This is somewhat a recommendation for Rudy Rucker's Mind Tools and somewhat a statement on mathematics, beauty, and freedom. If you've been to the house, there's a good chance I've shown you this. I showed it to disappearinjon last night and realized that it made sense to post it to LJ. Note that my math might be a bit rusty, I'm being careful in how I formulate my statements but it's possible I might make an improper word choice in some places. if so, I'd appreciate a correction.
Many of you are familiar with Gödel's incompleteness theorem:
The Principia Mathematica, or any other system within which arithmetic can be developed, is essentially incomplete. In other words, given any consistent set of arithmetical axioms, there are true mathematical statements that cannot be derived from the set… Even if the axioms of arithmetic are augmented by an indefinite number of other true ones, there will always be further mathematical truths that are not formally derivable from the augmented set. (from here)Or more generally, "Any formal system that is interesting enough to describe itself can never list all true statements possible by that system."
There are likely fewer of you familiar with Church's theorem:
There is no recursive (or Turing Machine) computation that can always be employed, and which will always yield an answer to the question "is X a theorem of Classical First Order Logic (FOL)?" If this is true, it follows that there is no effective computational way to determine whether an arbitrary formula of FOL is or is not a theorem.In other words, given a formal system, it's possible to list the statements possible in such a system but if the statement you are looking for hasn't shown up, you can't determine if it just hasn't been listed or if it never will be.
With an understanding of these two theorems, I'm ready to quote a section of Mind Tools that is the actual point of this post. Mr. Rucker does a more entertaining and more rigorous description of these theorems in the five pages preceding my quote and explained how these theorems apply not just to math but to people and society as well.
Midway through page 247, he goes into the implications of these two theorems:
A world with no Gödel's theorem would be a world where every property is listable — for any kind of human activity, there would be a programmatic description of how to carry i out. In such a world it would be possible to learn a hard and fast formula for "how to be an artist" or "how to be a scientist". It would just be a matter of learning the tricks of the trade.I'm just tickled pink that he uses mathematical theory to you're free and infinitely creative. Not what one usually expects from math. (Jump back if you jumped here without reading the middle.)
A world with no Gödel's theorem and no Church's theorem would be a world where every property is computable — for any kind of human activity, there would be a fixed code for deciding if the results were good. In such a world an Academy could pass judgment on what was art and what was science. Creativity would be a matter of measuring up to the Academy's rules, and the Salon des Refusès would contain only garbage.
But if there is one thing art history teaches us it is this: all tricks of the trade wear thin, and it's a good idea to keep an eye on the Salon des Refusès.
Our world is endlessly more complicated than any finite program or any finite set of rules. You're free, and you're really alive and there's no telling what you'll think of next, nor is there any reason you shouldn't kick over the traces and start a new life at any time.