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:

A core “Mathopedia” website, a wiki where each concept gets a page. A page’s subsections would go from more-intuitive/motivational/extensional-definition/multimedia/seizure explanations, to the more technical ones, ending with a ton of examples. In my head, this could involve a strong community of contributors.

A few desktop-software ideas that, if useful, seem (to me) too-powerful to give to non-alignment-researchers. I am probably wildly overestimating the utility of relatively-simple non-ML-based desktop software that hasn’t already been invented. Still, being careful.

[Reading the Arbital Postmortem while shaking my head so

~~other people know that~~I understand what went wrong there and how my “Mathopedia”” would do better.][Reading Paul Lockhart while alternately nodding and shaking my head so I agree with his emphasis on open-ended learning but dislike how mathematics is taught in US K-12 schools (as elaborated in Lockhart’s colorful examples/analogies).]

[Just cross-referencing 3 textbooks, Googling, and asking Discord, like every other mathematician since the days of Pythagoras. If the resources work for everyone else, shouldn’t they work for me?]

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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