Math notation is very consistent and it’s optimized for convenience. It’s been constantly refined over last centuries. People who complain about notation usually actually have problem with the substance, and the complaints about notation is just a coping mechanism, to deny hurtful reality that one can’t understand something hard, blaming external factors instead. Ask yourself: if notation was genuinely confusing, why would mathematicians make themselves suffer needlessly?
Now, to be sure, some people and some books are better at teaching than others, but it typically has nothing to do with notation used, and everything to do with the order of introduction of concepts, level of detail of explanation (which can be both too high and too low), amount and quality of examples, etc. However, the core issue here is that some things are actually genuinely hard, and people of average intelligence simply cannot grasp them without expending ludicrous amounts of effort.
If you have some concrete suggestions about mathematical notation, ways it could be improved in more than superficial manner, I (and the rest of mathematical community) is very much open to hear them. Improvements in notation do happen regularly, and when they are valuable, they reach wide acceptance. For example, in the second half of 20th century, the notation of commutative diagrams have been invented, and it spread like a wildfire, because it genuinely facilitates understanding.
>Ask yourself: if notation was genuinely confusing, why would mathematicians make themselves suffer needlessly?
I think its pretty clear that momentum causes a lot of nomenclature pain. You can't just redesign entire fields of understanding every couple decades. For instance, what other fields use single greek characters to label concepts in a seemingly unpredictable arrangement? Often, local maxima are found because concepts are added in the context of the field as it already existed.
And its not just math. Most sciences have this problem. It is what it is but its silly to say we're in the best of all possible worlds just because math has been around a long time.
> For instance, what other fields use single greek characters to label concepts in a seemingly unpredictable arrangement?
This is a perfect example of what I was talking about. Greek letters are typically used in opposition to Latin ones in order to distinguish what programmers can think of as “type” of concepts. For example, when doing geometry, you might want to designate angles with Greek letters, and points or line segments with Latin ones. This makes it much easier to mentally keep track of what name correspond to which object.
Yes, learning Greek letters for the first time is some amount of initial overhead (not much, as typically used ones are similar to Latin anyway, nobody starts with psi or xi). However, crucially, this overhead is paid once, and pales compared to the difficulty of learning the concepts being represented in the first place. It is never the case that replacing Greek letters with Latin makes students go “oh thank you, now I understand everything!”, instead, things are typically just as hard as before. However, replacing Latin with Greek might actually do that, by reducing the mental overhead through introduction of categories (types) of objects.
>Math notation is very consistent and it’s optimized for convenience.
Hard disagree. The most important thing I learned is that it is all made up on the spot to the point that the lecture material explicitly says that books have used 6 different forms of notation for the exact same concept. When you understand that you drop any pretense of "design" in the notation. That helps you abandon foolish ideas that it is "consistent" and that the only thing it is optimized for is the author. When you understand that then it's just a meaningless barrier to overcome but it also becomes easy to overcome precisely because it is that trivial. You just get used to it and e.g. learn the alphabets of the dozen languages (including klingon because the lecturer had to make that joke) from which the variable names where sourced from. Once you did the meaningless grind the barriers are gone.
> People who complain about notation usually actually have problem with the substance, and the complaints about notation is just a coping mechanism, to deny hurtful reality that one can’t understand something hard, blaming external factors instead.
No it is quite simple. You can't understand an easy or hard concept if you can't read it. I still remember how I understood nothing in the first semester. Then when I was preparing for the exam everything was extremely easy because the notation was understood by that point.
>If you have some concrete suggestions about mathematical notation, ways it could be improved in more than superficial manner, I (and the rest of mathematical community) is very much open to hear them.
As I already said that is meaningless because there is no universal notation. "The mathematical community" will adopt a fraction of proposals and further splinter into separate "factions".
> if notation was genuinely confusing, why would mathematicians make themselves suffer needlessly?
Because they are unable to change it. Just like with any thing evolved over a long time, like music notation, languages, even, to some extent, programming languages. Every change brings a lot of pushback, it's a monumental task to create a new one and even more so is getting any traction with it.
That doesn’t square well with the fact that notation keeps getting refined and improved. There is no pushback for genuine improvements. Biggest problem here is that there rarely are changes that clearly and meaningfully improve situation over status quo. I gave one example above, but overall, I am not going to take complaints about notation being obstacle to understanding seriously without concrete ways how to meaningfully improve it. You can of course keep complaining that it’s confusing, but without proposals for improvements, you’re actually complaining about the difficulty of substance, not the notation, and it says more about you than about notation.
That's a bad example: first, it's a question of definition, not of notation. Second, it's defined pretty consistently within physics, and pretty consistently within electrical engineering (none of which is mathematics).
> Ask yourself: if notation was genuinely confusing, why would mathematicians make themselves suffer needlessly?
Sometimes notation elides 'obvious details'. The details are obvious to those that have already gone through the learning curve, which is easier if one is strongly connected to other working mathematicians than for outsiders. Mathematicians barely notice inadequate or unusual notation. Outsiders struggle and spend significant energy just deciphering the notation.
Anecdote 1: Sometimes notation is too terse: single letters. Granted, efficient for whiteboard scribbling. Would be really nice to standardize an appendix for notation. E[X] = <expr>. Hmmm, what could E be? By the fifth paper, somebody bothers to write 'expected value' in plain English and the mystery in unambiguously clarified. In a voice-based interaction this is a non-issue, not so for those that only have text to deal with. This compounds as a novice has to juggle a set of mysterious symbols with tens of elements.
Anecdote 2: Long long time ago (before ubiquitous email or http:// took off) a young student spent some time working through a handful of type theory academic papers that somehow trickled into his corner of the Universe. The type inference rule notation (https://en.wikipedia.org/wiki/Type_rule), which is rather straightforward in retrospect, ate up more time that he's willing to admit. Γ is just a set of judgments and Γ |- expr is just a notation for 'Γ includes the judgement expr'? Then why don't they use the standard expr ∈ Γ, this is confusing? Are there some examples? Is there some code that one could possibly run through a debugger? The questions remained unanswered, as there was noone he knew in the same linguistic sphere interested in the topic. And the papers themselves never detailed such obvious notation details.
PS. I agree that the bigger obstacle is the lack of proper big picture 'why do we even bother with these concepts / theorems'. At an extreme, there is (used to be?) a certain style of math books consisting exclusively of a dry litany of 'Definition 1.2.3', 'Theorem 3.2.4', 'Corollary 2.3.1'. Very rigorous and very difficult to ascertain what problems they were trying to solve.
> Anecdote 1: Sometimes notation is too terse: single letters. Granted, efficient for whiteboard scribbling. Would be really nice to standardize an appendix for notation. E[X] = <expr>. Hmmm, what could E be? By the fifth paper, somebody bothers to write 'expected value' in plain English and the mystery in unambiguously clarified. In a voice-based interaction this is a non-issue, not so for those that only have text to deal with. This compounds as a novice has to juggle a set of mysterious symbols with tens of elements.
I really cannot conceive how one can learn what the concept of “expected value” of a “random variable” means, without encountering E[X] notation. This is a technical concept having a technical meaning, and any place that actually defines this meaning will teach you this notation. If you see this notation for the first time in some academic paper, but haven’t ever read any probability textbook, it means that you almost never actually learned the concept, which is my entire point. You might have some intuitive understanding derived purely from the literal meaning of the words “expected value”, but without actually getting technical, this intuitive understanding is mostly superficial, and, as such, not very useful. You won’t be, for example, be able to answer such fundamentally important questions like “is expected value of sum of random variables a sum of expected values of each? Is expected value of product a product of expectations?”. You can’t know answers to these questions without having ever seen E[X] notation, and if you don’t know the answers, your problem is with the actual concept, not notation.
> PS. I agree that the bigger obstacle is the lack of proper big picture 'why do we even bother with these concepts / theorems'. At an extreme, there is (used to be?) a certain style of math books consisting exclusively of a dry litany of 'Definition 1.2.3', 'Theorem 3.2.4', 'Corollary 2.3.1'. Very rigorous and very difficult to ascertain what problems they were trying to solve.
I agree that it very much often is a problem. It’s not a problem of notation, though.
Technically speaking, there are other cultural spaces than US/English. In one such space E[X] is written M(X) and called 'average value'. But that's quibbling. 'Expected value' is simply 'weighted sum' over possible values with respective probabilities as weights: weighted average value. Not exactly rocket science if one groks what a probability distribution is. But even that is quibbling. The more interesting point is that some would rather starting learning from concrete applications instead of pacing through a seemingly endless dry litany of definitions. Cryptic notation is unhelpful for this style of learning.
Ken Iverson thought mathematical notation was inconsistent, so he wanted to invent a better notation for thinking, which became the programming language APL. But APL syntax never captured the mainstream. Neither did Lisp for that matter. It was C-style syntax that ended up dominating.
