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
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 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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