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The other traditional problem with mathematics is terse and obscure notation.

It's only a problem if you are unfamiliar with the subject area, and if you are, you have very little chance to understand the paper or book anyway. Mathematics is just difficult, way more difficult than most other stuff people might be doing.

Here's an example: consider this classical paper[1], "Vector Bundles Over An Elliptic Curve" by M. Atiyah. The author is a well known and regarded mathematician, and the paper itself has around 1000 citations. Sections 1. and 2., "Generalities" and "Theorems A and B", just recall the basic notions and theorems that are absolutely necessary to have any grasp whatsoever as to what is going on in the paper.

If you know nothing about algebraic geometry, the classical way to learn these is Robin Hartshorne's famous textbook "Algebraic Geometry". It is famous for being both good, self-contained to a large degree, and having very well chosen exercises, but also for being quite terse and often difficult to follow. Here's an Amazon link[2]. You can look at the table of contents. To really fully understand these two sections from Atiyah's paper, you need to have very good understanding of Chapter 2. "Schemes", and at least first 5-6 sections of Chapter 3. "Cohomology". This is 200 pages of pretty terse mathematics. At a fast understanding pace of 4 pages a day, it will take you two months to even have some basic toolset to understand the Atiyah's paper.

But, if you try to understand the Harthshorne's textbook, you'll quickly find out that it also has some prerequisites of its own. Also, the "4 pages a day" pace is only possible if you've already spent 2-3 years learning how to learn mathematics.

I encourage anyone to try to understand even first 2-3 sentences of the "Generalities" section. Google and Wikipedia the unfamiliar terms, you can also try to look it up in Harthorne's textbook, or Vakil's lecture notes, or any other source. The notions used in these first 2-3 sentences are basic to anybody working in the field, and yet one needs to spend hours to fully understand these when starting from scratch.

Compare this to other famous paper from other field, "The Market for Lemons" by G. Akerlof[3]. This is also a very famous paper by a well regarded economist, who received a Nobel Memorial Prize In Economic Sciences for it. It is a much easier read, precisely because the economic sciences do not operate on nearly the same level of complexity as mathematics. Once you know what are some common sense notions like, supply, demand, utility etc., and some very basic calculus, you can easily follow the argument without too much training.

My point here is that it's not that it's difficult to read mathematics just because it uses terse and obscure notation. It all is just genuinely difficult and complex, and it is impossible to invent better notation that will transfer days and weeks worth of understanding straight to the reader's brain. I would love it to be the case, but then it would cease to be as fun and rewarding to really understand.

[1] - https://math.berkeley.edu/~nadler/atiyah.classification.pdf [2] - https://www.amazon.com/Algebraic-Geometry-Graduate-Texts-Mat... [3] - https://www.iei.liu.se/nek/730g83/artiklar/1.328833/AkerlofM...




> but then it would cease to be as fun and rewarding to really understand

Mathematics is fun to understand because it is [artificially] hard to understand?


No, it is fun and rewarding because it is genuinely hard to understand. Once you really understand it, it tends to become more obvious in hindsight, but good luck getting your understanding across to someone else who hasn't spent as much time thinking about this as you.


Economics would also be really hard to understand if it didn't operate on the real world: something very familiar to us.

I can imagine a mathematical version of economics operating on some abstract constructs designed to emulate an economic system. It would reuse no terminology from the real world for the sake of producing an abstract notion, completely (or at least artificially) decoupled from the system it was designed to emulate.

I would imagine that Akerlofs paper, encoded into this form, would be at least as hard and involved to understand as the one from Atiyah.


As long as the as the "abstract constructs" themselves weren't much more complicated than the current real-world economic concepts, it would only be slightly more difficult. You would need some background reading, but not any more than skimming relevant Wikipedia articles for definitions. And in that case, you'd have every reason to be pissed at economists who don't present their findings in much simpler and obvious way.

If, however, the abstract constructs were much more complicated then the current economic concepts, and if you were trying to solve problems on much higher level of abstraction than economists currently are, then it would just be mathematics, and indeed it would be more difficult.

I, personally, cannot imagine how you could rewrite Akerlof's paper to be as hard to understand as Atiyah's. I can, with great difficulty, follow Atiyah's paper only because I spent literally _years_ learning the necessary background material. I am completely unable to relay my understanding to someone who hasn't spent years doing the same. I wish I was -- I'd revolutionize algebraic geometry then, just like Alexander Grothendieck revolutionized it around when the Atiyah's paper was written. On the other hand, if someone rewrote Akerlof's paper in an intentionally obscure way, you could easily rewrite it back in a clear way, once you spend the effort to understand the obscure version yourself.


> Here's an example: consider this classical paper[1], "Vector Bundles Over An Elliptic Curve" by M. Atiyah. The author is a well known and regarded mathematician, and the paper itself has around 1000 citations.

Sir Michael Atiyah is an Abel prize and Fields medal winner. Apart from that, he has an excellent sense of humor. I had a chance to grab lunch with him last year and it was quite entertaining to say the least.




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