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..
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 understand that some degree of formalism is required to enable the sharing of knowledge amongst people across a variety of languages, but sometimes I'll read a white paper and think "wow, this could be written a LOT more simply".
Statistics is a major culprit of this.
I think you're confusing "I don't understand this" with "the man is keeping me down".
All fields develop specialized language and syntax because a) they handle specialized topics and words help communicate these specialized concepts in a concise and clear way, b) syntax is problem-specific for the same reason.
See for example tensor notation, or how some cultures have many specialized terms to refer to things like snow while communicating nuances.
> "wow, this could be written a LOT more simply"
That's fine. A big part of research is to digest findings. I mean, we still see things like novel proofs for the Pythagoras theorem. If you can express things clearer, why aren't you?
I'm surprised at how could you get at this conclusion. Formalisms, esoteric language and syntax are hard for everyone. Why would people invest in them if their only usefulness was gatekeeping? Specially when it's the same people who will publish their articles in the open for everyone to read.
A more reasonable interpretation is that those fields use those things you don't like because they're actually useful to them and to their main audience, and that if you want to actually understand those concepts they talk about, that syntax will end up being useful to you too. And that a lack of syntax would not make things easier to understand, just less precise.
OK, challenge accepted: find a way to write one of the following papers much more simply:
Fabian Hebestreit, Peter Scholze; A note on higher almost ring theory
https://arxiv.org/abs/2409.01940
Peter Scholze; Berkovich Motives
https://arxiv.org/abs/2412.03382
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What I want to tell you with these examples (these are, of course, papers which are far above my mathematical level) is: often what you read in math papers is insanely complicated; simplifying even one of such papers is often a huge academic achievement.
But I don’t believe it to be used as gatekeeping at all. At worst, hazing (“it was difficult for me as newcomer so it should be difficult to newcomers after me”) or intellectual status (“look at this textbook I wrote that takes great intellectual effort to penetrate”). Neither of which should be lauded in modern times.
I’m not much of a mathematician, but I’ve read some new and old textbooks, and I get the impression there is a trend towards presenting the material in a more welcoming way, not necessarily to the detriment of rigor.
The saying, "What one fool can do, another can," is a motto from Silvanus P. Thompson's book Calculus Made Easy. It suggests that a task someone without great intelligence can accomplish must be relatively simple, implying that anyone can learn to do it if they put in the effort. The phrase is often used to encourage someone, demystify a complex subject, and downplay the difficulty of a task.
Also, an additional thing is that videos are great are making people think they understand something when they actually don't.
What, as opposed to using ambiguous language and getting absolutely nothing done?
He separates conceptual understanding from notational understanding— pointing out that the interface of using math has a major impact on utility and understanding. For instance, Roman numerals inhibit understanding and utilization of multiplication.
Better notational systems can be designed, he claims.
That assumes it’s the language that makes it hard to understand serious math problems. That’s partially true (and the reason why mathematicians keep inventing new language), but IMO the complexity of truly understanding large parts of mathematics is intrinsic, not dependent on terminology.
Yes, you can say “A monad is just a monoid in the category of endofunctors” in terms that more people know of, but it would take many pages, and that would make it hard to understand, too.
A. Grothendieck
Understanding mathematical ideas often requires simply getting used to them
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...
Your rant would be akin to this if the sides are reversed: "It's surprising how many different ways there are to describe the same thing. Eg: see all the notations for dictionaries (hash tables? associative arrays? maps?) or lists (vectors? arrays?).
You don't have "the manual" of programming languages. "
Well, we kinda do when you can say "this python program" the problem with a lot of math is that you can't even tell which manual to look up.
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.
All of which is compounded by the desire to provide minimal "proofs from the book" and leave out the intuitions behind them.
Do you know the reason for that? The reason is that those problems are open and easy to understand. For the rest of open problems, you need an expert to even understand the problem statement.
Mathematics is old, but a lot of basic terminology is surprisingly young. Nowadays everyone agrees what an abelian group is. But if you look into some old books from 1900 you can find authors that used the word abelian for something completely different (e.g. orthogonal groups).
Reading a book that uses "abelian" to mean "orthogonal" is confusing, at least until you finally understand what is going on.
Hopefully interactive proof assistants like Lean or Rocq will help to mitigate at least this issue for anybody trying to learn a new (sub)field of mathematics.
If we are already venturing outside of scientific realm with philosophy, I'm sure fields of literature or politics are older. Especially since philosophy is just a subset of literature.
As far as anybody can tell, mathematics is way older than literature.
The oldest known proper accounting tokens are from 7000ish BCE, and show proper understanding of addition and multiplication.
The people who made the Ishango bone 25k years ago were probably aware of at least rudimentary addition.
The earliest writings are from the 3000s BCE, and are purely administrative. Literature, by definition, appeared later than writing.
That depends what you mean by "literature". If you want it to be written down, then it's very recent because writing is very recent.
But it would be normal to consider cultural products to be literature regardless of whether they're written down. Writing is a medium of transmission. You wouldn't study the epic of Gilgamesh because it's written down. You study it to see what the Sumerians thought about the topics it covers, or to see which god some iconography that you found represents, or... anything that it might plausibly tell you. But the fact that it was written down is only the reason you can study it, not the reason you want to.
That is what literature means: https://en.wiktionary.org/wiki/literature#Noun
Once someone writes it down, it is.
The person who hears that poem in circulation and records it in his notes has created literature; an anthology is literature but an original work isn't.
To use an example from functional programming, I could say:
- "A monad is basically a generalization of a parameterized container type that supports flatMap and newFromSingleValue."
- "A monad is a generalized list comprehension."
- Or, famously, "A monad is just a monoid in the category of endofunctors, what's the problem?"
The basic idea, once you get it, is trivial. But the context, the familiarity, the basic examples, and the relationships to other ideas take a while to sink in. And once they do, you ask "That's it?"
So the process of understanding monads usually isn't some sudden flash of insight, because there's barely anything there. It's more a situation where you work with the idea long enough and you see it in a few contexts, and all the connections become familiar.
(I have a long-term project to understand one of the basic things in category theory, "adjoint functors." I can read the definition just fine. But I need to find more examples that relate to things I already care about, and I need to learn why that particular abstraction is a particularly useful one. Someday, I presume I'll look at it and think, "Oh, yeah. That thing. It's why interesting things X, Y and Z are all the same thing under the hood." Everything else in category theory has been useful up until this point, so maybe this will be useful, too?)
You know the meme with the normal distribution where the far right and the far left reach the same conclusion for different reasons, and the ones in the middle have a completely different opinion?
So on the far right you have people on von Neumann who says "In mathematics we don't understand things". On the far left you have people like you who say "me no mats". Then in the middle you have people like me, who say "maths is interesting, let me do something I enjoy".
To date I have not met anyone who thought he summed the terms of the infinite series in geometric series term by term. That would take infinite time. Of course he used the expression for the sum of a geometric series.
The joke is that he missed a clever solution that does not require setting up the series, recognising it's in geometric progression and then using the closed form.
The clever solution just finds the time needed for the trains to collide, then multiply that with the birds speed. No series needed.
For example there's a story that von Neumann told Shannon to call his information metric entropy, telling S "nobody really understands entropy anyway." But if you've engaged with Shannon to the point of telling him that quantity seems to be the entropy, you really do understand something about entropy.
So maybe v N's worry was about really undertanding math concepts fully and extremely clearly. Going way beyond the point where I'd say "oh I get it!"
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.