Moving beyond both realist and anti-realist accounts of mathematics, Shapiro articulates a "structuralist" approach, arguing that the subject matter of a mathematical theory is not a fixed domain of numbers that exist independent of each other, but rather is the natural structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle.

"The availability of a paperback version of Boyle's philosophical writings selected by M. A. Stewart will be a real service to teachers, students, and scholars with seventeenth-century interests. The editor has shown excellent judgment in bringing together many of the most important works and printing them, for the most part, in unabridged form. The texts have been edited responsibly with emphasis on readability.... Of special interest in connection with Locke and with the reception of Descarte's Corpuscularianism, to students of (...) the Scientific Revolution and of the history of mechanical philosophy, and to those interested in the relations among science, philosophy, and religion. In fact, given the imperfections in and unavailability of the eighteenth-century editions of Boyle’s works, this collection will benefit a wide variety of seventeenth-century scholars." --Gary Hatfield, University of Pennsylvania. (shrink)

The computations involved in statistical learning have long been debated. Here, we build on work suggesting that a basic memory process, chunking, may account for the processing of statistical regularities into larger units. Drawing on methods from the memory literature, we developed a novel paradigm to test statistical learning by leveraging a robust phenomenon observed in serial recall tasks: that short‐term memory is fundamentally shaped by long‐term distributional learning. In the statistically induced chunking recall (SICR) task, participants are exposed to (...) an artificial language, using a standard statistical learning exposure phase. Afterward, they recall strings of syllables that either follow the statistics of the artificial language or comprise the same syllables presented in a random order. We hypothesized that if individuals had chunked the artificial language into word‐like units, then the statistically structured items would be more accurately recalled relative to the random controls. Our results demonstrate that SICR effectively captures learning in both the auditory and visual modalities, with participants displaying significantly improved recall of the statistically structured items, and even recall specific trigram chunks from the input. SICR also exhibits greater test–retest reliability in the auditory modality and sensitivity to individual differences in both modalities than the standard two‐alternative forced‐choice task. These results thereby provide key empirical support to the chunking account of statistical learning and contribute a valuable new tool to the literature. (shrink)

This book presents the framework for a new, comprehensive approach to cognitive science. The proposed paradigm, enaction, offers an alternative to cognitive science's classical, first-generation Computational Theory of Mind. _Enaction_, first articulated by Varela, Thompson, and Rosch in _The Embodied Mind_, breaks from CTM's formalisms of information processing and symbolic representations to view cognition as grounded in the sensorimotor dynamics of the interactions between a living organism and its environment. A living organism enacts the world it lives in; its embodied (...) action in the world constitutes its perception and thereby grounds its cognition. _Enaction_ offers a range of perspectives on this exciting new approach to embodied cognitive science. Some chapters offer manifestos for the enaction paradigm; others address specific areas of research, including artificial intelligence, developmental psychology, neuroscience, language, phenomenology, and culture and cognition. Three themes emerge as testimony to the originality and specificity of enaction as a paradigm: the relation between first-person lived experience and third-person natural science; the ambition to provide an encompassing framework applicable at levels from the cell to society; and the difficulties of reflexivity. Taken together, the chapters offer nothing less than the framework for a far-reaching renewal of cognitive science. Contributors: Renaud Barbaras, Didier Bottineau, Giovanna Colombetti, Diego Cosmelli, Hanne De Jaegher, Ezequiel A. Di Paolo. Andreas K. Engel, Olivier Gapenne, Véronique Havelange, Edwin Hutchins, Michel Le Van Quyen, Rafael E. Núñez, Marieke Rohde, Benny Shanon, Maxine Sheets-Johnstone, Adam Sheya, Linda B. Smith, John Stewart, Evan Thompson. (shrink)

This chapter provides broad coverage of the notion of logical consequence, exploring its modal, semantic, and epistemic aspects. It develops the contrast between proof-theoretic notion of consequence, in terms of deduction, and a model-theoretic approach, in terms of truth-conditions. The main purpose is to relate the formal, technical work in logic to the philosophical concepts that underlie reasoning.

This paper uses neo-Fregean-style abstraction principles to develop the integers from the natural numbers (assuming Hume’s principle), the rational numbers from the integers, and the real numbers from the rationals. The first two are first-order abstractions that treat pairs of numbers: (DIF) INT(a,b)=INT(c,d) ≡ (a+d)=(b+c). (QUOT) Q(m,n)=Q(p,q) ≡ (n=0 & q=0) ∨ (n≠0 & q≠0 & m⋅q=n⋅p). The development of the real numbers is an adaption of the Dedekind program involving “cuts” of rational numbers. Let P be a property (of (...) rational numbers) and r a rational number. Say that r is an upper bound of P, written P≤r, if for any rational number s, if Ps then either s<r or s=r. In other words, P≤r if r is greater than or equal to any rational number that P applies to. Consider the Cut Abstraction Principle: (CP) ∀P∀Q(C(P)=C(Q) ≡ ∀r(P≤r ≡ Q≤r)). In other words, the cut of P is identical to the cut of Q if and only if P and Q share all of their upper bounds. The axioms of second-order real analysis can be derived from (CP), just as the axioms of second-order Peano arithmetic can be derived from Hume’s principle. The paper raises some of the philosophical issues connected with the neo-Fregean program, using the above abstraction principles as case studies. (shrink)

It feels like there’s two of you inside—like there’s another half of you, which is my anorexia, and then there’s the real K [own name], the real me, the logic part of me, and it’s a constant battle between the two. The anorexia almost does become part of you, and so in order to get it out of you I think you do have to kind of hurt you in the process. I think it’s almost inevitable. We came to the (...) concept of authenticity belatedly, one might say. We had been talking to people who had a diagnosis of anorexia nervosa about their experiences of living with their condition, and though we had not raised issues of authenticity or identity ourselves, they often did. They struggled with questions of .. (shrink)

Meaning, Understanding, and Practice is a selection of the most notable essays of an eminent contemporary philosopher on a set of central topics in analytic philosophy. Barry Stroud offers penetrating studies of meaning, understanding, necessity, and the intentionality of thought, with particular reference to the thought of Wittgenstein.

This is a new release of the original 1924 edition.

There is a parallel between the debate between Gottlob Frege and David Hilbert at the turn of the twentieth century and at least some aspects of the current controversy over whether category theory provides the proper framework for structuralism in the philosophy of mathematics. The main issue, I think, concerns the place and interpretation of meta-mathematics in an algebraic or structuralist approach to mathematics. Can meta-mathematics itself be understood in algebraic or structural terms? Or is it an exception to the (...) slogan that mathematics is the science of structure? (shrink)

At the beginning of Die Grundlagen der Arithmetik [1884], Frege observes that “it is in the nature of mathematics to prefer proof, where proof is possible”. This, of course, is true, but thinkers differ on why it is that mathematicians prefer proof. And what of propositions for which no proof is possible? What of axioms? This talk explores various notions of self-evidence, and the role they play in various foundational systems, notably those of Frege and Zermelo. I argue that both (...) programs are undermined at a crucial point, namely when self-evidence is supported by holistic and even pragmatic considerations. (shrink)

The purpose of this paper is to assess the prospects for a neo‐logicist development of set theory based on a restriction of Frege's Basic Law V, which we call (RV): ∀P∀Q[Ext(P) = Ext(Q) ≡ [(BAD(P) & BAD(Q)) ∨ ∀x(Px ≡ Qx)]] BAD is taken as a primitive property of properties. We explore the features it must have for (RV) to sanction the various strong axioms of Zermelo–Fraenkel set theory. The primary interpretation is where ‘BAD’ is Dummett's ‘indefinitely extensible’.1 Background: what (...) and why?2 Framework3 GOOD candidates, indefinite extensibility4 The framework of (RV) alone, or almost alone5 The axioms6 Brief closing. (shrink)

Ethical codes have been hailed as an explicit vehicle for achieving more sustainable and defensible organizational practice. Nonetheless, when legal compliance and corporate governance codes are conflated, codes can be used to define organizational interests ostentatiously by stipulating norms for employee ethics. Such codes have a largely cosmetic and insurance function, acting subtly and strategically to control organizational risk management and protection. In this paper, we conduct a genealogical discourse analysis of a representative code of ethics from an international corporation (...) to understand how management frames expectations of compliance. Our contribution is to articulate the problems inherent in codes of ethics, and we make some recommendations to address these to benefit both an organization and its employees. In this way, we show how a code of ethics can provide a foundation for ethical sustainability, while addressing management intentions and employees’ ethical satisfaction. (shrink)

The purpose of this article is to assess the prospects for a Scottish neo-logicist foundation for a set theory. We show how to reformulate a key aspect of our set theory as a neo-logicist abstraction principle. That puts the enterprise on the neo-logicist map, and allows us to assess its prospects, both as a mathematical theory in its own right and in terms of the foundational role that has been advertised for set theory. On the positive side, we show that (...) our abstraction based theory can be modified to yield much of ordinary mathematics, indeed everything needed to recapture all branches of mathematics short of set theory itself. However, our conclusions are mostly negative. The theory will fall far short of the power of ordinary Zermelo-Fraenkel set theory. It is consistent that our set theory has models that are relatively small, smaller than the first cardinal with an uncountable index. More important, there is a strong tension between the idea that the iterative hierarchy is somehow ineffable, or indefinitely extensible, and the neo-logicist theme of capturing mathematical theories with abstraction principles. (shrink)

The purpose of this article is to explore the extent to which mathematics is subject to open texture and the extent to which mathematics resists open texture. The resistance is tied to the importance of proof and, in particular, rigor, in mathematics.

This chapter provides an overview of second-order logic and higher-order logic generally. It provides the basic formal languages, deductive systems, and model-theoretic semantics, including a brief account of George Boolos’s interpretation of second-order languages in terms of the plural construction. It then goes into some of the arguments in favor of second-order logic.

Hawe et al. raise concerns about Human Research Ethics Committees (HRECs) taking a risk-averse and litigation-sensitive approach to ethical review of research proposals. HRECs are tasked with reviewing proposals for compliance with the National Statement on Ethical Conduct in Human Research for the purpose of promoting the welfare of participants. While these guidelines intentionally include a significant degree of discretion in HREC decision making, there is also evidence that HRECs sometimes request changes that go beyond the guidance provided by the (...) National Statement. When HRECs request changes outside their remit, inconsistencies between individual HRECs become more common, contributing to delays in ethical review and reducing the quality of HREC decision making. Improvements to the HREC regulatory system are needed to promote transparency and accountability. (shrink)

A typical interpreted formal language has (first‐order) variables that range over a collection of objects, sometimes called a domain‐of‐discourse. The domain is what the formal language is about. A language may also contain second‐order variables that range over properties, sets, or relations on the items in the domain‐of‐discourse, or over functions from the domain to itself. For example, the sentence ‘Alexander has all the qualities of a great leader’ would naturally be rendered with a second‐order variable ranging over qualities. Similarly, (...) the sentence ‘there is a property that holds of all and only the prime numbers’ has a variable ranging over properties of natural numbers. Third‐order variables range over properties of properties, sets of sets, functions from properties to sets, etc. For example, according to some logicist accounts, the number 4 is the property shared by all properties that apply to exactly four objects in the domain. Accordingly, the number 4 is a third‐order item. Fourth‐order variables, and beyond, are characterized similarly. The phrase ‘higher‐order variable’ refers to the variables beyond first‐order. (shrink)

The purpose of this note is to examine the relationship between the practice of mathematics and the philosophy of mathematics, ontology in particular. One conclusion is that the enterprises are (or should be) closely related, with neither one dominating the other. One cannot 'read off' the correct way to do mathematics from the true ontology, for example, nor can one ‘read off’ the true ontology from mathematics as practiced.

In lieu of an abstract, here is a brief excerpt of the content:Response to CommentariesLauren Freeman (bio) and Heather Stewart (bio)We are grateful and humbled that this esteemed group of scholars and healthcare practitioners have dedicated the time to read and engage with our book. Their thoughtful, critical comments have given us new ways of reflecting on our own work, compelled us to develop and expand some of our claims, and have also nudged us to move in new directions (...) that we hope to explore in future research. In what follows, we reply to each of the commentaries, focusing on what we take to be the most central insights, critiques, and areas for further exploration from each contribution.Catherine Sherron (2024) offers an expansive commentary on chapter 4, "Epistemic Microaggressions," focusing on the nature of epistemic agency, situating epistemic agency within the bioethical context of the principle of autonomy, considering epistemic microaggressions and the phenomenon of silencing of patients, asking whose evidence counts when making claims within medical contexts, and questioning how we might develop a system of reward and admonishment to healthcare professionals who respectively mitigate and commit microaggressions. She presents far more content than we can adequately respond to here. We have chosen to focus on Sherron's specific point about why certain patients experience epistemic microaggressions and what they can do about it. Her line of reasoning echoes an argument found in popular articles about women's health and healthcare that, while seemingly reasonable at first glance, is fraught with problematic assumptions and implications that we would like to draw out and reflect on. Doing so helps pinpoint where the locus of our attention should be in our aim to diminish microaggressions in medical contexts, without placing that burden on the very people who experience and are harmed by them.Within the context of discussing possible contributing factors that perpetuate epistemic microaggressions for patients, Sherron mentions the reality that "patients often don't have good scientific or medical literacy, or education generally, so they might not be in a position to advocate for themselves as epistemic [End Page 169] agents" (150). We surmise that Sherron's more complete argument runs something like this. Many patients lack scientific or medical literacy, postsecondary and postgraduate education, and the language, concepts, and frameworks to be fully empowered epistemic agents and best communicate with healthcare professionals on their terms. As a consequence—and combined with other factors pertaining to their identities and the accompanying stereotypes and biases that healthcare professionals (and all of us) hold—patients who are members of marginalized groups are often not treated as epistemic equals or epistemic peers by healthcare professionals; they are also unable to best advocate for themselves or to speak up in the aftermath of experiencing microaggressions (or macroaggressions) in medical contexts. Thus, if patients were more literate, educated, and able to speak the language of healthcare professionals, they would be able to better advocate for themselves, and they would be better positioned to speak up and respond when epistemic microaggressions (and other injustices and harms) occur. As a result, their situation and relationships with healthcare professionals, their epistemic credibility, and consequently, their care (and maybe even their health outcomes) would improve. Thus, the line o argument implies that we ought to focus our attention on empowering patients: helping them learn how to better educate and advocate for themselves so that they can improve their epistemic situation and, consequently, receive better medical care.This line of argument is common in popular books and articles about women's health (see Dusenbery [2018], who criticizes the argument and Gainty [2023], Jacqueline [2024], Lichtman et al. [2015], Onque [2022], and Singh [2015] where the argument is promoted). The consensus seems to be that yes, there is a problem: women and members of marginalized groups are not getting adequate healthcare, and if only they took better control of and responsibility for their care, they could and would receive better care. We are highlighting this line of reasoning because it is commonplace, problematic, and dangerous.Such an argument is worrisome for at least two reasons. First, the suggested solution—that members of marginalized groups just need to become better advocates for themselves—is not only a form of (unintentional... (shrink)

Several defences, viewed in this study as specifications or 'offspring' of the 'parent' greater-good defence, have been formulated in response to the charge that Christianity is untenable because God's existence is incompatible with evil's existence. In this first book-length study of the parent defence, Stewart begins with careful definitions of the omni-attributes central to the dispute: omnipotence, omniscience, omnibenevolence. The parent defence is traced to tenets of theism and variant accounts of the defence considered. Plantinga's modal free-will defence and (...) Hick's and Yandell's soul-growth specifications are carefully analyzed and several difficulties resolved. An original formulation of a redemption specification and an original account of the origin of moral evil are offered. Stewart argues that the defences are rightly viewed as a family of defences falling under the greater-good heading, and that they comprise a complementary apologetic complex that refutes the claim of inconsistency. (shrink)

Since virtually every mathematical theory can be interpreted in Zermelo-Fraenkel set theory, it is a foundation for mathematics. There are other foundations, such as alternate set theories, higher-order logic, ramified type theory, and category theory. Whether set theory is the right foundation for mathematics depends on what a foundation is for. One purpose is to provide the ultimate metaphysical basis for mathematics. A second is to assure the basic epistemological coherence of all mathematical knowledge. A third is to serve mathematics, (...) by lending insight into the various fields and suggesting fruitful techniques of research. A fourth purpose of a foundation is to provide an arena for exploring relations and interactions between mathematical fields. While set theory does better with regard to some of these and worse with regard to others, it has become the de facto arena for deciding questions of existence, something one might expect of a foundation. Given the different goals, there is little point to determining a single foundation for all of mathematics. (shrink)

Famously, Michael Dummett argues that considerations concerning the role of language in communication lead to the rejection of classical logic in favor of intuitionistic logic. Potentially, this results in massive revisions of established mathematics. Recently, Neil Tennant (“The law of excluded middle is synthetic a priori, if valid”, Philosophical Topics 24 (1996), 205-229) suggested that a Dummettian anti-realist can accept the law of excluded middle as a synthetic, a priori principle grounded on a metaphysical principle of determinacy. This article shows (...) that the for the anti-realist, the law of excluded middle entails that humans have wildly implausible abilities. The proposed synthesis between anti-realism and classical mathematics thus fails. (shrink)

Since 1999 Thoemmes Press (now Thoemmes Continuum) has been engaged in a large-scale programme of biographical dictionaries of philosophy and related subjects. This volume on Irish philosophers follows the standard format of arranging entires alphabetically by thinker. It includes two forms of entry: (1) entries reproduced from previous editions of Thoemmes encyclopedias of British philosophy and (2) wholly new entries on early (renaissance-period) and_ modern (20th century) philosophers, together with some new entries on the intervening centuries. >.