In the Critique of Judgement, Kant offers a penetrating analysis of our experience of the beautiful and the sublime. He discusses the objectivity of taste, aesthetic disinterestedness, the relation of art and nature, the role of imagination, genius and originality, the limits of representation, and the connection between morality and the aesthetic. He also investigates the validity of our judgements concerning the degree in which nature has a purpose, with respect to the highest interests of reason and enlightenment. The work (...) profoundly influenced the artists, writers, and philosophers of the classical and romantic period, including Hegel, Schelling, Schopenhauer, and Nietzsche. In addition, it has remained a landmark work in fields such as phenomenology, hermeneutics, the Frankfurt School, analytical aesthetics, and contemporary critical theory. Today it remains an essential work of philosophy, and required reading for all with an interest in aesthetics. (shrink)

Extends the ideas of social constructivism to the philosophy of mathematics, developing a powerful critique of traditional absolutist conceptions of mathematics, and proposing a reconceptualization of the philosophy of mathematics.

Scientists and laypeople alike use the sense of understanding that an explanation conveys as a cue to good or correct explanation. Although the occurrence of this sense or feeling of understanding is neither necessary nor sufficient for good explanation, it does drive judgments of the plausibility and, ultimately, the acceptability, of an explanation. This paper presents evidence that the sense of understanding is in part the routine consequence of two well-documented biases in cognitive psychology: overconfidence and hindsight. In light of (...) the prevalence of counterfeit understanding in the history of science, I argue that many forms of cognitive achievement do not involve a sense of understanding, and that only the truth or accuracy of an explanation make the sense of understanding a valid cue to genuine understanding. (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)

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)

Auth: University College London, Distributed by Viking.

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)

Among the phenomena that make up the mind, cognitive psychologists and philosophers have postulated a puzzling one that they have called ?epistemic feelings.? This paper aims to (1) characterize these experiences according to their intentional content and phenomenal character, and (2) describe the nature of these mental states as nonconceptual in the cases of animals and infants, and as conceptual mental states in the case of adult human beings. Finally, (3) the paper will contrast three accounts of the causes and (...) mechanisms of epistemic feelings: the doxastic account; the mental scanner account; and the heuristic mechanism account. The paper will argue in favor of the heuristic mechanism account. (shrink)

In chapters ranging from "The Beautiful, the Dainty, and the Dumpy" to "Skin-deep or In the Eye of the Beholder?" Nick Zangwill investigates the nature of beauty as we conceive it, and as it is in itself. The notion of beauty is currently attracting increased interest, particularly in philosophical aesthetics and in discussions of our experiences and judgments about art. In The Metaphysics of Beauty, Zangwill argues that it is essential to beauty that it depends on the ordinary features of (...) things. He uses this principle to defend the notion of the aesthetic, to call for a version of aesthetic formalism, and to reconsider the reality of beauty. The Metaphysics of Beauty brings beauty to the center of intellectual consciousness in a manner informed by contemporary metaphysics and engages with beauty as an enduring object of human thought and experience. (shrink)

Mathematics is a subject we are all exposed to in our daily lives, but one which many of us fear. In this introduction, Timothy Gowers elucidates the most fundamental differences, which are primarily philosophical, between advanced mathematics and what we learn at school, so that one emerges with a clearer understanding of such paradoxical-sounding concepts as 'infinity', 'curved space', and 'imaginary numbers'. From basic ideas, through to philosophical queries, to common sociological questions about the mathematical community, this book unravels some (...) of the mysteries of space and numbers. (shrink)

In The Metaphysics of Beauty, Zangwill argues that it is essential to beauty that it depends on the ordinary features of things.

Philosophers agree that scientific explanations aim to produce understanding, and that good ones succeed in this aim. But few seriously consider what understanding is, or what the cues are when we have it. If it is a psychological state or process, describing its specific nature is the job of psychological theorizing. This article examines the role of understanding in scientific explanation. It warns that the seductive, phenomenological sense of understanding is often, but mistakenly, viewed as a cue of genuine understanding. (...) The article closes with a discussion of several new paths of research that tie the psychology of scientific explanation to cognate notions of learning, testimony, and understanding. (shrink)

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)

Dual Process Theories of reasoning posit that judgments are mediated by both fast, automatic processes and more deliberate, analytic ones. A critical, but unanswered question concerns the issue of monitoring and control: When do reasoners rely on the first, intuitive output and when do they engage more effortful thinking? We hypothesised that initial, intuitive answers are accompanied by a metacognitive experience, called the Feeling of Rightness, which can signal when additional analysis is needed. In separate experiments, reasoners completed one of (...) four tasks: conditional reasoning, a three-term variant of conditional reasoning, problems used to measure base rate neglect, or a syllogistic reasoning task. For each task, participants were instructed to provide an initial, intuitive response to the problem along with an assessment of the rightness of that answer. They were then allowed as much time as needed to reconsider their initial answer and provide a final answer. In each experiment, we observed a robust relationship between the FOR and two measures of analytic thinking: low FOR was associated with longer rethinking times and an increased probability of answer change. In turn, FOR judgments were consistently predicted by the fluency with which the initial answer was produced, providing a link to the wider literature on metamemory. These data support a model in which a metacognitive judgment about a first, initial model determines the extent of analytic engagement. (shrink)

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)

The purpose of this article is to integrate two outstanding problems within the philosophy of science. The first concerns what role aesthetics plays in scientific thinking. The second is the problem of how logically testable ideas are generated (the so-called "psychology of research" versus "logic of (dis)proof" problem). I argue that aesthetic sensibility is the basis for what scientists often call intuition, and that intuition in turn embodies (in a literal physiological sense) ways of thinking that have their own meta-logic. (...) Thus, aesthetics is a form of cognition. Scientists think not in equations or words or other logical abstractions, but emotionally and sensually, using visual and aural images, kinesthetic and other proprioceptive feelings, sensations, patterns, and analogies. These aesthetic forms of thinking have their own logics that I call "synosia", from the root words synaesthesia (a combining of senses) and gnosis, "to know". Synosia denotes understanding that integrates feeling that one knows with feeling what one knows. Eminent scientists universally describe an explicitly secondary process in which such personal knowledge must be "translated" into a formal language, such as words or equations, in order to be communicated to other people. Many of the unsolved problems that philosophers of science (as well as psychologists and artificial intelligence researchers) have had in making sense of scientific thinking have arisen from confusing the form and content of the final translations with the hidden means by which scientific insights are actually achieved. (shrink)

IN the right circumstances and the right frame of mind we are prepared to make aesthetic appraisals of almost anything, from hills, cottages and cars, to symphonies, people and poems. My problem is to try and set a boundary in at least one direction to the catholicity of this kind of judgement. I want to argue that when we use a word from our aesthetic vocabulary for appraising a theory in science or a proof in mathematics we are not properly (...) to be described as making an aesthetic appraisal, though there are close parallels with our normal use of these words. (shrink)

Peter Carruthers correctly claims that metacognition in humans may involve self-directed interpretations (i.e., may use the conceptual interpretative resources of mindreading). He fails to show, however, that metacognition cannot rely exclusively on subjective experience. Focusing on self-directed mindreading can only bypass evolutionary considerations and obscure important functional differences.

Review of Beauty and Revolution in Science, by JW McAllister.

Evidence and theory ranging from traditional philosophy to contemporary cognitive research support the hypothesis that consciousness has a two-part structure: a focused region of articulated experience surrounded by a field of relatively unarticulated, vague experience.William James developed an especially useful phenomenological analysis of this "fringe" of consciousness, but its relation to, and potential value for, the study of cognition has not been explored. I propose strengthening James′ work on the fringe with a functional analysis: fringe experiences work to radically condense (...) context information in consciousness; are vague because a more explicit representation of context information would overwhelm consciousness′ limited articulation capacity; help mediate retrieval functions in consciousness; and contain a subset of monitoring and control experiences that cannot be elaborated in focal attention and are "ineffable." In general, the phenomenology of the fringe is a consequence of its cognitive functions, constrained by consciousness′ limited articulation capacity. Crucial to monitoring and control is the feeling of "rightness." Rightness functions as a summary index of cognitive integration, representing, in the fringe, the degree of positive fit between a given conscious content and its parallel, unconsciously encoded context. Rightness appears analogous to the connectionist metric of global network integration, known variously as goodness-of-fit, harmony, or minimum energy. (shrink)

How do we go about weighing evidence, testing hypotheses, and making inferences? The model of " inference to the best explanation " -- that we infer the hypothesis that would, if correct, provide the best explanation of the available evidence--offers a compelling account of inferences both in science and in ordinary life. Widely cited by epistemologists and philosophers of science, IBE has nonetheless remained little more than a slogan. Now this influential work has been thoroughly revised and updated, and features (...) a new introduction and two new chapters. Inference to the Best Explanation is an unrivaled exposition of a theory of particular interest in the fields both of epistemology and the philosophy of science. (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)

The study of the feeling of knowing may have implications for some of the metatheoretical issues concerning consciousness and control. Assuming a distinction between information-based and experience-based metacognitive judgments, it is argued that the sheer phenomenological experience of knowing (''noetic feeling'') occupies a unique role in mediating between implicit-automatic processes, on the one hand, and explicit-controlled processes, on the other. Rather than reflecting direct access to memory traces, noetic feelings are based on inferential heuristics that operate implicitly and unintentionally. Once (...) such heuristics give rise to a conscious feeling that feeling can then affect controlled action. Examination of the cues that affect noetic feelings suggest that not only do these feelings inform controlled action, but they are also informed by feedback from the outcome of that action. (shrink)

What do mathematicians mean when they use terms such as ‘deep’, ‘elegant’, and ‘beautiful’? By applying empirical methods developed by social psychologists, we demonstrate that mathematicians' appraisals of proofs vary on four dimensions: aesthetics, intricacy, utility, and precision. We pay particular attention to mathematical beauty and show that, contrary to the classical view, beauty and simplicity are almost entirely unrelated in mathematics.

I argue that explanation should be thought of as the phenomenological mark of the operation of a particular kind of cognitive system, the theory-formation system. The theory-formation system operates most clearly in children and scientists but is also part of our everyday cognition. The system is devoted to uncovering the underlying causal structure of the world. Since this process often involves active intervention in the world, in the case of systematic experiment in scientists, and play in children, the cognitive system (...) is accompanied by a theory drive, a motivational system that impels us to interpret new evidence in terms of existing theories and change our theories in the light of new evidence. What we usually think of as explanation is the phenomenological state that accompanies the satisfaction of this drive. However, the relation between the phenomenology and the cognitive system is contingent, as in similar cases of sexual and visual phenomenology. Distinctive explanatory phenomenology may also help us to identify when the theory-formation system is operating. (shrink)

The feeling of error (FOE) is the subjective experience that something went wrong during a reasoning or calculation task. The main goal of the present study was to assess the accuracy of the FOE in the context of mental mathematical calculation. We used the number bisection task (NBT) to evoke this metacognitive feeling and assessed it by asking participants if they felt they have committed an error after solving the task. In the NBT participants have to determine whether the number (...) presented in the middle of a triplet corresponds to the arithmetic mean of the two outer numbers (e.g., 07_16_25) with a Yes/No answer. Our results show that FOE reports were strongly correlated with arithmetic errors and numerical properties of the NBT, suggesting that the FOE accurately represents the error. This finding indicates that even very fast metacognitive feelings are reliable when it comes to evaluating one’s own mental performance. Moreover, our results suggest that the occurrence of FOEs is determined by the fluency with which each triplet was solved and the post-decision evaluation processes that occurred after the NBT was solved. Additionally, we asked participants to report their confidence in the given answer for the cases where they did not report FOEs. Participants reported less confidence for the (objectively) incorrect answers than for the (objectively) correct ones, suggesting that in cases where they did not have a conscious FOE they still were able to implicitly detect their errors. Remarkably, confidence was also determined by the fluency of the NBT. (shrink)

Contrary to standard assumptions, reasoning is often an emotional process. Emotions can have good effects, as when a scientist gets excited about a line of research and pursues it successfully despite criticism. But emotions can also distort reasoning, as when a juror ignores evidence of guilt just because the accused seems like a nice guy. In _Hot Thought_, Paul Thagard describes the mental mechanisms -- cognitive, neural, molecular, and social -- that interact to produce different kinds of human thinking, from (...) everyday decision making to legal reasoning, scientific discovery, and religious belief, and he discusses when and how thinking and reasoning should be emotional. Thagard argues that an understanding of emotional thinking needs to integrate the cognitive, neural, molecular, and social levels. Many of the chapters employ computational models of various levels of thinking, including HOTCO models and the more neurologically realistic GAGE model. Thagard uses these models to illuminate thinking in the domains of law, science, and religion, discussing such topics as the role of doubt and reasonable doubt in legal and other contexts, valuable emotional habits for successful scientists, and the emotional content of religious beliefs. Identifying and assessing the impact of emotion, Thagard argues, can suggest ways to improve the process of reasoning. (shrink)

How do we go about weighing evidence, testing hypotheses, and making inferences? According to the model of _Inference to the Best Explanation_, we work out what to infer from the evidence by thinking about what would actually explain that evidence, and we take the ability of a hypothesis to explain the evidence as a sign that the hypothesis is correct. In _Inference to the Best Explanation_, Peter Lipton gives this important and influential idea the development and assessment it deserves. The (...) second edition has been substantially enlarged and reworked, with a new chapter on the relationship between explanation and Bayesianism, and an extension and defence of the account of contrastive explanation. It also includes an expanded defence of the claims that our inferences really are guided by diverse explanatory considerations, and that this pattern of inference can take us towards the truth. This edition of _Inference to the Best Explanation_ has also been updated throughout and includes a new bibliography. (shrink)