" Vivid . . . immense clarity . . . the product of a brilliant and extremely forceful intellect." — Journal of the Royal Naval Scientific Service "Still a sheer joy to read." — Mathematical Gazette "Should be read by any student, teacher or researcher in mathematics." — Mathematics Teacher The originator of algebraic topology and of the theory of analytic functions of several complex variables, Henri Poincare (1854–1912) excelled at explaining the complexities of scientific and mathematical ideas to lay (...) readers. Science and Method, written in 1908, has been appreciated by a wide audience of nonprofessionals and translated into many languages. It defines the basic methodology and psychology of scientific discovery, particularly in regard to mathematics and mathematical physics. Drawing on examples from many fields, it explains how scientists analyze and choose their working facts, and it explores the nature of experimentation, theory, and the mind. 1914 edition. Translated by Francis Maitland. (shrink)

The Critique of the Power of Judgment (a more accurate rendition of what has hitherto been translated as the Critique of Judgment) is the third of Kant's great critiques following the Critique of Pure Reason and the Critique of Practical Reason. This entirely new translation of Kant's masterpiece follows the principles and high standards of all other volumes in The Cambridge Edition of the Works of Immanuel Kant. This volume includes: for the first time the indispensable first draft of Kant's (...) introduction to the work; the only English edition notes to the many differences between the first (1790) and second (1793) editions of the work; and relevant passages in Kant's anthropology lectures where he elaborated on his aesthetic views. All in all this new edition offers the serious student of Kant a dramatically richer, more complete and more accurate translation. (shrink)

“Now, in calm weather, to swim in the open ocean is as easy to the practised swimmer as to ride in a spring-carriage ashore. But the awful lonesomeness is intolerable. The intense concentration of self in the middle of such a heartless immensity, my God! who can tell it? Mark, how when sailors in a dead calm bathe in the open sea—mark how closely they hug their ship and only coast along her sides.” (Herman Melville, Moby Dick, Chapter 94).

It has been observed that whereas painters and musicians are likely to be embarrassed by references to the beauty in their work, mathematicians instead like to engage in discussions of the beauty of mathematics. Professional artists are more likely to stress the technical rather than the aesthetic aspects of their work. Mathematicians, instead, are fond of passing judgment on the beauty of their favored pieces of mathematics. Even a cursory observation shows that the characteristics of mathematical beauty are at variance (...) with those of artistic beauty. For example, courses in art appreciation are fairly common; it is however unthinkable to find any mathematical beauty appreciation courses taught anywhere. The purpose of the present paper is to try to uncover the sense of the term beauty as it is currently used by mathematicians. (shrink)

This volume examines the limitations of mathematical logic and proposes a new approach to logic intended to overcome them. To this end, the book compares mathematical logic with earlier views of logic, both in the ancient and in the modern age, including those of Plato, Aristotle, Bacon, Descartes, Leibniz, and Kant. From the comparison it is apparent that a basic limitation of mathematical logic is that it narrows down the scope of logic confining it to the study of deduction, without (...) providing tools for discovering anything new. As a result, mathematical logic has had little impact on scientific practice. Therefore, this volume proposes a view of logic according to which logic is intended, first of all, to provide rules of discovery, that is, non-deductive rules for finding hypotheses to solve problems. This is essential if logic is to play any relevant role in mathematics, science and even philosophy. To comply with this view of logic, this volume formulates several rules of discovery, such as induction, analogy, generalization, specialization, metaphor, metonymy, definition, and diagrams. A logic based on such rules is basically a logic of discovery, and involves a new view of the relation of logic to evolution, language, reason, method and knowledge, particularly mathematical knowledge. It also involves a new view of the relation of philosophy to knowledge. This book puts forward such new views, trying to open again many doors that the founding fathers of mathematical logic had closed historically. (shrink)

This new translation is an extremely welcome addition to the continuing Cambridge Edition of Kant’s works. English-speaking readers of the third Critique have long been hampered by the lack of an adequate translation of this important and difficult work. James Creed Meredith’s much-reprinted translation has charm and elegance, but it is often too loose to be useful for scholarly purposes. Moreover it does not include the first version of Kant’s introduction, the so-called “First Introduction,” which is now recognized as indispensable (...) for an understanding of the work. Werner Pluhar’s more recent translation, which does include the First Introduction, is highly accurate when it confines itself to rendering Kant’s German. However, it is often more of a reconstruction than a translation, containing so many interpretative interpolations that it is often difficult to separate out Kant’s original text from the translator’s contributions. Paul Guyer and Eric Matthews have provided a translation that compares to or exceeds Pluhar’s in its literal approach to the German, but that confines all interpretative material to footnotes and endnotes, so that the text itself, with all its unclarities and ambiguities, lies open to view. In addition, Guyer, as editor of the volume, has provided a great deal of valuable supplementary material. This includes an introduction with an outline of the work and details of the history of its composition and publication, and a wealth of endnotes offering clarifications of the text, background information, and, most strikingly, many references to related passages in Kant’s voluminous writings, particularly in connection with Kant’s earlier writings related to aesthetics. The edition also records differences among the first three editions of the work, and—of particular interest—erasures from and additions to Kant’s manuscript of the First Introduction. Although the introduction and endnotes reflect interpretative views that are sometimes disputable, this supplementary material makes the present edition into a valuable resource even for those able to read the text in German. (shrink)

G.H. Hardy was one of this century's finest mathematical thinkers, renowned among his contemporaries as a 'real mathematician... the purest of the pure'. He was also, as C.P. Snow recounts in his Foreword, 'unorthodox, eccentric, radical, ready to talk about anything'. This 'apology', written in 1940, offers a brilliant and engaging account of mathematics as very much more than a science; when it was first published, Graham Greene hailed it alongside Henry James's notebooks as 'the best account of what it (...) was like to be a creative artist'. C.P. Snow's Foreword gives sympathetic and witty insights into Hardy's life, with its rich store of anecdotes concerning his collaboration with the brilliant Indian mathematician Ramanujan, his idiosyncrasies and his passion for cricket. This is a unique account of the fascination of mathematics and of one of its most compelling exponents in modern times. (shrink)

It is a common thought that mathematics can be not only true but also beautiful, and many of the greatest mathematicians have attached central importance to the aesthetic merit of their theorems, proofs and theories. But how, exactly, should we conceive of the character of beauty in mathematics? In this paper I suggest that Kant's philosophy provides the resources for a compelling answer to this question. Focusing on §62 of the ‘Critique of Aesthetic Judgment’, I argue against the common view (...) that Kant's aesthetics leaves no room for beauty in mathematics. More specifically, I show that on the Kantian account beauty in mathematics is a non-conceptual response felt in light of our own creative activities involved in the process of mathematical reasoning. The Kantian proposal I thus develop provides a promising alternative to Platonist accounts of beauty widespread among mathematicians. While on the Platonist conception the experience of mathematical beauty consists in an intellectual insight into the fundamental structures of the universe, according to the Kantian proposal the experience of beauty in mathematics is grounded in our felt awareness of the imaginative processes that lead to mathematical knowledge. The Kantian account I develop thus offers to elucidate the connection between aesthetic reflection, creative imagination and mathematical cognition. (shrink)

This paper discusses the viability of claims of mathematical beauty, asking whether mathematical beauty, if indeed there is such a thing, should be conceived of as a sub-variety of the more commonplace kinds of beauty: natural, artistic and human beauty; or, rather, as a substantive variety in its own right. If the latter, then, per the argument, it does not show itself in perceptual awareness – because perceptual presence is what characterises the commonplace kinds of beauty, and mathematical beauty is (...) not among these. I conclude that the reference to mathematical beauty merely expresses the awe in the mathematician about the intricate complexities and simplicity of certain proofs, theorems or mathematical “objects.”. (shrink)

" Vivid . . . immense clarity . . . the product of a brilliant and extremely forceful intellect." — Journal of the Royal Naval Scientific Service "Still a sheer joy to read." — Mathematical Gazette "Should be read by any student, teacher or researcher in mathematics." — Mathematics Teacher The originator of algebraic topology and of the theory of analytic functions of several complex variables, Henri Poincare (1854–1912) excelled at explaining the complexities of scientific and mathematical ideas to lay (...) readers. Science and Method, written in 1908, has been appreciated by a wide audience of nonprofessionals and translated into many languages. It defines the basic methodology and psychology of scientific discovery, particularly in regard to mathematics and mathematical physics. Drawing on examples from many fields, it explains how scientists analyze and choose their working facts, and it explores the nature of experimentation, theory, and the mind. 1914 edition. Translated by Francis Maitland. (shrink)

Science has more to offer than just knowledge of nature; it can give us understanding of nature as well. Epistemology of science is usually focused on knowledge and the criteria of justification, while paying little attention to understanding. In a reversal of this emphasis, this article is more about scientific understanding. I argue that the hallmarks of understanding are similar to an aesthetic feature associated with literature, music, and the visual arts. It is the feature described as coherence, harmony, and (...) inevitability of fit. Aesthetics thus plays an epistemic role in science as an indication of understanding. (shrink)

A young scientist and mathematician explores the mystery and complexity of human thought processes from an interdisciplinary point of view.

First published in 1818, The World as Will and Representation contains Schopenhauer's entire philosophy, ranging through epistemology, metaphysics, philosophy of mind and action, aesthetics and philosophy of art, to ethics, the meaning of life and the philosophy of religion, in an attempt to account for the world in all its significant aspects. It gives a unique and influential account of what is and is not of value in existence, the striving and pain of the human condition and the possibility of (...) deliverance from it. This new translation of the first volume of what later became a two-volume work reflects the eloquence and power of Schopenhauer's prose and renders philosophical terms accurately and consistently. It offers an introduction, glossary of names and bibliography, and succinct editorial notes, including notes on the revisions of the text which Schopenhauer made in 1844 and 1859. (shrink)

... as 'logicism') that the content expressed by true propositions of arithmetic and analysis is not something of an irreducibly mathematical character, ...

Presents general information about meteorology, weather, and climate and includes more than thirty activities to help study these topics, including making a ...

In this article I examine the status of putative aesthetic judgements in science and mathematics. I argue that if the judgements at issue are taken to be genuinely aesthetic they can be divided into two types, positing either a disjunction or connection between aesthetic and epistemic criteria in theory/proof assessment. I show that both types of claim face serious difficulties in explaining the purported role of aesthetic judgements in these areas. I claim that the best current explanation of this role, (...) McAllister's 'aesthetic induction' model, fails to demonstrate that the judgements at issue are genuinely aesthetic. I argue that, in light of these considerations, there are strong reasons for suspecting that many, and perhaps all, of the supposedly aesthetic claims are not genuinely aesthetic but are in fact 'masked' epistemic assessments. (shrink)

THE VALUE OF SCIENCE INTRODUCTION The search for truth should be the goal of our activities; it is the sole end worthy of them. Doubtless we should first bend our efforts to assuage human suffering, but why ? Not to suffer is a negative ...

Can aesthetic judgements legitimately be linked to the success of scientific theories? I suggest that a satisfactory answer to this question should account for the persistent attraction that aesthetic considerations seem to have for scientists, while also explaining the apparent instability of the link between the beauty of a theory and its truth. I argue that two widespread tendencies in the literature, Pythagorean and subjectivist approaches, have difficulties meeting this twofold challenge. I propose a Kantian conception of aesthetic judgements as (...) second‐order considerations, related to our own intellectual capacities for making sense of the world, and argue that it fares better. (shrink)

From antiquity several philosophers have claimed that the goal of natural science is truth. In particular, this is a basic tenet of contemporary scientific realism. However, all concepts of truth that have been put forward are inadequate to modern science because they do not provide a criterion of truth. This means that we will generally be unable to recognize a scientific truth when we reach it. As an alternative, this paper argues that the goal of natural science is plausibility and (...) considers some characters of plausibility. (shrink)