on 18 April 2015
Godel's Incompleteness Theorems were a revolution in mathematics and there were repercussions and misunderstandings that rippled out into other fields. The main theorem first appeared in an Austrian journal in 1931 and can be stated very simply.
In any consistent formal system S within which it is possible to perform a minimum amount of elementary arithmetic, there are statements that can neither be proved nor disproved.
The consequences are enormous, in that it means that in any system that can be used to perform arithmetic, there will be theorems that can never be verified as either true or false. In other words, some knowledge will forever be unattainable within that system. Of course, this does not preclude adding additional axioms that will allow other theorems to be proved.
Franzen does an excellent job in explaining the incompleteness theorems in a manner that can be understood by people with a limited knowledge of mathematics. While there are few places where a high school mathematics education is not sufficient to understand a more technical argument, it will be enough to understand and appreciate the theorems.
My favorite parts of the book were the sections devoted to "applications" of the incompleteness theorem outside of mathematics. Some examples are from religion, political science and philosophy. Godel's theorems are used to "prove" that no religion can contain a complete set of answers and that any constitution must of necessity be incomplete. Human thought is also interpreted in the context of the incompleteness theorems. The statement is:
Insofar as humans attempt to be logical, their thoughts form a formal system and are necessarily bound by Godel's theorem.
This statement and others related to the nature of human thought are examined in detail. The philosophy of Ayn Rand is also examined as a system that must of necessity be incomplete. This book would be an excellent supplemental text for a philosophy course where the nature of truth is examined. It would also be a very good choice for a course in the philosophy of mathematics.
Published in Journal of Recreational Mathematics, reprinted with permission.
on 8 August 2009
Godel's theorem is tossed about with wild abandon, particularly by people who don't really understand it - and I certainly would not claim to understand all it's subtleties - and this is a great little book to help you clear some of the fog. It is worth noting that Godel's theorem is quite limited in practical import, a large number of the theories in Mathematics do not fit the criteria for Godel's theorem and one can safely say that Physics has far from complete in terms of working out the consequences of its theories. For example there is a million dollars waiting for you if you can mathematically prove there is mass in the universe.
A quick example of the sort of things that this book helps to make clear. For ages I didn't understand the flaw in the standard philosophical argument about how Godel's theorem "shows" that you can "know" a statement is true but not prove it... I always thought it was due to the lack of non-contradiction between the statement and the axioms but somehow this didn't seem to work. The claim is that as Godel's theorem states that if T is a consistent theory then there is a statement P that says it cannot be proved within T. Well it must be true because otherwise there would be a proof! Simple right? Well no, because it is only true if T is consistent which of course you can't.... worth the price just for clarifying that.