Comment by rramadass
9 hours ago
I have tried to read some articles and watch some videos "explaining" Godel but never really understood it. Everyone seems to be explaining the mechanics of what Godel did but explaining the Why is lacking i.e. What was it in mathematics that got him even thinking on these lines in the first place? Can this problem be demonstrated with a simple toy axiomatic formal system? How did he hit upon his approach? Answers to these sort of questions is what seems to me the most important thing to understand before following his arguments.
I recently came to know of The Annotated Godel: A Reader's Guide to his Classic Paper on Logic and Incompleteness by Hal Prince which i think i need to sit with :-)
I think that Gödel did all of this stuff because of David Hilbert basically posing the challenge to make maths' foundations consistent and complete.
https://en.wikipedia.org/wiki/Hilbert's_program
And the reason for Hilbert's program? The problem of "Russel's Paradox" which is a contradiction in naive set theory - https://en.wikipedia.org/wiki/Russell%27s_paradox (Note that there were other paradoxes too).
Hilbert's idea was that by completely formalizing mathematics on a axiomatic/deductive basis, one can mechanically derive proofs so that you don't run into paradoxes/contradictions.
But then Godel showed such a formal system applied to basic mathematics can never be complete (if consistent) and never prove its own consistency.
To me the mechanics and the why are closely intertwined. If you feel like self-referentiality is a way to demonstrate a problem (this is the “why”), it is not a long step to the mechanics of encoding.
The work is in creating the theorem / contradiction from that point, but in the big picture, the approach doesn’t have to come from nowhere.
> the mechanics and the why are closely intertwined
But the "why and what" must always be explained first even if it is incomplete; since that is the problem we are trying to solve. With logic it is even more important since you can follow a proof from one step to the next but by the time you reach the conclusion you have lost the connecting thread to the starting point (for most general folks) i.e. "you have missed the forest for the trees". This is why many folks feel lost when doing mathematics as mere symbol-pushing.