[pierre_d528]

"“I understood nothing, but it was really fascinating,” he said. So Scholze worked backward, figuring out what he needed to learn to make sense of the proof. “To this day, that’s to a large extent how I learn,” he said. “I never really learned the basic things like linear algebra, actually — I only assimilated it through learning some other stuff.”"

https://www.quantamagazine.org/peter-scholze-and-the-future-...

[tzs]

I've long wanted a series of interactive math ebooks that work that way. Each would take one interesting theorem, such as the prime number theorem, and work backward.

When you start the book, it would give the theorem and proof at a level that would be used in a research journal. For each step of the proof, you would have two options for getting more detail.

The first option would be at the same level, but less terse. E.g., if the proof said something like "A implies B", asking for more detail might change that to "A implies B by the Soandso theorem". Asking for more detail there might elaborate on how you use the Soandso theorem with A".

The second expansion options gives you the background to understand what is going on. In the above example, doing this kind of expansion on the Soandso theorem would explain that theorem and how to prove it.

Both types of expansion can be applied to the results of either type of expansion. In particular, you can use the second type to go all the way down to high school mathematics.

If you started with just high school math, and used one of these books, you would get the basics...but only those parts of the basics you need to understand the starting theorem.

Pick a different starting theorem, and you get a different subset of the basics. It should be possible to pick a set of theorems to treat this way that together end up covering most of the basics.

That might be a more engaging way to teach mathematics, because you are always working directly toward some interesting theorem.

[Blazespinnaker]

Yes, you and absolutely everyone else in the world that loves math, didn't have time to get a phd and isn't elitist wants this.

Sadly, the monetization of this is tricky. Probably has to be an open source effort. Need some visionary like wales or khan, but they are very very rare.

[pierre_d528]

It's a great idea and I think it's much bigger than maths. If you do not already know about it, searching around what a "Dynabook" is cannot be a waste of time.

You may be interested in this kind of laying out a proof: https://lamport.azurewebsites.net/pubs/proof.pdf

[frostwhale]

Yeah. Reading the post I see a guy overwhelmed by a bunch of equations and numbers. Which isn't to say he shouldn't learn them, but math is always far more intimidating when you don't understand it than other subjects.

[pierre_d528]

> "intimidating"

Exactly.

/speculation

There is a point where one starts to see "behind" the symbols. It's a strange sensation, as if one could understand the ideas in a non-verbal way. The symbols become optional. Intimidation crawls back before curiosity at this point.

An amazing book on the subject is:

  "Hadamard - The psychology of invention in the mathematical field"

/speculation

[SomeStupidPoint]

What took me a long time -- and is still a skill I'm developing -- is to both verify and "read" the math at the same time, to see the proof and the story at the same time.

At one level, you're observing a technical construction and trying to ensure that it's (mostly) sound; but at another level, you're trying to understand the broader picture of how it fits in, what the builder was trying to accomplish or what perspective of the world they're trying to share.

Mathematics is -- like any language -- just the articulation of an experience, of an insight, of an understanding. As you get further into mathematics (and possess more technical skills of your own), it becomes more important to see "Oh, he's trying to apply the machinery of homotopy to type theories as a means of discussing equivalence" than it is to get bogged down in the technical details. Often, the details are wrong in the first draft, but in a fixable way. (This is extremely common in major proofs.)

> There is a point where one starts to see "behind" the symbols. It's a strange sensation, as if one could understand the ideas in a non-verbal way

I think at some point, you have to compile mathematics to non-verbal ideas for computational reasons -- your verbal processing skills are simply too slow and too simple compared to other systems. Your visual and motor systems are way more powerful and (in the case of motor systems) operate in high dimensions. Much like GPUs in computers, if you can find a representation of a problem that works on a specialized system, you can often get a big computational boost; in mathematics, we have to push our understanding of self and experience to the limits to find more efficient representations of ideas, so we can operate on more interesting or complex ones.

I think most mathematicians work in extremely personal, non-portable internal representations, and then use the symbols as a way to create an external representation that the other mathematicians can compile into their own internal representations.

If you see mathematics as extremely high level code meant to be compiled to equivalent internal representations on thousands of slightly different compilers, I think the language starts to make more sense -- it's meant to be a reverse compilation target for machine code that's been under revision for ~3000 years, so of course it looks a little funky.

Ed:

I will say this --

One thing I've noticed as I've gotten older is that we do a really poor job of teaching students the story of mathematics -- the human motivations, the community, the long standing projects (some have gone on for hundreds of years; some are still ongoing).

I sincerely believe that for young kids (less than, say 10), it would be better for their development to teach skills 4 days a week and simply tell them part of the story on the 5th. It would make mathematics much more relatable and understandable.

[pierre_d528]

> Ed: ...

A few people have thought about this very idea. You may take a look at:

  "http://www.vpri.org/html/words_links/links_ifnct.htm"

[matt_j]

I liked but didn't love mathematics in high school and as such I just did what I had to do and moved on. A decade later I worked through a CS degree and gravitated towards books about mathematicians and now I have a deep fascination with mathematics and I wish I read these books when I was in high school!

[hyperpallium]

A survey of how mathematicians think about mathematics [citation needed] found 80% visually, 15% kinesthetically, and 5% symbolically (i.e. in terms of notation).

[amelius]

> math is always far more intimidating when you don't understand it than other subjects.

In a way it is like a magic trick. Frustrating when you don't know how it works, but when you find out it's like: oh was that all there's to it? However, unlike a magic trick, math leaves you with something that can be actually useful.

[frostwhale]

And once you understand it you can't see how you didn't understand it before.

[cJ0th]

Hmm that's very interesting. I just don't understand how he made it through university. When I was enrolled in CS I somewhat got along with Algebra and was completely lost when it came to Analysis and so I dropped out. Back then I was working so hard at my courses I felt that I simply had no time to even consider "other stuff". I would like to know how it was obvious to him what he had to do.

[pierre_d528]

> Scholze started teaching himself college-level mathematics at

> the age of 14

And also that a few people have exceptional intellectual abilities, built-in.