This is a wide-ranging 2004 book about arguments for and against beliefs in God. The arguments for the belief are analysed in the first six chapters and include ontological arguments from Anselm to Gödel, the cosmological arguments of Aquinas and Leibniz, and arguments from evidence for design and miracles. The next two chapters consider arguments against belief. The last chapter examines Pascalian arguments for and against belief in God. There are discussions of Cantorian problems for omniscience, of challenges to divine (...) omnipotence, and of the compatibility of everlasting complete knowledge of the world with free-will. There are appendices that present formal proofs in a system for quantified modal logic, a theory of possible worlds, notes on Cantorian set theory, and remarks concerning non-standard hyperreal numbers. This book will be a valuable resource for philosophers of religion and theologians and will interest logicians and mathematicians as well. (shrink)

"Kant and the Capacity to Judge" will prove to be an important and influential event in Kant studies and in philosophy.

In this collection of essays written over a period of twenty years, Solomon Feferman explains advanced results in modern logic and employs them to cast light on significant problems in the foundations of mathematics. Most troubling among these is the revolutionary way in which Georg Cantor elaborated the nature of the infinite, and in doing so helped transform the face of twentieth-century mathematics. Feferman details the development of Cantorian concepts and the foundational difficulties they engendered. He argues that the freedom (...) provided by Cantorian set theory was purchased at a heavy philosophical price, namely adherence to a form of mathematical platonism that is difficult to support. -/- Beginning with a previously unpublished lecture for a general audience, Deciding the Undecidable, Feferman examines the famous list of twenty-three mathematical problems posed by David Hilbert, concentrating on three problems that have most to do with logic. Other chapters are devoted to the work and thought of Kurt Gödel, whose stunning results in the 1930s on the incompleteness of formal systems and the consistency of Cantors continuum hypothesis have been of utmost importance to all subsequent work in logic. Though Gödel has been identified as the leading defender of set-theoretical platonism, surprisingly even he at one point regarded it as unacceptable. -/- In his concluding chapters, Feferman uses tools from the special part of logic called proof theory to explain how the vast part--if not all--of scientifically applicable mathematics can be justified on the basis of purely arithmetical principles. At least to that extent, the question raised in two of the essays of the volume, Is Cantor Necessary?, is answered with a resounding no. -/- This volume of important and influential work by one of the leading figures in logic and the foundations of mathematics is essential reading for anyone interested in these subjects. (shrink)

Kurt Gödel’s version of the ontological argument was shown by J. Howard Sobel to be defective, but some plausible modifications in the argument result in a version which is immune to Sobel’s objection. A definition is suggested which permits the proof of some of Godel’s axioms.

In his text ‘The modern development of the foundations of mathematics in the light of philosophy’ from around 1961, Gödel announces a turn to Husserl's phenomenology to find the foundations of mathematics. In Gödel's archive there are two draft letters that shed some further light on the exact strategy that he formulated for himself in the early 1960s. Transcriptions of these letters are presented, together with some comments.

Kurt Godel was the most outstanding logician of the twentieth century, famous for his work on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum hypothesis. He is also noted for his work on constructivity, the decision problem, and the foundations of computation theory, as well as for the strong individuality of his writings on the philosophy of mathematics. Less well-known is his discovery of unusual cosmological models for Einstein's (...) equations, permitting "time-travel" into the past. This second volume of a comprehensive edition of Godel's works collects together all his publications from 1938 to 1974. Together with Volume I (Publications 1929-1936), it makes available for the first time in a single source all of his previously published work. Continuing the format established in the earlier volume, the present text includes introductory notes that provide extensive explanatory and historical commentary on each of the papers, a facing English translation of the one German original, and a complete bibliography. Succeeding volumes are to contain unpublished manuscripts, lectures, correspondence, and extracts from the notebooks. Collected Works is designed to be accessible and useful to as wide an audience as possible without sacrificing scientific or historical accuracy. The only complete edition available in English, it will be an essential part of the working library of professionals and students in logic, mathematics, philosophy, history of science, and computer science. These volumes will also interest scientists and all others who wish to be acquainted with one of the great minds of the twentieth century. (shrink)

In this new book, Michael Friedman argues that Kant's continuing efforts to find a metaphysics that could provide a foundation for the sciences is of the utmost ...

It is by now well known that Gödel first advocated the philosophy of Leibniz and then, since 1959, that of Husserl. This raises three questions:1.How is this turn to Husserl to be interpreted? Is it a dismissal of the Leibnizian philosophy, or a different way to achieve similar goals?2.Why did Gödel turn specifically to the later Husserl's transcendental idealism?3.Is there any detectable influence from Husserl on Gödel's writings?Regarding the first question, Wang [96, p.165] reports that Gödel ‘[saw] in Husserl's work (...) a method of refining and consolidating Leibniz' monadology’. But what does this mean? In what for Gödel relevant sense is Husserl's work a refinement and consolidation of Leibniz' monadology?The second question is particularly pressing, given that Gödel was, by his own admission, a realist in mathematics since 1925. Wouldn't the uncompromising realism of the early Husserl's Logical investigations have been a more obvious choice for a Platonist like Gödel?The third question can only be approached when an answer to the second has been given, and we want to suggest that the answer to the first question follows from the answer to the second. We begin, therefore, with a closer look at the actual turn towards phenomenology.Some 30 years before his serious study of Husserl began, Gödel was well aware of the existence of phenomenology. Apart from its likely appearance in the philosophy courses that Gödel took, it reached him from various directions. (shrink)

The best known and most widely discussed aspect of Kurt Gödel's philosophy of mathematics is undoubtedly his robust realism or platonism about mathematical objects and mathematical knowledge. This has scandalized many philosophers but probably has done so less in recent years than earlier. Bertrand Russell's report in his autobiography of one or more encounters with Gödel is well known:Gödel turned out to be an unadulterated Platonist, and apparently believed that an eternal “not” was laid up in heaven, where virtuous logicians (...) might hope to meet it hereafter.On this Gödel commented:Concerning my “unadulterated” Platonism, it is no more unadulterated than Russell's own in 1921 when in the Introduction to Mathematical Philosophy … he said, “Logic is concerned with the real world just as truly as zoology, though with its more abstract and general features.” At that time evidently Russell had met the “not” even in this world, but later on under the infuence of Wittgenstein he chose to overlook it.One of the tasks I shall undertake here is to say something about what Gödel's platonism is and why he held it.A feature of Gödel's view is the manner in which he connects it with a strong conception of mathematical intuition, strong in the sense that it appears to be a basic epistemological factor in knowledge of highly abstract mathematics, in particular higher set theory. Other defenders of intuition in the foundations of mathematics, such as Brouwer and the traditional intuitionists, have a much more modest conception of what mathematical intuition will accomplish. (shrink)

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 modal ontological argument is the centerpiece of an extensive examination of intensional logic. First, classical type theory is presented semantically, tableau rules for it are introduced, and the Prawitz/Takahashi completeness proof is given. Then modal machinery is added to produce a modified version of Montague/Gallin intensional logic. Finally, various ontological proofs for the existence of God are discussed informally, and the Gödel argument is fully formalized. Parts of the book are mathematical, parts philosophical.

The paper is devoted to the discussion of some philosophical and historical problems connected with the theorem on the undefinability of the notion of truth. In particular the problem of the priority of proving this theorem will be considered. It is claimed that Tarski obtained this theorem independently though he made clear his indebtedness to Gödel’s methods. On the other hand, Gödel was aware of the formal undefinability of truth in 1931, but he did not publish this result. Reasons for (...) that are also considered. (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)

We describe a KB Gödelian ontological system, and some other weak systems, in a fully formal way using theory of types and natural deduction, and present a completeness proof in its main and specific parts. We technically and philosophically analyze and comment on the systems (mainly with respect to the relativism of values) and include a sketch of some connected aspects of Gödel's relation to Kant.

Hao Wang was one of the few confidants of the great mathematician and logician Kurt Gödel. _A Logical Journey_ is a continuation of Wang's _Reflections on Gödel_ and also elaborates on discussions contained in _From Mathematics to Philosophy_. A decade in preparation, it contains important and unfamiliar insights into Gödel's views on a wide range of issues, from Platonism and the nature of logic, to minds and machines, the existence of God, and positivism and phenomenology. The impact of Gödel's theorem (...) on twentieth-century thought is on par with that of Einstein's theory of relativity, Heisenberg's uncertainty principle, or Keynesian economics. These previously unpublished intimate and informal conversations, however, bring to light and amplify Gödel's other major contributions to logic and philosophy. They reveal that there is much more in Gödel's philosophy of mathematics than is commonly believed, and more in his philosophy than his philosophy of mathematics. Wang writes that "it is even possible that his quite informal and loosely structured conversations with me, which I am freely using in this book, will turn out to be the fullest existing expression of the diverse components of his inadequately articulated general philosophy." The first two chapters are devoted to Gödel's life and mental development. In the chapters that follow, Wang illustrates the quest for overarching solutions and grand unifications of knowledge and action in Gödel's written speculations on God and an afterlife. He gives the background and a chronological summary of the conversations, considers Gödel's comments on philosophies and philosophers, and his attempt to demonstrate the superiority of the mind's power over brains and machines. Three chapters are tied together by what Wang perceives to be Gödel's governing ideal of philosophy: an exact theory in which mathematics and Newtonian physics serve as a model for philosophy or metaphysics. Finally, in an epilog Wang sketches his own approach to philosophy in contrast to his interpretation of Gödel's outlook. (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 his text 'The modern development of the foundations of mathematics in the light of philosophy' from around 1961, Go¨del announces a turn to Husserl's phenomenology to find the foundations of mathematics. In Go¨del's archive there are two draft letters that shed some further light on the exact strategy that he formulated for himself in the early 1960s. Transcriptions of these letters are presented, together with some comments.

One of the cornerstone books of Western philosophy, Critique of Pure Reason is Kant's seminal treatise, where he seeks to define the nature of reason itself and builds his own unique system of philosophical thought with an approach known as transcendental idealism. He argues that human knowledge is limited by the capacity for perception and attempts a logical designation of two varieties of knowledge: a posteriori, the knowledge acquired through experience; and a priori, knowledge not derived through experience. This accurate (...) translation by J. M. D. Meiklejohn offers a simple and direct rendering of Kant's work that is suitable for readers at all levels. (shrink)

Skeptical problems arising for Kant's version of transcendental idealism have been raised from Kant's own time to the present day. By focussing on how such problems originally arose in the wake of Kant's work, and on the first formulations of absolute idealism by Schelling, I argue that the skeptical problems in question ultimately depend on fundamental features of Kant's philosophy of natural science. As a result, Naturphilosophie and the organic conception of nature cannot easily be separated from the deep and (...) insightful response to these problems offered by absolute idealism. (shrink)

This is a very important book. It has already become required reading for researchers on the relation between the exact sciences and Kant’s philosophy. The main theme is that Kant’s continuing program to find a metaphysics that could provide a foundation for the science of his day is of crucial importance to understanding the development of his philosophical thought from its earliest precritical beginnings in the thesis of 1747, right through the highwater years of the critical philosophy, to his last (...) unpublished writings in the Opus postumum. In the course of articulating this theme, Friedman has made extensive use of detailed historical information about their scientific and mathematical background to illuminate Kant’s texts. Over and over again, such information is used to suggest interesting and quite subtle interpretations for texts that may have seemed puzzling or just wrong-headed. (shrink)

The essay centers on Gödel's views on the place of our intuitive concept of time in philosophy and in physics. It presents my interpretation of his work on the theory of relativity, his observations on the relationship between Einstein's theory and Kantian philosophy, as well as some of the scattered remarks in his conversations with me in the seventies — namely, those on the philosophies of Leibniz, Hegel and Husserl — as a successor of Kant — in relation to their (...) conceptions of time. (shrink)

It is a widely known but little considered fact that Albert Einstein and Kurt Gödel were best friends for the last decade and a half of Einstein's life. The two walked home together from Princeton's Institute for Advanced Study every day; they shared ideas about physics, philosophy, politics, and the lost world of German science in which they had grown up. By 1949, Gödel had produced a remarkable proof: In any universe described by the Theory of Relativity, time cannot exist (...) . Einstein endorsed this result-reluctantly, since it decisively overthrew the classical world-view to which he was committed. But he could find no way to refute it, and in the half-century since then, neither has anyone else. Even more remarkable than this stunning discovery, however, was what happened afterward: nothing. Cosmologists and philosophers alike have proceeded with their work as if Gödel's proof never existed -one of the greatest scandals of modern intellectual history. A World Without Time is a sweeping, ambitious book, and yet poignant and intimate. It tells the story of two magnificent minds put on the shelf by the scientific fashions of their day, and attempts to rescue from undeserved obscurity the brilliant work they did together. (shrink)

Various parallels between Kant's critical program and Hilbert's formalistic program for the philosophy of mathematics are considered.

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)

This is an expansion of the author's 1991 work which investigates the implications of Gödel's writings on Einstein's theory of relativity as they relate to the fundamental questions of the nature of time and the possibilities for time travel.

It is unfortunate for the philosophical community generally, and for those philosophers who pursue various versions of idealism in particular, that a logician of Kurt Goedel’s genius published very little of non-mathematical philosophical interest. Amongst his unpublished papers at Princeton there are, however, several versions of a paper he wrote on the relevance of contemporary relativity to the philosophy of Kant. The purpose of the present paper is to give a partial exposition and defence of Goedel’s view that contemporary relativity (...) supports a neo-Kantian idealistic position with regard to time. (shrink)