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cousin_it comments on Completeness, incompleteness, and what it all means: first versus second order logic - Less Wrong
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We form beliefs about mathematics the same way we form beliefs about everything else: heuristic-based learning algorithms. We typically accept things based on intuition and inductive inference until trained to rely on proof instead. There is nothing stopping a computer from forming mathematical beliefs based on statistical inference rather than logical inference.
Have a look at experimental mathematics or probabilistic number theory for some related material.
Saying "heuristic-based learning algorithms" doesn't seem to compress probability mass very much. It feels like skipping over the mysterious part. How exactly do we write a program that would arrive at the axioms of PA by using statistics? I think if we did that, then the program probably wouldn't stop at PA and would come up with many other interesting axioms, so it could be a useful breakthrough.
Yes, but the point is that we are learning features from empirical observations, not using some magic deduction system that our computers don't have access to. That may only be one bit of information, but it's a very important bit. This skips over the mysterious part in the exact same way that "electrical engineering" doesn't answer "How does a CPU work?" -- it tells you where to look to learn more.
I know far less about empirical mathematics than about logic. The only thing along these lines I'm familiar with is Douglas Lenat's Automated Mathematician (which is only semi-automated). A quick search for "automated mathematician" on Google Scholar gives a lot of more recent work, including a 2002 book called "Automated theory formation in pure mathematics."