I don't remember this much. BTW the author's name is quite something. You Learn Math?!
Speak of math related books read in high school days... one of the books I remember well was a very short Chinese version of "Gödel, Escher, Bach: An Eternal Golden Braid" or "GEB - 永恆的金帶". It was in A5 format only 1 cm thick. It was my first time seeing Escher's paintings. I did not understand much of the book. I recently found the English version. It's A4 size about 3 cm thick. I read a few page and still do not understand what's it says.
Yeah, I remember that book, you borrowed the Chinese version to me in Form 3 or 4? ... At that time I didn't understand most of it.
In about 10 years ago, I finally read the original version. It's all about recursion, no matter in Gödel's maths, Escher's painting, or in Bach's music (in particular the Canon), these recursion are present. Combined with the illustration of AI, these could be applied to what we called 'goal seeking'. But I dislike the part where the author use a lot of space for story telling, which was not my favorite at all times.
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[ 21st Oct update on GEB]
By reading this thread I decided to 'scan' the book again from my bookshelf and see whether I can pick something out of it and share among all here.
Recursion, I think most of you may be familiar with them, the earliest time I understand this was to calculate the factorial. Well, factorial (n) = n x (n-1) x .... x 1 which can be written in f(n) = n x f(n-1). A big problem reduced to a small problem, a small problem yet to reduced to a smaller problem, etc. And finally it's solved in 'trivial' condition, recombining everything we got the answer. But in fact, recursion is the foundation of the computer(also human?) language, when a parser tries to understand your (computer) language, it tries to resolve your syntax in a recursive way.
Well, I saw Escher's self reference paints, Bach's Canon. (In fact I love Pachelbel's Canon more) So, who is the guy Gödel? The book didn't mention a lot of him directly. In fact he is one of the most important Mathematician in recent mathematics.
What Gödel told us was whenever we have a formal system, there exists some truths where we cannot prove or disprove. When this system embrace all these truths, it cannot remain consistent any further. In other words, completeness and consistent cannot both exist.
Err, too difficult already? Taking the Geometry as an example, here we all learnt Parallel Postulate. //-Postulate says that given a straight line and a point not on this line, we could ONLY find one line that past thro' point and didn't intercept the first line. Simple? But for centuries people were having trouble proving / disproving this using the geometry.
So there are some truths that cannot be derived from the original truths.
How about we throw in //-Postulate to the list of axioms of geometry to make it really cover everything in geometry. It's Gödel telling us that you can't just simply do that, by doing so you will discover some new truths that couldn't be proved/disproved. If you keep throwing in new axioms to perfect geometry, finally you will arise in an inconsistent situation in geometry, and will break it down. (See? This is a good way to destroy something)
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[ 22nd Oct update on GEB]
Understanding the Gödel's infamous Incompleteness Theorem one needs to understand the predicate logic, involving something like "This statement is false", and construct a statement of the logic system referring the logic system itself, i.e. a self-referencing mechanism. With the help of another numbering system, the proves were able to map to numbers. Proving is then reduced to proving on this numbers, well, does this kind of meta-proving make sense?
Self-referencing is another interesting things that differentiate conscious one and non-conscious one. For a conscious/mindful body, he can jump out of the system and talk about himself. Like we see a doctor and tell them we're sick. "I think therefore I am" is an example that you take yourself out of the system and judge it whether it's conscious or not. Couldn't a robot somehow act like this, I bet nop. Seeing this I'd rather believe there should be other way for AI.
