How do expert mathematicians think?
How do expert mathematicians think?
We want to understand how extremely good mathematicians think and work – real, live, human mathematicians at the top of their fields, or on their way there. We have lots of public data on this group (e.g., research talks, the papers they write, sometimes the blogs they create) but this huge volume is also a problem, and we need to hire people to help us make sense of things – people who are themselves expert mathematicians, who can work with us to help validate our analyses, develop stimuli for psychological experiments, and expand our understanding.
Mathematicians have ways of thinking and seeing that go well beyond our standard psychometric accounts of what we mean by intelligence (e.g., the pattern-recognition story in Raven's Progressive Matrices). Expert mathematicians, when they take the time to report their own mental processes, describe experiences, ways of thinking, and long-duration concentration and attention events that far exceed what psychological studies have been able to probe in any systematic fashion. Distinctions such as algebraic-geometric, the use of examples and mental experiments, and the notion of ontological transfer are just a few cases where we have reliable evidence to support their existence, but very little systematic knowledge of how these are deployed.
Mathematics is not the only domain in which extreme cognition happens, but it is a particularly fruitful one because it is cross-cultural (compared to, say, artistic works, mathematical proofs do not express any particular national culture), transhistorical (proof-based mathematics has existed for more than two thousand years), and largely free of capital constraints (in contrast to scientific experiments such as CERN or Hubble, proof-making does not require large equipment, or specific government of corporate investment).
We are not sure what we’re going to find, but we think this will be a first scientific portrait of humans thinking at this extreme edge of our species’ limits. Understanding the weird ways they do things will expand our understanding of what intelligence is, and help us develop new ways to think about the next stages of AGI.
Our work is important for AGI because the intense mental activity associated with proof-making is in many ways the closest we will get to "superintelligence" in the human population.
There is an enormous amount of speculation, beginning with classic works such as Bostrom, about the "IQ 3000" machine. But we don't actually have any idea what intelligence actually looks like above the standard university samples (IQ 100 to 120 or so).
This is a huge gap and blindspot in our understanding. It means that attempts to mitigate x-risk are working from extrapolations that a wide variety of qualitative evidence (e.g., a text such as Grothendieck's Reapings and Sowings) already suggests are invalid — rendering them ineffective. Our project addresses this problem directly, probing an area that is today dominated mostly by intuitions, rather than scientific realities.
We may very well find that that qualitative evidence is a bit too poetic to be real, and that standard psychometric stories are valid up to the new limits that this project can probe. But, any violations of the standard story will have important implications for how we think about x-risk, will help reveal where mitigation strategies are working from implicit metaphors that we expect to break in real-world situations, and will help those in x-risk fields develop more effective and powerful tools.
Team Member
$10k level: costs would go to short-term hires of graduate-level mathematicians at CMU.
$50k level: costs would go to partial salary support of a postdoctoral fellow (including benefits, equipment costs, etc).
$100k level: costs would go to support one or two years of a postdoctoral fellow.
No comments yet. Be the first to share your thoughts.
