Goedel, Escher, Bach - Lecture 5

A few months ago I started to watch the MIT video lecture series on Goedel, Escher, Bach. Due to time constraints I wasn't able to complete watching the entire series. Today I continued with watching lecture 5. I have learned quite a lot on the subject through M381 and I am about to really 'get it' as far as the Goedel Incompleteness Theorems are concerned. My first reading of GEB took months and now parts of the book begin to look simple. If you don't know what I mean browse through a mathematics book you thought was hard, a few years ago. It often seems if there is 'nothing in the book'. The odd thing with Goedel ( and with all mathematics, I suppose ) is that in your mind you think you can explain it to a laymen in one or two sentences. ( It is -that- simple, I am afraid. ) The power of mathematics is that it can capture an entire knowledge tree in a single word. That word remains meaningless without understanding of all the words in the knowledge tree.

A bit about lecture 5.

Dress shows arrogance

I wonder if Justin Curry would go to a job interview in that Club Med outfit. Students are paying customers ( and a pool of cheap labor for lucrative research deals the university makes ) deserving respect from teaching staff.

Formal number theory ( as in M381 )

Justin talks about Typographic Number Theory, ( formal number theory in M381 ). For example $$\forall x ( \neg x = \mathbf{0} ( \exists y x = y') )$$ can be interpreted as

"Every x that is not equal to 0 is the successor of some y."

Leibniz was the first to propose a formal language for number theory. He asked whether it was true that an algorithm could decide if a statement in number theory was true. - Although in M381 this question is answered negatively that does not mean computers can not play a role in proving mathematical propositions. There is an abundance of ( open source ) software for proving theorems.

( Not in video: ) Isabelle a formal proof theory assistant has been used in testing an operating system kernel written in C and assembler. It not only verified that the spec was implemented correctly but it also discovered hundreds (...) of programming and design (...) errors which were not found by traditional testing methods.

Previous posts on the series:
- Lecture 1
- Lecture 2
- Lecture 3
- Lecture 4