TITLE: Could computers someday discover theorems or find demonstrations?
QUESTION [15 upvotes]: Cloud computing and quantum computers bring computers to what seems like a limitless calculation power?
If one sees all the mathematical operations and theorems as a toolset that a computer can use, bringing the adecuate automatic or interactive algorithm, it's easy to imagine that some day computers could "brute-force" the discovery of new theorems, for instance.
Take the Pythagorean theorem, someone could measure the sides of a right-angled triangle, and give the input to the computer. Then the computer could give the theorema's formula, and some demostrations.

REPLY [8 votes]: In a loose sense they already have:

Prover9, for example, can be fed in "axioms" and "goals" and can (hopefully) deduce the goal from the input axioms.  Importantly, this still requires human input.

For important results (which are usually out of the reach for automated theorem provers directly), usually a human operator divides the result into a sequence of intermediate goals that the software can handle.
This method can be useful when you have to prove a large number of trivial theorems (which would be too tedius for humans to do).  For example, when you want to classify varieties that satisfy short identities (e.g. this; PDF warning).


Doron Zeilberger has taken the idea of automated theorem proving even further, and has programmed the computer to automatically conjecture and prove results (e.g. 3x+1-like conjectures: ref.).
I think, in the not-too-distant future, we will see computers automatically (a) finding counter-examples to human-created "theorems", and (b) proving respectable conjectures.  Two major obstacles are:

Automated theorem provers are somewhat limited to first order logic, whereas humans are not.
Humans, when proving things, can consult the literature, which is written in "human".

I think, in the future, advances in machine learning, data mining, and natural language processing will enable automated theorem provers to "learn" lemmata (and find conjectures) from the internet, and automatically use them in a proof.  This would be a way computers could "cheat", so they don't need to limit themselves to theorems proved only by first-order logic.
I think the most likely area that automated theorem provers will begin to compete with humans in non-trivial ways will be abstract algebra.  It's somewhat already started (see this; PDF warning).