The biggest problem is that it is very unfriendly to uninitiated newcomers and makes insufficient effort to draw people in. You end up with a culture that is unfortunately insular and has trouble engaging with even engineers and scientists, much less the general public. It’s also not very friendly to people who approach problems in different ways: symbol pushing has been elevated and anyone who has difficulty with symbol pushing (for whatever reason) ends up at least partly excluded.
Students who have a lot of practice/experience by the time they get to be teenagers (often via extra-curricular help and support) are much better prepared than those without that practice. Which is of course not a problem per se, you see the same in any field and it’s great if kids want to learn ahead of their peers. But then the content, curricular design, and pedagogy of mathematics courses leave students with the impression that those differences in preparation are due to innate differences in aptitude (“I suck at math”; “she’s just a math person”; ...), toss less well prepared students into the deep end to sink without enough support, and ultimately chase a huge number of people away who might otherwise find the subject beautiful and interesting, and could meaningfully contribute.
Well, we can't know that until we find (or won't find) the more effective way of teaching (or a way to do math without "symbol pushing" for that matter).
Until then it will not be wise to break what works (even for a minority of students).
I don’t think it’s fair to say they are set up to do that. They weren’t conceived with that purpose. It’s just a fact of life that once we’ve invested a huge amount of effort in one set of conventions it’s very costly to change those conventions.
I don’t mean that some secret committee got together to “set up” all of the social incentives of the entire school system, university system, textbook industry, scientific publication system, engineering fields, etc.
What I mean is that there are incentives for the people involved in those systems which are extremely difficult to reform, and as long as the current incentives prevail it is all but impossible for anyone to refactor things like basic mathematical notions and notations.
Switching and retraining costs are high, gaps in inter-operability are expensive, and there is almost nobody who will achieve any career advancement through promoting changes to the high school and early undergraduate curriculum.
Mathematicians are generally most interested in pushing on the shiny boundaries of the field rather than trying to clean up the centuries-old material for novices. Teachers have their hands full enough with their students to do much new research in pedagogy. Practitioners in industry have their own problems to solve.
Such as? Many branches of mathematics have their own mutually unintelligible dialects of notation. Many longer papers or Ph.D thesis will just create notation just for the context of the paper.
An obvious one is that traditional mathematical convention requires single glyph variable names due to the unfortunate decision to save paper by denoting the product by juxtaposition. That’s an admittedly trivial example though, even though the higher order consequences are considerable.
Computing science is when notation came into its own. Younger mathematicians have taken those lessons to heart, but as the old saying goes, progress comes one funeral at a time.
Being forced to mechanically parse and interpret a syntax has a way of really bringing out any ambiguity.
> that traditional mathematical convention requires single glyph variable names due to the unfortunate decision to save paper by denoting the product by juxtaposition.
It's not to save paper or because of the product. You don't know the solution to the problem you are working on from the beginning and most of the time is spent writing and writing and writing in a scratchpad trying to solve what you need. Anything longer than a single glyph for variables would be too tedious so everyone evolved to use single letters. And then the papers are written with the same convention since it's natural. You have variable names though with the use of subscripts with the added benefits that it can be (and is) used to elegantly group relevant variables together giving you some sort of abstraction
I once wrote a comment about it here on HN - language in maths is not a programming language used to tell a computer how to go from A to B, but a natural language used to talk about maths between peers. Every natural language have idioms, inconsistences and other quirks. Polish will not change for you so it's easier for you to learn it, it will change in the way that let's polish people communicate better with each other which also include a lot of historical and cultural happenstances. Same with maths
There are attempts like Esperanto and other artificial languages like that and I think any attempts at 'codification' of maths to use some programming language has the same chance of success of wide adoption
> They are attempts like Esperanto and other artificial languages like that and I think any attempts at 'codification' of maths to use some programming language has the same chance of success of wide adoption
Aren't existing programming languages already types of codified artificial math dialects which have seen wide adoption
Programming languages are more for humans than for computers. Otherwise we’d be writing our programs in 1s and 0s, and extending our editors in Emacs Binary and VSCode BinaryScript.
> language in maths is not a programming language used to tell a computer how to go from A to B, but a natural language
Right, we're on the same page, I just think this is a bad thing and you evidently think it's a good thing. I'm well aware many mathematicians don't, because it's how they were trained and unlearning is the hardest kind of learning. The ambiguity[1] of natural language is observably ill-suited for formal reasoning, and the experience of computing science has shown this conclusively.
Do bear in mind that the pioneers in our field were virtually all trained mathematicians. They were well aware of the historic faults of the field because having to make programs actually work forced them to be.
The legacy fuzzy pencil and paper approach of traditional mathematics is going to end up being to proper formal mathematics just as what's now called philosophy is to formal logic.
> single glyph variable names due to the unfortunate decision to save paper by denoting the product by juxtaposition.
Programmers tend to have this lack of fluency with written math that they completely miss: the concise names are not to save paper or make writing easier or anything like that. They're because they make the structure of expressions easier to visually identify and parse. The shapes of expressions are an incredibly important feature of the language and often contain implicit structural analogies. You need to be able to see those analogies to correctly read mathematics, and long variable names would obscure that part of the language.
I suppose it's similar to having enough fluency in a natural language to mechanically translate the words of a poem, but you can't properly read things like the metre, so you've unknowingly missed half of what the author originally wrote and lost it all in translation.
> An obvious one is that traditional mathematical convention requires single glyph variable names due to the unfortunate decision to save paper by denoting the product by juxtaposition.
Generally you'd use upright text in square brackets to denote longer variable names, the notation is often seen in applied fields. But this quickly becomes clunky with longer expressions.
A bit of that even though apl arguably push things too far (maybe that's what you implied). Parametric types also help making point, abstract combinators, recursive schemes.. All very helpful to define and manipulate ideas.
Limits are not an inherent part of calculus. You can do all calculus relevant for the physical world just fine with nilpotent infinitesimals if you but give up excluded middle.
I've heard of this constructivist approach to calculus, but hadn't made the connection with nilpotents. that's really interesting, could you explain why nilpotenxy and forgoing the law of the excluded middle relate to each other?
You can use nilpotents with classical logic and the excluded middle. This is called dual numbers and it's already a good model for "calculus without limits". They are like complex numbers, but instead of x^2=-1 you set x^2=0.
However, if you want to get really serious about that, you'll need that zero plus an infinitessimal be equal to zero. This is impossible in classical logic due to the excluded middle (which forces each number to be either equal to zero or non-zero).
My point is not that limits are an inherent part of calculus. My point is that calculus as currently taught mixes infinitesimal-like notation that predates limits with limit-based calculus.
Students who have a lot of practice/experience by the time they get to be teenagers (often via extra-curricular help and support) are much better prepared than those without that practice. Which is of course not a problem per se, you see the same in any field and it’s great if kids want to learn ahead of their peers. But then the content, curricular design, and pedagogy of mathematics courses leave students with the impression that those differences in preparation are due to innate differences in aptitude (“I suck at math”; “she’s just a math person”; ...), toss less well prepared students into the deep end to sink without enough support, and ultimately chase a huge number of people away who might otherwise find the subject beautiful and interesting, and could meaningfully contribute.