Tag: Gödel’s incompleteness theorems

I have mentioned this book (or it’s author) several times in previous posts, so I was surprise to find out yesterday that I never sat down and wrote a review for it. The time has now come :-).

From my understanding, Hofstadter‘s I Am a Strange Loop is a shortened and simpler version of his “Gödel, Escher, Bach: an Eternal Golden Braid“. Since I have not finished this book (I is like a million pages long, and very complex), I cannot say anything about this. But it is a relatively simple book to read, hiding in its simplicity very complex concepts and ideas that take a while to sink in and become real.

The book tries to analyze what is “I”, what is consciousness, what is meaning. He does this by telling us a tale of his life, and by using mathematical and philosophical tools, experiences and mental games. What is “I” is not a simple question, and it doesn’t have one answer, surely not one that can be proven by any means. But the ideas posted in the book are simple awesome. The book has such depth that I don’t even know where to start, so I’ll simply write some of the things that came to my mind while reading the book and from there until now.

One mathematical proof that has impressed me since I did my BSc in computer science is Goedel’s Incompleteness Theorem. In short (and in my words) – Goedel proved that given a theory with a consistent set of axioms (truths) and methods to derive theorems from these axioms, there will always be a theorem in this theory that can neither be proven true or false. Thing about it for a second: the task of science is to find an explanation for everything, but here comes a mathematician, and with rigorous and meticulous care tells us that no matter how much we try, there are things that we simply cannot explain! WOW. But how is Goedel’s theorem related to consciousness? this is the interesting connection that Hofstadter makes: to proof his theorem, Goedel used self referential systems – systems that can reference themselves. Informally, here is the theorem that can never be proven: “I am a theorem that cannot be proven in this theory”, where “I” is encoded in a symbol that references itself. So if the theorem can be proven, the theory is inconsistent – since the theorem states that it cannot be proven. But although the theory must be able to probe all theorems that are true. Q.E.D. (all mathematicians in the audience, please don’t shoot me). And what other system is incredibly self-referential? We, us, I. The human brain. We are thinking machines that unlike computers, can think about ourselves. We don’t just run a pre-defined program until it is finished, but are an infinite loop that can think about our actions, and how we would behave in such and such circumstances, and think about how this would affect us… all of this inside our brains, without even one thing happening outside. And that is what is so special about us – that we can think about ourselves, just like Goedel’s system could reference itself.

Another interesting point explained in the book is about how we think. On the one hand our brain/body is just a set of chemicals that interact. Like balls in a billiard table. But if you had millions and millions of balls running around a very large table, and you could look from far away you would start to see patterns. These is our consciousness. On the one hand it is just the interaction of the parts, but on the other it is the emergent property that comes out from these interactions. He does a much better job of explaining this.

This is an excellent book, for anyone who is the search for… meaning? well, something like that. And what I liked most of the book is the tone of the author – he is just telling his story, like he felt it and saw it. And this makes the book an even greater peace of work. Enjoy it.