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Dreams of “Mathopedia”

Discuss this on LessWrong and/or the Effective Altruism Forum.

In high school, I held up a pretty-decent-level calculus class because I was confused about something. Specifically that thing where you rotate a curve around some spatial axis (like sculpting pottery) and calculate the volume of the resulting enclosed 3D shape.

I kept being confused, and the teacher (who was super nice and knowledgeable and good-at-teaching 1)… her explanations kept not-getting-through to my brain.

“How do we know y=x^2’s ‘vase’ volume? Wouldn’t it be infinite since it’s open at the top?” –> [explanation involving rotating around the Z-axis so it’s like y=sqrt(x), or something idk] –> “But that doesn’t seem very principled! What’s the rule law for how to turn the shapes?”

Then some other student in class said like 1-2 sentences, but the only key info I needed was the phrase “domain and range”.

Then I was like “oh, I get it completely now, thanks!” And then the class laughed/sighed/was somewhat exasperated.

I developed a maybe-seemingly-trivial hypothesis, that if someone receives explanation E_1 of a concept C, and they’re paying attention, and they still don’t intuitively grok C, then they need at least one more different explanation E_2.

An idea immediately came to mind: Could you teach someone any advanced math concept, by throwing every explanation at once at them? Could this work on anybody without more-straightforward mental disabilities? 2

So I’ve long had a back-of-my-mind idea, which I labeled “Mathopedia”. This is not to be confused with any other existing math website that someone would find useful, including MathWorld, Khan Academy, MathOverflow, Wikipedia, Mathematics Stack Exchange, YouTube, Arbital, the OEIS, Metamath, Tricki, ProofWiki, nLab, Hypertextbook, and… uh… at least one literally called Mathopedia. Might need a new name then…

The idea was simple: a math-learning tool that explains advanced uni/graduate/research-level mathematical concepts by gathering a huge number of explanations per concept, and putting them together in an extremely-multimodal (bordering on seizure-inducing) format.

This led me to a few more trains of thought:

I’m still not sure of whether a real “Mathopedia” is worth the effort to build, in some kinds of short-AI-timelines. (Here I’m wanting such a website/tool to mainly be of use for technical AI alignment research, though if it worked it would aid many causes). Then again, when some people entering the field still lack linear algebra on arrival, maybe it is worth it.

Despite the clear emotional/self-serving/imposter-syndrome biases at play, I’m still legitimately unsure as to whether “make advanced maths easier to grok” is secretly the same activity as “stop filtering for the intelligence/conscientiousness needed to wade through terse jargon-heavy not-always-standardly-written-or-correct explanations quickly, in a way that would kneecap any sub/field that actually did make it easy to metaphorically inject concepts into one’s brain without wading through terse jargon-heavy not-always-standardly-written-or-correct explanations quickly”.

How does my original hypothesis look? What, if any, marginal value is there in this sort of project? Does “making math understandable quicker” make things worse? And, of course, can any of this be tested and/or used within a decade or less?

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