Recently, the book on my bedside cupboard is Gödel, Escher, Bach(aka GEB). The first time I saw this book on Douban, it was not that attractive to me. Cause I am not interested either in Bach or in classical music, and have never heard of the name Escher. Admittedly, the title does strike a chord with me. It reminds me of an old movie called the good the bad and the ugly. While the rev...
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Recently, the book on my bedside cupboard is Gödel, Escher, Bach(aka GEB). The first time I saw this book on Douban, it was not that attractive to me. Cause I am not interested either in Bach or in classical music, and have never heard of the name Escher. Admittedly, the title does strike a chord with me. It reminds me of an old movie called the good the bad and the ugly. While the reviews of this book on amazon and Douban are surprisingly positively given. Most people give five star and seems totally fascinated by the author’s charisma. With the fact that I had to take Math 414 in the next semester(which involves Godel’s theorem) and the book is really cheap, I decided to give it a try and ordered one on amazon.
Front Cover of GEB
Generally, reading a book with over 1000 pages is out of comfort zone. But I find this one is ok. Chapter one and two talks about the inner connection between Bach’s fugue and Escher’s painting, of which I am not interested at all. Anyway I finished reading them.
Bach, the great German composer of the Baroque period, famous for his strictly structured canons, fugues and contrapuntal techniques.
Escher, Duntch painter, known for his often mathematically inspired woodcuts, lithographs, and mezzotints.(Wikipedia)
Endlessly rising and repetition of loops are the structural similarities of these two great artists’ works. While repetition of loops involves the most dangerous thing in logic and mathematics---self reference.
Endlessly rising in Escher’s painting
Endlessly rising in Bach’ fugue
It is the power of self reference that lead to the the Foundational Crisis of Mathematics. I don’t want to talk about what exactly that crisis is. After Godel Incompleteness Theorem(which is based on the paradox of self reference. Russell’s Barber Paradox is a famous example. ) was proved, mathematicians started to doubt the basis of mathematics. This crisis had great influence on other fields such as computer science and cognitive science. One of the famous problems is halting problem. In 1936, Allan Turing proved that a general algorithm to solve the halting problem for all possible program-input pairs cannot exist. The general idea is that if there exists a plugin(a program) in the compiler that can tell whether a piece of code goes on forever in an infinite loop or not(if yes, output is a boolean value true. otherwise no), use the plugin on itself. If the plugin itself does not halt, then it halts(because it will eventually halt and give the answer true). If it halts, then it is in an infinite loop. A PERFECT SELF REFERENCE!
Actually, if these exists a solution to halting problem, we can easily prove or disprove Goldbach’s conjecture. We can just build a program to test if every even number can be expressed as a sum of two prime numbers. Then use the “halting” plugin on this program. If it halts, Goldbach’s conjecture is false. Otherwise, it’s true. QED. Unfortunately, we can never do it in this way.
A variation of halting problem
In the book, the author gave another wild idea that the elegant structure of self reference is happening everyday in our daily life. Our conciousness is referencing itself all the time! So maybe the key to Artificial Intelligience is self reference. I was astonished by the idea!
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