Diversity Reading List

Abstract: This paper argues that it is scientific realists who should be most concerned about the issue of Platonism and anti-Platonism in mathematics. If one is merely interested in accounting for the practice of pure mathematics, it is unlikely that a story about the ontology of mathematical theories will be essential to such an account. The question of mathematical ontology comes to the fore, however, once one considers our scientific theories. Given that those theories include amongst their laws assertions that imply the existence of mathematical objects, scientific realism, when construed as a claim about the truth or approximate truth of our scientific theories, implies mathematical Platonism. However, a standard argument for scientific realism, the 'no miracles' argument, falls short of establishing mathematical Platonism. As a result, this argument cannot establish scientific realism as it is usually defined, but only some weaker position. Scientific 'realists' should therefore either redefine their position as a claim about the existence of unobservable physical objects, or alternatively look for an argument for their position that does establish mathematical Platonism.

Summary: Defends an account of mathematical knowledge in which mathematical knowledge is a kind of modal knowledge. Leng argues that nominalists should take mathematical knowledge to consist in knowledge of the consistency of mathematical axiomatic systems, and knowledge of what necessarily follows from those axioms. She defends this view against objections that modal knowledge requires knowledge of abstract objects, and argues that we should understand possibility and necessity in a primative way.

Abstract: I propose a framework that explicates and distinguishes the epistemic roles of data and models within empirical inquiry through consideration of their use in scientific practice. After arguing that Suppes' characterization of data models falls short in this respect, I discuss a case of data processing within exploratory research in plant phenotyping and use it to highlight the difference between practices aimed to make data usable as evidence and practices aimed to use data to represent a specific phenomenon. I then argue that whether a set of objects functions as data or models does not depend on intrinsic differences in their physical properties, level of abstraction or the degree of human intervention involved in generating them, but rather on their distinctive roles towards identifying and characterizing the targets of investigation. The paper thus proposes a characterization of data models that builds on Suppes' attention to data practices, without however needing to posit a fixed hierarchy of data and models or a highly exclusionary definition of data models as statistical constructs.

Publisher's Note: Conventional wisdom has it that the sciences, properly pursued, constitute a pure, value-free method of obtaining knowledge about the natural world. In light of the social and normative dimensions of many scientific debates, Helen Longino finds that general accounts of scientific methodology cannot support this common belief. Focusing on the notion of evidence, the author argues that a methodology powerful enough to account for theories of any scope and depth is incapable of ruling out the influence of social and cultural values in the very structuring of knowledge. The objectivity of scientific inquiry can nevertheless be maintained, she proposes, by understanding scientific inquiry as a social rather than an individual process. Seeking to open a dialogue between methodologists and social critics of the sciences, Longino develops this concept of "contextual empiricism" in an analysis of research programs that have drawn criticism from feminists. Examining theories of human evolution and of prenatal hormonal determination of "gender-role" behavior, of sex differences in cognition, and of sexual orientation, the author shows how assumptions laden with social values affect the description, presentation, and interpretation of data. In particular, Longino argues that research on the hormonal basis of "sex-differentiated behavior" involves assumptions not only about gender relations but also about human action and agency. She concludes with a discussion of the relation between science, values, and ideology, based on the work of Habermas, Foucault, Keller, and Haraway.

Publisher's Note: Our much-valued mathematical knowledge rests on two supports: the logic of proof and the axioms from which those proofs begin. Naturalism in Mathematics investigates the status of the latter, the fundamental assumptions of mathematics. These were once held to be self-evident, but progress in work on the foundations of mathematics, especially in set theory, has rendered that comforting notion obsolete. Given that candidates for axiomatic status cannot be proved, what sorts of considerations can be offered for or against them? That is the central question addressed in this book. One answer is that mathematics aims to describe an objective world of mathematical objects, and that axiom candidates should be judged by their truth or falsity in that world. This promising view - realism - is assessed and finally rejected in favour of another - naturalism - which attends less to metaphysical considerations of objective truth and falsity, and more to practical considerations drawn from within mathematics itself. Penelope Maddy defines this naturalism, explains the motivation for it, and shows how it can be helpfully applied in the assessment of candidates for axiomatic status in set theory. Maddy's clear, original treatment of this fundamental issue is informed by current work in both philosophy and mathematics, and will be accessible and enlightening to readers from both disciplines.

Abstract: This talk surveys a range of positions on the fundamental metaphysical and epistemological questions about elementary logic, for example, as a starting point: what is the subject matter of logic - what makes its truths true? how do we come to know the truths of logic? A taxonomy is approached by beginning from well-known schools of thought in the philosophy of mathematics - Logicism, Intuitionism, Formalism, Realism - and sketching roughly corresponding views in the philosophy of logic. Kant, Mill, Frege, Wittgenstein, Carnap, Ayer, Quine, and Putnam are among the philosophers considered along the way.

Summary: A clear introduction to mathematical naturalism and its Quinean roots; developing and defending Maddy's own naturalist philosophy of mathematics. Maddy claims that the Quinian ignores some nuances of scientific practice that have a bearing on what the naturalist should take to be the real scientific standards of evidence. Historical studies show that scientists sometimes do not take themselves to be committed to entities that are indispensably quantified over in their best scientific theories, hence the Quinian position that naturalism dictates that we are committed to entities that are indispensably quantified over in our best scientific theories is incorrect.

Abstract: The paper challenges Williamson's safety based explanation for why we cannot know the cut-off point of vague expressions. We assume throughout (most of) the paper that Williamson is correct in saying that vague expressions have sharp cut-off points, but we argue that Williamson's explanation for why we do not and cannot know these cut-off points is unsatisfactory. In sect 2 we present Williamson's position in some detail. In particular, we note that Williamson's explanation relies on taking a particular safety principle ('Meta-linguistic belief safety' or 'MBS') as a necessary condition on knowledge. In section 3, we show that even if MBS were a necessary condition on knowledge, that would not be sufficient to show that we cannot know the cut-off points of vague expressions. In section 4, we present our main case against Williamson's explanation: we argue that MBS is not a necessary condition on knowledge, by presenting a series of cases where one's belief violates MBS but nevertheless constitutes knowledge. In section 5, we present and respond to an objection to our view. And in section 6, we briefly discuss the possible directions a theory of vagueness can take, if our objection to Williamson's theory is taken on board.

Abstract: Val Plumwood charged classical logic not only with the invalidity of some of its laws, but also with the support of systemic oppression through naturalization of the logical structure of dualisms. In this paper I show that the latter charge - unlike the former - can be carried over to classical mathematics, and I propose a new conception of inconsistent mathematics - queer incomaths - as a liberatory activity meant to undermine said naturalization.

Abstract:

Recent works about ecumenical systems, where connectives from classical and intuitionistic logics can co-exist in peace, warmed the discussion on proof systems for combining logics. This discussion has been extended to alethic modalities using Simpson’s meta-logical characterization: necessity is independent of the viewer, while possibility can be either intuitionistic or classical. In this work, we propose a pure, label free calculus for ecumenical modalities, nEK, where exactly one logical operator figures in introduction rules and every basic object of the calculus can be read as a formula in the language of the ecumenical modal logic EK. We prove that nEK is sound and complete w.r.t. the ecumenical birelational semantics and discuss fragments and extensions.

Abstract: The highest level of mathematics has traditionally been seen as a solitary endeavour, to produce a proof for review and acceptance by research peers. Mathematics is now at a remarkable inflexion point, with new technology radically extending the power and limits of individuals. Crowdsourcing pulls together diverse experts to solve problems; symbolic computation tackles huge routine calculations; and computers check proofs too long and complicated for humans to comprehend. The Study of Mathematical Practice is an emerging interdisciplinary field which draws on philosophy and social science to understand how mathematics is produced. Online mathematical activity provides a novel and rich source of data for empirical investigation of mathematical practice - for example the community question-answering system mathoverflow contains around 40,000 mathematical conversations, and polymath collaborations provide transcripts of the process of discovering proofs. Our preliminary investigations have demonstrated the importance of “soft” aspects such as analogy and creativity, alongside deduction and proof, in the production of mathematics, and have given us new ways to think about the roles of people and machines in creating new mathematical knowledge. We discuss further investigation of these resources and what it might reveal. Crowdsourced mathematical activity is an example of a “social machine”, a new paradigm, identified by Berners-Lee, for viewing a combination of people and computers as a single problem-solving entity, and the subject of major international research endeavours. We outline a future research agenda for mathematics social machines, a combination of people, computers, and mathematical archives to create and apply mathematics, with the potential to change the way people do mathematics, and to transform the reach, pace, and impact of mathematics research.

Publisher's Note: There is hardly another principle in physics with wider scope of applicability and more far-reaching consequences than Pauli's exclusion principle. This book explores the principle's origin in the atomic spectroscopy of the early 1920s, its subsequent embedding into quantum mechanics, and later experimental validation with the development of quantum chromodynamics. The reconstruction of this crucial historic episode provides an excellent foil to reconsider Kuhn's view on incommensurability. The author defends the prospective rationality of the revolutionary transition from the old to the new quantum theory around 1925 by focusing on the way Pauli's principle emerged as a phenomenological rule 'deduced' from some anomalous phenomena and theoretical assumptions of the old quantum theory. The subsequent process of validation is historically reconstructed and analysed within the framework of 'dynamic Kantianism'

Summary: In this article Massimi discusses the important role that history and philosophy of science plays or ought to play within philosophy. The aim of the paper is to offer a historical reconstruction and a possible diagnosis of why the long marriage between philosophy and the sciences was eventually wrong after Kant. Massimi examines Kant's view on philosophy and the sciences, from his early scientific writings to the development of critical philosophy and the pressing epistemological he felt the need to address in response to the sciences of his time.

Abstract: In The Structure of Scientific Revolutions, Kuhn famously advanced the claim that scientists work in a different world after a scientific revolution. Kuhn's view has been at the center of a philosophical literature that has tried to make sense of his bold claim, by listing Kuhn's view in good company with other seemingly constructivist proposals. The purpose of this paper is to take some steps towards clarifying what sort of constructivism (if any) is in fact at stake in Kuhn's view. To this end, I distinguish between two main (albeit not exclusive) notions of mind-dependence: a semantic notion and an ontological one. I point out that Kuhn's view should be understood as subscribing to a form of semantic mind-dependence, and conclude that semantic mind-dependence does not land us into any worrisome ontological mind-dependence, pace any constructivist reading of Kuhn.

Abstract: The low visibility and specialised languages of mathematical work pose challenges for the ethnographic study of communication in mathematics, but observation-based study can offer a real-world grounding to questions about the nature of its methods. This paper uses theoretical ideas from linguistic pragmatics to examine how mutual understandings of diagrams are achieved in the course of conference presentations. Presenters use shared knowledge to train others to interpret diagrams in the ways favoured by the community of experts, directing an audience’s attention so as to develop a shared understanding of a diagram’s features and possible manipulations. In this way, expectations about the intentions of others and appeals to knowledge about the manipulation of objects play a part in the development and communication of concepts in mathematical discourse.