Review: Godel’s Proof

This was something that I had to stop reading because of the exam season, and finally finished within 24 hrs of the exam getting over.

It is a rightly acknowledged classic in the field of popular mathematics (Stuff which makes abstruse mathematical ideas accessible to the (almost) general public).

Like the title says, it deals with a set of proofs that Godel published which demolished one of the most ambitious projects of modern times, to axiomatize all of mathematics.

It was believed that given a finite set of axioms and a set of rules to operate on them, one can mindlessly manipulate symbols to generate every known theorem. A simple example would be translating geometry into algebra ($x^2 + y^2 = c^2$ is how a circle would look in algebraic terms ). So, we can say that if  algebra is consistent, then geometry is consistent.

Hilbert developed a method to provide proofs of consistency, which was then taken over by the logicians of the time to try and provide similar proofs to certain parts and eventually all of  mathematics.

That is, until Godel came along. How he did it is explained beautifully in the book and is unecessary to reproduce here. Godel proved that any system for number theory that is consistent is incomplete, i.e,  you can produce a true theorem which cannot be proved within the system. Thus, we cannot have a formal system describing all of mathematics. We can have formal systems with finite set of axioms, but not for something like number theory, where Godel proved it was impossible to generate a finite set of axioms.

The book also has a section on the philosophical implications of this proof, which essentially says that present day A.I is a far cry from imitating the human brain, and unless we can actually describe the human brain as a logical system with a finite set of axioms.

There are many approaches to imitate human intelligence, like neural networks, but they run on a system with a very limited set of axioms and rules of inference commonly known as a digital computer. Maybe we need some new kind of ‘computer’. Researchers have a lot of work to do!!

4 thoughts on “Review: Godel’s Proof”

1. I took a class about Godel’s theorem and it is a pretty powerful blow not only against Strong AI, but also against empiricism. He believed that although finite formal systems were incomplete, there are abstract entities that are complete, and we are able to understand how to complete mathematical truths beyond a finite formal system. Even if there aren’t mathematical entities, it is still implausible that numbers are nothing more than a made-up human invention.

2. Yup, I remember reading that Godel himself believed that something like the Platonic Ideal existed which can be manifest only imperfectly.

3. This sounds a bit similar to another book I have read.

Douglas Hofstadter wrote “I Am a Strange Loop”, which tries to explain how human’s develop conciousness despite being made up of physical inanimate material. He also looks at why we have fallen short in developing AI etc.

What Hofstadter does is try to explain the human mind as a self-referential system, based on Gödel’s Incompleteness Theorem.

To be honest, I didn’t particularly enjoy the book as I am not a fan of Hofstadter’s indirect, meandering and repetitive writing style, but it did raise some interesting points and made me think.

So it MAY be something you want to check out if you enjoyed this one, as they sound similar. I’ll have to find myself a copy of “Gödel’s Proof”

4. Well, this one just has a intro by Hofstadter, and is not as much about epistemology as about the proof itself, which was fleshed out by the original authors. I have GEB by hofstadter waiting in my lib, so will try him out anyways :)