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)

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 ...

"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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

Gödel has argued that we can cultivate the intuition or 'perception' of abstract concepts in mathematics and logic. Gödel's ideas about the intuition of concepts are not incidental to his later philosophical thinking but are related to many other themes 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 his 1951 Gibbs Lecture Gödel formulates the central implication of the incompleteness theorems as a disjunction: either the human mind infinitely surpasses the powers of any finite machine or there exist absolutely unsolvable diophantine problems (of a certain type). In his later writings in particular Gödel favors the view that the human mind does infinitely surpass the powers of any finite machine and there are no absolutely unsolvable diophantine problems. I consider how one might defend such a view in (...) light of Gödel's remark that one can turn to ideas in Husserlian transcendental phenomenology to show that the human mind ‘contains an element totally different from a finite combinatorial mechanism’. (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)

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.

In 1928 Edmund Husserl wrote that “The ideal of the future is essentially that of phenomenologically based (“philosophical”) sciences, in unitary relation to an absolute theory of monads” (“Phenomenology”, Encyclopedia Britannica draft) There are references to phenomenological monadology in various writings of Husserl. Kurt Gödel began to study Husserl’s work in 1959. On the basis of his later discussions with Gödel, Hao Wang tells us that “Gödel’s own main aim in philosophy was to develop metaphysics—specifically, something like the monadology of (...) Leibniz transformed into exact theory—with the help of phenomenology.” (A Logical Journey: From Gödel to Philosophy, p. 166) In the Cartesian Meditations and other works Husserl identifies ‘monads’ (in his sense) with ‘transcendental egos in their full concreteness’. In this paper I explore some prospects for a Gödelian monadology that result from this identification, with reference to texts of Gödel and to aspects of Leibniz’s original monadology. (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 in variants 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)

In this paper I contrast Husserlian transcendental eidetic phenomenology with some other views of what phenomenology is supposed to be and argue that, as eidetic, it does not admit of being ‘naturalized’ in accordance with standard accounts of naturalization. The paper indicates what some of the eidetic results in phenomenology are and it links these to the employment of reason in philosophical investigation, as distinct from introspection, emotion or empirical observation. Eidetic phenomenology, unlike cognitive science, should issue in a ‘logic’ (...) of consciousness. Instead of being derived from empirical investigations its results should consist of high-level background conditions that are necessary for cognitive science to be possible in the first place. To negate these conditions is to be faced with certain types of ‘material’ contradictions. Some analogies with science – mathematical science – are used to develop the argument. (shrink)

This paper provides examples in arithmetic of the account of rational intuition and evidence developed in my book After Gödel: Platonism and Rationalism in Mathematics and Logic . The paper supplements the book but can be read independently of it. It starts with some simple examples of problem-solving in arithmetic practice and proceeds to general phenomenological conditions that make such problem-solving possible. In proceeding from elementary ‘authentic’ parts of arithmetic to axiomatic formal arithmetic, the paper exhibits some elements of the (...) genetic analysis of arithmetic knowledge that is called for in Husserl’s philosophy. This issues in an elaboration on a number of Gödel’s remarks about the meaning of his incompleteness theorems for the notion of evidence in mathematics. (shrink)

In this paper I argue that it is more difficult to see how Godel's incompleteness theorems and related consistency proofs for formal systems are consistent with the views of formalists, mechanists and traditional intuitionists than it is to see how they are consistent with a particular form of mathematical realism. If the incompleteness theorems and consistency proofs are better explained by this form of realism then we can also see how there is room for skepticism about Church's Thesis and the (...) claim that minds are machines. (shrink)

In this paper the reader is asked to engage in some simple problem-solving in classical pure number theory and to then describe, on the basis of a series of questions, what it is like to solve the problems. In the recent philosophy of mind this “what is it like” question is one way of signaling a turn to phenomenological description. The description of what it is like to solve the problems in this paper, it is argued, leads to several morals (...) about the epistemology and ontology of classical pure mathematical practice. Instead of simply making philosophical judgments about the subject matter in advance, the exercise asks the reader to briefly engage in a mathematical practice and to then reflect on the practice. (shrink)

(No abstract is available for this citation).

Although there is no consensus on what distinguishes analytic from Continental philosophy, I focus in this paper on one source of disagreement that seems to run fairly deep in dividing these traditions in recent times, namely, disagreement about the relation of natural science to philosophy. I consider some of the exchanges about science that have taken place between analytic and Continental philosophers, especially in connection with the philosophy of mind. In discussing the relation of natural science to philosophy I employ (...) an analysis of the origins of natural science that has been developed by a number of Continental philosophers. Awareness and investigation of interactions between analytic and Continental philosophers on science, it is argued, might help to foster further constructive engagement between the traditions. In the last section of the paper I briefly discuss the place of natural science in relation to global philosophy on the basis of what we can learn from analytic/Continental exchanges. (shrink)

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.

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)

Kurt Gödel began to study the philosophy of Edmund Husserl in 1959. In this paper I present an overview of central themes in Gödel’s study of Husserl’s phenomenology. Since many of Gödel’s ideas concerning Husserl were never put into a systematic form by Gödel himself, I quote fairly extensively in the paper from several sources in order to inform the reader of the nature of Gödel’s interest in Husserl. Gödel prepared one manuscript specifically on Husserl, as we will see below, (...) and many of Gödel’s comments on Husserl are included in the books of Hao Wang. I will also quote some relevant texts from the Gödel Nachlass. In accordance with these various sources, I provide a brief overview in a later section of the paper of Gödel’s interest in eidetic transcendental phenomenology as a new type of monadology. The relationship of Gödel’s incompleteness theorems to Husserl’s notion of ‘definite’ axiom systems is also discussed briefly. (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)

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)