There is a cliche that there are two types of mathematicians: “theory developers” and “problem solvers”. Similarly Dyson’s “birds and frogs”, and Robin Hanson divides the production of knowledge into “framing” and “filling”.

It seems to me there are actually three sorts of information in the world:

• “Ideas”: math/science theories and models, inventions, business ideas, solutions to open-ended problems
• “Answers”: math theorems, experimental observations, results of computations
• “Proofs”: math proofs, arguments, evidence, digital signatures, certifications, reputations, signalling

From a strictly Bayesian perspective, there seems to be no “fundamental” difference between these forms of information. They’re all just things you condition your prior on. Yet this division seems to be natural in quite a variety of informational tasks. What gives?

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adding this from replies for prominence–

Yes, I also realized that “ideas” being a thing is due to bounded rationality – specifically they are the outputs of AI search. “Proofs” are weirder though, and I haven’t seen them distinguished very often. I wonder if this is a reasonable analogy to make:

• Ideas : search
• Answers : inference
• Proofs: alignment