Published in 1891, Edmund Husserl’s first book, Philosophie der Arithmetik, aimed to “prepare the scientific foundations for a future construction of that discipline.” His goals should seem reasonable to contemporary philosophers of mathematics: . . . through patient investigation of details, to seek foundations, and to test noteworthy theories through painstaking criticism, separating the correct from the erroneous, in order, thus informed, to set in their place new ones which are, if possible, more adequately secured. [7, p. 5]2 But the (...) ensuing strategy for grounding mathematical knowledge sounds strange to the modern ear. For Husserl cast his work as a sequence of “psychological and logical investigations,” providing a psychological analysis.. (shrink)

Marcus Giaquinto presents an investigation into the different kinds of visual thinking involved in mathematical thought, drawing on work in cognitive psychology, philosophy, and mathematics. He argues that mental images and physical diagrams are rarely just superfluous aids: they are often a means of discovery, understanding, and even proof.

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)

Marcus Giaquinto tells the compelling story of one of the great intellectual adventures of the modern era: the attempt to find firm foundations for mathematics. From the late nineteenth century to the present day, this project has stimulated some of the most original and influential work in logic and philosophy.

Marcus Giaquinto traces the story of the search for firm foundations for mathematics. The nineteenth century saw a movement to make higher mathematics rigorous; this seemed to be on the brink of success when it was thrown into confusion by the discovery of the class paradoxes. That initiated a period of intense research into the foundations of mathematics, and with it the birth of mathematical logic and a new, sharper debate in the philosophy of mathematics. The Search for (...) Certainty focuses mainly on two major twentieth-century programmes: Russell's logicism and Hilbert's finitism. Giaquinto examines their philosophical underpinnings and their outcomes, asking how successful they were, and how successful they could be, in placing mathematics on a sound footing. He sets these questions in the context of a clear, non-technical exposition and assessment of the most important discoveries in mathematical logic, above all Gödel's underivability theorems.More than six decades after those discoveries Giaquinto asks what our present perspective should be on the question of certainty in mathematics. Taking recent developments into account, he gives reasons for a surprisingly positive response. (shrink)

Visual thinking in mathematics is widespread; it also has diverse kinds and uses. Which of these uses is legitimate? What epistemic roles, if any, can visualization play in mathematics? These are the central philosophical questions in this area. In this introduction I aim to show that visual thinking does have epistemically significant uses. The discussion focuses mainly on visual thinking in proof and discovery and touches lightly on its role in understanding.

This paper investigates the following question: when can one reliably infer the existence of an intersection point from a diagram presenting crossing curves or lines? Two cases are considered, one from Euclid's geometry and the other from basic real analysis. I argue for the acceptability of such an inference in the geometric case but against in the analytic case. Though this question is somewhat specific, the investigation is intended to contribute to the more general question of the extent and limits (...) of reliable diagrammatic reasoning in mathematics. (shrink)

Can visual thinking be a means of discovery in elementary analysis, as well as a means of illustration and a stimulus to discovery? The answer to the corresponding question for geometry and arithmetic seems to be ‘yes’ (Giaquinto [1992], [1993]), and so a positive answer might be expected for elementary analysis too. But I argue here that only in a severely restricted range of cases can visual thinking be a means of discovery in analysis. Examination of persuasive visual routes (...) to two simple theorems (Rolle, Bolzano) shows that they are not ways of discovering the theorems; the type of visual thinking involved can never be used to discover analytic theorems of a certain generality. The hypothesis that visual thinking is never a means of discovering the existence or nature of the limit of some infinite process is considered, and a likely counter-example is set out. It is still possible that restricted theorems can be discovered visually: an example from Littlewood is examined in detail and not found wanting. Even when visual thinking is not a means of discovery it can provide the idea for a proof in a direct way; an example is presented (Intermediate Value Theorem). In conclusion: it may be possible to discover theorems in elementary real analysis by visual means, but only theorems of a restricted kind; however, visual thinking in analysis can be very useful in other ways. (shrink)

How can we acquire a grasp of cardinal numbers, even the first very small positive cardinal numbers, given that they are abstract mathematical entities? That problem of cognitive access is the main focus of this paper. All the major rival views about the nature and existence of cardinal numbers face difficulties; and the view most consonant with our normal thought and talk about numbers, the view that cardinal numbers are sizes of sets, runs into the cognitive access problem. The source (...) of the problem is the plausible assumption that cognitive access to something requires causal contact with it. It is argued that this assumption is in fact wrong, and that in this and similar cases we should accept that a certain recognise-and-distinguish capacity is sufficient for cognitive access. We can then go on to solve the cognitive access problem, and thereby support the set-size view of cardinal numbers, by paying attention to empirical findings about basic number abilities. To this end some selected studies of infants, pre-school children and a trained chimpanzee are briefly discussed. (shrink)

Mathematicians sometimes judge a mathematical proof to be beautiful and in doing so seem to be making a judgement of the same kind as aesthetic judgements of works of visual art, music or literature. Mathematical proofs are also appraised for explanatoriness: some proofs merely establish their conclusions as true, while others also show why their conclusions are true. This paper will focus on the prima facie plausible assumption that, for mathematical proofs, beauty and explanatoriness tend to go together. To make (...) headway we need to have some grip on what it is for a proof to be beautiful, and for that we need some account of judgements of beauty in general. That is the concern of the first section. The second section faces the problem that it is far from obvious how abstract entities, such as mathematical proofs, can be beautiful, strictly and literally speaking. Reasons are given for the view that they can be. The third section introduces the distinction between proofs which explain their conclusions and proofs which do not. Finally, the question whether, for mathematical proofs, the beautiful and the explanatory tend to coincide is addressed. It is argued that we have reason to doubt that explanatory proofs tend to be beautiful, and insufficient reason to believe or disbelieve that beautiful proofs tend to be explanatory. (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)

Published in 1891, Edmund Husserl's first book, Philosophie der Arithmetik, aimed to ‘prepare the scientific foundations for a future construction of that discipline’. His goals should seem reasonable to contemporary philosophers of mathematics: "…through patient investigation of details, to seek foundations, and to test noteworthy theories through painstaking criticism, separating the correct from the erroneous, in order, thus informed, to set in their place new ones which are, if possible, more adequately secured. 1"But the ensuing strategy for grounding mathematical knowledge (...) sounds strange to the modern ear. For Husserl cast his work as a sequence of ‘psychological and logical investigations’, providing a psychological analysis "…of the concepts multiplicity, unity, and number, insofar as they are given to use authentically and not through indirect symbolizations. "This emphasis on psychology is a reflection of Husserl's training. As a teenager studying in Leipzig, he attended the lectures of Wilhelm Wundt, a seminal figure in the field of experimental psychology. Wundt held that, via introspection, we can study and classify our inner experiences, in much the same way that scientists study the natural world. 2 People working in his laboratory were therefore trained in procedures for observing and reporting on their own thought processes, as a means of gathering scientific data regarding our cognitive faculties. Bridging the gap between psychology and epistemology, Wundt felt that the results of such inquiry could have normative consequences, since the principles of reasoning employed in the sciences not only have their origins in psychological processes, but, moreover, are justified by the fundamental role they play in thought. His two-volume work, Logik , thus combined empirical considerations with a Kantian emphasis on the way that knowledge depends on our cognitive faculties. 3In The Philosophy of …. (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)

Marcus Giaquinto claims that finite cardinal numbers are sensible properties, and that the smallest ones are known by acquaintance. In this paper I compare Giaquinto’s epistemology to the Russellian one with which it invites comparison, before showing how it is subject to a version of Jody Azzouni’s “epistemic role” objection. Then I argue that the source of this problem is Giaquinto’s misconception that numbers, like quantities, are sensible properties. Finally, I offer a sketch of a theory (...) of how we grasp finite cardinals on the assumption that they are not sensible, and show why this is not subject to the epistemic role objection. (shrink)

Consider the following three claims. (i) There are no truths of the form ‘p and ~p’. (ii) No one holds a belief of the form ‘p and ~p’. (iii) No one holds any pairs of beliefs of the form {p, ~p}. Irad Kimhi has recently argued, in effect, that each of these claims holds and holds with metaphysical necessity. Furthermore, he maintains that they are ultimately not distinct claims at all, but the same claim formulated in different ways. I find (...) his argument suggestive, if not entirely transparent. I do think there is at least an important kernel of truth even in (iii), and that (i) ultimately explains what’s right about the other two. Consciousness of an impossibility makes belief in the obtaining of the corresponding state of affairs an impossibility. Interestingly, an appreciation of this fact brings into view a novel conception of inference, according to which it consists in the consciousness of necessity. This essay outlines and defends this position. A central element of the defense is that it reveals how reasoners satisfy what Paul Boghossian calls the Taking Condition and do so without engendering regress. (shrink)

The dominant theory of the evolution of moral cognition across a variety of fields is that moral cognition is a biological adaptation to foster social cooperation. This chapter argues, to the contrary, that moral cognition is likely an evolutionary exaptation: a form of cognition where neurobiological capacities selected for in our evolutionary history for a variety of different reasons—many unrelated to social cooperation—were put to a new, prosocial use after the fact through individual rationality, learning, and the development and transmission (...) of social norms. This chapter begins with a brief overview of the emerging behavioral neuroscience of moral cognition. It then outlines a novel theory of moral cognition that I have previously argued explains these findings better than alternatives. Finally, it shows how the evidence for this theory of moral cognition and human evolutionary history together suggest that moral cognition is likely not a biological adaptation. Instead, like reading sheet music or riding a bicycle, moral cognition is something that individuals learn to do—in this case, in response to sociocultural norms created in our ancestral history and passed down through the ages to enable cooperative living. (shrink)

Kant’s revolutionary new approach to philosophy was accompanied by the introduction of a largely novel terminology. With the Kant-Lexikon, a lexical reference gives the modern reader access to his work on the basis of present-day editions and takes into account 20th century and contemporary research and advances in lexicology. The Kant-Lexikon includes 2395 entries authored by 221 scholars.

The foundation of historical materialism.--A study on authority.--Sartre's existentialism.--Karl Popper and the problem of historical laws.--Freedom and the historical imperative.

This classic collection, newly revised and with a foreword by classicist Spencer Klavan, includes the famed original introduction by Russell Kirk, the full text of the Meditations of Marcus Aurelius, the complete Enchiridion of Epictetus, and key selections from Seneca and Hierocles of Alexandria in one compact volume.

The paper offers an account of luck, fate, and fortune. It begins by showing that extant accounts of luck are deficient because they do not identify the genus of which luck is a species. That genus of properties, the tychic, alert an agent to occasions on which the external world cooperates with or frustrates their goal-achievement. An agent’s sphere of competence is the set of goals that it is possible for them to reliably achieve. Luck concerns occasions on which there (...) is a mismatch between attempt and result; in bad luck the external world thwarts goal-achievement within the agent’s sphere of competence, in good luck the external world assists goal-achievement beyond the agent’s sphere of competence. Fateful events are those where, more passively, the agent finds the external world achieving or frustrating their goals. Fortune concerns the contraction and expansion of the agent’s sphere of competence. Eight reasons are given for accepting the account; its theoretical virtues and various things it explains. Lastly, three objections are answered; that the tychic properties relate to well-being rather than agency, that there are alternative theories of fortune available in the contemporary literature, that the account draws arbitrary distinctions between synonyms. (shrink)

Meditations offers timeless guidance for troubled times. Renowned for his principled leadership, Aurelius kept private notes detailing his philosophy on life and leadership. Meditations is a collection of those private notes, filled with insights on responding well to hardship both in thought and in action. His writings are a cornerstone of the Stoic philosophy, embraced by leaders throughout history and across the world for its emphasis on collaboration, rationality, and striving for the good of all people. George Long's elegant 1862 (...) translation balances accessibility with preserving the classic tone of the text. With archival printing, this pocket-sized new hardcover edition is designed for both portability and longevity. "Motions and changes are continually renewing the world, just as the uninterrupted course of time is always renewing the infinite duration of ages." -Marcus Aurelius. (shrink)

Marcus Aurelius was Emperor of Rome from 121 to 180. Thoughts of Marcus Aurelius was written for school age children. The author believed that children should be given the wisdom of great leaders from all eras. Marcus Aurelius believed that human happiness arises in part from man's acceptance of his duties and responsibilities. He believed that one should accept calmly what cannot be avoided and perform one's duties as well as possible. "It was the doctrine of (...) class='Hi'>Marcus Aurelius that most of the ills of life come to us from our own imagination, that it was not in the power of others seriously to interfere with the calm, temperate life of an individual, and that when a fellow being did anything to us that seemed unjust he was acting in ignorance, and that instead of stirring up anger within us it should stir our pity for him. Oftentimes by careful self-examination we should find that the fault was more our own than that of our fellow, and our sufferings were rather from our own opinions than from anything real.". (shrink)

A deluxe special edition of the ancient classic written by the Roman Emperor known as “The Philosopher” Meditations is a series of personal journals written by Marcus Aurelius, Emperor of Rome from 169 to 180 AD. The last of the “Five Good Emperors,” he was the most powerful and influential man in the Western world at the time. Marcus was one of the leaders of Stoicism, a philosophy of personal ethics which sought resilience and virtue through personal action (...) and responsibility. Stoicism, viewed as a foundation of modern self-help, has inspired many personal development and psychotherapy approaches through to the present day. Meditations is perhaps the most important source of our modern understanding of Stoic philosophy. Its twelve books chronicle different stages of Marcus Aurelius’ life and ideas. Although he ruled during the Pax Romana, the age of relative peace and stability throughout the empire, his reign was marked by near-constant military conflict and a devastating plague which killed upwards of five million people. Aurelius’ writings give modern readers an unprecedented look into the “spiritual exercises” which helped him through his tumultuous life and strengthened his patience, empathy, generosity, self-knowledge and emotional health. The private reflections recorded in the Meditations were never meant to be published, rather they were a source for Marcus’ own guidance and self-improvement, and jotted down by campfires or in military tents on the Roman front. The lessons, insights and perspectives contained within this remarkable work are just as relevant today as they were two millennia ago. This volume: Presents the timeless wisdom of Emperor Marcus Aurelius and his Stoic philosophy, with new research on his life and times Contains valuable insights on topics such as resilience, moderation and emotional control Discusses how to live “in agreement with nature” and abide by strong ethical principles Part of the bestselling Capstone Classics Series edited by Tom Butler-Bowdon, this attractive, high-quality hardcover volume includes: An original Introduction by Marcus Aurelius authority and Stoicism expert Donald Robertson, author of How To Think Like A Roman Emperor. A modernised, up to date version of the classic George Long translation. Meditations: The Philosophy Classic is a volume which will occupy a prominent place in any library for years to come. (shrink)

-- Douglas Kellner, University of Texas, Austin.

We argue that many intuitions do not have conscious propositional contents. In particular, many of the intuitions had in response to philosophical thought experiments, like Gettier cases, do not have such contents. They are more like hunches, urgings, murky feelings, and twinges. Our view thus goes against the received view of intuitions in philosophy, which we call Mainstream Propositionalism. Our positive view is that many thought-experimental intuitions are conscious, spontaneous, non-theoretical, non-propositional psychological states that often motivate belief revision, but they (...) require interpretation, in light of background beliefs, before a subject can form a propositional judgment as a consequence of them. We call our view Interpretationalism. We argue (i) that Interpretationalism avoids the problems that beset Mainstream Propositionalism and (ii) that our view meshes well with empirical results in contemporary cognitive science. (shrink)

This new edition brings Farquharson's authoritative 1944 translation up to date and includes a helpful introduction and notes for the student and general reader. Rutherford includes a selection of letters from Marcus to his tutor Fronto--most of which date from his earlier years--that offer personal detail and help to fill out the somber portrait of the emperor that is found in the Meditations.

Discover the ancient wisdom of Roman Emperor Marcus Aurelius Antoninus, as collected and edited by Walter Lee Brown. These helpful thoughts and meditations offer timeless insight on how to live a virtuous and fulfilling life. This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work is in the "public domain in the United States of America, and possibly other nations. Within the United States, (...) you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work.This work is in the public domain in (...) the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work.As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

Existential-Ontologie und historischer Materialismus bei Herbert Marcuse, von A. Schmidt.--Das Ganze und das ganz Andere; zur Kritik der reinen revolutionären Transzendenz, von W.F. Haug.--Technik und Eindimensionalität; eine Version der Technokratiethese? Von C. Offe.--Technologische Rationalität und spätkapitalistische ökonomie, von J. Bergmann.--Die geschichtliche Dimension des Realitätsprinzips, von H. Berndt und R. Reiche.--Marcuse and the New Left in America, by P. Breines.--Ausgewählte Bibliographie der Schriften Herbert Marcuses (p. 155-[161]).

The nature of light is a focus of Thomas Hobbes’s natural philosophical project. Hobbes’s explanation of the light of lucid bodies differs across his works, from dilation and contraction in Elements of Law to simple circular motions in De corpore. However, Hobbes consistently explains perceived light by positing that bodily resistance generates the phantasm of light. In Letters I.XIX–XX of Philosophical Letters, fellow materialist Margaret Cavendish attacks the Hobbesian understanding of both lux and lumen by claiming that Hobbes has illicitly (...) made abstractions from matter. In this paper, I argue that Cavendish’s criticisms rely on an incorrect understanding of the nature of Hobbesian geometry and the role it plays in Hobbes’s natural philosophy. Rather than understanding geometry as wholly abstract, Hobbes attempts to ground geometry in different ways of considering bodies and their motions. Furthermore, Hobbes’s own criticisms of abstraction suggest that he would share many of the worries she raises but deny that he falls prey to them. (shrink)

This book is concerned with helping you improve your approach to decision-making. The author examines judgement in a selection of managerial contexts and provides important understanding that can help you make better leadership decisions. The book also pinpoints the in-house politics of organisational decision-making. Drawing on the very latest research, it introduces practical techniques that show you how to analyse and develop your own decision-making style. It will help you to deliver sharp and insightful analyses of your business and develop (...) effective solutions. In addition, it presents simple checklists that will give you vital insights throughout the decision-making process. Students and practitioners of leadership, management, and allied fields will find this book useful in order to understand and implement useful methods. (shrink)