rejects the First Commandment of empiricist analytic philosophy, namely, Ockham’s razor. Such a standpoint opens the door to the clarification of such fundamental issues and to possible new solutions to each of them.