Terry Tao on some desirable properties of mathematical notation
mathoverflow.net
mathoverflow.net
For a (even more) subjective point of view, as part of my studies I had oral practicals every 2 weeks or so where I was basically standing at a whiteboard along with 2 classmates, each solving problems given to us by an examiner for ~1 hour. It's simply easier for a lot of students to draw a symbol that's legible from 5 meters away than some words without making them huge.
1. Write a good narrative
and then
2. Write good code
Which is to say that a mathematics text is an instance of literate programming.
For a mathematician it is the opposite, they just wish it would be written with symbols so that they could know precisely what the book is trying to say
It helps somewhat, but it isn’t a panacea.
We still have to comment our code and add context.
> speed = distance / time
is less clear than
> speed is the ratio of distance over time
? To me the first equation is genuinely easier to read for the purposes of understanding, not just for formal manipulation. For larger equations the difference is only more stark, not less. I have a maths PhD so your comment about symbols being for novices doesn't apply.
I suppose the beauty of the first equation is that the objects (speed, distance, time) are visually very distinct from the operations and relationships. In the second form, there's a bit of a word soup so you need to "manually" parse the sentence rather than letting your eyes (really the visual cortex) do that bit of the processing for you.
* Usually you don't come across a journal article or even blog post with a single isolated equation. So any new or unusual notation can be explained once and reused many times.
* Even if you did have an isolated equation with unusual notation, I still think it's more clear to define the notation and then show the equation rather than spelling it out it words! (I'm sure you could find terrible counterexamples!) The visual benefit seems so great to me that it would be worth splitting it into two parts. A bit like a diagram or a graph can make things clearer even if it needs a bit of explanation.
This sentence seems to contradict the rest of your comment; did you mean "I find it quite unusual in practice for genuinely new symbolic notation to NOT be introduced by an author."?
Most academic texts I've read will use something like
s = d/t
Some will even make up their own notation:
s = d(t)
Some even take it a step further and use random greek letters (without exhausting the English alphabet first ofc.), where small gamma and big gamma mean completely different and unrelated things.
Edit: Though I agree, mathematical notation lends itself to using single letters to denote objects. This can of course be problematic. I'm a fan of 'pseudocode' - somewhere between natural language and rigorous notation - where possible.
Slightly worse than that. That equation is always written as v = s/t. v represents velocity and s represents distance, for some reason.
- Any two distinct points in the space have disjoint neighbourhoods.
- ∀x,y ∈ X with x ≠ y, ∃U,V ⊂ X s.t. x ∈ U, y ∈ V and U ∩ V = ∅.
Another example would be Navier-Stokes equations, where they're much easier to understand in words than in symbols. Symbols are ok when you don't have to search too much to see what they mean and when the idea you're trying to transmit with them is relatively simple, but trying to build complicated phrases and definitions with symbols, for me, ends up being a mess.
edit: fixed hausdorff space definition, points should be distinct
Within a certain domain, it is assumed you know basic definitions within it that can be used to talk about more complex things.
Since there is in the modern world almost no use case where you're heavily space contrained (maybe the final print version of an article for some publications ? I'm not that familiar with the research world), I don't see why you'd try to choose one instead of including both where needed.
Communication is hard, and unfortunately doing it well requires experience and thought rather than simplistic rules like "always use symbols rather than words" or vice-versa.
Amusingly and to go in the direction of your argument, this is not a very rigorous definition of a Hausdorff space. You should introduce the topology τ associated with X and specify that U and V are open sets of τ.
- speed is the derivative of position with respect to time.
and
- Let x(t) be the position of an object at time t then its speed v(t) is:
v(t) = (dx/dt)(t)
For example, reading 5-7, I don’t have to translate the - symbol to the word “subtract”. I know this is -2. And I don’t translate the - in that to the word “negative,” and certainly not to the word “subtract”. And it’s vastly faster to agree 5-7=-2 is correct than “5 subtract 7 equals negative 2.”
Symbols are how mathematicians think.
Sure you can take simple English statements, and write them with quantifiers, and claim the English is simpler. But going the other way, expressing complex items in English, is a non-starter. English is far too sloppy, whereas the quantifier version is mathematically precise.
Try converting something professional, such as Godel's incompleteness proofs, into English. Without precise quantifiers you'd quickly get lost, make mistakes, and take forever to get anywhere.
For example, look at page 17 of the proof [1], at AG(6)(a) (after the "Thus"), where there is a long statement in logic. Convert to English (near impossible, certainly not possible without something like parentheses) and tell me it's easier for a logician to read. It's not. As written it's concise and parseable without confusion for a logician.
The simple English sentence case is a tiny part of what mathematicians do.
At the very least, you will struggle to persuade me that the use of \wedge is easier to understand than the English word "and" with line breaks.
Also you've picked a specific example where the objects of study are these long strings of symbols. Of course any paper worth its salt is going to use them - they're literally the things that the paper is there to examine. It's the metamathematical statements in this context, not the quotation of the formulas under study, that I want to replace with English.
No - just pointing out that no matter how hard you studied that an English equivalent of such a terse expression will be a mess, vastly harder to understand.
>At the very least, you will struggle to persuade me that the use of \wedge is easier to understand than the English word "and" with line breaks.
You're making my case for me :)
The word "and" and "or" are ambiguous in common English, and have no common or even technical well-defined precedence. "Or" in English both can mean "inclusive or" or "exclusive or," yet most people simple write "or".
In math they have well defined precedence, and math has parentheses to order correctly, unlike English.
The only reason you find English easier to understand is you have used more than math symbols at a ratio that makes that true for you. It's not true for everyone, especially professionals, that use some set of notation a lot.
>Also you've picked a specific example where the objects of study are these long strings of symbols
Avoiding the point. The math does not have to self-referential to use complex expressions not amenable to writing in English. This expression is not complex because it's a meta argument. Such expressions can occur in all sorts of places.
At this point I'm sure your present these red herrings to avoid considering that your opinion is not common among professionals that can read symbols much faster and more accurately than can be done in English.
No sense in continuing.
On this we agree, at least!
And other times in some fields there is a lack of well-defined symbols and notations. I don't recall the specifics, but in game theory there is some notation that is ambiguous. There are two methods or forms of representation, and the field is split on which version the notation stands for. It's also frequently not mentioned explicitly in papers, so you have to dig around a bit to see if you can find a usage that makes the meaning clear.
I don't know why we can't just have both.
For example, Wikipedia describes a 'field' like this:
"In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do."
It doesn't make sense to me. What does it mean if an operation 'is defined' on a set? Does it mean that any 2 elements combined together using that operation always need to output an element which is also in the same set? But if that was the case then "behave as the corresponding operations on rational and real numbers do" would mean that the fields would always need to be of infinite size (have an infinite number of elements) wouldn't it? Because if the field had a limited number of elements and you added the last two (highest) elements together, the property which requires that the result also be present in the same set could not be met because the result would be greater than the highest element in that set...
The problem is that if you start with a highly abstracted math concept and you dig through all the links and definitions of sub-concepts, they all have huge gaps like this... So when you try to combine all the definitions together to make sense of that original highly abstracted concept, you end up with tens or hundreds of possible interpretations. But in fact, Math should only have 1 interpretation for each concept so this is a very bad situation to be in.
I think math definitions should be more elaborate and repetitive if necessary. They should not try to sound terse and clever. They should not assume that the reader can fill in the gaps. The most rational readers will not be able to fill in the gaps because rational people know the dangers of making assumptions.
> Formally, a field is a set F together with two binary operations on F called addition and multiplication. A binary operation on F is a mapping F × F → F, that is, a correspondence that associates with each ordered pair of elements of F a uniquely determined element of F. The result of the addition of a and b is called the sum of a and b, and is denoted a + b. Similarly, the result of the multiplication of a and b is called the product of a and b, and is denoted ab or a ⋅ b. These operations are required to satisfy the following properties, referred to as field axioms. In these axioms, a, b, and c are arbitrary elements of the field F. [...]
Still, i don't know how to say it in another way but you probably don't have much experience in mathematics, even the first quote is arguably quite accurate.
> Does it mean that any 2 elements combined together using that operation always need to output an element which is also in the same set?
Yes, unless told otherwise an operation is an internal binary operation, it's really the most common form. When it is not the output is notable enough to be specified.
> But if that was the case then "behave as the corresponding operations on rational and real numbers do" would mean that the fields would always need to be of infinite size wouldn't it? Because if the field had a limited number of elements and you added the last two (highest) elements together [...]
No, the important point here is your use of "highest". A set by default only has equality (and mappings, in and out) but no order relationship. So the most conservative interpretation of "behave as the corresponding operations on rational" would be to only include stuff that can be written using the 4 operations and equality, not ordering.
---
Maybe i'm biaised by the fact that i know what a field is, but still, this particular intro is also how i would present a field: give the most common example and say which operations it has. It sure can create false intuitions like yours about the size, but this will always be the case when we use non-normalized language.
For a binary operation f to defined on a set, f(x,y) must exist for every x and y in the set. There is no requirement that f(x,y) itself is in the set. Adding that requirement would mean that the set is "closed" under the operation f. So if we take Z+ = {1, 2, 3, ...}, ordinary division is defined on Z+, but Z+ is not closed under division, since we can get results like 2/3 that are not in Z+. Whereas division is not defined on Z0+ = {0, 1, 2, ...} because we can get undefined results like 2/0.
However, some definitions of "binary operation" include the "closed" property, so under such definitions, division would not be considered a binary operation on Z0+.
>Does it mean that any 2 elements combined together using that operation always need to output an element which is also in the same set?
Specifically in the case of a field, yes; addition needs to be defined, closed and invertible for the set; multiplication needs to be defined, closed and invertible for the set excluding the additive identity (zero).
In the first place, what you quoted is not a formal, precise definition; it is not a substitute for such a definition, nor is it intended to be one. The Wikipedia page you mention has a precise definition further down the page.
So what, then, is the purpose of the description you quoted? Why include it at all?
Because it's how mathematicians conceptualize of what a field is. It is the peg we hang our hat on; it is what we remember. A mathematician who has seen fields would be able to fill in the details; and if not, they would know to look up the precise definition in a textbook.
In short, these definitions are how we keep track of the forest at the same time as the trees.
I should note that taste differs among mathematicians, and you can find different styles of exposition in math books. Some are very formal and precise; whereas others are more informal and have lots of handwavy statements along the lines of the one you quoted.
> "In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do."
This isn't a mathematical definition of a field. This is an encyclopedia's intuitive definition. It is a good one, in my opinion, as it'll evoke the correct definition in a person who is trained at mathematics, and hopefully convey the gist of the idea to someone who is not. Very good for one sentence! But a mathematical definition it is not!
A mathematical definition of a field would be as follows:
BEGIN DEFINITION
A field is a set F together with two functions A:F×F→F and M:F×F→F and two elements z∈F and o∈F that together satisfy the following properties:
ASSOCIATIVITY OF A: A(A(a, b), c) = A(a, A(b, c)) for all a,b,c∈F.
ASSOCIATIVITY OF M: Same as above with M in place of A.
COMMUTATIVITY OF A: A(a, b) = A(b, a) for all a,b∈F.
COMMUTATIVITY OF A: Same as above with M in place of A.
NEUTRALITY OF z WRT A: A(a, z) = a for all a∈F.
NEUTRALITY OF e WRT M: M(a, o) = a for all a∈F.
INVERSE FOR A: For all a∈F, there exists an element na∈F such that A(a, na) = z.
INVERSE FOR M: For all a∈F such that a≠z, there exists an element ra∈F such that M(a, ra) = o.
M DISTRIBUTES OVER A: M(a, A(b, c)) = A(M(a, b), M(a, c)) for all a,b,c∈F.
END DEFINITION.
(This definition presupposes that one knows what a set is under a standard framework.)
Since this notation is cumbersome, it is common to write A(a,b) as a+b and M(a,b) as a·b or ab. Likewise, z if often written 0 and and o is often written 1 (or e). Similarly, na is often written -a and ra is often written 1/a, but do take care to recall that "-a" and "1/a" are just symbols. One typically compresses notation even further, and writes "a + -b" as "a - b" (not to be confused for the juxtaposition of a and -b).
Do you feel better about this definition than Wikipedia's informal one? I invite you to scribble out a verification that for example the reals form a field under this definition (with ordinary addition as A, ordinary multiplication as M, ordinary 0 as z, ordinary 1 as o).
> Because if the field had a limited number of elements and you added the last two (highest) elements together, the property which requires that the result also be present in the same set could not be met because the result would be greater than the highest element in that set...
You're reading too much into "behave like the corresponding operations on real numbers do". One does not demand that addition preserves order. Notice how there is no reference to ordering or elements being "larger" or "smaller" in the definition above. It is for example the case that in the field of two elements, 1+1=0. That's fine.
> The problem is that if you start with a highly abstracted math concept and you dig through all the links and definitions of sub-concepts, they all have huge gaps like this
Not at all. You are trying to do rigorous mathematics using informal statements (not to detract from the informal statement; someone who has seen the formal definition of a lot of mathematical structure can almost surely reconstruct the correct formal definition of a field in a second from the informal one).
> I think math definitions should be more elaborate and repetitive if necessary.
So mathematicians should make communication between ourselves more cumbersome in order to please outsiders? I'm sorry, I don't mean to sound like a gatekeeper, but this is ridiculous.
> They should not try to sound terse and clever. They should not assume that the reader can fill in the gaps.
Do you demand this of musicians and engineers and chefs and mechanics and pilots and doctors and nurses too?
> The most rational readers will not be able to fill in the gaps because rational people know the dangers of making assumptions.
The most rational readers who have studied mathematics will. To a point, of course. There is often a tradeoff to be made, but your blanket statement is just plain wrong.
I don't see why certain knowledge should be out of reach of those who are not involved directly in that field. I could explain complex software engineering concepts to a layman. They wouldn't be able to use that knowledge to implement the software themselves, but they would be able to use the knowledge to make good high level decisions about it; for example to decide which of two solutions is better given a specific problem.
Sure, his definition is also a good one. It leaves out a lot, though. Which is fine, if one can assume the reader knows the context. My definition, too, leaves out a lot (it assumes set theory), and rests on the informal language known as English.
> I don't see why certain knowledge should be out of reach of those who are not involved directly in that field.
It isn't. The content of mathematics research papers may well be out of reach, but that's quite natural, don't you think? I, as a mathematician, do not expect to be able to read research papers on chemistry without putting in a lot of work.
> I could explain complex software engineering concepts to a layman.
OK. It does not follow from that that everything can be explained to a layman. Some things are easier to explain with layman analogies and mental images than others. However, to keep this fair, I think you should see how many laymen can follow Wikipedia articles on complex software engineering topics with ease! That is, afterall, where we started this discussion.
I actually like informal definitions a lot. I think they serve two different purposes:
1. For beginners they usually soften the blow of a fully rigorous definition, letting them get an idea of the concept before getting it exactly.
2. For experts they can often suggest what the exact definition is faster than it would be to read a precise definition!
But if you don't get anything out of them, you can skip 'em. Definitions are more important: informal descriptions are there to help you grasp the definition faster and to help you develop an intuition for the concept being defined. If you can do those things faster from just a rigorous definition, you don't need an informal description. (But as I said, I feel informal descriptions benefit both beginners and experts, so I'd also suggest practicing reading them more to get a feeling for what kind of details people tend to omit or emphasize.)
You're projecting (in the psychological rather than set-theoretic way).
Type theory is a little like that, at least from some authors that - although the are is well suited for symbols - kind of goes a little off the rails using seven different kind of arrows and all the symbols in at least three different alphabets, instead of maybe just write covariant, or even an abbreviated form adjecent to the arrow?
I've had / have trouble and misconceptions many times, but when discussing with someone fluent in the notation and the field, 100% of the times things cleared up, and I couldn't think of a better notation to use.
The biggest issue I've encountered is borrowing lecture slides from different universities/lecturers and the notation changes between slides on the same topic (even small things like using square brackets instead of parentheses).
There’s a hybrid notation that I prefer, for example “Pr_X(x)” for the density function of random variable X at point x; you drop X if the random variable is clear from the context, and you drop x if you’re referring to the entire distribution. Or Pr_X(x|Y=y) for a conditional density. But this notation still has problems when you’re working with hairier conditional distributions, or with distributions that are neither discrete nor continuous.
(Source: used to be a mathematical probabilistic, now working in ML.)
It's definitely worthwhile everyone using the full notation at least once so they can get a feel for what's really going on. I've spoken to Bayesian ML professionals who are especially unconfortable with that because it conditions on a zero-probability event (if Y is continuous)... of course p(x|y) does too, they just weren't thinking about it before! And (as I think you're getting at) the appreviated p(x|y) simply throws away information e.g. there's no way to represent the identity p_Y(x)=p_X(x) without adding back some sort of subscript.
But on the other hand p(x|y) is obviously much visually cleaner. If you're writing out a more complex identity and the abbreviated notation isn't ambiguous then it generally communicates the idea much more clearly because there's so much less visual noise.
v_after = A_3 A_2 A_1 v
not (what I would consider more straightforward)
v_after = v A_1 A_2 A_3
Usually Tao's posts are so insightful, and crystallise some idea so perfectly that it feels like I was just on the cusp of discovering it myself—a rare talent, and hard to cultivate since it goes against the ego. In this case, though: I'm a professional mathematician, and as prone as anyone in my discipline to use mathematical language to describe not strictly mathematical things, but the pseudo-mathematisation here ("Notation^{-1}(C)", for example) seems more like wit than clarity. Not that there's anything wrong with wit, but in this case it seems to me that it's at the expense of, rather than a pleasant addition to, the central point.
I'd like to hear especially from anyone who isn't a professional mathematician: did you feel that this post improved your understanding of the purpose and function of good notation?
(EDIT: I was scared about making this post, since there's rightfully a lot of respect and appreciation for Tao—and I hope it's clear that I concur on both counts—and I wasn't sure how my reticence on his post would go over; but I'm super glad I asked. Thanks so much to everyone downthread; these are wonderful responses and I feel that it benefited me a lot to read them.)
A large part of math is knowing notation and definition. When I started studying higher levels of Mathematics a big part of me that was keeping my understanding was figuring out the notation and the definitions.
Graph theory is so notorious for this that I can remember a joke that everyones notation for graph theory is slightly different.
No, but I did learn about new dialects and conceptualize mathematical notations better. I think for someone new to languages/notations the first set of detailed bullet points are useful. For more experienced practitioners I think it sums up to "use the right tool for the job".
What I found very interesting was the latter set of bullet points where he presents a sort of Rosetta Code (http://rosettacode.org/wiki/Rosetta_Code) of math. I am familiar with thousands of computer languages but not a mathematician and not familiar with what I'm sure are hundreds (thousands/more?) of Mathematical notations, so the few examples he lists here have me intrigued.
I am bookmarking the post as I found it valuable for next time I invent notation.
What I found particularly insightful here is applying this mathematical notation to programming languages and their syntaxes. His notation described as ``` Notation:{well-formed expressions}→{abstract objects in 𝑋} ``` isn't too far off from what most programming languages implement at some level.
As a result of this, what properties do programming languages share with mathematical notations, and why are some languages deemed more "expressive" than others? How much does the "expressiveness" of a certain language in a domain lead to better understanding of the abstractions beneath them?
To answer your question; I don't think this post explicitly increased my understanding of notations (especially not in the context of mathematics), but rather led me to ponder the importance of them in communicating & extending abstract concepts effectively across domains.
Yes
I skipped past the long list of properties and went straight for his long list of ways to write the inner product of two vectors and his illustrations of why you want to use different notation in different cases.
Here's a question for you, do you think Tao's primary mission is to be a "professional mathematician" or is he just enjoying the math, enjoying teaching, enjoying notation...?
I definitely think that he's earned the right to contribute to the mathematical community in whatever way he sees fit, and, especially if it benefits other people, which it clearly does, then my opinion of it shouldn't and doesn't matter a hill of beans. I just usually feel that he combines those missions—not that he has to, but that he does—and felt that he didn't here; but, again, that is, if at all, very much my problem and not at all his.
A good notation should make this map Notation (and its inverse) as close to a (natural) isomorphism as possible.
Having said he want an isomorphism for Notation sums up all the following points he made explicitly (and more), so we could argue that we can keep the Notation and the isomorphism part but remove all the following point 1..9+
I think that in this way the post reach more people :) some needs only to read the first sentences, some the properties and finally some the example.
Isn't it a nice way to understand what he is trying to say in as much different ways as possible?
(Preservation of quality, I) Every "natural" concept in 𝑋 X should be easily expressible using the notation.
So, if he is going to commit to his idea of the notation-to concept-isomorphism, it would have been weirder if he did NOT introduce the notion of the inverse map.
Framed this way, the properties listed are (1) (part of) functorality, (2) surjectiveness, (3/4) smallness, (5) renormalizability (!!), (6) smoothness/continuity of some sort, and (8) the ability to have natural transformations applied. Only (7) is culturally dependent. (1) and (8) are free for all functors!
To address your question on the head (I am not a mathematician), this post gave me some properties of good notation which I hadn't considered before. On one hand, yeah, it's kind of obvious that notation should somehow be able to surjectively reach all possible objects of interest, while still being small enough to definitely not reach everything. On the other hand, renormalization is kind of a dense topic, so it's surprising to see it arise here.
For example in Mathematica there is a (mostly deprecated) package called Notation`[0] that does just this kind of stuff. I have to admit that it's not really used in production code anymore as MakeBoxes and MakeExpression are more fine-grained and robust.
Thus said I have to admit that the transformation between 2-D boxes and M-expression is not as foundational as what Tao is talking about, however the whole field of designing programming languages is deep down an exercise in defining notation, the transformation mentioned above just make this a bit more explicit.
[0] http://reference.wolfram.com/language/Notation/guide/Notatio...
I definitely don't think that notation is pseudo-mathematisation; good notation is inordinately powerful in enabling good mathematics (and bad notation can make even simple mathematics hard). What I meant to describe as pseudo-mathematisation was the discussion of notation in what seemed to me in a (to me) unnecessarily formally mathematical way.
One bugbear is that mathematical writing leaks into engineering science. I wouldn't ask professional mathematicians to start caring about units or change their style while communicating amongst themselves, but in my view textbooks ought to define notation before they use it, and clearly define the units used in all expressions.
c ≡ u +.× v
I wish they taught it to math majors.
APL's a wonderful rabbit hole to fall down, and J was my eso-lang of choice last year.
I'm not disagreeing, but I'm curious how you would back up that assertion.
that's the thing though, you can never grapple with the subject itself, only representations thereof. this is a mix of feelings/images/movements inside your head & mechanical manipulations of the notation; furthermore, the notation itself influences our internal model/feeling of the subject.
I consistently have run into "perceived or real" confusing mathematical notation as an impediment to learning in a way that programming languages have never, ever caused me.
Does no one else feel this way? I can't be alone, and like the others responding to you have said, your claim does not seem substantiated.
I would say this trouble comes about in programming though, in the form of DSLs. Anyone who's worked with Rails knows the pain of having to go look up the non-obvious syntax for xyz specific file that you haven't touched in months.
Maybe you could say this is a problem with large software systems in general - the problems they solve often aren't that complicated (especially if you've worked on other similar software), but learning all the names and patterns and existing features creates a high barrier to entry.
Say, you come across this symbol: ζ(s)
And you asked me, what does it mean?
Oh, that's probably the Riemann zeta function (it could be other things, but that's probably its most famous meaning).
Would you be like, ah, now I understand it all!
I sincerely doubt just knowing the English words for the symbols (and all mathematical notation is just shorthand for English, or whatever other natural language; it can all be read out loud) would get you that much further along. I could give you one of several possible equivalent definitions, but you still wouldn't really know much of anything. I could start explaining some things about primes and how they relate to the Riemann zeta function, and you'd still be in the dark. I'd have to talk about the prime number theorem and estimates and complex analysis and a whole bunch of other things before the Riemann hypothesis even started making sense. In fact, people have written entire books just trying to explain in the most accessible way possible what that one symbol is:
I believe what's going on is that a lot of people who find notation to be the first stumbling block are trying to learn mathematics in isolation, without the benefit of a mathematical community to be able to talk to. You pick up a book, see a bunch of weird symbols, blame the notation, you put it down. But people who train for mathematics specifically rarely learn that way. Mathematics is a social activity, you talk to other people, you read the symbols out loud to each other on chalkboards or side-by-side as you scribble notes on paper or on a napkin. You talk to each other a lot, pronouncing the symbols as much as you write them down.
The notation is a very superficial obstacle. Notation is very incidental, malleable, almost irrelevant (or as mathematicians might say, non-canonical or not natural). If notation is your biggest obstacle, there's a lot more underneath that you probably don't have enough experience with.
Are you saying that it was the notation alone which caused you to completely stop reading the book? I can't see that I share that experience. There are certainly times where I don't fully understand a proof or a lemma due to uncertainty about the notation, but that's rarely a _blocker_. I just skip it, continue reading and come back to it later.
I agree with the GP here: The main thing which blocks _me_ on a book is the subject itself and how the author presents it. The problem is rarely the notion by itself, but the fact that the author uses concepts which I don't know about.
EDIT: It should also be pointed out that the GP didn't say that notation was _never_ a problem, only that it's a much _smaller_ problem than "those presented with the subject itself". The big question about your experience would be if you would have been unblocked if the notation was explained in English, or if it would just uncover that it uses concepts which haven't been fully explained to you.
However, I fully agree that reading a text that uses notation you are not familiar with without introducing it is almost futile. Even the simple fact of not knowing the names of the operators, which makes it impossible to read the formulas cursively in you mind (for example, when you encounter 'a + b' you can read it in your mind as 'a plus b', but for something like 'a <<=> b', even if given `a <<=> b = 2*a+b`, reading the text becomes a mental chore).
Don't worry: you are not alone in making that huge mistake, plenty of people do. People that don't give up when confused about notation usually quickly learn that is the concepts and relations between them that require careful thought (and also, usually, that the notation was advantages they didn't realize when at first blush they found it confusing).
The C pointer syntax can be confusing even to experienced C programmers.
If programming languages strive for the same elegance via notation, you end up with something like Perl.
This is a good goal, but I'm not sure it's the primary goal; the phrase 'abuse of notation' exists precisely to describe its breakage, with even the best mathematicians and expositors engaging in it, and I think insisting on no abuse of notation leads us rapidly to the style of impenetrable Principia-style logic, or of modern formal proofs—both of which have their place (at least the latter …), but neither of which should govern all mathematical discourse.
As with all writing, I think that part of being a good mathematical writer is knowing the rules so that you can figure out when to break them unintentionally, rather than stumbling into it accidentally.
My biggest pet peeve with mathematical symbols is the difficulty of looking them up when you don’t know them already. If I’m reading a text on a topic I'm unfamiliar with, I can at least google the keywords I don't know. This is difficult with symbols.
https://detexify.kirelabs.org/classify.html.
Then just need to look up the Latex on Wikipedia:
One that I like is that in Einstein notation you can't have 3 of the same index, e.g., u_i u_i u_i is invalid.
Another good example is the Liebnitz notation for derivatives. Proper application of the chain rule visually resembles how fractions cancel: dy/dz dz/dx = dy/dx. It's very easy for the eye to follow and make sure that the cancellation is valid. Newton's notation doesn't make that as easy.
I'm going to claim that string diagrams have reasonably good error-detection properties, too, again because they lay facts out in space in a way that humans are quite well optimised for.
I would add
1. Clearly telegraphing notations. Not hiding them in the middle of long paragraphs or even, and yes I have seen this a few times, defining essential notation in an optional exercise.
2. Having a glossary of notations, so people don't have to remember every single notation and to read every word of the book sequentially.
3. Not creating low value notations that may be used only once and then, possibly forgotten. I have read books with > 1 new notation or definition per page, mostly forgotten thereafter but some random subset needed later, and you are not to know which.
Programming languages are notations within this framework - and domain specific languages, while much more efficient are unpopular as the costs of changing notation, in terms of training people, are too high.
The cost of communicating the notation is captured in a few of the desiderata (e.g. 1,7) but practically it is most important. If we want to be easily understood we should speak a common language!
JSX seems pretty popular, and when XML was popular similar XML embeddings were, as well. Templating languages are popular. Heck, the relative popularity of “general purpose” programming languages is not consistent across domains, with domain fit being a factor even for general purpose languages.
I don't think any of these things are popular; in my experience people using them mostly hate them, or at best grudgingly accept that they're the least-bad option.
Of course it's got its fair share of dumb gotchas, but I found it far easier to learn than, say, any of the myriad Rails DSLs.
c = 0
for i in range(u.size):
c = c + u[i] * v[i]
c = u.T @ v
- As a regular array (for someone with Python experience but no knowledge of linear algebra). In this case you can just loop over it as with any iterable.
- As a vector, with the associated mathematical properties. In this case you can operate on the entire vector, which is much faster; this happens to be because of implementation details (using C structures instead of Python lists, parallelisation, whatever), but is also just highly intuitive.
I'd argue that this is exactly what Tao is saying, about how different notations suit different contexts, and that allowing both methods in no way violates the Zen of Python (in reference to jimhefferon's comment).
c = 0
for i in range(u.size):
if u[i] < u[i + 1]:
c = c + u[i] * v[i]
else:
c = 0
Saying that the Python interpreter should reinterpret the iterative code as if it were vectorised isn't increasing the expressivity, it's reducing it, by overriding the user's intentions.
I suggest that should one strive to 'fine tune' notation N to possess the above two qualities for a family of objects in X, other categories of objects in X will become opaque and difficult to express, i.e. a domain specific notation.
I think it can be agreed the present grub documentation is very broad, with only a few undocumented features, but still not very deep even on the default operation. Grub was also undergoing more rapid change at the time.
Came across a message where Tao had explained a concept like no other, and after what I had seen it was clear to me he understood it like no other, so it was purely logical. He knew more than the documentation. I didn't know he was a widely recognized mathematician or anything, I just thought he was a very bright computer scientist on a message board.
There were only a couple sentences that nearly applied to my system, within a couple paragraphs on his solution to a different problem.
There was no useful answer for me yet so I moved on.
Googled to exhaustion that session with no code changes to make, but I looked at it again and it was the only tab I kept open, even though Tao did not show the direct way forward for me at all, everything else was actually useless.
Next day I read it again, scrutinizing it over & over for an action I could follow through with, wishing someone had posted equally straightforward advice for my particular situation.
No such luck, but it inspired me to go forward in a similar fashion.
Mostly use both grub and syslinux as separate alternatives to boot my distributions ever since.
For years I've felt like I couldn't have done it otherwise.
And with Windows, grub or not, I ended up bringing more reliability to my employer.
Fairly recently I found out Tao had started out as a mathematical child.
That was incredibly helpful.
He actually communicated the unique abstract concept I needed without even knowing the problem and without intentionally trying.
I imagine he made up his own notations quite a bit before he carefully adopted the various professional terminologies.
I think in the future programming will force all mathematicians to code or give out simulations. Most mathematical notation was intended to be throwaway by the original authors, thats why there are so many notations. Trying to find relevance in them is a pointless exercise. Much like 80x20, tabs vs spaces ... most of the original intent is lost and what survives is guff meant for ceremonious purposes.
A triangle has a sum of 180 ? Well how about if you push the triangle inside out. In code you can easily run a more complex simulation which gives you all possible values of the sum ... which is why ascertaining useful facts like 180 ad-nausea is boring at best. In fact most mathematics if it can't be simulated can't exist.
> Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap.
https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html
I see simulating as a part of the experiment. If the proof is wrong it wouldn't last a seconds worth of simulation. I suppose a proof in essence is a pattern or an invariant of the system ... but most proofs have really no meat to them. The notation is merely intimidating like obfuscated code.
It's kind of amazing how quickly that article got so wrong. Math isn't a subset of physics. Physics is the estranged brother of Math, always needing to borrow some money from him or else the power goes out or he can't afford food or some other sob story.
Yes. But most of us don't use those to prove anything; rather, a lot of us use it to implement computations based on those proofs (and do some exploratory "could this possibly be tru?" kind of work). Useful tools, for sure, but not something that remotely proofs your point. Most mathematicians also eat bread. That does not mean that math is a baked good.
> The notation is medieval ...
It is not. Read Gauß or Euler from the 18th and 19th century, and the notation is nothing like modern mathematical notation. I can't even imagine what medieval mathematics notation looks like!
> https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html
That is indeed the opinion of Arnold, a giant of mathematics. An opinion that, I daresay, does not reflect the majority opinion on mathematics.
> I see simulating as a part of the experiment.
Sure. Simulating is a valuable experimental tool to many mathematicians (where available; of course it isn't always).
> If the proof is wrong it wouldn't last a seconds worth of simulation.
At face value this statement betrays how little you know about this matter. There can very well be errors in proofs that cannot be uncovered without thousands of years of simulation, if at all.
Now, even interpreting your statement in the best possible light, namely something along the light of "simulation can often uncover mistakes in proofs", I would say: fine, but what about the converse?
> but most proofs have really no meat to them. The notation is merely intimidating like obfuscated code.
Are you insane? Take something that is patently "useful" and patently "real world", like the fundamental theorem of calculus. Meatless?
1. Socrates is mortal
2. Mortals die
3. Socrates dies
Deduction is really like amazing. Holy shit we really proved something spectacular here. I guess you would be really impressed if I used tau and sigma and defined death with vietnamese alphabet.
Almost the entirety of calculus was derived from problems related to physics. Volumes were calculated for doing engineering. Mathematics != Thinking. The last time I checked both logic and critical thinking were branches of philosophy.
All good mathematicians are physicists or engineers. Heck some even learnt maths on their own. All mediocre mathematicians write textbooks and hide behind notations. Come to think of it they remind me of OO programmers in their utter arrogant mediocrity. Most abstract mathematics is like the definition of protocols/interfaces and other platonic garbage. I suppose this debate will never end. Plato vs Aristotle, Deduction vs Induction, Analytic vs Synthetic ....
Further, that phrase is bigoted. What you are implying is that someone with OCD is "lesser" or "other" as you are using the phrase to discount the person you are talking with. Hence it is bigoted.
In fact, it is obvious that you have no idea what you are talking about. Mathematics is not "just notation" in the same way software engineering is not "just programming language syntax", music is not just "notes on a piece of paper", and literature is not just "grammar rules".
If you cannot see that, I suggest you read more and expand your view of the world. Don't hurl insults at others.
If you want a more concrete example, show that the sum:
1 + (1/2)^2 + (1/3)^2 + (1/4)^2 + ... = pi^2/6
That is, first define what it means to take a sum of an infinite number of terms, prove that your definition is consistent with a sum of a finite number of terms, and then show that the sum is exactly pi^2/6. Showing that they agree to 100 billion decimal places is not enough. You need to show they are exact.
When you are done with that, find an exact closed form for the sum:
1 + (1/2)^3 + (1/3)^3 + (1/4)^3 + ...
He won't. I keep running into people like this all the time. They are hellbent on the idea that anything they don't understand must be meaningless, useless, or the fault of others. If you get a reply at all, I suspect it will be something like "pi is just a meaningless approximation to a real physical concept" or "infinite series don't actually exist in real life, I'll sum the first 1000000 terms on a computer and that's all that exists".
The internet makes it easy for pedantic losers to have a loud opinion. Hell I have even run in to pedantic losers who have the time to create multiple new and fake accounts and use old sock puppets to create the illusion of an audience because these friendless, loveless losers literally have no one to talk to IRL.
> I keep running into people like this all the time.
Psychological attacks, amazing! I'm guessing you are one of those deeply insecure symbol twiddlers. Let me guess, as kid you were crap at everything, especially sports except symbol twiddling so you latched onto those praises your teacher gave you and as an adult that is the only source of your self-esteem. And you can't handle it when someone on the internet thinks abstract mathematicians are full of shit.
Mathematicians will not say anything like "infinity is real" or "infinity is not real". We are careful creatures, and will ask what you mean by "infinity". In this subthread we've been discussing infinite series. What part of the definition of those do you have a problem with? (Prediction: you'll never answer this, but instead go on ranting with no ability to focus on the topic at hand. I can definitely see why math is hard for you, you have a severe problem with focus).
> The internet makes it easy for pedantic losers to have a loud opinion.
I can see that.
> Hell I have even run in to pedantic losers who have the time to create multiple new and fake accounts and use old sock puppets to create the illusion of an audience because these friendless, loveless losers literally have no one to talk to IRL.
That's pretty sad. It's also very sad that this is the conclusion you jump to when someone speaks out against your insane ravings in an entirely logical and coherent way.
> Psychological attacks, amazing!
It's a bit entertaining that you can go from what you wrote above (and what you write below) straight into accusing me of this.
> I'm guessing you are one of those deeply insecure symbol twiddlers.
I am indeed quite insecure. I'm working on managing that. If you by "symbol twiddler" mean mathematician, then yes – and quite proud of it too. You'll do well to get back on track to the topic at hand though, seeing as you're currently coming off a bit like the people one sometimes see yelling incoherent nonsense on subway trains.
> Let me guess, as kid you were crap at everything, especially sports except symbol twiddling so you latched onto those praises your teacher gave you and as an adult that is the only source of your self-esteem.
Not at all. While I was quite mediocre at sports (though far from crap), I did really well in most things. I was not a favorite of the teachers, because I had (and probably still have) a bit of problem with authority. Are you done derailing the discussion now? I'll remind you: we're discussing the usefulness of mathematics, not my childhood or athletic abilities.
> And you can't handle it when someone on the internet thinks abstract mathematicians are full of shit.
I can handle it just fine, primarily because what raving lunatics believe has no influence on the extreme actual power and usefulness of mathematics. The reason I care to have the discussion is to set the record straight for third parties' sake.
In the face of what? No system for writing numbers? Sure. Arabic numerals? No way, they're far superior to Roman ones.
> Almost the entirety of calculus was derived from problems related to physics.
No it is not. It is/was largely motivated by problems in physics. One may obtain a lot of intuition about calculus from physical intuition, but one does not obtain calculus from it.
> Volumes were calculated for doing engineering.
Yes. And?
> Mathematics != Thinking.
I don't see anyone in this thread claiming that.
> The last time I checked both logic and critical thinking were branches of philosophy.
I thought we were talking about math?
> All good mathematicians are physicists or engineers.
This is patently not true. Tell me how many Abel prize winners of Fields medalists are physicists or engineers. There are indeed some, but they are a minority – your claim is absurd.
> Heck some even learnt maths on their own.
Absolutely. Can you elaborate on how this is relevant?
> All mediocre mathematicians write textbooks and hide behind notations.
What are you on about? As a mediocre mathematician myself, I must admit I have never written a textbook.
> Most abstract mathematics is like the definition of protocols/interfaces and other platonic garbage.
You have made it abundantly clear that you haven't that slightest grasp of abstract mathematics. Would you at least humor me as to provide a few examples?
By construction.
I think that he has a good idea for the most part and if you did formalize this notation there is a good chance that someone in the computer science domain would eventually program something that could interpret it. Lisp comes to mind.
Now imagine if those words I gave you were in Vietnamese, or some language you don't speak. Suddenly the task becomes much more confusing. You aren't remembering a small handful of objects and ideas, but instead trying to juggle the individual syllables in your head.
Math notation sucks because none of it maps to things non-mathematicians know. Every time a new symbol is introduced, whether it be a greek letter or a operator, it's one more mapping your brain has to create to remember it. And on top of this, you also have to remember the English names too. Yes I said names - most math concepts have so many different names it's crazy. Even basic arithmetic can't escape this. There are two names for multiplication (multiply, product) and four common notations for representing it (*, x, ·, and whatever you call it when two variables are next to each other).
This doesn't mean maths notation sucks any more than vim's user interface sucks because it doesn't make sense to non-vim-users.
Mathematical notation presumably mostly makes perfect sense to the kind of people who deal with mathematical notation all day long.
Maybe the overuse of opaque names leads to self-selection of who becomes a mathematician?
Single-letter non-descriptive variable and functions names would “make sense” to programmers who use it all day long too — but that alone doesn’t make it a good idea.
Now, formalising that above sentence even with symbols is quite difficult.
Also, no better notation could help you understand most modern mathematics. There's no better choice of symbols or names that will help you understand Galois theory. You simply need to understand high-school algebra, then group theory, then Galois theory.
This probably isn't completely true—as will be clear to anyone, even a subject expert, who tries to go back and read the original papers. At least part of this is due to fads in notation—in my field, I can easily read the papers of people who do similar work to mine, and struggle with the papers of those who don't, even when they're talking about exactly the same thing—but some of it must be due to genuine improvement.
Being able to recognize things like "oh, that's a polynomial!" is extremely useful, especially in the context of learning new math rather than a more applied context. And I think it would be extremely hard to spot an opportunity to do integration by parts when working with Java-style verbose variable names.
We're trying to derive that a^x = b^(x * log_b(a)).
Or in verbose descriptive terms, oldBase `exponentiate` oldExponent = newBase `exponentiate` ( oldExponent `multiply` inverseExponentiationInNewBase (oldBase) ).
We could maybe argue about the verbosity and clarity of each statement, but I think math's notation shines when you compare derivations instead of statements:
a^x
= b^( log_b(a^x) )
= b^( ln(a^x)/ln(b) )
= b^( x * ln(a)/ln(b) )
= b^( x * log_b(a) )
oldBase `exponentiate` oldExponent
= newBase `exponentiate` ( inverseExponentiationInNewBase(oldBase `exponentiate` oldExponent) )
= newBase `exponentiate` ( inverseNaturalExponentiation(oldBase `exponentiate` oldExponent) `divide` inverseNaturalExponentiation(newBase) )
= newBase `exponentiate` ( oldExponent `multiply` inverseNaturalExponentiation (oldBase) `divide` inverseNaturalExponentiation(newBase) )
= newBase `exponentiate` ( oldExponent `multiply` inverseExponentiationInNewBase (oldBase) )
This is a ridiculous over the top example, since I'm avoiding using everyday symbols like * and / and ^ and log, but for working mathematicians and applied mathematicians, their notation is just as familiar to them and I'm sure it'd be equally annoying to write out descriptively as it is for us to write out basic math symbols descriptively.
Programming notations are to be written and read. Math notation is to be manipulated.
But there are alternative ways of describing that derivation that are not as symbol-manipulation heavy; you would certainly not communicate this proof in words to another mathematician — or to a non-mathematician — by simply reading your derivation (or your alternate, verbose derivation) symbol by symbol. Instead you would more likely rely on the meanings of the symbols.
Even in your verbose derivation, however, it’s worth noting that your new names capture the fact that b^x and log_b(x) are inverses, a key piece that someone unfamiliar with logarithms and their relationship to exponentiation now has a hope to understand.
Because it is better communicated in notation than in words.
Imagine you would write the code on paper by hand and have to repeat some long var or function name all the time . Even with IDEs I see programming languages using shorter keywords like "var","const","int" then the full word.
Because you are agains single letter symbols let me know what are your suggestions for PI, or X(in polynomials or equations) and show me a screenshot on how some real world physics equation solving would look like with camel case named symbols.
Sure, you could write
-CoefficientInFrontOfXToTheOne/2
But then each row of your calculation will need line breaks and will be hard to understand. It will take forever to write and it will take forever to read. No thank you.
Vim, like math notation, is optimized for efficiency. That's great for power users, but it's not what you would use if you wanted to teach someone how to word process.
(BTW, try imaging Vim where each file mandates its own special keybindings. That's math notation for ya.)
Vim is a text editor, not a word processor. Additionally, Vim is a tool designed for power users. Mathematical notation is also meant for power users. When we teach mathematics, we introduce the notation gradually, so students have time to pick it up. This is a process which takes decades from Kindergarten through PhD. Just as a Vim user would scoff at being forced to write in Notepad, a mathematics PhD would scoff at being forced to express their ideas in Kindergarten-level mathematical notation.
try imaging Vim where each file mandates its own special keybindings. That's math notation for ya.
That's how Emacs works, and how Vim works when you install filetype-specific plugins.
I have too many things to remember, and there are plenty of options that give me equal power but don't demand that investment from me.
Math isn't like that - I do more math than the average person and I hate having to decipher the annotation. The number of times I've tried to work out if this sigma is the same as the sigma in this other paper frustrates me enormously.
Or the paper that used a accuracy^bar metric as their primary reporting metric, and we couldn't get near it until we found an obscure footnote in an appendix that explained the ^bar metrics were temporally averaged at test time around a known labelled standard (ie, test data was used to optimise the performance).
It seems like your complaints are with academic papers, not with mathematical notation specifically. This is a problem that is universal to academic papers. Try reading a critical theory paper, for example, and you'll find it's extremely dense with critical theory jargon the authors don't bother explaining at all.
The problem with academic papers is that they aren't intended for a general audience. The authors of papers are often operating in subfields that are so small that they've actually met most of the other people who will be reading their papers. In that situation, conventions and jargon arise naturally among colleagues. Variable names form a part of these conventions such that, in a more common example, sigma will mean standard deviation among statisticians but mean singular value among linear algebraists.
The other problem with academic papers is that the authors generally don't care about reproducibility, consistency, clarity, pedagogy, or even intelligibility. They're optimizing for quantity of papers published, not quality. As long as their expert peer reviewers understand and give the green light for publication, that's good enough.
It's really about the lazy habits of many who work at the intersection of math and computer science, and use maths to express themselves without defining things.
> The other problem with academic papers is that the authors generally don't care about reproducibility, consistency, clarity, pedagogy, or even intelligibility. They're optimizing for quantity of papers published, not quality. As long as their expert peer reviewers understand and give the green light for publication, that's good enough.
Having worked around this field, I think this (common) perception of publishing doesn't quite capture what is happening. Peer review isn't anything like code review, and software engineers (separate from computer scientists and mathematicians) think that it is.
A lot of mathematics is optimised for pencil efficiency for a reason, and it's not at odds with learning mathematics.
That is like trying to learn an instrument by looking at it but never touching it. Notation like other tools is made to useful. Also if you never played an instrument you would think they are each their own little completely different world. That's totally wrong for ya.
I think this nails it perfectly. There are plenty of good programmers who refuse to use vim. In math those people can't operate.
I dropped out again and haven't looked back. That system wasn't for me and didn't care about my success or its ability to transfer knowledge.
Mathematics is a deep subject. It requires many hours of study over decades to develop. Students who are struggling deserve to have a patient teacher explain things to them in terms they already understand. Not every student has this experience, however. I have tutored a number of my peers in undergrad and they have complained of similar treatment from professors or TAs. I do my best to help them and make sure they really understand the topics they're studying.
Funny enough, that grad student was almost certainly making less than that. Universities are screwed up.
As a novice programmer, it took years to get out of the habit of using non-descriptive single-letter variable and function names. It’s a terrible habit, and makes code incomprehensible — and programmers know it.
Math, as taught to kids, is full of single-letter variable names (or worse, as you point out, using non-Roman letters and other symbols) that are devoid of any hints to the uninitiated of what their meaning might be.
I wonder what it would be like if math were taught with descriptive names — starting with the basics, for functions and variables, and used those names consistently.
Would it be more verbose? Absolutely — but I’m pretty sure it would also be a huge kindness to learners.
If they did, it would be a lot easier to talk about trains speeding through the night!
A person can wonder if it would be better to start with understanding. (Personally, I only briefly taught that age but my oracle for how to teach it, Mr Barton, does say that experience shows you have to have down the class's ability to mechanically solve before you introduce the trains. The other order, apparently, doesn't work when you try it with actual students in practice.)
Note that if you programmed in FP languages, you'd have to unlearn the "descriptive variable names" thing you were taught. Once you deal with abstractions, names cease to be meaningful. If the most you can say about a parameter to a higher order function is that it is "a function", then "f" is the best name for it. Some very good programmers argue that long/descriptive names often obscure the shape of an abstraction.
Of course. But even in FP languages you don’t name all your functions “f”! At some level of abstraction, sure, you use “i” for index variables, etc., but once you back out of that low level, you are going to be naming things that are less abstract.
Any time a new symbol is used in a text, it will typically be introduced. The phrase "Let A be the subgraph composed of G" for example defines A in relation to G. Typically, the author will also define their notation at the start of a paper with: "A graph G is defined as the set of edges E and vertices V such that...". To be frank, I always see math papers and communications written in this style.
More often than not, I think folks just say "SINGLE VARIABLE NAMES BAD. MATH HAVE SINGLE VARIABLE NAMES". It's really dense. In any given math paper, the author typically defines any such variables. And if those definitions don't make sense, it's because the underlying mathematical objects are beyond you, and you need further background.
Yes, that’s my point.
Imagine learning a foreign language by being given a dictionary and a set of grammar rules. Is that enough to learn the language? Not really. You’d be missing idioms, patterns of speech, common phrases & compositions - the same is true in math.
If you already halfway understand the language, then yes, you can do with a reference / definitions and rules for application — but even then, understanding how to use them is not trivial.
Also, you are talking about “texts”, but there are plenty of domains in which conference and journal papers assume a certain set of notation, and that notation is not defined, nor even consistent; you need to be well-read in the field to even have a hope of understanding what’s going on. AI/robotics comes to mind, but ML in general is sometimes guilty of this.
Math notation sucks because none of it maps to things non-mathematicians know.
I'm struggling to see how someone could conclude the latter statement from the former. Why is it reasonable for Vietnamese to be unintelligible to non-Vietnamese speakers, yet unreasonable for mathematics to be unintelligible to non-mathematicians?
Programming language syntax sucks because none of it maps to things that non-programmers know. Every time a new token is introduced, whether it be an ascii symbol or a word it's one more mapping your brain has to create to remember it. And on top of this you have to remember how these actually execute too. Yes I said execute - most programming concepts have so many implementations it's crazy. Even basic conditionals can't escape this. There are several tokens for if-statements (if, guard, case, (^:), etc.) and countless common implentations (gcc, ghc, guile, J, and whatever actual processors end up doing).
Deep, complex things are hard, my friend. The pain is real; the payoff is totally worth it though.
I think a good example is: https://xkcd.com/936/
"Correct Horse Battery Staple" is easier to remember than "Tl)7x" because the latter maps into plumbing we already have. Likewise, print "Hello world" is easier to remember than printf("Hello World"), which is easier to remember than std::cout << "Hello world"; This is why we recommend beginners start with a language like Python or BASIC and not C++.
Also, my point wasn't about ideas, but rather the notation used to represent these ideas. For example, you might not be able to read "Ăn quả nhớ kẻ trồng cây", but if I tell you it means "When eating a fruit, think of the person who planted the tree" you understand the idea. Notation is just a means of transferring ideas. Having deep complex ideas is not an excuse for poor notation.