https://www.quantamagazine.org/peter-scholze-and-the-future-...
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.
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.
You may be interested in this kind of laying out a proof: https://lamport.azurewebsites.net/pubs/proof.pdf
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"
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.
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"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.
> the age of 14
And also that a few people have exceptional intellectual abilities, built-in.