Topic: Godel
Most of us have heard about Gödel via Douglass Hofstadter, someone who has been utterly mystified by Hofstadter, or someone who has been inspired by Hofstadter to invent their own Gödel mystery cult. Being susceptible to mysticism myself I knew I'd need to be strong when I set out to get to the bottom of Gödel a few years back. It really is a temptation to extrapolate too far. I mean, take possibly the most important result in logic - ever - and combine it with the one subject the general public would have an interest in where Gödel's theorem might have an application, the mind - which just in fact might conceal the greatest mystery ever - and how could the average person not be seduced? Hell, how could the average person not want to be seduced? While I've personally only scratched the surface of mathematical logic, I think I learned enough in my little excursion to benefit others who have become interested in Gödel through popular literature understand his theorem and how to skeptically consider the applications. I think the best way to present Gödel given the walls I ran into is, after the remainder of this brief introduction, forgo all the context and just spell it out in grossly oversimplified terms. Then I'll add some context, along with considerations to the philosophy of mind. I'll have some final notes that step through his actual proof and translate the main ideas.
To over-oversimplify, Gödel will show that from no formal system, or rather,
list of assumptions and rules for making deductions from those assumptions, can
we produce all true statements of the formal system. In other words, Gödel
will discover a statement that is true within a formal system, but this
statement can't be derived from the formal system. Why that's important to
math and logic I'll save for later. But a brief note on the importance of the
result to the philosophy of mind, since that's what's the interesting part for
most people. It would seem the only analog we have for describing how the
brain works, or how the mind works, is a machine. And probably a fast one, like
a computer. But a machine is just the physical embodiment of rules.
So if Kurt Gödel's brain is a machine, then how could he have ever come
up with his "Gödel sentence?"
