The Limits of Mathematics: A course on information theory and the limits of formal reasoning (Discrete Mathematics and Theoretical Computer Science) 2nd Printing Edition

by Gregory J. Chaitin (Author)

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ISBN-13: 978-9813083592

ISBN-10: 981308359X

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The LIMITS of MATHEMATICS: A Course on Information Theory and the Limits of Formal Reasoning (Discrete Mathematics and Theoretical Computer Science)
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This book is the final version of a course on algorithmic information theory and the epistemology of mathematics and physics. It discusses Einstein and Goedel's views on the nature of mathematics in the light of information theory, and sustains the thesis that mathematics is quasi-empirical. There is a foreword by Cris Calude of the University of Auckland, and supplementary material is available at the author's web site. The special feature of this book is that it presents a new "hands on" didatic approach using LISP and Mathematica software. The reader will be able to derive an understanding of the close relationship between mathematics and physics. "The Limits of Mathematics is a very personal and idiosyncratic account of Greg Chaitin's entire career in developing algorithmic information theory. The combination of the edited transcripts of his three introductory lectures maintains all the energy and content of the oral presentations, while the material on AIT itself gives a full explanation of how to implement Greg's ideas on real computers for those who want to try their hand at furthering the theory." John Casti, Santa Fe Institute

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Editorial Reviews

Review

"...a very personal and idiosyncratic account of Greg Chaitins entire career in developing algorithmic information theory. The combination of the edited transcripts of his three introductory lectures maintains all the energy and content of the oral presentations, while the material on AIT itself gives a full explanation of how to implement Gregs ideas on real computers ..." John Casti, Santa Fe Institute

Product details

Publisher : Springer; 2nd Printing edition (November 14, 1997)
Language: : English
Hardcover : 148 pages
ISBN-10 : 981308359X
ISBN-13 : 978-9813083592
Item Weight : 1.25 pounds
Dimensions : 9.58 x 6.47 x 0.58 inches

Best Sellers Rank: #1,638,438 in Books (See Top 100 in Books)
- #283 in Counting & Numeration
- #935 in Mathematical Logic
- #3,113 in AI & Machine Learning

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Cellier Cedric

Rather a collection of introductory speeches than a course

Reviewed in the United States on February 9, 2011

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The book is composed of a small set of speeches given at various occasions to maths students.
Each speech share the same topic and is build almost according to the same plan, and merely
introduces the research area of G. Chaitin from a very high standpoint, without giving any details.

So this can hardly be considered as a course.

The book is very frustrating because at several places the author jump over many expected details because he says he lacks time, yet in the following speech he again skip over details because he still lacks time. One would rather have an actual course that's diging from chapter to chapter into the core of the matter, instead of having to repeatedly read an introduction.

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Gary L. Herstein

A subjective estimation

Reviewed in the United States on March 3, 2007

I wish that I could give this book a higher rating than I have, for the subject matter is one that I find of enormous intrinsic interest. Moreover, Dr. Chaitin is one of the most important contributors to this field of the last 30+ years.

My reasons for being disappointed in this book may well be the reasons others enthusiastically endorse it. Dr. Chaitin himself, in his preface, places this volume as the one thing he would most wish to save were a disaster to wipe out the rest of his oeuvre.

The sub-title of the book is "A Course on Information Theory and the Limits of Formal Reasoning." This sub-heading I find to be quite misleading. The book is not a "course" on any thing -- rather, it is a collection of a very small number of informal papers that Dr. Chaitin has given in recent years, and a very large number of pages devoted to LISP programs that can be used to demonstrate aspects of his extensions to the results of Turing and Goedel. The collection of articles seem largely redundant to me; any one of the articles by itself would be sufficient to summarize the rest of the book's contents. As for the programming, that should either have been provided in the form of a CD-ROM (as only someone of a genuinely "special" nature would actually sit down and manually type in all those instructions) or a functioning URL (such URL's as do appear in the book do not seem to be working as of this writing, Mar. 2007).

I was hoping to get something more comprehenive, and that could function as a stand-alone text. This book seems to be neither. The technical details are all to be found elsewhere, and the functional aspects that might translate into an actual course of study are simply not to be found at all. Dr. Chaitin notes that the original technical work of his, published in the 60's, had a formal error that has since been corrected. Quite frankly, I would rather have that work plus a footnote regarding the later developments, than this volume which (sadly) I find of no real help. (I have since ordered and received a used, 1987 imprint of his "Algorithmic Information Theory" as printed by Cambridge.)

Alternatively, and perhaps more importantly, I would very much liked to have seen this "course" developed as a *COURSE*, rather than as three more or less popular, and largely independent, lectures. These lectures seem, at best, only to minimally build upon one another. A more integrated and coherent work that developed its subject in a step-wise manner, rather than repeating itself with only slightly different glosses, is something that I would have liked much more.

My background in logic and computer science, while not trivial, remains that of a studious and committed autodidact. It is possible that someone with less of a background in topics of formal reasoning than myself would find this book of enormous value. For me, however, it lacked both the technical details to make it a worthy struggle, and the pedagogical depth to make it of significant value.

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Dr. T.

Spielt Gott Würfel mit der Mathematik?

Reviewed in Germany on July 27, 2016

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Das vorliegende Bändchen enthält drei Vorlesungen von Gregory J. Chaitin zu den Grenzen der Mathematik, die sich aus der algorithmischen Informationstheorie ergeben, zu der der Autor, aufbauend auf Ideen von Ray Solomonoff und Andrei Kolmogorov, wesentliche Beiträge geleistet hat. Die erste Arbeit “Randomness in Arithmetics...“ basiert auf einem Kolloquiumvortrag, in dem Chaitin seine grundlegenden theoretischen Ideen allgemein verständlich entwickelt. In den beiden folgenden Kursen zeigt der Autor, wie man LISP einsetzt, um die Algorithmen seiner Theorie zu modellieren.

David Hilbert entwickelte rund um 1900 sein Programm zur Formalisierung der gesamten Mathematik, damit sollten, die sich rasant entwickelten neuen Zweige der Mathematik, auf soliden Grund gestellt werden, der garantiert, dass die Theorien frei von Widersprüchen und vollständig sind. Nach vielversprechenden Entwicklungen von Zermelo, Faenkel, Russel, Withehead, v. Neumann u.a., erschütterten Kurt Gödel 1931 und Alan Turing 1936 dieses Weltbild mit der Entdeckung ihrer Unvollständigkeits- und Nicht- Berechenbarkeits- Resultate. Turings negative Antwort auf die Frage, ob man algorithmisch entscheiden kann, ob eine gegebenes Programm terminiert oder endlos weiterläuft, dem Halteproblem, ist Chaitins Ausgangspunkt zur Konstruktion der Haltewahrscheinlichkeit Omega, einer wohldefinierte reellen Zahl, die zugleich der Inbegriff der Unberechenbarkeit ist – nicht nur, dass ihre Bits ab einer gewissen Stelle alle unberechenbar sind, Omega ist auch eine algorithmische Zufallszahl, d.h. algorithmisch irreduzibel, in den Anfangsbitfolgen von Omega sind schließlich die Antworten auf alle möglich mathematischen Probleme kodiert.

Der Autor spricht metaphorische davon, dass die 0en und 1en von Omega für die Mathematik wie eine steinerne Mauer sind, hier erlahmt jegliche weitere Argumentation. Die Quantenphysik gab Anlass, entgegen Einsteins Überzeugung, festzustellen, dass Gott doch Würfeln mit dem Universum spielt, so scheinen die hier skizzierten Ergebnisse anzudeuten, dass ER dass auch – in gewissen Bereichen – mit der Mathematik tut.

Im zweiten Paper zeigt Chaitin, dass es möglich ist, Unvollständigkeits- Resultate auch ohne rekursive Funktionen, Turing Maschinen oder Fixpunkt Sätze ableitbar sind, einfach in Rahmen der real existierenden Programmiersprache LISP, und dass die so entwickelten Algorithmen auf heutigen Computern ausgeführt werden können. Dazu betrachtet der Autor den harmlos aussehenden Begriff des Eleganten LISP Programms, d.h. eines Programms, zu dem es kein kürzeres Programm gibt, dass den gleichen Output erzeugt; trivialer Weise gibt es zu jedem LISP Programm mindestens ein zugehöriges elegantes LISP Programm. Chaitin zeigt aber – zunächst hand waving –, dass man algorithmisch nicht entscheiden kann, ob ein gegebenes Programm mit einer bestimmten Mindestlänge elegant ist oder nicht. Schließlich zeigt der Autor, dass man diesen Plan tatsächlich auch als LISP Programm realisieren kann, dazu er das Standard LISP geringfügig um eine eval- Funktionsvariante erweitern, die die Angabe eines runtime- Limits erlaubt, und die der Autor 'try' nannte. In der Tat ist das Resultat über Elegante List Programme eine Variante von Chaitins Unvollständigkeitssatz, aus dem Gödels erster Unvollständigkeitssatz als Korollar folgt.

Im dritte Beitrag 'An invitation to algorithmic information theory' wird Omega in Termen von LISP Algorithmen untersucht und diskutiert, insbesondere wird ein Beweis, dass Omega der algorithmischen Irreduzibel ist, gegeben.

Der vierte Teil stellt den Source Code verschiedener LISP Programme zusammen, über die der Autor bei seinen Erörterungen immer wieder gesprochen hat. Der Anhang enthält den Mathematica Code des Interpreters der LISP Variante des Autors.

Das Buch präsentiert eine exzellente kurze Einführung in dieses interessante Thema aus erster Hand, dem Autor gelingt es, die Kernideen allgemein verständlich zu skizzieren; schon auf Grund der Kürze des Textes, bleiben Details natürlich ausgelassen. Der interessierte Leser findet etwa in dem einführenden Buch 'Grenzen der Mathematik' von Dirk Hoffmann auch ein Kapitel über Chaitins Theorie mit formalen Definitionen und Beweisen. Ein schönes Interview des Autor mit dem argentinischen Mathematiker und Schriftsteller Guillermo Martinez über das vorliegende Buch findet sich in dessen Essayband “Borges and Mathematics“. Die im Buch angegeben Links zu den Programm- Sourcen sind veraltet, eine naheliegende Internet Suche fördert aber einen Github Server mit den genannten Quellcodes zu Tage.

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Giovanni M. Mazza

A Text to do mathematics as mathematician.

Reviewed in Italy on November 22, 2014

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Text very valid about mathematics and expositive form, very useful the examples of using of LISP.
Testo molto valido per i contenuti e la forma espositiva, molto utili gli esempi dell'uso di LISP.

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