If the idea is that the right notation will make getting insights easier, that's a futile path to go down on. What really helps is looking at objects and their relationships from multiple viewpoints. This is really what one does both in mathematics and physics.

Someone quoted von Neumann about getting used to mathematics. My interpretation always was that once is immersed in a topic, slowly it becomes natural enough that one can think about it without getting thrown off by relatively superficial strangeness. As a very simple example, someone might get thrown off the first time they learn about point-set topology. It might feel very abstract coming from analysis but after a standard semester course, almost everyone gets comfortable enough with the basic notions of topological spaces and homeomorphisms.

One thing mathematics education is really bad at is motivating the definitions. This is often done because progress is meandering and chaotic and exposing the full lineage of ideas would just take way too long. Physics education is generally far better at this. I don't know of a general solution except to pick up appropriate books that go over history (e.g. https://www.amazon.com/Genesis-Abstract-Group-Concept-Contri...)

A little off topic perhaps, but out of curiosity - how many of us here have an interest in recreational mathematics? [https://en.wikipedia.org/wiki/Recreational_mathematics]

1. Can we reinvent notation and symbology? No superscripts or subscripts or greek letters and weird symbols? Just functions with input and output? Verifiable by type systems AND human readable

2. Also, make the symbology hyperlinked i.e. if it uses a theorem or axiom that's not on the paper - hyperlink to its proof and so on..

One last rant point is that you don't have "the manual" of math in the very same way you would go on your programming language man page and so there is no single source of truth.

Everybody assumes...

This fundamental truth is embedded in the common symbols of arithmetic...

+ ... one line combined with another ...linear...line wee

- ...opposite of + one line removed

x ...eXponential addition, combining groups

•/• ... exponential breaking into groups ...also hints at inherent ratio

From there it's symbols that describe different objects and how to apply the fundamental arithmetic operations; like playing over a chord in music

The interesting work is in physical science not the notation. Math is used to capture physics that would be too verbose to describe in English or some other "human" language. Which IMO should be reserved for capturing emotional context anyway as that's where they originate from.

Programming languages have senselessly obscured the simple and elegant reality of computation, which is really just a subset of math; the term computer originated to describe humans that manually computed. Typescript, Python, etc don't exist[1]. They are leaky abstractions that waste a lot of resources to run some electromagnetic geometry state changes.

Whether it's politics, religion or engineering, "blue" language, humans seem obsessed with notation fetishes. Imo it's all rather prosaic and boring

[1] at best they exist as ethno objects of momentary social value to those who discuss them

A. Grothendieck

Understanding mathematical ideas often requires simply getting used to them

All of which is compounded by the desire to provide minimal "proofs from the book" and leave out the intuitions behind them.

This type of resoning becomes void if instead of "AI" we used something like "AGA" or "Artificial General Automation" which is a closer description of what we actually have (natural language as a programming language).

Increasingly capable AGA will do things that mathematitians do not like doing. Who wants to compute logarithmic tables by hand? This got solved by calculators. Who wants to compute chaotic dynamical systems by hand? Computer simulations solved that. Who wants to improve by 2% a real analysis bound over an integral to get closer to the optimal bound? AGA is very capable at doing that. We just want to do it if it actually helps us understand why, and surfaces some structure. If not, who cares it its you who does it or a machine that knows all of the olympiad type tricks.

Probably they are trying to romanticize something that may not sound good if told plainly.

Face it. Mathematics is one of fields strongly affected by AI, just like programming. You need to be more straight forward about it rather than beating around the bush.

To simply put, it appears to be a struggle for redefining new road map, survival and adoption in AI era.

For example, Dvoretzky-Rogers theorem in isolation is hard to understand.

While more applications of it appear While more generalizations of it appear While more alternative proofs of it appear

it gets more clear. So, it takes time for something to become digestible, but the effort spent gives the real insights.

Last but not least is the presentation of this theorem. Some authors are cryptic, others refactor the proof in discrete steps or find similarities with other proofs.

Yes it is hard but part of the work of the mathematician is to make it easier for the others.

Exactly like in code. There is a lower bound in hardness, but this is not an excuse to keep it harder than that.

> "few of us"

You see, if you plebs are unable to understand our genius its solely due to your inadequacies as a person and as an intellect, but if we are unable to understand our genius, well, that's a lamentable crisis.

To make Mathematics "understandable" simply requires the inclusion of numerical examples. A suggestion 'the mathematics community' is hostile to.

If you are unable to express numerically then I'd argue you are unable to understand.

1. Study predicate logic, then study it again, and again, and again. The better and more ingrained predicate logic becomes in your brain the easier mathematics becomes.

2. Once you become comfortable with predicate logic, look into set theory and model theory and understand both of these well. Understand the precise definition of "theory" wrt to model theory. If you do this, you'll have learned the rules that unify nearly all of mathematics and you'll also understand how to "plug" models into theories to try and better understand them.

3. Close reading. If you've ever played magic the gathering, mathematics is the same thing--words are defined and used in the same way in which they are in games. You need to suspend all the temptation to read in meanings that aren't there. You need to read slowly. I've often only come upon a key insight about a particular object and an accurate understanding only after rereading a passage like 50 times. If the author didn't make a certain statement, they didn't make that statement, even if it seems "obvious" you need to follow the logical chain of reasoning to make sure.

4. Translate into natural english. A lot of math books will have whole sections of proofs and /or exercises with little to no corresponding natural language "explainer" of the symbolic statements. One thing that helps me tremendously is to try and frame any proof or theorem or collection of these in terms of the linguistic names for various definitions etc. and to try and summarize a body of proofs into helpful statements. For example "groups are all about inverses and how they allow us to "reverse" compositions of (associative) operations--this is the essence of "solvability"". This summary statement about groups helps set up a framing for me whenever I go and read a proof involving groups. The framing helps tremendously because it can serve as a foil too—i.e. if some surprising theorem contravene's the summary "oh, maybe groups aren't just about inversions" that allows for an intellectual development and expansion that I find more intuitive. I sometimes think of myself as a scientist examining a world of abstract creatures (the various models (individuals) of a particular theory (species))

5. Contextualize. Nearly all of mathematics grew out of certain lines of investigation, and often out of concrete technical needs. Understanding this history is a surprisingly effective way to make many initially mysterious aspects of a theory more obvious, more concrete, and more related to other bits of knowledge about the world, which really helps bolster understanding.