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 These previously unpublished intimate and informal conversations, however, bring to light and amplify Gödel's other major contributions to logic and philosophy.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)

In this first extended treatment of his life and work, Hao Wang, who was in close contact with Godel in his last years, brings out the full subtlety of Godel's ideas and their connection with grand themes in the history of mathematics and ...

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)

Noted logician and philosopher addresses various forms of mathematical logic, discussing both theoretical underpinnings and practical applications. After historical survey, lucid treatment of set theory, model theory, recursion theory and constructivism and proof theory. Place of problems in development of theories of logic, logic’s relationship to computer science, more. Suitable for readers at many levels of mathematical sophistication. 3 appendixes. Bibliography. 1981 edition.

This cogent and knowledgeable critique of the tradition of modern analytic philosophy focuses on the work of its central figures -- Russell, Carnap, and Quine -- and finds it wanting. In its place, Hao Wang unfolds his own original view of what philosophy could and should be. The base of any serious philosophy, he contends, should take as its point of departure the actual state of human knowledge. He explains the relation of this new tradition to mathematical logic and reveals (...) the crucial transitions and mistakes in mainstream Anglo-American philosophy that make a new approach so compelling.Equally at home in philosophy and mathematics, Wang is uniquely qualified to take on the task of critically examining modern philosophy. He carefully traces the path of ideas from Russell and Wittgenstein through the Vienna Circle to modern British and American philosophy, and makes use of his familiarity with the profound thought of Kurt Gödel with whom he has had numerous discussions. He also presents the broader significance of Russell's philosophy, provides a comprehensive and unified treatment of Quine's work in logic and in philosophy, and delineates what is common between Carnap and Quine. (shrink)

I propose to sketch my views on several aspects of the philosophy of mathematics that I take to be especially relevant to philosophy as a whole. The relevance of my discussion would, I think, become more evident, if the reader keeps in mind the function of (the philosophy of) mathematics in philosophy in providing us with more transparent aspects of general issues. I shall consider: (1) three familiar examples; (2) logic and our conceptual frame; (3) communal agreement and objective certainty; (...) (4) the transcommunal universality of mathematics; (5) the big jump to the potential infinite; (6) the reconciliation of local creation with the hypothesis of discovery; (7) Platonism as realism plus conceptualism; (8) foundational studies and mathematical practice; and (9) the decomposition of philosophical disagreements. The views of Gödel and Wittgenstein are emphasized in order to add specificity to the discussions. (shrink)

This essay discusses a number of questions which arise from attempts to reduce the mental to the physical or the mental and the physical to the computational. It makes, in an organized way, several basic distinctions between different kinds of accounts of the mind. It reconstructs and elaborates many discussions between Gödel and the author on the nature of the human mind, with special emphasis on its mathematical capabilities.

Apoptosis proteins play an essential role in regulating a balance between cell proliferation and death. The successful prediction of subcellular localization of apoptosis proteins directly from primary sequence is much benefited to understand programmed cell death and drug discovery. In this paper, by use of Chou’s pseudo amino acid composition , a total of 317 apoptosis proteins are predicted by support vector machine . The jackknife cross-validation is applied to test predictive capability of proposed method. The predictive results show that (...) overall prediction accuracy is 91.1% which is higher than previous methods. Furthermore, another dataset containing 98 apoptosis proteins is examined by proposed method. The overall predicted successful rate is 92.9%. (shrink)

A survey is made of work since 1879 on foundational problems viewed as an analysis, by reduction and formalization, of the concepts proof, feasible, number, set, and constructivity. It is suggested that there are five domains of concepts and methods, viz., anthropologism, finitism, intuitionism, predicativism, and platonism. It is also suggested that the central problem is to characterize these domains by formalization and to determine their interrelations by different forms of reduction. Finally, the range of logic in the narrower sense (...) is discussed, and applications of mathematical logic are briefly outlined.ZusammenfassungDie Entwicklung der Grundlagenforschung seit dem Jahre 1879 wird hier im Überblick dargestellt und zwar vom Standpunkt einer Analyse durch Reduktion and Formalisierung der Begriffe Beweis, Zahl, Menge und Konstruktivität aus gesehen. Fünf Bereiche von Begriffsbildungen und Methoden werden zu Grunde gelegt, nämlich: Anthropologismus, Finitismus, Intuitionismus, Prädikativismus und Platonismus. Als zentrale Aufgabenstellung wird hier die Charakterisierung dieser Bereiche durch Formalisierung betrachtet und weiterhin die Bestimmung ihrer gegenseitigen Zusammenhänge durch verschiedene Arten von Reduktionen. Schliesslich wird das Gebiet der Logik im engeren Sinne diskutiert und Anwendungen der mathematischen Logik skizziert.RésuméL'article réexamine les travaux faits depuis 1879 sur les problèmes du fondement des mathématiques, que P'on considère comme des analyses par réduction et formalisation des concepts de démonstration, de nombre, d'ensemble, et de constructivité. On suggère qu'il existe cinq domaines de concepts et de méthodes, à savoir: anthropologisme, finitisme, intuitionisme, « predicativism » et platonisme. On suggère aussi que le problème essentiel consiste à caractériser ces domaines par formalisation et à déterminer leurs relations réciproques par différentes formes de réduction. Finalement, on étudie la portée de la logique prise dans son sens le plus étroit et on esquisse rapidement les applications de la logique mathématique. (shrink)