TITLE: What areas of math can be tackled by artificial intelligence?
QUESTION [20 upvotes]: Artificial intelligence is nearing, with image/speech recognition, chess/go engines etc. My question is, what areas of math that are interesting to mathematicians, is likely to be the first to be able to be tackled by artificial intelligence? Is there some areas of math where some open conjectures or similar was solved by AI? Has AI been of use in math at all yet?

REPLY [2 votes]: There are two bottlenecks in the flowering of Mathematics:  

The gap between intuitively 'knowing' a mathematical truth and rigorous proof.
The dissemination of rigorously proven mathematical truths to the world at large.

In both cases, opportunities for the application of AI aren't obvious.

Intuition and rigor (induction and deduction) aren't obvious bedfellows but both are needed for important mathematical advancements. According to Leonhard Euler, as quoted by Polya: 

The properties of numbers have been mostly discovered by observation, and discovered long before their truth has been confirmed by rigid demonstrations. The kind of knowledge which is supported only by observations and is not yet proved must be carefully distinguished from the truth; it is gained by induction, as we usually say. Yet we have seen cases in which mere induction led to error. 

Intuition isn't only the first step to proof. It generates the interesting ideas that capture the imagination and fuel both mathematics research and 'cranks'. This capturing of imagination is undoubtedly important: See Fermat's Last Theorem, Hilbert Problems, Guy's Unsolved Problems in Number Theory, The Clay Millennium Prize Problems, et.al. In fact, humans have the realm of intuition covered quite well. However, for every hyper-intuitive Ramanujan, there are many more intuitive would-be mathematicians whose ideas will die on the vine, or be wrong, or be published and never thought of again. The same will be true of populations of highly intuitive math-AIs. The missing ingredient is not induction but rather deduction.
Now if AIs achieve superhuman deduction, they could be assigned to attacking outstanding problems. This was the gist of Hilbert's 23 Problems, a list of the most important unproven conjectures in 1900. The second problem was essentially this: prove that Math isn't a house of cards, built on false premises. In 1931, Godel's Incompleteness Theorems gave an unsettling answer: we might never know. Godel showed that in any axiomatic system there are undecidable propositions (they cannot be proven true or false) and that given only the axioms of arithmetic, the consistency of arithmetic is undecidable. In other words, if you assign an open question to the best AI, there is no guarantee that a proof is even possible.
Not only is the gap from intuition to proof a potential dead-end, very few humans are willing and able to parse even their own intuitively generated ideas. Grigori Perelman can no longer be bothered to explain his Fields Medal winning proof. Shinichi Mochizuki claimed a proof of the abc-conjecture in 2012 and six years later it has yet to be dismissed or accepted. Maybe they both feel like an average professor, trying to explain mathematics to an eager amateur. Maybe Mochizuki is the crank and academia, with its norms of conferences and peer-reviewed journals, isn't capable of discerning the difference.
Despite the popular fascination with 'genius' as a synonym for inexplicable intuition, the opportunities in math research are actually the leaps from induction to deduction and again to dissemination. Although AI is progressing, (DeepMind's Alpha Zero is looking pretty good), AI is seemingly very far from contributing to mathematics. 

Computers are pedants.

Pedantry and mastery are opposite attitudes toward rules. To apply a rule to the letter, rigidly, unquestioningly, in cases where it fits and in cases where it does not fit, is pedantry. … To apply a rule with natural ease, with judgment, noticing the cases where it fits, and without ever letting the words of the rule obscure the purpose of the action or the opportunities of the situation, is mastery.   -Polya, How To Solve It

But humans are fallible.

On how many chess moves ahead he can see:
  Just one. The best one.
  - either Capablanca (per Kasparov) or Jaffe, after defeating Capablanca.