Offering a collection of fifteen essays that deal with issues at the intersection of phenomenology, logic, and the philosophy of mathematics, this 2005 book is divided into three parts. Part I contains a general essay on Husserl's conception of science and logic, an essay of mathematics and transcendental phenomenology, and an essay on phenomenology and modern pure geometry. Part II is focused on Kurt Godel's interest in phenomenology. It explores Godel's ideas and also some work of Quine, Penelope Maddy and (...) Roger Penrose. Part III deals with elementary, constructive areas of mathematics. These are areas of mathematics that are closer to their origins in simple cognitive activities and in everyday experience. This part of the book contains essays on intuitionism, Hermann Weyl, the notion of constructive proof, Poincaré and Frege. (shrink)

Volume 1 of this biography of L. E. J. Brouwer was published in 1999.1 The volume under review here covers the period from the early nineteen twenties until Brouwer's death in 1966. It also includes a short epilogue that discusses the disposition of Brouwer's estate after his death, his influence on others, the paths of some of his students and colleagues, and other matters. Van Dalen notes in the Preface that in preparing this volume he consulted some historical studies that (...) appeared after the first volume was published. He also used new material from various archives. The biography contains interesting quotations from unpublished materials in the Brouwer Archive and from correspondence. The bibliographical references to Brouwer's publications, it should be noted, are somewhat different in this volume. This volume, like the first, contains some nice photographs and reproductions. I noted that there were many typographical errors in the earlier book but Volume 2 is relatively free of them.As is the case in Volume 1, the discussion of Brouwer's mathematical and philosophical work is woven into the narrative of Brouwer's life and times. The story in this volume starts with Brouwer's first contacts with Paul Alexandrov and Paul Urysohn in 1923. The interaction began when Urysohn announced that he had found a mistake in Brouwer's definition of dimension in Brouwer's 1913 paper on natural dimension. Was it just a slip of the pen, as Brouwer always maintained , or something more substantial? Urysohn and Alexandrov ultimately came to agree with Brouwer on the matter, and Urysohn was prepared to grant Brouwer priority for the definition of dimension. There were, however, ups and downs along the way. Karl Menger, through his own work in topology and dimension theory, soon got into the picture, and Brouwer and Menger were to …. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:After Gödel: Platonism and Rationalism in Mathematics and LogicMark C. R. SmithRichard Tieszen. After Gödel: Platonism and Rationalism in Mathematics and Logic. Oxford-New York: Oxford University Press, 2011. Pp. xi + 245. Cloth, $75.00.Tieszen’s new book offers a synthesis and extension of his longstanding project of bringing the method of Husserl’s phenomenology to bear on fundamental questions—both epistemological and ontological—in the philosophy of mathematics. Gödel held Husserl’s philosophy (...) in high regard, and thought that it offered the most promising avenue for an adequate understanding of the foundations of mathematical thought and practice, but Gödel himself did not give a full systematic statement of how phenomenology might carry out that project. Tieszen takes up the task in this book, which is structured around two main guiding themes. The first is to locate the sources of Gödel’s philosophical views—principally [End Page 303] in Plato, Leibniz, and of course Husserl, along with Kant as a sort of backcloth—and the second is to argue that from these materials, a promising rationalism in mathematics and logic can be consistently articulated. Along the way we also encounter a good many of Gödel’s arguments against an array of reductive positions, Hilbert’s formalism and Carnap’s “logical syntax” project among them, as well as arguments against the prospects of a mechanical or purely computational understanding of the human mind. Many of the arguments (both negative and positive) stem from a reflection on the consequences of Gödel’s landmark incompleteness results of 1931.The core of Tieszen’s positive proposals comes in chapters 4, 5, and 6: here is where we meet with “constituted Platonism,” Tieszen’s name for the rationalist, realist synthesis of Husserl’s method and Gödel’s mathematical insights. Methodologically, the starting point is the actual practice of mathematics as a human undertaking, and its conditions of possibility. At the most general level, it is characteristic of thought to be intentional, that is, to be (or try to be) about something, to have a target, to be a bearer of meaning. So it goes for thought about items in sensory awareness, about fictional objects, or about mathematical subject-matter. Concerning the latter, what we encounter in mathematical experience are frequently “phenomena that resist our will, so that we are ‘forced’ in certain directions” (170); what accounts for this resistance to our will are invariants, which have the further characteristics of repeatability—both for a single subject, and among many subjects—and availability for convergence (221), which, I take it, means availability as structures or concepts upon which many independent rational investigators will come to rest. Thinking mathematical thoughts is quite unlike thinking thoughts about fictional items, because fictions are indefinitely plastic in the face of human will. It is also unlike thought about sensory items, in that the intentional objects at issue are not available to outer sense; but from another angle it is more like thought about real material items than imaginary beings, because our mathematical thoughts are exposed to the possibility of error or misrepresentation in various ways. (An elementary example: most people think that 1 is distinct from 0.9999... repeating: this is a misrepresentation, because they are in fact identical.)The phenomenology of mathematical practice is such that we assay various conjectures and proposals about invariants already given to consciousness by previous practice, and discover what further invariant structures or concepts are revealed as necessary constraints upon the various possibilities that we can experimentally float for ourselves. Sometimes our conjectures or intentions are fulfilled, sometimes they are frustrated; but it is the very fact of fulfillment and frustration which points us to a Platonic reality. It is not the noumenal realm of the “naive” Platonist, in Tieszen’s phrase, but nevertheless an objective reality marked off “on the basis of what stabilizes or becomes invariant in our experience, on the basis of what has a history and a sedimentation that permits of further elaboration, development, constraints, surprises, and so on” (101). In something like this way, human reason and consciousness “constitute the meaning of being” of mathematical objects... (shrink)

Richard Tieszen's new book1 is a collection of fifteen articles and reviews, spanning fifteen years, presenting the author's approach to philosophical questions about logic and mathematics from the point of view of phenomenology, as developed by Edmund Husserl in the later phase2 of his philosophical thinking known as transcendental phenomenology, starting in 1907 with the Logical Investigations and characterized by the introduction of the notions of ‘reduction’. Husserlian transcendental phenomenology as philosophy of mathematics is described as one that ‘cuts across’ (...) different philosophical positions, such as platonism, nominalism, fictionalism, Hilbertian formalism, etc. but, at the same time, as having built in the conceptual tools which allow one not to incur the kinds of problems which are usually related to one's preferred approach. Phenomenology centers around the notion of intentionality3 or aboutness, i.e. the characteristic of acts of cognition of being about something. The ‘something’ a cognitive act is about is called its ‘intentional object’, meaning the object of an intentional act. Any kind of object can be seen as an intentional object irrespective of whether it is concrete, illusory, abstract, etc. Indeed the emphasis is on the intentional act, as it is in the intentional act that the object—which need not be claimed to exist—is present. In order to achieve knowledge of an object the phenomenologist investigates the consciousness of the knowing subject when performing the act directed to that particular object.Characterized in this way, it is not difficult to see that phenomenology has a straightforward bearing on almost everything, philosophy of mathematics included. As Tieszen puts it , ‘[…] our mathematical beliefs are always about something. They are about certain objects, such as numbers, sets, functions or groups […]’. …. (shrink)

Weyl's inclination toward constructivism in the foundations of mathematics runs through his entire career, starting with Das Kontinuum. Why was Weyl inclined toward constructivism? I argue that Weyl's general views on foundations were shaped by a type of transcendental idealism in which it is held that mathematical knowledge must be founded on intuition. Kant and Fichte had an impact on Weyl but HusserFs transcendental idealism was even more influential. I discuss Weyl's views on vicious circularity, existence claims, meaning, the continuum (...) and choice sequences, and the intuitive-symbolic distinction against the background of his transcendental idealism and general intuitionism. (shrink)

Gödel has argued that we can cultivate the intuition or ‘perception’ of abstractconcepts in mathematics and logic. Gödel's ideas about the intuition of conceptsare not incidental to his later philosophical thinking but are related to many otherthemes in his work, and especially to his reflections on the incompleteness theorems.I describe how some of Gödel's claims about the intuition of abstract concepts are related to other themes in his philosophy of mathematics. In most of this paper, however,I focus on a central (...) question that has been raised in the literature on Gödel: what kind of account could be given of the intuition of abstract concepts? I sketch an answer to this question that uses some ideas of a philosopher to whom Gödel also turned in this connection: Edmund Husserl. The answer depends on how we understand the conscious directedness toward ‘objects’ and the meaning of the term ‘abstract’ in the context of a theory of the intentionality of cognition. (shrink)

Michael Dummett has interpreted and expounded upon intuitionism under the influence of Wittgensteinian views on language, meaning and cognition. I argue against the application of some of these views to intuitionism and point to shortcomings in Dummett's approach. The alternative I propose makes use of recent, post-Wittgensteinian views in the philosophy of mind, meaning and language. These views are associated with the claim that human cognition exhibits intentionality and with related ideas in philosophical psychology. Intuitionism holds that mathematical constructions are (...) mental processes or objects. Constructions are, in the first instance, forms of consciousness or possible experience of a particular type. As such, they must be understood in terms of the concept of intentionality. This view has a historical basis in the literature on intuitionism. In a famous 1931 lecture Heyting in fact identifies constructions with fulfilled or fulfillable mathematical intentions. I consider some of the consequences of this identification and contrast them with Dummett's views on intuitionism. (shrink)

Edmund Husserl has argued that we can intuit essences and, moreover, that it is possible to formulate a method for intuiting essences. Husserl calls this method ‘ideation’. In this paper I bring a fresh perspective to bear on these claims by illustrating them in connection with some examples from modern pure geometry. I follow Husserl in describing geometric essences as invariants through different types of free variations and I then link this to the mapping out of geometric invariants in modern (...) mathematics. This view leads naturally to different types of spatial ontologies and it can be used to shed light on Husserl's general claim that there are different ontologies in the eidetic sciences that can be systematically related to one another. The paper is rounded out with a consideration of the role of ideation in the origins of modern geometry, and with a brief discussion of the use of ideation outside of pure geometry. What would be the study that would draw the soul away from the world of becoming to the world of being?… Geometry and arithmetic would be among the studies we are seeking … a philosopher must learn them because he must arise out of the region of generation and lay hold on essence or he can never become a true reckoner … they facilitate the conversion of the soul itself from the world of generation to essence and truth… they are knowledge of that which always is and not of something which at some time comes into being and passes away. (shrink)

Reason and Value collects 15 new papers by leading contemporary philosophers on themes from the work of Joseph Raz. Raz has made major contributions in a wide range of areas, including jurisprudence, political philosophy, and the theory of practical reason; but all of his work displays a deep engagement with central themes in moral philosophy. The subtlety and power of Raz's reflections on ethical topics make his writings a fertile source for anyone working in this area. Especially significant are his (...) explorations of the connections between practical reason and the theory of value, which constitute a sustained and penetrating treatment of a set of issues at the very center of moral philosophy as it is practiced today. The contributors to the volume acknowledge the importance of Raz's contributions by engaging critically with his positions and offering independent perspectives on the topics that he has addressed. The volume aims both to honour Raz's accomplishments in the area of ethical theorizing, and to contribute to an enhanced appreciation of the significance of his work for the subject. (shrink)

This chapter contains section titled: Arithmetic, Geometry, Logic, and the Science of All Possible Sciences Phenomenological Philosophy and the Foundation of the Sciences The Crisis of the Modern Sciences Conclusion.

Offering a collection of fifteen essays that deal with issues at the intersection of phenomenology, logic, and the philosophy of mathematics, this 2005 book is divided into three parts. Part I contains a general essay on Husserl's conception of science and logic, an essay of mathematics and transcendental phenomenology, and an essay on phenomenology and modern pure geometry. Part II is focused on Kurt Godel's interest in phenomenology. It explores Godel's ideas and also some work of Quine, Penelope Maddy and (...) Roger Penrose. Part III deals with elementary, constructive areas of mathematics. These are areas of mathematics that are closer to their origins in simple cognitive activities and in everyday experience. This part of the book contains essays on intuitionism, Hermann Weyl, the notion of constructive proof, Poincaré and Frege. (shrink)

Richard Tieszen presents an analysis, development, and defense of a number of central ideas in Kurt Gödel's writings on the philosophy and foundations of mathematics and logic. Tieszen structures the argument around Gödel's three philosophical heroes - Plato, Leibniz, and Husserl - and his engagement with Kant, and supplements close readings of Gödel's texts on foundations with materials from Gödel's Nachlass and from Hao Wang's discussions with Gödel. He provides discussions of Gödel's views, and develops a new type of platonic (...) rationalism that requires rational intuition, called 'constituted platonism'. (shrink)

This collection of new essays offers a 'state-of-the-art' conspectus of major trends in the philosophy of logic and philosophy of mathematics. A distinguished group of philosophers addresses issues at the centre of contemporary debate: semantic and set-theoretic paradoxes, the set/class distinction, foundations of set theory, mathematical intuition and many others. The volume includes Hilary Putnam's 1995 Alfred Tarski lectures, published here for the first time.

"Intuition" has perhaps been the least understood and the most abused term in philosophy. It is often the term used when one has no plausible explanation for the source of a given belief or opinion. According to some sceptics, it is understood only in terms of what it is not, and it is not any of the better understood means for acquiring knowledge. In mathematics the term has also unfortunately been used in this way. Thus, intuition is sometimes portrayed as (...) if it were the Third Eye, something only mathematical "mystics", like Ramanujan, possess. In mathematics the notion has also been used in a host of other senses: by "intuitive" one might mean informal, or non-rigourous, or visual, or holistic, or incomplete, or perhaps even convincing in spite of lack of proof. My aim in this book is to sweep all of this aside, to argue that there is a perfectly coherent, philosophically respectable notion of mathematical intuition according to which intuition is a condition necessary for mathemati cal knowledge. I shall argue that mathematical intuition is not any special or mysterious kind of faculty, and that it is possible to make progress in the philosophical analysis of this notion. This kind of undertaking has a precedent in the philosophy of Kant. While I shall be mostly developing ideas about intuition due to Edmund Husser! there will be a kind of Kantian argument underlying the entire book. (shrink)

Gödel's relation to the work of Plato, Leibniz, Kant, and Husserl is examined, and a new type of platonic rationalism that requires rational intuition, called ...

From the vantage point of comparative philosophy, this anthology explores how analytic and "Continental" approaches in the Western and other philosophical traditions can constructively engage each other and jointly contribute to the contemporary development of philosophy.

Edmund Husserl has argued that we can intuit essences and, moreover, that it is possible to formulate a method for intuiting essences. Husserl calls this method 'ideation'. In this paper I bring a fresh perspective to bear on these claims by illustrating them in connection with some examples from modern pure geometry. I follow Husserl in describing geometric essences as invariants through different types of free variations and I then link this to the mapping out of geometric invariants in modern (...) mathematics. This view leads naturally to different types of spatial ontologies and it can be used to shed light on Husserl's general claim that there are different ontologies in the eidetic sciences that can be systematically related to one another. The paper is rounded out with a consideration of the role of ideation in the origins of modern geometry, and with a brief discussion of the use of ideation outside of pure geometry. (shrink)

Godel began to seriously study Husserl's phenomenology in 1959, and the Godel Nachlass is known to contain many notes on Husserl. In this paper I describe what is presently known about Godel's interest in phenomenology. Among other things, it appears that the 1963 supplement to "What is Cantor's Continuum Hypothesis?", which contains Godel's famous views on mathematical intuition, may have been influenced by Husserl. I then show how Godel's views on mathematical intuition and objectivity can be readily interpreted in a (...) phenomenological theory of intuition and mathematical knowledge. (shrink)

The thesis is a study of the notion of intuition in the foundations of mathematics which focuses on the case of natural numbers and hereditarily finite sets. Phenomenological considerations are brought to bear on some of the main objections that have been raised to this notion. ;Suppose that a person P knows that S only if S is true, P believes that S, and P's belief that S is produced by a process that gives evidence for it. On a phenomenological (...) view the relevant evidence is provided by intuition , and this should be the case in either ordinary perceptual knowledge or in mathematical knowledge. Intuition is to be understood in terms of fulfillments of intentions. Knowledge is a product of intuition and intention. In the case of mathematical knowledge it is said that there is a construction for a mathematical statement S if and only if the intention expressed by S is fulfilled . Constructions are thus viewed as intuition processes that could actually or possibly be carried out. In elementary parts of mathematics they might be characterized in terms of certain classes of recursive functions. This view is discussed in the case where S is taken to be a singular statement about natural numbers or finite sets, and also where S is taken to be a general statement about such objects. The distinction between intuition of and intuition that is also investigated in this context. ;It is pointed out how on a phenomenological view a number of central problems about mathematical intuition can be avoided: problems about the analogousness of perceptual and mathematical intuition, about causal accounts of knowledge in mathematics, and about structuralism in mathematics. The bearing of the account on issues concerning constructivism and platonism is also discussed. (shrink)

Gödel has argued that we can cultivate the intuition or perception of abstractconcepts in mathematics and logic. Gödel's ideas about the intuition of conceptsare not incidental to his later philosophical thinking but are related to many otherthemes in his work, and especially to his reflections on the incompleteness theorems.I describe how some of Gödel's claims about the intuition of abstract concepts are related to other themes in his philosophy of mathematics. In most of this paper, however,I focus on a central (...) question that has been raised in the literature on Gödel: what kind of account could be given of the intuition of abstract concepts? I sketch an answer to this question that uses some ideas of a philosopher to whom Gödel also turned in this connection: Edmund Husserl. The answer depends on how we understand the conscious directedness toward objects and the meaning of the term abstract in the context of a theory of the intentionality of cognition. (shrink)

In a lecture manuscript written around 1961, Gödel describes a philosophical path from the incompleteness theorems to Husserl's phenomenology. It is known that Gödel began to study Husserl's work in 1959 and that he continued to do so for many years. During the 1960s, for example, he recommended the sixth investigation of Husserl's Logical Investigations to several logicians for its treatment of categorial intuition. While Gödel may not have been satisfied with what he was able to obtain from philosophy and (...) Husserl's phenomenology, he nonetheless continued to recommend Husserl's work to logicians as late as the 1970s. In this paper I present and discuss the kinds of arguments that led Gödel to the work of Husserl. Among other things, this should help to shed additional light on Gödel's philosophical and scientific ideas and to show to what extent these ideas can be viewed as part of a unified philosophical outlook. Some of the arguments that led Gödel to Husserl's work are only hinted at in Gödel's 1961 paper, but they are developed in much more detail in Gödel's earlier philosophical papers. In particular, I focus on arguments concerning Hilbert's program and an early version of Carnap's program.§1. Some ideas from phenomenology. Since Husserl's work is not generally known to mathematical logicians, it may be helpful to mention briefly a few details about his background. (shrink)

Brouwer and Weyl recognized that the intuitive continuum requires a mathematical analysis of a kind that set theory is not able to provide. As an alternative, Brouwer introduced choice sequences. We first describe the features of the intuitive continuum that prompted this development, focusing in particular on the flow of internal time as described in Husserl's phenomenology. Then we look at choice sequences and their logic. Finally, we investigate the differences between Brouwer and Weyl, and argue that Weyl's conception of (...) choice sequences is defective on several counts. (shrink)

Despite All Critique (2014) -- World Politics and Western Reason (1980) -- The Doubled Outsides of the Modern International (2005) -- The Subject of Security (1995) -- The Protection of Nature and the Nature of Protection (2005) -- Social Movements/World Politics (1994) -- Europe is Not Where It is Supposed to Be (2000) -- They Seek it Here, They Seek it There : Looking for Politics in Clayoquot Sound (2003) -- Violence, Modernity, Silence : From Weber to International Relations (1993) (...) -- Hobbes, Origins, Limits (2011) -- War, Terror, Judgement (2002) -- International, Imperial, Exceptional (2005) -- Which Democracy for Which Demos? (2013) -- The Political Theory of Boundaries and the Boundaries of Political Theory : Interview With Raia Prokhovnik (2012). (shrink)

This book focuses upon the perennial question of the existence and nature of an obligation to obey the law. Leading writers have, at one time or another, emphasized considerations such as gratitude, 'divine ordering, ' prudence, contract, autonomy, and utility in seeking to justify, or to deny any justification for, some sort of obligation to obey the positive law. The book provides relevant selections from a sampling of the historical approaches to legal obligation taken by writers such as Plato, Augustine, (...) Aquinas, Hobbes, Locke, Hume, Rousseau, Thoreau, Bentham, Marx and Engels, and Martin Luther King, Jr. These classical discussions are augmented by critical questions and commentary, by independent discussions of the question of legal obligation by a wide range of contemporary writers, and by relevant judicial cases discussing matters such as conscientious objection and civil disobedience. (shrink)