In the present article two possible meanings of the term mathematical structure are discussed: a formal and a nonformal one. It is claimed that contemporary mathematics is structural only in the nonformal sense of the term. Bourbaki's definition of structure is presented as one among several attempts to elucidate the meaning of that nonformal idea by developing a formal theory which allegedly accounts for it. It is shown that Bourbaki's concept of structure was, from a mathematical point of view, a (...) superfluous undertaking. This is done by analyzing the role played by the concept, in the first place, within Bourbaki's own mathematical output. Likewise, the interaction between Bourbaki's work and the first stages of category theory is analyzed, on the basis of both published texts and personal documents. (shrink)

The history of Pythagoreanism is littered with different and incompatible interpretations. This observation directs this book towards a fundamentally historiographical rather than philological approach, setting out to reconstruct the way in which the tradition established Pythagoreanism s image.".

Mathematics has a great variety ofapplications in the physical sciences.This simple, undeniable fact, however,gives rise to an interestingphilosophical problem:why should physical scientistsfind that they are unable to evenstate their theories without theresources of abstract mathematicaltheories? Moreover, theformulation of physical theories inthe language of mathematicsoften leads to new physical predictionswhich were quite unexpected onpurely physical grounds. It is thought by somethat the puzzles the applications of mathematicspresent are artefacts of out-dated philosophical theories about thenature of mathematics. In this paper I argue (...) that this is not so.I outline two contemporary philosophical accounts of mathematics thatpay a great deal of attention to the applicability of mathematics and showthat even these leave a large part of the puzzles in question unexplained. (shrink)

Mathematics has a great variety ofapplications in the physical sciences.This simple, undeniable fact, however,gives rise to an interestingphilosophical problem:why should physical scientistsfind that they are unable to evenstate their theories without theresources of abstract mathematicaltheories? Moreover, theformulation of physical theories inthe language of mathematicsoften leads to new physical predictionswhich were quite unexpected onpurely physical grounds. It is thought by somethat the puzzles the applications of mathematicspresent are artefacts of out-dated philosophical theories about thenature of mathematics. In this paper I argue (...) that this is not so.I outline two contemporary philosophical accounts of mathematics thatpay a great deal of attention to the applicability of mathematics and showthat even these leave a large part of the puzzles in question unexplained. (shrink)

This book not only outlines the indispensability argument in considerable detail but also defends it against various challenges.

This exploration of a notorious mathematical problem is the work of the man who discovered the solution. Written by an award-winning professor at Stanford University, it employs intuitive explanations as well as detailed mathematical proofs in a self-contained treatment. This unique text and reference is suitable for students and professionals. 1966 edition. Copyright renewed 1994.

This volume of essays brings together leading commentators from both sides of the Atlantic to provide an introduction to Badiou's work through critical studies ...

Any discussion of the possibility of ‘science and religion’ as a distinct field of study represented a clear step forward from the dominant prejudice of an earlier age. By contrast, it seems hard to deny that a new area of study has emerged, one devoted to the study of the complex and multifaceted relationships between science and religion. The text in this book testifies to the existence of a distinct field of inquiry. One can hope that carefully studying how differently (...) the various religions conceive their relationship with the sciences will help to overcome the tendency, until recently dominant in Western scholarship, to equate ‘religion and science’ with ‘Christianity and science’. (shrink)

This book is concerned with `the problem of existence in mathematics'. It develops a mathematical system in which there are no existence assertions but only assertions of the constructibility of certain sorts of things. It explores the philosophical implications of such an approach through an examination of the writings of Field, Burgess, Maddy, Kitcher, and others.

The Dappled World: A Study of the Boundaries of Science. nancy cartwright. Plato's Reception of Parmenides. john a. palmer.

In this sequence of philosophical essays about natural science, the author argues that fundamental explanatory laws, the deepest and most admired successes of modern physics, do not in fact describe regularities that exist in nature. Cartwright draws from many real-life examples to propound a novel distinction: that theoretical entities, and the complex and localized laws that describe them, can be interpreted realistically, but the simple unifying laws of basic theory cannot.

At a time when the analytic/continental split dominates contemporary philosophy, this ambitious work offers a careful and clear-minded way to bridge that divide. Combining conceptual rigor and clarity of prose with historical erudition, A Thing of This World shows how one of the standard issues of analytic philosophy—realism and anti-realism—has also been at the heart of continental philosophy. Using a framework derived from prominent analytic thinkers, Lee Braver traces the roots of anti-realism to Kant's idea that the mind actively organizes (...) experience. He then shows in depth and in detail how this idea evolves through the works of Hegel, Nietzsche, Heidegger, Foucault, and Derrida. This narrative presents an illuminating account of the history of continental philosophy by explaining how these thinkers build on each other's attempts to develop new concepts of reality and truth in the wake of the rejection of realism. Braver demonstrates that the analytic and continental traditions have been discussing the same issues, albeit with different vocabularies, interests, and approaches. By developing a commensurate vocabulary, his book promotes a dialogue between the two branches of philosophy in which each can begin to learn from the other. (shrink)

In this paper a constructive empiricist account of scientific change is put forward. Based on da Costa's and French's partial structures approach, two notions of empirical adequacy are initially advanced (with particular emphasis on the introduction of degrees of empirical adequacy). Using these notions, it is shown how both the informativeness and the empirical adequacy requirements of an empiricist theory of scientific change can then be met. Finally, some philosophical consequences with regard to the role of structures in this context (...) are drawn.Now, we daily see what science is doing for us. This could not be unless it taught us something about reality; the aim of science is not things themselves, as the dogmatists in their simplicity imagine, but the relations between things; outside those relations there is no reality knowable. (shrink)

We examine, from the partial structures perspective, two forms of applicability of mathematics: at the “bottom” level, the applicability of theoretical structures to the “appearances”, and at the “top” level, the applicability of mathematical to physical theories. We argue that, to accommodate these two forms of applicability, the partial structures approach needs to be extended to include a notion of “partial homomorphism”. As a case study, we present London's analysis of the superfluid behavior of liquid helium in terms of Bose‐Einstein (...) statistics. This involved both the introduction of group theory at the top level, and some modeling at the “phenomenological” level, and thus provides a nice example of the relationships we are interested in. We conclude with a discussion of the “autonomy” of London's model. (shrink)

The aim of this paper is to provide a unified account of scientific and mathematical change in a thoroughly empiricist setting. After providing a formal modelling in terms of embedding, and criticising it for being too restrictive, a second modelling is advanced. It generalises the first, providing a more open-ended pattern of theory development, and is articulated in terms of da Costa and French's partial structures approach. The crucial component of scientific and mathematical change is spelled out in terms of (...) partial embeddings. Finally, an application of this second pattern is made, with the examination of the early formulation of set theory. (shrink)

A work in the history of systematic philosophy that is itself animated by a systematic philosophic aspiration, this book by one of the most prominent American ...

One establishes oneself within science from the start. One does not reconstitute it from scratch. One does not found it. Alain Badiou, Le Concept de modèle1 [T]here are no crises within science, nor can there be, for science is the pure affirmation of difference. Alain Badiou, "Marque et manque" 2.

This paper explains the nature and origin of what I am calling Transgressive Realism, a middle path between realism and anti-realism which tries to combine their strengths while avoiding their weaknesses. Kierkegaard created the position by merging Hegel’s insistence that we must have some kind of contact with anything we can call real (thus rejecting noumena), with Kant’s belief that reality fundamentally exceeds our understanding; human reason should not be the criterion of the real. The result is the idea that (...) our most vivid encounters with reality come in experiences that shatter our categories, the way God’s commandment to kill Isaac irreconcilably clashes with the best understanding of ethics we are capable of. I explain the genesis of this idea, and then show it at work in Heidegger and Levinas’ thought. Understanding this position illuminates important aspects of the history of continental philosophy and offers a new perspective on realism. (shrink)

A stellar line-up of leading philosophers from around the world offer new treatments of a topic which has long been central to philosophical debate, and in ...

Author Max Black argues that language should conform to the discovered regularities of experience it is radically mistaken to assume that the conception of language is a mirror of reality.

For thousands of years, people have used nature to justify their political, moral, and social judgments. Such appeals to the moral authority of nature are still very much with us today, as heated debates over genetically modified organisms and human cloning testify. The Moral Authority of Nature offers a wide-ranging account of how people have used nature to think about what counts as good, beautiful, just, or valuable. The eighteen essays cover a diverse array of topics, including the connection of (...) cosmic and human orders in ancient Greece, medieval notions of sexual disorder, early modern contexts for categorizing individuals and judging acts as "against nature," race and the origin of humans, ecological economics, and radical feminism. The essays also range widely in time and place, from archaic Greece to early twentieth-century China, medieval Europe to contemporary America. Scholars from a wide variety of fields will welcome The Moral Authority of Nature, which provides the first sustained historical survey of its topic. Contributors: Danielle Allen, Joan Cadden, Lorraine Daston, Fa-ti Fan, Eckhardt Fuchs, Valentin Groebner, Abigail J. Lustig, Gregg Mitman, Michelle Murphy, Katharine Park, Matt Price, Robert N. Proctor, Helmut Puff, Robert J. Richards, Londa Schiebinger, Laura Slatkin, Julia Adeney Thomas, Fernando Vidal. (shrink)

The Fate of Reason is the first general history devoted to the period between Kant and Fichte, one of the most revolutionary and fertile in modern philosophy.

This book argues against the view that mathematical knowledge is a priori,contending that mathematics is an empirical science and develops historically,just as ...

William P. Alston formulates and defends a realist conception of truth, which he calls alethic realism (from "aletheia", Greek for "truth").

In a late discussion of Kant’s essay, “Was ist Aufklärung?,” Foucault credits Kant with posing “the question of his own present” and positions himself as an inheritor of this Kantian legacy.1 Foucault has high praise for the critical tradition that emerges from Kant’s historical-political reflections on the Enlightenment and the French Revolution; Kant’s concern in these writings with “an ontology of the present, an ontology of ourselves” is, he says, characteristic of “a form of philosophy, from Hegel, through Nietzsche and (...) Max Weber, to the Frankfurt School,” a form of philosophy in which Foucault, perhaps surprisingly, situates his own work. (shrink)

Epistemic infinitism is the view that infinite series of inferential relations are productive of epistemic justification. Peirce is explicitly infinitist in his early work, namely his 1868 series of articles. Further, Peirce's semiotic categories of firsts, seconds, and thirds favors a mixed theory of justification. The conclusion is that Peirce was an infinitist, and particularly, what I will term an impure infinitist. However, the prospects for Peirce's infinitism depend entirely on the prospects for Peirce's early semantics, which are not good. (...) Peirce himself revised the semantic theory later, and in so doing, it seems also his epistemic infinitism. (shrink)

I will assume here the defenses of epistemic infinitism are adequate and inquire as to the variety standpoints within the view. I will argue that infinitism has three varieties depending on the strength of demandingness of the infinitist requirement and the purity of its conception of epistemic justification, each of which I will term strong pure, strong impure, and weak impure infinitisms. Further, I will argue that impure infinitisms have the dialectical advantage.

What is it to be scientific? Is there such a thing as scientific method? And if so, how might such methods be justified? Robert Nola and Howard Sankey seek to provide answers to these fundamental questions in their exploration of the major recent theories of scientific method. Although for many scientists their understanding of method is something they just pick up in the course of being trained, Nola and Sankey argue that it is possible to be explicit about what this (...) tacit understanding of method is, rather than leave it as some unfathomable mystery. They robustly defend the idea that there is such a thing as scientific method and show how this might be legitimated. This book begins with the question of what methodology might mean and explores the notions of values, rules and principles, before investigating how methodologists have sought to show that our scientific methods are rational. Part 2 of this book sets out some principles of inductive method and examines its alternatives including abduction, IBE, and hypothetico-deductivism. Part 3 introduces probabilistic modes of reasoning, particularly Bayesianism in its various guises, and shows how it is able to give an account of many of the values and rules of method. Part 4 considers the ideas of philosophers who have proposed distinctive theories of method such as Popper, Lakatos, Kuhn and Feyerabend and Part 5 continues this theme by considering philosophers who have proposed naturalised theories of method such as Quine, Laudan and Rescher. This book offers readers a comprehensive introduction to the idea of scientific method and a wide-ranging discussion of how historians of science, philosophers of science and scientists have grappled with the question over the last fifty years. (shrink)

This book is concerned with the history of metaphysics since Descartes. Taking as its definition of metaphysics 'the most general attempt to make sense of things', it charts the evolution of this enterprise through various competing conceptions of its possibility, scope, and limits. The book is divided into three parts, dealing respectively with the early modern period, the late modern period in the analytic tradition, and the late modern period in non-analytic traditions. In its unusually wide range, A. W. Moore's (...) study refutes the tired old cliché that there is some unbridgeable gulf between analytic philosophy and philosophy of other kinds. It also advances its own distinctive and compelling conception of what metaphysics is and why it matters. Moore explores how metaphysics can help us to cope with continually changing demands on our humanity by making sense of things in ways that are radically new. (shrink)