Mathematics is hard for mathematicians to understand too

I find software engineers spend too much time focused on notation. Maybe they are right to do so and notation definitely can be helpful or a hindrance, but the goal of any mathematical field is understanding. It's not even to prove theorems. Proving theorems is useful (a) because it identifies what is true and under what circumstances, and (b) the act of proving forces one to build a deep understanding of the phenomenon under study. This requires looking at examples, making a hypothesis more specific or sometimes more general, using formal arguments, geometrical arguments, studying algebraic structures, basically anything that leads to better understanding. Ideally, one understands a subject so well that notation basically doesn't matter. In a sense, the really key ingredient are the definitions because the objects are chosen carefully to be interesting but workable.

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...)

I think this would be extremely valuable: “We need to focus far more energy on understanding and explaining the basic mental infrastructure of mathematics—with consequently less energy on the most recent results.” I’ve long thought that more of us could devout time to serious maths problems if they were written in a language we all understood.

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]

I love math but the symbology and notations get in my way. 2 ideas:

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..

I was writing a small article about [Set, Set Builder Notation, and Set Comprehension](https://adropincalm.com/blog/set-set-builder-natatio-set-com...) and while i was investigating it surprised me how many different ways are to describe the same thing. Eg: see all the notation of a Set or a Tuple.

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...

A lot of people here suggesting they'd be great mathematicians if only it wasn't for the pesky notation. What they are missing is that the notation is the easy part..

> As Venkatesh concludes in his lecture about the future of mathematics in a world of increasingly capable AI, “We have to ask why are we proving things at all?” Thurston puts it like this: there will be a “continuing desire for human understanding of a proof, in addition to knowledge that the theorem is true.”

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.

Just the other day I was listening to EconTalk on this: https://www.econtalk.org/a-mind-blowing-way-of-looking-at-ma...

Mathematics is such an old field, older than anything except arguably philosophy, that it's too broad and deep for anyone to really understand everything. Even in graduate school I often took classes in things discovered by Gauss or Euler centuries before. A lot of the mathematical topics the HN crowd seems to like--things like the Collatz conjecture or Busy Beavers--are 60, 80 years old. So, you end up having to spend years specializing and then struggle to find other with the same background.

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

The views quoted are just as cryptic as modern mathematics. Did mathematicians lose the ability to convey stuff tin plain simple ways?

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.

I thought we were well past trying to understand mathematics. After all, John von Neumann long ago said "In mathematics we don't understand things. We just get used to them."

"The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration... the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it... yet finally it surrounds the resistant substance."

A. Grothendieck

Understanding mathematical ideas often requires simply getting used to them

I recently came to realize the same things about physics. Even physicists find it hard to develop an intuitive mental picture of how space-time folds or what a photon is.

> Venkatesh argued that the record on this is terrible, lamenting that “for a typical paper or talk, very few of us understand it.”

> "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.

As someone who has always struggled with mathematics at the calculational level, but who really enjoys theorems and proofs (abstract mathematics), here are some things that help me.

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.

Mathematics is hard when there is not much time invested in processing the core idea.

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.