Godel's Theorem in Focus

Godel's Theorem in Focus

Godel's Theorem in Focus

Godel's Theorem in Focus

Synopsis

A layman's guide to the mechanics of Gouml;del's proof together with a lucid discussion of the issues which it raises. Includes an essay discussing the significance of Gouml;del's work in the light of Wittgenstein's criticisms.

Excerpt

An ever-growing number of sub-disciplines in philosophy—ranging from the philosophies of science and language to the philosophy of mind and aesthetics—now demand a working acquaintance with Kurt Gödel’s incompleteness theorems from their students. For Gödel’s theorems raise issues which lie at the very heart of modem attempts to revitalise metaphysics and/or the Mechanist Thesis. Unfortunately, given the highly technical nature of Gödel’s proof these debates have remained relatively inaccessible to those not trained in mathematical logic. The present book has been designed to meet these needs by providing a lucid introduction to the mechanics and mathematical import of Gödel’s proof. We begin with a short biographical sketch of Kurt Gödel by John W. Dawson, Jr., followed by Stephen Kleene’s overview of Gödel’s work in mathematical logic. With this background in place we then address the mounting controversy in the philosophy of mathematics surrounding the philosophical significance of Gödel’s theorems.

Some will no doubt regard the latter phenomenon as a reflection of the inevitable time-lag between scientific and mathematical discoveries versus philosophical comprehension. Perhaps it will even be seen to corroborate the increasingly popular thesis in the sociology of knowledge that it is only once a science has digested the full implications of a breakthrough that it becomes the property of philosophers to exaggerate and distort. But unlike standard mathematical results Gödel’s theorems are inextricably linked to the epistemological disputes which they have sparked off; indeed, nowhere could this be more evident than in the writings of Gödel himself. It is not surprising, therefore, that Gödel’s theorems should present us with a catalogue of philosophical problems, many of which we are only just beginning to recognise, let alone resolve.

To begin with there is the anomalous reception of Gödel’s first incompleteness theorem; in the words of John W. Dawson, Jr., how was it that ‘one of the most profound discoveries in the history of logic and mathematics was assimilated promptly and almost without objection by Gödel’s contemporaries’? As Solomon Feferman shows us, this issue is intimately connected with the larger question of how Gödel’s subsequent work in . . .

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