Kant's attempt to establish the principles behind the faculty of judgment remains one of the most important works on human reason.

This is a critical examination of the astonishing progress made in the philosophical study of the properties of the natural numbers from the 1880s to the 1930s. Reassessing the brilliant innovations of Frege, Russell, Wittgenstein, and others, which transformed philosophy as well as our understanding of mathematics, Michael Potter places arithmetic at the interface between experience, language, thought, and the world.

This book departs from much of the scholarship on Kant by demonstrating the centrality of imagination to Kant's philosophy as a whole. In Kant's works, human experience is simultaneously passive and active, thought and sensed, free and unfree: these dualisms are often thought of as unfortunate byproducts of his system. Gibbons, however, shows that imagination performs a vital function in "bridging gaps" between the different elements of cognition and experience. Thus, the role imagination plays in Kant's works expresses his fundamental (...) insight into the complexity of cognition for finite rational beings such as ourselves. (shrink)

In this new book, Michael Friedman argues that Kant's continuing efforts to find a metaphysics that could provide a foundation for the sciences is of the utmost ...

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)

I try to identify the characteristic and distinguishing features of a theory of natural beauty (as opposed to the sublime) that can be found in Kant's Critique of Judgement. Lest this may seem superfluous, I argue first that, contrary to a common view, Kant's theory does not take the experience of beauty in nature as theoretically basic and that he does not deal with beauty in art only as a derivative case of aesthetic experience. I then try to understand what (...) it means to require, as Kant famously does, that beautiful nature has to look as if it were art. Does this commit him to a projectionist view, according to which we appreciate nature aesthetically only for what it really is not (namely art) rather than for what it is? Pursuing this question we find that for Kant nature can be the object of pure judgements of taste only insofar as nature can be explained scientifically, which in Kant's framework means purely mechanically. This ‘blind mechanism’ of nature plays an important role in Kant's explication of the moral significance the experience of beautiful nature has for us. (shrink)

This important book by a major American philosopher brings together eleven essays treating problems in logic and the philosophy of mathematics.

This is Werner S. Pluhar's translation of Immanuel Kant's Critique of Judgment (Kritik der Urtheilskraft) for Hackett Publications (Indianapolis, Indiana). ISBN 9780872200258 (paperback).

Kant and the Claims of Taste, published here for the first time in paperback in a revised version, has become, since its initial publication in 1979, the standard commentary on Kant's aesthetic theory. The book offers a detailed account of Kant's views on judgments of taste, aesthetic pleasure, imagination and many other topics. For this new edition, Paul Guyer has provided a new foreword and has added a chapter on Kant's conception of fine art. This re-issue will complement the author's (...) companion volume, Kant and the Experience of Freedom, which places Kant's aesthetics in its historical context and examines the fundamental connection between Kant's aesthetics and his moral theory. (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)

According to Kant, “mathematical knowledge is the knowledge gained by reason from the construction of concepts.” In this paper, I shall make a few suggestions as to how this characterization of the mathematical method is to be understood.

" 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)

Visual thinking -- visual imagination or perception of diagrams and symbol arrays, and mental operations on them -- is omnipresent in mathematics. Is this visual thinking merely a psychological aid, facilitating grasp of what is gathered by other means? Or does it also have epistemological functions, as a means of discovery, understanding, and even proof? By examining the many kinds of visual representation in mathematics and the diverse ways in which they are used, Marcus Giaquinto argues that visual thinking in (...) mathematics is rarely just a superfluous aid; it usually has epistemological value, often as a means of discovery. Drawing from philosophical work on the nature of concepts and from empirical studies of visual perception, mental imagery, and numerical cognition, Giaquinto explores a major source of our grasp of mathematics, using examples from basic geometry, arithmetic, algebra, and real analysis. He shows how we can discern abstract general truths by means of specific images, how synthetic a priori knowledge is possible, and how visual means can help us grasp abstract structures. Visual Thinking in Mathematics reopens the investigation of earlier thinkers from Plato to Kant into the nature and epistemology of an individual's basic mathematical beliefs and abilities, in the new light shed by the maturing cognitive sciences. Clear and concise throughout, it will appeal to scholars and students of philosophy, mathematics, and psychology, as well as anyone with an interest in mathematical thinking. (shrink)

This landmark book is now reissued in a new edition that has been vastly rewritten and updated to respond to recent Kantian literature.

The first systematic study of the aesthetic evaluations that scientists pass on their theories.

"Sir Hermann Bondi, NatureThe late S. Chandrasekhar was best known for his discovery of the upper limit to the mass of a white dwarf star, for which he received ...

In lieu of an abstract, here is a brief excerpt of the content:Kant on the Method of MathematicsEmily Carson1. INTRODUCTIONThis paper will touch on three very general but closely related questions about Kant’s philosophy. First, on the role of mathematics as a paradigm of knowledge in the development of Kant’s Critical philosophy; second, on the nature of Kant’s opposition to his Leibnizean predecessors and its role in the development of the Critical philosophy; and finally, on the specific role of intuition (...) in Kant’s philosophy of mathematics. One of the points that I want to make is that recognizing the importance of these first two issues is essential to giving an adequate account of the third. That is, only by appreciating how Kant uses the example of mathematical knowledge, and in particular the objections he had to previous accounts of such knowledge, can we understand the philosophical role that he ascribes to intuition in his own account of mathematics.I don’t propose to explain the details of Kant’s philosophy of mathematics, but rather to compare his views on mathematics in the pre-Critical Prize Essay of 1764 to those of the Critique of Pure Reason, with the aim of showing how Kant’s doctrine of construction in pure intuition arises out of a combination of two factors: first, his attempt to ground the distinction between the respective methods of mathematics and philosophy, and second, his concern to defend the reality of mathematical knowledge against the views of those he called ‘the metaphysicians.’ The first factor demands an account of the certainty of mathematics, the second requires an account of its content. I shall argue that there is a possible conflict between Kant’s accounts of these features in the Prize Essay, a possibility which is finally ruled out only by means of the Critical doctrine of pure intuition. So I hope to show that there is a significant philosophical role for intuition in Kant’s account of mathematical knowledge (over and above the more logical role attributed to it on readings [End Page 629] such as those of Friedman, Beth, and Hintikka1) in reconciling the content of mathematics and its certainty.2. THE MATHEMATICIANS VS. THE METAPHYSICIANSThroughout his early writings, Kant appealed to the evidence and certainty of mathematics; his attitude towards metaphysics, however, underwent a dramatic shift. In the Physical Monadology of 1756, Kant asked how metaphysics and geometry can be united,[f] or the former peremptorily denies that space is infinitely divisible, while the latter, with its usual certainty, asserts that it is infinitely divisible. Geometry contends that empty space is necessary for free motion, while metaphysics hisses the idea off the stage. Geometry holds universal attraction or gravitation to be hardly explicable by mechanical causes but shows that it derives from the forces which are inherent in bodies at rest and which act at a distance, whereas metaphysics dismisses the notion as an empty delusion of the imagination [1:475–6].2He attempted in this work to reconcile the respective positions of metaphysics and geometry regarding infinite divisibility, thereby demonstrating that “it is neither the case that the geometer is mistaken nor that the opinion to be found among metaphysicians deviates from the truth” [1:480]. But in a series of works from 1762–3, Kant reveals a markedly different attitude towards metaphysics. For example, in “The only possible argument in support of a demonstration of the existence of God,” Kant says:the mania for method and the imitation of the mathematician, who advances with a sure step along a well-surfaced road, have occasioned a large number of such mishaps on the slippery ground of metaphysics. These mishaps are constantly before one’s eyes, but there is little hope that people will be warned by them, or that they will learn to be more circumspect as a result [2:71].In contrast, Kant describes the geometer as uncovering “with the greatest certainty” the most secret properties of that which is extended [2:70]. But the less conciliatory attitude to metaphysics comes out most clearly in Kant’s actual procedure in “The only possible argument.” In the Seventh Reflection, he presents an attempt to explain the origin of... (shrink)

This book constitutes one of the most important contributions to recent Kant scholarship. In it, one of the pre-eminent interpreters of Kant, Henry Allison, offers a comprehensive, systematic, and philosophically astute account of all aspects of Kant's views on aesthetics. The first part of the book analyses Kant's conception of reflective judgment and its connections with both empirical knowledge and judgments of taste. The second and third parts treat two questions that Allison insists must be kept distinct: the normativity of (...) pure judgments of taste, and the moral and systematic significance of taste. The fourth part considers two important topics often neglected in the study of Kant's aesthetics: his conceptions of fine art, and the sublime. (shrink)

Diese erste, umfassende Darstellung von Kants Philosophie der Mathematik unternimmt es, die Aussagen Kants über Mathematik im Kontext seiner jeweiligen philosophischen Problemkonstellation zu verorten. Darüber hinaus werden die Ergebnisse dieser Zuweisungen einer systematischen Evaluation unterworfen. Der Befund: Kants Mathematikbegriff ist durch Prämissen geprägt, die sich aus seinen generellen philosophischen Einstellungen ergeben. Es sind also nicht so sehr die immanenten Probleme der Mathematik, die Kants Bild vom Wesen des Mathematischen bestimmten, sondern vielmehr die systematischen Probleme der Philosophie, die ihn zu einer (...) Auseinandersetzung mit den epistemischen und methodologischen Aspekten der Mathematik nötigten. (shrink)

Immanuel Kant was a German philosopher. He is a central figure of modern philosophy, and set the terms by which all subsequent thinkers have had to grapple. He argued that human perception structures natural laws, and that reason is the source of morality. His thought continues to hold a major influence in contemporary thought, especially in fields such as metaphysics, epistemology, ethics, political philosophy, and aesthetics.

In _An Introduction to Kant’s Aesthetics_, Christian Wenzel discusses and demystifies Kant’s Critique of the Power of Judgment, guiding the reader each step of the way and placing key points of discussion in the context of Kant’s other work. Explains difficult concepts in plain language, using numerous examples and a helpful glossary. Proceeds in the same order as Kant’s text for ease of reference and comprehension. Includes an illuminating foreword by Henry E. Allison. Offers twenty-six further-reading sections, commenting briefly on (...) books and articles from the English, German, and French, that are relevant for each topic Provides an extensive bibliography and a chapter summarizing Kant's main points. (shrink)

Science is often thought of as a methodical but dull activity. But the finest science, the breakthroughs most admired and respected by scientists themselves, is characterized by elegance." "What does elegance mean in the context of science? Economy is a considerable part of it; creativity too. Sometimes, a suggested solution is so simple and neat that it elicits an exclamation of wonder from the observer. The greatest science, whether primarily theoretical or experimental, reflects a creative imagination." "In this book, the (...) distinguished physiologist and writer Ian Glynn explores some of the finest examples of elegance in science: the delightful simplicity of the laws discovered by Kepler and Newton in the motion of the planets; Galileo's experiments with an inclined plane; the remarkable work of Thomas Young on light; dazzling insights into the counterintuitive workings of our perception demonstrated by Richard Gregory and Vilayanur Ramachandran; and the several breakthroughs that led to Crick and Watson's unravelling of the structure and function of DNA. The result is a fascinating tour through some of the most important episodes in the history of science."--BOOK JACKET. (shrink)

In the chapter of the Critique of Pure Reason entitled ‘The Discipline of Pure Reason in Dogmatic Use’, Kant contrasts mathematical and philosophical knowledge in order to show that pure reason does not (and, indeed, cannot) pursue philosophical truth according to the same method that it uses to pursue and attain the apodictically certain truths of mathematics. In the process of this comparison, Kant gives the most explicit statement of his critical philosophy of mathematics; accordingly, scholars have typically focused their (...) interpretations and criticisms of Kant’s conception of mathematics on this small section of the Critique. (shrink)

_Philosophy of Mathematics_ is an excellent introductory text. This student friendly book discusses the great philosophers and the importance of mathematics to their thought. It includes the following topics: * the mathematical image * platonism * picture-proofs * applied mathematics * Hilbert and Godel * knots and nations * definitions * picture-proofs and Wittgenstein * computation, proof and conjecture. The book is ideal for courses on philosophy of mathematics and logic.

First published in 1938. The aim of this book is to expound Kant’s _Critique of Judgement _by interpreting all the details in the light of what Kant himself declares to be his fundamental problem. _A Commentary on Kant’s Critique of Judgement _provides an excellent introduction to Kant’s third critique, and will be of interest to students of philosophy.

The study is carried out in five chapters, with the first two offering a reconsideration of the function of the imagination in the Transcendental Deduction and Schematism of the first Critique. The last three follow the order of topics discussed by Kant in the third Critique in regard to judgments of taste, the sublime, and teleology; they conclude with an interpretation of "productive imagination" as a "model for the ideal of intellectual intuition". The comparison between "human and divine spontaneity" is (...) introduced in the first chapter, where Gibbons notes that while we have "Kant's explicit and repeated emphasis on the difference" between these, "in considering the similarity of these types of spontaneity," she wishes "to correct a possible imbalance in Kant's exposition": there are, she suggests, "formal similarities between these types of spontaneity". The main thrust of the first chapter is to respond to a particular reading of Kant—namely, to those unsympathetic interpreters who see in the Deductions an "imaginary subject of transcendental psychology" and also to those who give a "sympathetic reading" but do so in terms of an identification of "all aspects of synthesis with concept application," thereby reducing the "role of the transcendental synthesis of the imagination to that of unifying a manifold of intuition under a concept". Gibbons holds instead that the "theory of synthesis... focuses on the nature of human cognition and what is required for us to know anything about objects" and, further, that it presupposes "the broader function of the imagination as grounding the possibility of concept-application in the first place". (shrink)

Our visual experience seems to suggest that no continuous curve can cover every point of the unit square, yet in the late 19th century Giuseppe Peano proved that such a curve exists. Examples like this, particularly in analysis received much attention in the 19th century. They helped to instigate what Hans Hahn called a ‘crisis of intuition’, wherein visual reasoning in mathematics came to be thought to be epistemically problematic. Hahn described this ‘crisis’ as follows : " Mathematicians had for (...) a long time made use of supposedly geometric evidence as a means of proof in much too naive and much too uncritical a way, till the unclarities and mistakes that arose as a result forced a turnabout. Geometrical intuition was now declared to be inadmissible as a means of proof … "Avoiding geometrical evidence, Hahn continued, mathematicians aware of this crisis pursued what he called ‘logicization’, ‘when the discipline requires nothing but purely logical fundamental concepts and propositions for its development’. On this view, an epistemically ideal mathematics would minimize, or avoid altogether, appeals …. (shrink)

This introduction to the special issue on "Aesthetics of Science" reviews recent philosophical research on aesthetic aspects of science. Topics represented in this research include the aesthetic properties of scientific images, theories, and experiments; the relation of science and art; the role of aesthetic criteria in scientific practice and their effect on the development of science; aesthetic aspects of mathematics; the contrast between a classic and a Romantic aesthetic; and the relation between emotion, cognition, and rationality.