Customer Reviews

Forever Undecided: A Puzzle Guide to Godel

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Do you believe that you know what you believe? Is it possible to believe only those things that are true? These questions are but two of the many addressed in this book by Professor Smullyan, and the answers will surprise you!
With most of the material in the form of puzzles, and nearly all of them solved, it is possible for a logical novice to read and understand the book. However, it is recommended that you find another source for explanations of the standard logical operators and read it first. Even then, you should not be surprised to find yourself rereading a solution thinking that the answer simply cannot be correct. But keep trying, as eventually you will begin to understand, and come to appreciate some of the unresolvable consequences of mathematics, the most precise of disciplines. For while Professor Smullyan may not be logically consistent, he is consistently logical. (This is of course a bad pun, but it is no worse than some of the groaners in the book.)
Finally, as the following example shows:

"The following dialogue ensues between a student and his theology professor:

Student: If I believe that God exists, then I will also believe that I will be saved?
Professor: It that is true, then God exists.
Student: If I believe that God exists, then will I be saved?
Professor: If God exists, then that is true.


Prove that if the professor is accurate and if the student believes the professor, then God must exist and the student will be saved"

You may also be given insight into one of the most fundamental questions regarding human existence.

Published in Journal of Recreational Mathematics, reprinted with permission.

This book was recommended (an aeon ago) in my undergraduate philosophy class, with the recommendation that if one will bother to think through each example, then one will become very good at formal logic. Many years later I have found time to do this. So many things can keep me awake at night, idle thoughts, terror at the credit crisis, and so on. Now I read one puzzle from Smullyan, and gently tug my mind back to it if it wanders as I lie in bed. This has been a revolution for me. Either I learn something or I sleep better.
This book hits the happy medium between on the one hand an overly formal approach to logic such that it is unsuitable for anything but hard work or formal study and on the other a text that is watered down to the extent of being useless.
I do not recommend my particular way of reading this as the only way or the best way, but merely as an example of how it has worked very well for me. The practical benefit of using this book and learning from it is that one can grasp arguments and any problem in them much more rapidly than before, both in work and non-work life.

I recommend this book very highly to anyone at any level who is interested in logic, language, problem solving or reasoning.

EPICTETUS

This is a great book for anyone wanting to understand Go:del's theorem, which basically proves that logic is inherently flawed. The book is very readable. Just make sure that you don't let your head explode.

In the every day practice of mathematics or viewing the works of others, I have never ran across or derived a self-referential statement of the kind that Goedel used in his proof of the incompleteness theorems. The appearance of these kinds of statements takes place only in the context of mathematical logic, and their construction is is somewhat artificial, involving the use of 'diagonalization'. It is for this reason that I don't find the incompleteness theorems in any way troubling for the "truth" of mathematics. If they kept arising in the everyday practice of mathematics, this would lend support to the incompleteness theorems. As such though there is no "empirical" support for them, and until they do arise they can be safely ignored.

The results of Goedel have been used to cast doubt on the "foundations" of mathematics and the "strong AI" problem. But they have also been used to support "strong AI", as it is felt that the existence of self-referential statements are an indication that a machine is conscious. All of these arguments are interesting, but they have yet to help in the practice of mathematics or in the construction of intelligent machines. In fact, too long an emphasis on these results has probably retarded the advancement of artificial intelligence research. The incompleteness theorems though have stimulated research in the field of 'automatic theorem proving' and in this respect they can be said to have had some value.

This book gives an overview of Goedel's incompleteness theorems and its corollaries from a "semi-popular" point of view, meaning that readers are expected to have some background in elementary logic as well as philosophy, in order to appreciate the contents. The author is eloquent and enthusiatic throughout the book, and this serves to make the book more palatable for the absolute beginner. It would be unfortunate maybe that readers will begin to doubt the "truth" of mathematics after reading this book, but that doubt could also be viewed as a virtue, in that it would motivate further thinking and research. But again, to this date there has been no self-referential statement that has appeared in the everyday practice of mathematics....that I know of anyway.