This question as to whether machines can, or could, be made to think, has become familiar in recent years since the renewed outburst of interest that has taken place in the development of Cybernetics. The notion of servo–mechanisms and the like has a history in remote antiquity but the form of its fundamental question has recently taken on a new and especially acute significance.

I Would like to draw attention to the basic defect in the argument used by Mr J. R. Lucas.Mr Lucas there states that Gödel's theorem shows that any consistent formal system strong enough to produce arithmetic fails to prove, within its own structure, theorems that we, as humans, can nevertheless see to be true. From this he argues that ‘minds’ can do more than machines, since machines are essentially formal systems of this same type, and subject to the limitation implied (...) by Godel's theorem. (shrink)

The March 2002 symposium Human Dignity and Reproductive Technology brought together philosophers, theologians, scientists, lawyers, and scholars from across the United States. The essays of this book are the contributions of the symposium's participants.

The purpose of this valuable book is to consider recent cultural trends in bioethics from a Catholic perspective. Bioethics is intended for a lay audience interested in understanding bioethical issues from a Catholic perspective.

I would like to make some further clarifying remarks about the nature of learning machines, or finite automata as they are more generally known these days. It is clear from much that has recently been written on this subject that there are still many misunderstandings about their capacity and significance.

Professor A. D. Ritchie's remarks cannot go without some reply, since otherwise they would only have the effect of increasing the already considerable confusion on the subject of Cybernetics.

In an interesting work ‘The Ethical Animal’ Professor C. H. Waddington valiantly attempts to bridge the gap between ‘ought’ and ‘is’ without, it seems, succeeding in doing so. Notwithstanding his erudition, honesty of purpose and charm in exposition, the gulf remains unbridged. Indeed there are passages where it is difficult to be certain whether the author considers that he has bridged it or even what standpoint he finally adopts.

I Would like to draw attention to the basic defect in the argument used by Mr J. R. Lucas.Mr Lucas there states that Gödel's theorem shows that any consistent formal system strong enough to produce arithmetic fails to prove, within its own structure, theorems that we, as humans, can nevertheless see to be true. From this he argues that ‘minds’ can do more than machines, since machines are essentially formal systems of this same type, and subject to the limitation implied (...) by Godel's theorem. (shrink)

Two contradictions are in contradiction. One is tragic, or insurmountable, contradiction: two equally founded demands oppose each other without any possibility of reconciliation (e.g., Antigone's conflict). The other is dialectical contradiction, which guarantees its own overcoming. Hegel's use of the German word Aufhebung, which means both preservation and suppression, is well known. Translators have proposed equivalents such as “replacement” or “sublimation.” I would suggest incorporation, indeed, digestion. Such a translation is not without consequences because it prompts one to think that (...) even if the dialectic has a strong stomach, its functioning is not free of aftereffects: one has to wonder if there is such a thing as dialectical waste. (shrink)

Artificial intelligence and the interrogation game; Scientific method and explanation; Godel's incompleteness theorem; Determinism and uncertainty; Axioms, theorems and formalisation; Creativity; Consciousness and free will; Pragmatics; A ...

This book presents the foundations of a general theory of algebras. Often called “universal algebra”, this theory provides a common framework for all algebraic systems, including groups, rings, modules, fields, and lattices. Each chapter is replete with useful illustrations and exercises that solidify the reader's understanding. The book begins by developing the main concepts and working tools of algebras and lattices, and continues with examples of classical algebraic systems like groups, semigroups, monoids, and categories. The essence of the book lies (...) in Chapter 4, which provides not only basic concepts and results of general algebra, but also the perspectives and intuitions shared by practitioners of the field. The book finishes with a study of possible uniqueness of factorizations of an algebra into a direct product of directly indecomposable algebras. There is enough material in this text for a two semester course sequence, but a one semester course could also focus primarily on Chapter 4, with additional topics selected from throughout the text. (shrink)