(I) MATHEMATICAL LECTURES. LECTURE I. Of the Name and general Division of the Mathematical Sciences. BEING about to treat upon the Mathematical Sciences, ...

In lieu of an abstract, here is a brief excerpt of the content:Isaac Barrow on the Mathematization of Nature: Theological Voluntarism and the Rise of Geometrical OpticsAntoni MaletIntroductionIsaac Newton’s Mathematical Principles of Natural Philosophy embodies a strong program of mathematization that departs both from the mechanical philosophy of Cartesian inspiration and from Boyle’s experimental philosophy. The roots of Newton’s mathematization of nature, this paper aims to demonstrate, are to be found in Isaac Barrow’s (1630–77) philosophy of (...) the mathematical sciences.Barrow’s attitude towards natural philosophy evolved from his earnest interest in medicine of around 1650, when a young Cambridge graduate, to natural philosophy (apparently under Henry More’s influence); from his thesis on the insufficiency of the Cartesian hypothesis to geometrical optics and the strong program of mathematization of natural philosophy of the middle 1660s; from Lucasian professor of Mathematics to Chaplain of his Majesty and eminent Restoration divine. Contemporary accounts of Barrow’s life suggest that he grew ever more skeptical about the worth of natural philosophy and mathematics. 1 In his last years he became a prolific [End Page 265] author of sermons and theological works. Published shortly after his death by John Tillotson (1630–94), later archbishop of Canterbury, they occupy over two thousand folio pages. Overloaded with involved philosophical arguments, Barrow’s sermons were apparently not very popular, but they were highly regarded by scholars and the Anglican hierarchy. 2 It is on certain of his sermons, as well as on the philosophical discussions contained in the Mathematical Lectures that our account of Barrow’s philosophy of the mathematical sciences will rest. 3 Barrow’s understanding of the mathematical sciences will allow us to discuss together three issues often analyzed independently: the theological background to English natural philosophy, the changing notion of mixed mathematical sciences during the seventeenth century, and finally the philosophical foundations of modern geometrical optics.It has long been recognized that significant relationships exist between theological voluntarism or intellectualism and views on natural philosophy. In particular Robert Boyle’s theological voluntarism is seen as grounding his experimentalist approach to natural philosophy. It is not quite so clear, however, how well theological voluntarism may relate to a strong program of mathematization such as the one embodied in Newton’s Principia Mathematica. In fact it has been suggested that the relationship is a negative one. This is derived from the necessary character of mathematical laws, which would put unwanted restrictions on God’s absolute dominion over nature, and also from the notion that theological intellectualism is conducive to a deductive, a priori science—the paradigm of which is of course geometry. However, Barrow’s theological voluntarism lead him to heighten the role of mathematics within natural philosophy.During the seventeenth century the so-called mixed or subalternate mathematical sciences changed profoundly. In the Enlightenment mixed mathematics—meaning above all rational mechanics—became one of the most prestigious and influential disciplines. The mixed mathematics of the Enlightenment, however, was markedly different from the Aristotelian [End Page 266] mixed mathematical sciences. The differences are noticeable both in the subject matter and in the substitution of mathematical infinitesimal analysis for geometrical synthesis, but also in the way of grounding mathematical theory on empirical evidence. Barrow’s Mathematical Lectures (delivered at Cambridge from 1664 to 1666) offer a fresh insight into the metamorphosis of these sciences just when Newton’s “mathematical principles” were in the making. Not the least interesting feature of Barrow’s discussion is that God’s omnipotence allows an evaluation of the truth of mathematical theories that do not apply to this world. Therefore, Barrow is led to introduce the distinction between the internal consistency, or mathematical truth, of a mathematical theory and its physical truth. This, in turn, leads him to the notion that theories need testing.Barrow on Matter and GodRecent literature has established significant correlations between theological voluntarism and empiricism, as well as between theological intellectualism and rationalism. As E. B. Davis writes, “the Christian doctrine of creation is a dialogue between God’s unconstrained will, which utterly transcends the bounds of human comprehension, and God’s orderly intellect, which serves as the model for the human mind.” Intellectualist theology considers God’s omniscience His... (shrink)

Published among the posthumous Opuscula of Isaac Barrow in 1687, the Epitome fidei et religionis Turcicae offers an exposition of the main tenets and practices of Islam that is unusually accurate for its time. The Epitome has been noted in passing by Barrow’s biographers and by scholars of seventeenth-century Oriental studies; but it is here firmly identified as the work of the Polish-born Ottoman dragoman and musician Ali Ufki, known in Latin as Albertus Bobovius (Wojciech Bobowski). As (...) the Epitome has never before been studied in detail, this article considers its nature, its importance and its circulation in the European Republic of Letters. The first part treats the evidence of Barrow’s amateur interest in Oriental studies, including his acquisition of the Epitome, and establishes Ufki’s authorship of the treatise. The second part examines the Epitome itself and traces its dissemination and influence through such figures as John Worthington, Samuel Hartlib, William Sancroft, Claes Rålamb and Johan Ulrich von Wallich. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Barrow and Newton E. W. STRONG As E. A. Buxrr HAS ADDUCED,Isaac Barrow (1630-1677) in his philosophy of space, time, and mathematical method strongly influenced the thinking of Newton: The recent publication of an early paper written by Newton (his De gravitatione et aequipondio fluidorum)2 affords evidence not known to Burtt of Newton's indebtedness in philosophy to Barrow, his teacher. Prior to its publication in (...) 1962, this paper was utilized by Alexandre Koyrd in his essay, "Newton and Descartes," 3 a study based on the third Horblit Lecture in the History of Science which he gave at Harvard University, March 8, 1961. Koyr8 maintains that there is a radical opposition in Newton's Principia not only to Descartes' purely scientific theories but also to the Cartesian philosophy. Yet, as he remarks, "we do not find in the Principia an open criticism of the philosophy of Descartes.... " He attributes this fact principally "to the very structure of the Principia: it is essentially a book on rational mechanics, which provides principles for physics and astronomy. In such a book there is a normal place for the discussion of Cartesian optics, but not of the conception of the relations of mind and body, and other such things." Koyrd asserts, nonetheless, that the criticism is not absent. It lurks in Newton's carefully worded definitions of fundamental concepts--those of space, time, motion, and matter--and becomes more apparent in the Optice of 1706 and in the second odition of the Principia. In support of this argument, he makes much of young Newton's unfini.~hed paper. He holds that this essay is of exceptional value "as it enables us to get some insight into the formation of Newton's thought, and to recognize that preoccupation with philosophic problems was not an external additatmentum but an integral dement of his thinking." 4 As concerns "young Newton's conception of space in its being and in its relation to God and time," Koyrd remarks that we learn from this essay The Metaphysical Foundations of Modern Physical Science (New York, 1927), p. 144. A. Rupert Hall and Marie Boas Hall, Unpublished Scientific Papers of Isaac Newton (Cambridge, 1962), Introduction, pp. 75-88; Latin text, pp. 90-121; English translation, pp. 121156. " Newtonian Studies (Harvard University Press, 1965), pp. 53-114. " Newtonian Studies, pp. 82-83. In his De gravitatione et aequipondio fluidorum, Newton is indeed engrossed with philosophic problems. With regard, however, to the announced purpose of his paper, namely, "to treat of the science of gravity and solid bodies in fluids by two methods," does Newton represent kis metaphysical "explanation of the nature of body" as an element of his science essential to its completeness7 He does not do so. At the end of his explanation he remarks: "I have already digressed enough; let us return to the main theme." He proceeds to set forth fifteen more definitions and then turns to the demonstration of two "Propositions on Non-Elastic Fluids." [155] 156 HISTORY OF PHILOSOPHY "that space is necessary, eternal, immutz~ble, and unmovable, that though we can imagine that there is nothing in space we cannot think space is not.., and that if there is no space, God would be nowhere. We learn also that all points of space are simultaneous and that, therefore, the divine omnipresence does not introduce composition in God.... " What we here learn from Newton about space in confutation of Descartes is to be found in Barrow's tenth mathematical lecture, "'Of Space, and Impenetrability." 5 In this lecture, Barrow opposes both Descartes and Hobbes. He objects, as does Newton, to the position taken by Descartes in his Principia Philosophiae, namely, "it is necessary.forMatter to be infinitely extended." He proceeds, as does Newton, to show how "very much Cartesius's great Subtility has failed him in this Case" to conclude that there is space empty of matter and distinct from magnitude from which an infinity of matter cannot be deduced. The correspondence of arguments tendered by Barrow and Newton indicates that the tenth lecture constituted a source from which Newton gleaned principal objections to Descartes' Principia Philosophiae. Although Koyr~ makes reference to Burtt's... (shrink)

SummaryIn 1693, Gottfried Wilhelm Leibniz published in the Acta Eruditorum a geometrical proof of the fundamental theorem of the calculus. It is shown that this proof closely resembles Isaac Barrow's proof in Proposition 11, Lecture 10, of his Lectiones Geometricae, published in 1670. This comparison provides evidence that Leibniz gained substantial help from Barrow's book in formulating and presenting his geometrical formulation of this theorem. The analysis herein also supports the work of J. M. Child, who in (...) 1920 studied the early manuscripts of Leibniz and concluded that he had frequently copied his diagrams from Barrow's book, but without acknowledgement. It is also shown that the diagram of Leibniz associated with his 1693 proof has often been reproduced with errors that make some aspects of his text difficult to comprehend. (shrink)

ABSTRACT Few texts in the history of science and philosophy have achieved the level of interpretative indeterminacy as a short manuscript tract by Isaac Newton, known as ‘De gravitatione’. On the basis of some new evidence, this article argues that it is an introductory fragment of some lectures on hydrostatics delivered in the of spring 1671. Taking seriously the possibility of a pedagogical purpose, it is then argued that the famous digression on space, far from articulating a sophisticated metaphysics (...) that may have owed something to Henry More, was a simple piece of mixed-mathematical prolegomena designed to facilitate the subsequent geometrical argumentation. In this regard, Newton was doing the same as his mentor, Isaac Barrow, had done in his own mathematical lectures; both drew heavily on the explicitly anti-metaphysical approach of Pierre Gassendi. It is shown that More himself would have almost certainly opposed Newton’s approach. The excesses of metaphysical readings of Newton’s intentions are challenged; there is no warrant for reading the digression as directly relevant to the Principia. (shrink)

In ?155 of his New Theory of Vision Berkeley explains that a hypothetical ?unbodied spirit? ?cannot comprehend the manner wherein geometers describe a right line or circle?.1The reason for this, Berkeley continues, is that ?the rule and compass with their use being things of which it is impossible he should have any notion.? This reference to geometrical tools has led virtually all commentators to conclude that at least one reason why the unbodied spirit cannot have knowledge of plane geometry is (...) because it cannot manipulate a ruler or a compass. In this article I will show that such an interpretation is flawed. I will instead argue that Berkeley's understanding of Euclidian geometry was based on Isaac Barrow's account of the foundations of geometry. On this view geometrical objects are conceived in terms of the idealized motion that generates the objects of geometry. Consequently, that what the unbodied spirit cannot do in this context is to form an idea of motion rather than being unable to handle geometrical tools. 1All references to Berkeley are from, A. A. Luce and T. E. Jessop (eds.), The Works of George Berkeley, Bishop of Cloyne (London: Thomas Nelson and Sons, Ltd., 1948) The following abbreviations are used: An Essay Towards A New Theory of Vision, section x = New Theory x; Philosophical Commentaries, entry x = Commentaries x; Part I of A Treatise concerning the Principles of Knowledge, section x = Principles x. All other references to Berkeley's works are of the form The Works of George Berkeley, Bishop of Cloyne, volume x, page y = Works, x, y. (shrink)

In the 4th edition of this best-selling textbook, the authors introduce students to the business of philosophizing, thereby inducting them into the art of reasoning and analyzing key concepts in education. This introductory text, continuously in print for more than thirty years, is a classic in its field. It shows, first and foremost, the importance of philosophy in educational debate and as a background to any practical activity such as teaching. What is involved in the idea of educating a person (...) or the idea of educational success? What, if anything, can be known and how should we organize what we know for curriculum purposes? What are the criteria for establishing the optimum balance between formal and informal teaching techniques? How trustworthy is educational research? In addition to these questions, which strike to the heart of the rationale for the educative process as a whole, the authors explore such concepts as culture, creativity, autonomy, indoctrination, needs, interests and learning by discovery. In this new updated edition, the authors draw on the latest research in genetics to argue that education is uniquely human and is essentially what develops us as humans. Resisting modern tendencies to equate knowledge with opinion, and value judgments with taste, this book leads the reader into the business of philosophizing and champions the cause of reason in education. (shrink)

Ever since Copernicus, scientists have continually adjusted their view of human nature, moving it further and further from its ancient position at the center of Creation. But in recent years, a startling new concept has evolved that places it more firmly than ever in a special position. Known as the Anthropic Cosmological Principle, this collection of ideas holds that the existence of intelligent observers determines the fundamental structure of the Universe. In its most radical version, the Anthropic Principle asserts that (...) "intelligent information-processing must come into existence in the Universe, and once it comes into existence, it will never die out." This wide-ranging and detailed book explores the many ramifications of the Anthropic Cosmological Principle, covering the whole spectrum of human inquiry from Aristotle to Z bosons. Bringing a unique combination of skills and knowledge to the subject, John D. Barrow and Frank J. Tipler--two of the world's leading cosmologists--cover the definition and nature of life, the search for extraterrestrial intelligence, and the interpretation of the quantum theory in relation to the existence of observers. The book will be of vital interest to philosophers, theologians, mathematicians, scientists, and historians, as well as to anyone concerned with the connection between the vastness of the universe of stars and galaxies and the existence of life within it on a small planet out in the suburbs of the Milky Way. (shrink)

I propose a `stage theory’ of groups: a group is a fusion of group-stages, where a group-stage is a plurality of individuals at a world and a time. The stage theory consists of existence conditions, identity conditions, and parthood conditions for groups.

Motus quidem veros corporum singulorum cognofcere , & ab apparentibus actu diícriminare, difficillimum est ; propterca quod partes ípatij illius immobilis in quo corpora vere moventur, non incurrunt in sensus.

In this enlightening study, a noted scholar elucidates the distinguishing characteristics of the works of several Jewish thinkers of the Middle Ages. In addition to summaries of the main arguments and teachings of Moses Maimonides, Isaac Israeli, Judah Halevi, Abraham Ibn Daud, Hillel ben Samuel, Levi ben Gerson, Joseph Albo, and many others, the author offers insightful analyses and commentary. Of particular value to beginners, this volume is also an ever-relevant resource for many issues of scholarly debate.

When schools react to contemporary events and focus on complex moral problems they commonly fail to make basic distinctions between the morally serious and trivial, the moral and the non-moral, and problems and dilemmas. We need to teach the distinction between moral and other values, and between what is intrinsically good, what is right in practice and what is justifiable. Moral theory seeks to delineate an ideal situation. Different circumstances give rise to different particular practices; but the principles of freedom, (...) fairness, respect for persons, truth and felicity constitute a widely recognized view of the essence of morality. These principles, and the nature of education itself, are often ignored in programs of moral education, which should proceed by providing an understanding of the nature and limits of the moral sphere. (shrink)

We present a novel model of individual people, online posts, and media platforms to explain the online spread of epistemically toxic content such as fake news and suggest possible responses. We argue that a combination of technical features, such as the algorithmically curated feed structure, and social features, such as the absence of stable social-epistemic norms of posting and sharing in social media, is largely responsible for the unchecked spread of epistemically toxic content online. Sharing constitutes a distinctive communicative act, (...) governed by a dedicated norm and motivated to a large extent by social identity maintenance. But confusion about this norm and its lack of inherent epistemic checks lead readers to misunderstand posts, attribute excess or insufficient credibility to posts, and allow posters to evade epistemic accountability—all contributing to the spread of epistemically toxic content online. This spread can be effectively addressed if people and platforms add significantly more context to shared posts and platforms nudge people to develop and follow recognized epistemic norms of posting and sharing. (shrink)

For the past fifty years anxiety over naturalism has driven debates in social theory. One side sees social science as another kind of natural science, while the other rejects the possibility of objective and explanatory knowledge. _Interpretation and Social Knowledge_ suggests a different route, offering a way forward for an antinaturalist sociology that overcomes the opposition between interpretation and explanation and uses theory to build concrete, historically specific causal explanations of social phenomena.

My aim in this paper is to give a philosophical analysis of the relationship between contingently available technology and the knowledge that it makes possible. My concern is with what specific subjects can know in practice, given their particular conditions, especially available technology, rather than what can be known “in principle” by a hypothetical entity like Laplace’s Demon. The argument has two parts. In the first, I’ll construct a novel account of epistemic possibility that incorporates two pragmatic conditions: responsibility and (...) practicability. For example, whether subjects can gain knowledge depends in some circumstances on whether they have the capability of gathering relevant evidence. In turn, the possibility of undertaking such investigative activities depends in part on factors like ethical constraints, economical realities, and available technology. In the second part of the paper, I’ll introduce “technological possibility” to analyze the set of actions made possible by available technology. To help motivate the problem and later test my proposal, I’ll focus on a specific historical case, one of the earliest uses of digital electronic computers in a scientific investigation. I conclude that the epistemic possibility of gaining access to scientific knowledge about certain subjects depends (in some cases) on the technological possibility for making responsible investigations. (shrink)

Why would God institute the practice of efficacious petitionary prayer? Why would God not simply give us what we need before we ask? I examine recently proposed solutions to this puzzle and argue that they are inadequate to explain why an omniscient and perfectly good God would act differently in response to prayer. I propose that God has reasons to not always maximize a creature’s good, even in a sinless world, and that petitionary prayer functions as a means to reward (...) those who trust God, to enable us to actively love those we cannot otherwise help, and to give the petitioner personal evidence of God’s existence and care for her, creating a virtuous cycle of increasing faith. I refine this proposal by responding to several objections involving human responsibility and the epistemology of divine action. Along the way, I offer several ways petitioners can recognize God’s having answered a prayer and how God might help us with some common obstacles to prayer. (shrink)

When artificial intelligence (AI) is used to make high-stakes decisions, some worry that this will create a morally troubling responsibility gap—that is, a situation in which nobody is morally responsible for the actions and outcomes that result. Since the responsibility gap might be thought to result from individuals lacking knowledge of the future behavior of AI systems, it can be and has been suggested that deploying explainable artificial intelligence (XAI) techniques will help us to avoid it. These techniques provide humans (...) with certain forms of understanding of the systems in question. In this paper, I consider whether existing XAI techniques can indeed close the responsibility gap. I identify a number of significant limits to their ability to do so. Ensuring that responsibility for AI-assisted outcomes is maintained may require using different techniques in different circumstances, and potentially also developing new techniques that can avoid each of the issues identified. (shrink)

An increasingly prevalent approach to studying human cognition is to construe the mind as optimally allocating limited cognitive resources among cognitive processes. Under this bounded rationality approach (Icard in Philos Sci 85(1):79–101, 2018; Simon in Utility and probability, Palgrave Macmillan, 1980), it is common to assume that resource-bounded cognitive agents approximate normative solutions to statistical inference problems, and that much of the bias and variability in human performance can be explained in terms of the approximation strategies we employ. In this (...) paper, we argue that this approach restricts itself to an unnecessarily narrow scope of cognitive models, which limits its ability to explain how humans flexibly adapt their representations to novel environments. We argue that more attention should be paid to how we form our cognitive representations in the first place, and advocate for pluralistic framework which jointly optimizes over both representations and algorithms for manipulating them. We identify several fundamental trade-offs that manifest in this joint optimization, and draw on recent work to motivate a unified formal framework for this analysis. We illustrate a simplified version of this analysis with a case study in social cognition, and outline several new directions for research that this approach suggests. (shrink)

Our decision-theoretic states are not luminous. We are imperfectly reliable at identifying our own credences, utilities and available acts, and thus can never be more than imperfectly reliable at identifying the prescriptions of decision theory. The lack of luminosity affords decision theory a remarkable opportunity — to issue guidance on the basis of epistemically inaccessible facts. We show how a decision theory can guarantee action in accordance with contingent truths about which an agent is arbitrarily uncertain. It may seem that (...) such advantages would require dubiously adverting to externalist facts that go beyond the internalism of traditional decision theory, but this is not so. Using only the standard repertoire of decision-theoretic tools, we show how to modify existing decision theories to take advantage of this opportunity. These improved decision theories require agents to maximize conditional expected utility — expected utility conditional upon an agent’s actual decision situation. We call such modified decision theories ‘self-confident’. These self-confident decision theories have a distinct advantage over standard decision theories: their prescriptions are better. (shrink)

In books such as The World Within the World and The Anthropic Cosmological Principle, astronomer John Barrow has emerged as a leading writer on our efforts to understand the universe. Timothy Ferris, writing in The Times Literary Supplement of London, described him as "a temperate and accomplished humanist, scientist, and philosopher of science--a man out to make a contribution, not a show." Now Barrow offers the general reader another fascinating look at modern physics, as he explores the quest (...) for a single, unifying theory that will unlock nature's secrets. Theories of Everything is more than a history of science, more than a popular report on recent research and discoveries. Barrow provides a reflective, intelligent commentary on what a true Theory of Everything would be--its ingredients, its limitations, and what it could tell us about the universe. Never before, he writes, have physicists been so confident and so eager in the hunt for this "cosmic Rosetta Stone," as he calls it: "a single all-embracing picture of all the laws of nature from which the inevitability of all things seen must follow with unimpeachable logic." He lays out eight essential ingredients for a Theory of Everything and then explores each in turn, tracing how our knowledge has developed and how scientific discovery relates to our changing philosophy and religious thought in each area. Some of these ingredients are obvious--the laws of nature must be explained, for example, as well as its organizing principles--but others may be surprising, such as broken symmetries and selection biases. A Theory of Everything must account for the fact that the universe is "messy and complicated," he tells us, and for the limitations imposed by the questions we ask and the information we can obtain. The key lies in the remarkable capacity of mathematics to express the fundamental workings of the physical world--a language that the human mind is uniquely equipped to understand and manipulate. Barrow examines what mathematics actually is and describes how it makes the universe intelligible and provides a path to the underlying coherence in nature--which has led, in fact, to arguments that the universe itself is a vast computer. Yet even the most complete theory, even the most comprehensive mathematical explanation, cannot account for the uncomputable varieties of human experience and thought. "No non-poetic account of reality," he writes, "can be complete." In a field where the authorities converse in equations and mathematical notations, John Barrow speaks with the voice of thoughtful and knowledgeable humanist. Written with eloquence and expertise, Theories of Everything establishes a new perspective on humanity's efforts to explain the universe. (shrink)

Presents Newton's unifying idea of gravitation and explains how he converted physics from a science of explanation into a general mathematical system.

This major work challenges some widely held positions in epistemology - those of Peirce and Popper on the one hand and those of Quine and Kuhn on the other.

This book draws on fields as diverse as biochemistry, physiology, pharmacology, psychology, psychiatry, and ethology, to form a fascinating synthesis of information on the nature of fear and of panic and anxiety disorders. Dr. Marks offers both a detailed discussion of the clinical aspects of fear-related syndromes and a broad exploration of the sources and mechanisms of fear and defensive behavior. Dealing first with normal fear, he establishes a firm, scientific basis for understanding it. He then presents a thorough analysis (...) of the development, symptoms and treatment of fear-related syndromes. Phobic and obsessive-compulsive disorders are examined in detail. The book is illustrated with examples of fear and defensive behavior in other living organisms. By drawing provocative analogies between animal and human behavior, it sheds new light on the origins of fears, phobias, and obsessive-compulsive problems, as well as on their treatment by drugs and psychological means. Clinical psychologists, ethologists, and anyone interested in the mechanisms of behavior will be fascinated by this authoritative study. The text is intriguing and informative, and the bibliography of over 2,100 entries makes it an invaluable reference. (shrink)

David Chalmers thinks his iPhone exemplifies the extended mind thesis by meeting the criteria ‎that he and Andy Clark established in their well-known 1998 paper. Andy Clark agrees. We take ‎this proposal seriously, evaluating the case of the GPS-enabled smartphone as a potential mind ‎extender. We argue that the “trust and glue” criteria enumerated by Clark and Chalmers are ‎incompatible with both the epistemic responsibilities that accompany everyday activities and the ‎practices of trust that enable users to discharge them. Prospects (...) for revision of the original ‎criteria are dim. We therefore call for a rejection of the trust criterion and a reevaluation of the ‎extended mind thesis.‎. (shrink)

John Barrow is increasingly recognized as one of our most elegant and accomplished science writers, a brilliant commentator on cosmology, mathematics, and modern physics. Barrow now tackles the heady topic of impossibility, in perhaps his strongest book yet. Writing with grace and insight, Barrow argues convincingly that there are limits to human discovery, that there are things that are ultimately unknowable, undoable, or unreachable. He first examines the limits on scientific inquiry imposed by the deficiencies of the (...) human mind: our brain evolved to meet the demands of our immediate environment, Barrow notes, and much that lies outside this small circle may also lie outside our understanding. Barrow investigates practical impossibilities, such as those imposed by complexity, uncomputability, or the finiteness of time, space, and resources. Is the universe finite or infinite? Can information be transmitted faster than the speed of light? The book also examines the deeper theoretical restrictions on our ability to know, including Godel's theorem--which proved that there were things that could not be proved--and Arrow's Impossibility theorem about democratic voting systems. Finally, having explored the limits imposed on us from without, Barrow considers whether there are limits we should impose upon ourselves. For instance, if the secrets of the atom are to be found only by recreating extreme environments at great financial cost, just how much should we devote to that quest? Weaving together this intriguing tapestry, he illuminates some of the most profound questions of science, from the possibility of time travel to the very structure of the universe. (shrink)