This final unit brings together all the ideas introduced in the course. These ideas constitute the technical machinery that enables us to prove some very important theorems which answer what we have called Leibniz's and Hilbert's Questions. These theorems, Goedel's Incompleteness Theorems, are among the most profound intellectual discoveries of the the twentieth century. Thus you should not be surprised if you find this unit hard going in places.
M381 - Unit 8.
In Goedel, Escher, Bach (GEB) Hofstadter asks the question: what happens when 'things' start referencing themselves? ( Like people do who are in essence not more than a set of linked molecules. )
At last I took the time to watch video 1 of the GEB series.
The teacher is Justin Curry. He started by telling that most undergraduates don't get through GEB in less than 13 weeks and that it took him seven years to get through the book. I am not sure but I think my first attempt in reading GEB was in 2007 or 2008. It took me almost six months to get through it. Which I thought was really bad. When I finished the book and still didn't understand what he was talking about I started to seriously doubt my learning abilities. I have to admit that I still don't get it but I made progress. And I am getting closer, thanks to M381 Mathematical Logic ( read: Nigel Cutland ).
Anyway, to the point: the lecture.
He starts with the concept of isomorphism. In GEB, Hofstadter explains isomorphism as a map between structures that maps parts with similar purpose to similar purpose ( my words ). This is different than the mathematical definition which states that an isomorphic map is both surjective and injective. Hofstadters definition can be understood immediately, whereas the mathematical definition needs understanding of layer upon layer upon layer. Since what is a map in mathematical sense? What does surjective mean? What does injective mean? Analyzing a mathematical sentence always creates a ( large ) tree structure.
Recursion. The concept of recursive definition. A fascinating concept which I use a lot, since I am a programmer by profession. Curry uses the example of the Fibonacci sequence 1,1,2,3,5,8,13,... and translates it to f(n) = f(n-1) + f(n-2) and the Sierpinski triangle ( fractal ).
|Drawing the Sierpinski triangle|
( To be continued in the next post. ) Edit: Nope. I'll make a last note about lecture 1 here and continue with lecture 2 next time.
Some remarks, tips for if you want to give it a try ( like myself ). I was not in continuous awe while watching this lecture. You know when like you are watching the latest BBC Horizon or similar. It's not like that. I don't have the feeling as if I have wasted my time, not at all. I am going to watch lecture 2 soon.
- You definitely need the 720+ pages ( 20 chapters ) book. ( Details on the course site. )
- You need to be ( somewhat ) familiar with Bach's music, or at least -know- someone who is. ( What are forums for anyway? ) To fully grasp the genius of Hofstadter's work.
- If you are a religuous person than GEB might not be for you.
There is an audio set in the lecture room. Near the end of the lecture a piece of Bach is played. Students familiar with that music could elaborate on it. Since I am ignorant to most classical music I must have missed a lot of what Hofstadter said. It might be an opportunity to start listening to some Bach, who knows what happens.,
So far for lecture 1,