Goedel's Way: Exploits into an undecidable world

Kurt Gödel is one of the most important personalities that most people have never heard of.  He is known for his incompleteness theorems, of which much of mathematical logic of the last 80 years is built on.

Gödel become somewhat of a household word in 1979 with the publication of Gödel, Escher, Bach: An Eternal Golden Braid, which went on to win the 1980 Pulitzer Prize. 

With that, Goedel's Way: Exploits into an undecidable world, is a fascinating book.  Authors Gregory Chaitin, Newton da Costa and Francisco Antônio Dória cover a huge amount in but 130 pages. 

All three of the authors are world-class mathematicians and it is worth noting that Dória and Da Costa have published papers with conditional proofs of the consistency of the P versus NP problem.

When it comes to Gödel, Chaitin also has his own interpretation of the incompleteness theorem (Gödel-Chaitin), and is also the discoverer of the Chaitin constant, of which the book terms and references as the omega number

Gödel’s incompleteness theorem has traditionally been used in the realm of mathematical logic.  The authors premise is that Gödel can be extended into nearly every field; from biology, ecology, to economics, computer science and more.  In fact, their hypothesis is that undecidability and incompleteness is everywhere in mathematics. 

Goedel's Way: Exploits into an undecidable world is a hard book to classify.  Part of it includes numerous vignettes into the life of Gödel, part of it a detailed explanation of the incompleteness theorems, and a lot more. 

What piqued my interest in the title is that the book is described as an accessible book gives a new, detailed and elementary explanation of the Gödel incompleteness theorems and presents the Chaitin results and their relation to the da Costa-Doria results, which are given in full, but with no technicalities.   

The truth is that about 40% of the book requires the reader to have a much more sophisticated understanding of higher mathematics (including 4th–year calculus, advanced number theory, set theory and more), something I don’t have.  It is likely that the authors understanding of accessible andno technicalities means something quite different to them than to the average reader. 

For those that don’t mind reading a book where almost half of it is beyond their comprehension, then Goedel's Way is a book worth reading. 

The books 6 chapters touch upon an array of fascinating topics including: Turing machines, complexity and randomness, halting functions, entropy and more.  When not engaging in mind-numbing mathematics, the authors throw in snippets about mathematical personalities such as Leibniz, Shannon, Hilbert and others.  The authors note that this is not a standard textbook, and they add these stories as a human interest feature.  The authors write in the prologue that this book discusses a piece of their idea, but is certainly not the entire picture.  

Goedel's Way: Exploits into an undecidable world is a fascinating, albeit challenging book.  Those with a degree in mathematics will likely find more enjoyment out of the book.  For the rest of us, the book gives them a glimpse into one of the most important logicians in recent memory and the remarkable work he did, which is still extremely relevant today.

Posted on December 1, 2011

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