TITLE: Application of computers in higher mathematics
QUESTION [10 upvotes]: Currently the main application of computers in mathematics seems to be to compute things, i.e. to solve equations, evaluate integrals, etc.
It is at all possible to delegate the thinking of a mathematician to a computer, such that a computer can prove something like a mathematician would, instead of merely computing things? Is it technically possible?
Could a computer be programmed to solve Millennium Prize Problems on its own for example? If that is too ambitious, then could it assist a mathematician in solving them in a more useful way than merely computing?
Considering that mathematics is a deductive system, if a computer became able to prove things like a mathematician would, starting with axioms and deriving new properties, where could this lead if a computer worked on deriving new theorems non-stop for sometime? Will it ever hit a brick wall and become unable to make progress?
Much advancement is happening in AI and many companies/universities are trying to delegate human abilities like driving a car, walking, conversing, etc. to computers so why wouldn't they be able to delegate the skills of a mathematician?
One last question: what is the nature of the thinking of a mathematician? What exactly is the thinking of a mathematician?

REPLY [2 votes]: http://en.wikipedia.org/wiki/Automated_theorem_proving
http://en.wikipedia.org/wiki/First-order_logic
The short answer to your question is, "yes" there are many computer programs that can solve and check mathematical proofs for validity of the proposition and of the proof itself. Mathematicians have been designing programs like this pretty much as long as computers have been around. Computer programs are especially good at solving first-order logic problems and there are many ways (described in the article) to reduce higher-order logic problems down to first-order. So from a purely conjecture-proof standpoint computers are quite capable of solving problems. When it comes to putting in a research question and getting a PhD thesis or journal article out of a computer with no human assistance, well that would put a lot of researchers out of work so it probably wouldn't be in their best interest to figure out how to do that lol.
"Gödel's completeness theorem states that the theorems (provable statements) are exactly the logically valid well-formed formulas, so identifying valid formulas is recursively enumerable: given unbounded resources, any valid formula can eventually be proven." However, one problem is that if the conjectures are not valid then the computer would just keep working and neither the computer (nor the operator perhaps) would know when to terminate the program.
One of my old professors has a theorem solver for some specific sets of axioms: http://math.boisestate.edu/~holmes/marcel.html
Journal of Automated Reasoning:
http://link.springer.com/journal/10817
The stuff in that journal is beyond me, but it's interesting to read the abstracts.