The Limits of Mathematics: A Course on Information Theory and the Limits of Formal Reasoning (Springer Series in Discrete Mathematics and Theoretical Computer Science)

Rating:
Summary: Belongs up there with Godel and Jantsch...
Review: Chaitin has spent his life working on translating Godel's work (and Turing's too) into a version that is much more generally applicable: the limits of mathematical reasoning. I'm not sure why this book is not more generally known (probably because we don't like to have our balloon of arrogant rationality popped so easily) and read but it should be required for anyone contemplating further education in science.

Over the years the message from mathematics, physics and cybernetics (and AI for that matter) has simply been one of showing all formal forms of Aristotle's logic cannot ever capture the TRUTH.

Chaitin's easy delivery (this book is based on taped lectures) makes the book a gentle read even for those without the math background to fully understand the few equations. I highly reccommed buying a copy and reading it.

Rating:
Summary: Belongs up there with Godel and Jantsch...
Review: Chaitin has spent his life working on translating Godel's work (and Turing's too) into a version that is much more generally applicable: the limits of mathematical reasoning. I'm not sure why this book is not more generally known (probably because we don't like to have our balloon of arrogant rationality popped so easily) and read but it should be required for anyone contemplating further education in science.

Over the years the message from mathematics, physics and cybernetics (and AI for that matter) has simply been one of showing all formal forms of Aristotle's logic cannot ever capture the TRUTH.

Chaitin's easy delivery (this book is based on taped lectures) makes the book a gentle read even for those without the math background to fully understand the few equations. I highly reccommed buying a copy and reading it.

Rating:
Summary: Beautiful attempt to understand the nature of mathematics
Review: Greg Chaitin has written a beautiful and entertaining book about his work to understand the nature of mathematics. He describes his results in a highly informal and entertaining way which is still rigorous. Clearly he sincerly wants to _know_ what mathematics is. I wish that all math books were written this way.

Chaitin shows us the fairly simple computer programs he has written to demonstrate theorems about the limits of the axiomatic method. He shows results which are similar to Go"del's and Turing's results and in fact imply them. He has a simple and striking method of arriving at his results. It is all done with both brains and heart.

Chaitin defines a number which represents the probablility that a computer program halts. He shows how this number cannot be computed with a computer program which contains fewer bits than the number itself. Moreover, no set of mathematical axioms can compute this number with more precision than there are information bits in the axioms.

Since no set of axioms can enable us to fully compute the halting probability, and since axioms enable us to write proofs, and since proofs give us the reason why a mathematical statement is true, then there are some mathematical truths which are true for no reason, i.e. they are random.

But I cannot agree with this conclusion. Chaitin, and many others who say similar things, assume that a proof is a reason. But a proof is only a chain of implication. Our faith in the statement, in the light of the proof, rests on our faith in the axioms and logic.

Imagine a culture somewhere that has a mathematics like ours, except they don't have the distibutive property. They could prove some things that we know, but not others. They play around with their computers and discover some facts which they could easily show if they only had the distributive property, but they can't prove them.

Would it be right for these people to claim that these facts are random, that they are true for no reason? I don't think so. Instead, I think that these people should say, and we should too, that there are some things which are true, they are not random, but they are beyond our ability to prove. They are true for a reason, but we just don't see it yet.

Rating:
Summary: Beautiful attempt to understand the nature of mathematics
Review: Greg Chaitin has written a beautiful and entertaining book about his work to understand the nature of mathematics. He describes his results in a highly informal and entertaining way which is still rigorous. Clearly he sincerly wants to _know_ what mathematics is. I wish that all math books were written this way.

Chaitin shows us the fairly simple computer programs he has written to demonstrate theorems about the limits of the axiomatic method. He shows results which are similar to Go"del's and Turing's results and in fact imply them. He has a simple and striking method of arriving at his results. It is all done with both brains and heart.

Chaitin defines a number which represents the probablility that a computer program halts. He shows how this number cannot be computed with a computer program which contains fewer bits than the number itself. Moreover, no set of mathematical axioms can compute this number with more precision than there are information bits in the axioms.

Since no set of axioms can enable us to fully compute the halting probability, and since axioms enable us to write proofs, and since proofs give us the reason why a mathematical statement is true, then there are some mathematical truths which are true for no reason, i.e. they are random.

But I cannot agree with this conclusion. Chaitin, and many others who say similar things, assume that a proof is a reason. But a proof is only a chain of implication. Our faith in the statement, in the light of the proof, rests on our faith in the axioms and logic.

Imagine a culture somewhere that has a mathematics like ours, except they don't have the distibutive property. They could prove some things that we know, but not others. They play around with their computers and discover some facts which they could easily show if they only had the distributive property, but they can't prove them.

Would it be right for these people to claim that these facts are random, that they are true for no reason? I don't think so. Instead, I think that these people should say, and we should too, that there are some things which are true, they are not random, but they are beyond our ability to prove. They are true for a reason, but we just don't see it yet.

Rating:
Summary: This book is a piece of junk. Do not buy it.
Review: Years ago I read a fascinating article in Scientific American
"Randomness and Mathematical Proof" by G Chaitin. (Of course
that was back in the good old days before SciAm switched to the
kindergarten market). Anyway, the article was fascinating and
well written. It was followed up by Martin Gardiner in his
"Mathematical Recreations" column, I think. This left me
with a desire to learn more. On the basis of the article and
the excellent reputation of Springer. I bought this book 2
years ago sight unseen.

What a big mistake. I feel cheated every time I see it on my
book shelf! The worst thing is that I simply cannot believe
that a reputable publisher like Springer actually published
this piece of junk.

I also cannot believe the same person wrote the SciAm article
and this book. Chaitin comes across in his book completely
self-obsessed and full of his own importance: comparing himself
(favourably) to K Godel and Einstein.

The book claims to be "a course in information theory". False.
It is not a course in anything and is written by an illiterate.
It has no value as a text book - indeed its value is negative
becasue it will turn readers away.

The subject of algorithmic information theory is worth learning
about. But please do look for another book to learn from - any
book except this one.

Rating:
Summary: This book is a piece of junk. Do not buy it.
Review: Years ago I read a fascinating article in Scientific American
"Randomness and Mathematical Proof" by G Chaitin. (Of course
that was back in the good old days before SciAm switched to the
kindergarten market). Anyway, the article was fascinating and
well written. It was followed up by Martin Gardiner in his
"Mathematical Recreations" column, I think. This left me
with a desire to learn more. On the basis of the article and
the excellent reputation of Springer. I bought this book 2
years ago sight unseen.

What a big mistake. I feel cheated every time I see it on my
book shelf! The worst thing is that I simply cannot believe
that a reputable publisher like Springer actually published
this piece of junk.

I also cannot believe the same person wrote the SciAm article
and this book. Chaitin comes across in his book completely
self-obsessed and full of his own importance: comparing himself
(favourably) to K Godel and Einstein.

The book claims to be "a course in information theory". False.
It is not a course in anything and is written by an illiterate.
It has no value as a text book - indeed its value is negative
becasue it will turn readers away.

The subject of algorithmic information theory is worth learning
about. But please do look for another book to learn from - any
book except this one.