This is a special post for quick takes by cdwhite. Only they can create top-level comments. Comments here also appear on the Quick Takes page and All Posts page.
Mathematics has something I think I've heard called "mathematical culture". It's less the concepts & results than a sort of sprachgefuhl for what kinds of results one might expect or want to prove, combined with an appreciation for the history---how we got to those results, how one led to the next, who the personalities involved were.
Claude has done a fantastic job of teaching me math, but I wonder if I would be better off with more of this "mathematical culture". Concretely---I've been wanting to learn about concentration of measure (https://en.wikipedia.org/wiki/Concentration_of_measure) for ages. (Concentration of measure is a set of results where smoothish functions lie close to their averages with overwhelming probability.) recently I (1) had a problem where that was unambiguously the right tool; (2) had Claude suggest a route to the result I wanted, and (3) had Claude make me a problem set working through the things I needed to know.
This worked great! Turns out these things aren't so hard, when you have somebody to break them down step-by-step for you and you have confidence that they'll get you to the result you need for the paper you're trying to write. I know way more math than I did before! That's a wonderful thing! And the problem sets, notes, and conversations with Claude were full of little hooks: "this is related to <that>, which leads to <another thing> that is important in <this other context>."
But I still worry. Claude took me through one particular route to these results---in this case, a sort of differential-geometric diffusion-on-manifolds route. I've always heard people talk about concentration of measure in completely different way---via isoperimetric inequalities and Levy's lemma. I'm sure these are equivalent, but I worry that the particular route I took has cut me off from the rest of my community, a little bit.
It feels a little bit like the slow demise of physical stacks in libraries---probably the thing I miss most about my old university job is access to a good research library---or print journals. People talk about how they came up with important ideas by reading the next article in some journal, which they didn't realize connected to something else they'd been thinking about. Likewise I've found all sorts of cool books by lookng in the general vicinity of something I know is relevant.
Like I said, I unambiguously know way more math because of Claude, and that's a great thing. But I want to be thoughtful about what we're losing.
It seems like technology that allows more direct execution of your specific intention gets rid of a lot of serendipity (e.g. another cover catching your eye while you search for a specific book, being forced to interact with unfamiliar people in your immediate vicinity instead of being able to call your existing friends all the time.) We can probably reproduce this with good design, but this loss is at least partially inherent to improving the ability to directly get what you want, so we’d have to pay a collective price for it.