Comment by practal
1 day ago
Hmmh. I like motivated explanations, but, as acknowledged in the text, this is a subjective thing to measure. What is a great motivated explanation for Tao, might be hard to grasp for me. So I guess judging how well an explanation motivates something depends on two things: 1) My way of thinking, and 2) what I already know and how well I recall it in this context.
There is a third thing: how well does the motivation chime with or go against my current belief system? You would think this is not much of an issue in mathematics, but it can be, and I had my fair share of frustrations because of it.
Anyway, all of the above points to one thing: the best motivated explanation will be generated by an AI, knowing the subject and you in a deep way that no other human will, and being able to interact with you during the explanation.
> 1) My way of thinking, and 2) what I already know and how well I recall it in this context.
I don't think we're discussing pedagogy. Good _research_ exposition is instead related to communicate your intuition and way of seeing things. The conceptualization of a given situation or problem is what is valuable, how you connect it with other stuff, etc. It is then up to you to memorize and interiorize it.
Good research exposition is the same as good "pedagogy" just for a different audience. Both have to consider didactics. When writing a paper, you teach something to the fellow researchers who know less about a thing than you.
Seems to me to be two different sides of the same coin. Pedagogy is about finding a way to communicate to me an idea based on my intuition and seeing things. A research exposition is about presenting the idea in terms of your intuition and seeing things.
What's an example of your belief system conflicting with a motivated example? I'd love to understand that a bit more.
One example is what currently plays out, see the previous guest post on Tao's page: https://terrytao.wordpress.com/2026/09/12/after-math/
The blog post says that the statement "AI really did solve a problem in mathematics." is wrong. But a formal proof showing that Navier-Stokes equations can blow up is certainly such a solution, by AI. There is not much in this world that is more objective than a formal proof, so any disagreement on this is based on how we see the world. Michael Harris will agree with the statement being wrong, Jacob Tsimerman will not.
Another example, Hilbert famously battled Brouwer's view of mathematics. From my point of view, Hilbert was right: intuitionistic logic is certainly interesting; but I like to study it using "normal" (= classical) mathematics.
Finally, my personal frustrations are about how hard it is to publish my work on abstraction logic. I would never have thought it is that difficult, mathematics being objective and all. It seems essential to take out as much motivation out of your paper as possible, because it might offend your reviewers and their belief system. By now my papers come with full Isabelle/HOL formalisations, let's see if that helps.