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We could have written = as "equals", + as "plus", $\exists$ as "thereExists" and so on. Supplemented with some brackets everything would be just as precise.

$$\exists x,y,z,n \in \mathbb{N}: n>2 \land x^n+y^n=z^n$$

could equally be written as:

ThereExists x,y,z,n from theNaturalNumbers suchThat 
     n isGreaterThan 2 and x toThePower n plus y toThePower n equals z toThePower n

What is the reason that we write these words as symbols (almost like a Chinese word system?)

Is it for brevity? Clarity? Can our visual system process it better?

Because not only do we have to learn the symbols, in order to understand it we have to say the real meaning in our heads.

If algebra and logic had been invented in Japan or China, might the symbols actually have just been the words themselves?

It almost seems like for each symbol there should be an equivalent word-phrase that it corresponds to that is accepted.

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    $\begingroup$ Symbols have two big advantages: clarity; and good symbology allows manipulation that makes things easier. Books even a few hundred years ago were "rhetorical": even algebra was explained in words, rather than symbols. Trying to solve an equation described in words, by performing algebraic manipulations in words, is extremely difficult. As to Japan/China, as it turns out positional system and some of the algorithms for arithmetic did arise in China... and they did not use ideograms. $\endgroup$ – Arturo Magidin Jun 28 at 18:56
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    $\begingroup$ Partly for brevity, and partly for ease of manipulation. It's much easier to do algebra on $x + 2 = 5$ than on "two more than $x$ is five." There is also a balance that needs to be struck between brevity and clarity when writing, as replacing too many words with symbols definitely takes away from the clarity of the statement. $\endgroup$ – DMcMor Jun 28 at 18:56
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    $\begingroup$ There is of course the classic limerick "A dozen, a gross, and a score / plus three times the square root of four / divided by seven / plus five times eleven / is nine squared and not a bit more." $\endgroup$ – aschepler Jun 29 at 4:11
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    $\begingroup$ Why did you write "2" instead of "two"? Why did you write "x,y,z,n" instead of "an integer, another integer, yet another integer, still yet another integer"? $\endgroup$ – Joel Reyes Noche Jun 29 at 12:30
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    $\begingroup$ "ThereExists", "NaturalNumbers" "suchThat" are easily understood by someone speaking English, but maybe not as easily by someone who speaks a different language. Math symbols have also the advantage of being universal. $\endgroup$ – Andrea Mori Jun 29 at 13:40

12 Answers 12

106
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"Integral From a x squared Plus b Plus c To Infinity of a Times LeftParenthesis x Plus LeftParenthesis x Plus c RighParenthesis Over LeftParenthesis x Plus One RightParenthesis Differential x IsGreater Integral From a x squared Plus b Minus c To Infinity of a Times LeftParenthesis x Plus LeftParenthesis x Plus c RighParenthesis Over LeftParenthesis x Minus One RightParenthesis Differential x Implies Integral From a x squared Minus b Plus c To Infinity of a Times LeftParenthesis x Plus LeftParenthesis x Plus c RighParenthesis Over LeftParenthesis x Plus One RightParenthesis Differential x IsGreater Integral From a x squared Plus b Minus c To Infinity of a Times LeftParenthesis x Plus LeftParenthesis x Plus c RighParenthesis Over LeftParenthesis x Minus One RightParenthesis Differential x" ?


Quiz:

Do you recognize this one ?

Summation On j From 1 Upto N Of y Index k Times the Product On k From 1 And k NotEqual to j Upto N Of x Minus x Index k Over x Index j Minus x Index k.

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    $\begingroup$ @HenningMakholm . And typos "Righ" . Almost inevitable in this style. Medieval European math in the original actually looks something like this, for example Tartaglia's poem (!) presenting the solution to the cubic equation.... And at one time you had to write $xxxxx$ prior to a general theory of exponents, until someone got tired of it and wrote $x^5$ $\endgroup$ – DanielWainfleet Jun 28 at 19:36
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    $\begingroup$ @DanielWainfleet: furthermore, some parenthesis are not balanced... $\endgroup$ – Yves Daoust Jun 28 at 19:37
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    $\begingroup$ The examples intend to show that this writing style is inferior, but I think it mainly shows that we are not used to it. If the words would be in Russian, I would have an even harder time to find out what was meant, but I don't think that Russian is inherently inferior to English. Maybe someone growing up with this system would find it easy and would understand it better than we do our system. (l don't think so, but I have no good arguments.) $\endgroup$ – Pakk Jun 29 at 7:02
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    $\begingroup$ @Pakk: it is not just "slightly inferior", it is completely unusable because it is lacking a geometric layout. $\endgroup$ – Yves Daoust Jun 29 at 8:03
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    $\begingroup$ @Pakk it takes an entire page to write a one-liner for no specific gain. This is indeed an inferior notation system! $\endgroup$ – David Jul 1 at 15:19
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Consider this problem, taken from The Evolution of Algebra in Science, vol. 18, no. 452 (Oct 2, 1891) pp. 183-187 (taken from JSTOR, itself translated from work of Nesselman on a problem by Mohammed ibn Musa):

A square and ten of its roots are equal to nine and thirty units, that is, if you add ten roots to one square, the sum is equal to nine and thirty. The solution is as follows: halve the number of roots, that is, in this case, five; then multiply this by itself, and the result is five and twenty. Add this to the nine and thirty, which gives four and sixty; take the square root, or eight, and subtract from it half the number of roots, namely five, and there remains three: this is the root of the square which was required and square itself is nine.

This is how algebra used to be done; you have similar descriptions in Babylonian scribe tablets, Egyptian papyrii, Middle Age textbooks, etc.

Using symbols, the problem becomes, first, to solve $x^2+10x = 39$. The process is to complete the square: $$\begin{align*} x^2 + 10x &= 39\\ x^2 + 10x + 25 &= 64\\ (x+5)^2 &= 64\\ x+5 &= 8\\ x &= 3 \end{align*}$$ Something that is much easier to do without too much thought, and certainly much less effort, than the decription. Also, the idea of completing the square is much simpler to explain in symbols than it is to do so rhetorically.

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    $\begingroup$ @zooby: So, here’s the thing. Symbols are meant to make it easier to convey and manipulate the information. Leibnitz used to say that having good notation can do as much as solve half the problem for you. The notation of the Legendre symbol, for example, makes quadratic reciprocity simpler to use, understand, and prove, than Gauss’s original notation for it. But nobody forces you to use symbology, especially as a solving tool. You should use whichever method seems better to you. What you should not do, however, is confuse the difficulty of starting to use symbols with that of using them. $\endgroup$ – Arturo Magidin Jun 29 at 0:17
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    $\begingroup$ @zooby: That is: there is a “start up cost” that can be pretty steep, but it is also possible that once that price is paid, things are much easier. An example of that might be touch typing, or LaTeX: it is difficult to start using them, and the initial investment of effort to be able to use them is often much higher than that needed to solve any particular problem. But once you make a one-time investment to learn to use them, you never have to pay the (smaller) price for the one-time problems, and so if you do a lot of them, it’s worth the investment. $\endgroup$ – Arturo Magidin Jun 29 at 0:18
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    $\begingroup$ @L.F. Negative numbers weren't a thing when the problem was written. $\endgroup$ – eyeballfrog Jul 1 at 2:33
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    $\begingroup$ @ArturoMagidin It's probably because, a long time ago, English numerals followed the same pattern. I was just making fun of that. I don't know why you are taking this as a personal offence $\endgroup$ – David Jul 2 at 15:36
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    $\begingroup$ @David: It seemed more like you were criticising either me or the author I quoted for messing up and translating from the German literally, rather than "realizing" that "zwei und zwantzih" should be "twenty two" and not "two and twenty". My apologies if I misinterpreted your tone. $\endgroup$ – Arturo Magidin Jul 2 at 15:59
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Others have already answered on why one should use symbols. I want to add that one shouldn't overuse symbols, as people sometimes do.

With too many symbols, statements get clustered and confusing. Out of lazyness, many like to write $\exists$ quantors in the middle of a sentence. Others overuse e.g. $\land$, etc. ($\land$ is not a synonym for "and"!)

So basically, one shouldn't overuse symbols.

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    $\begingroup$ Example : " $ \exists >1$ example of over-use"............+1 $\endgroup$ – DanielWainfleet Jun 28 at 19:45
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    $\begingroup$ +1 I've written several answers on this site suggesting that the OP use words, with symbols only when they are the best way to convey meaning. That's not nearly as often as people think. $\endgroup$ – Ethan Bolker Jun 28 at 23:21
  • $\begingroup$ ∧ could on occasion be useful as a means to express "and", because "and" in a mathematical context could be interpreted as set union. $\endgroup$ – Rosie F Jun 29 at 6:29
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    $\begingroup$ @DanielWainfleet What's that even supposed to mean? "There exists more than one"? $\endgroup$ – Tanner Swett Jun 29 at 9:21
  • $\begingroup$ I guess it's "There exist more than one exemple of over-use". Also note that in mathematical articles, there's often many words outside of formula, sometime more than symbols. $\endgroup$ – Tryss Jun 29 at 14:10
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Quoting Robert Recorde, inventor of the equals sign:

"And to avoid the tedious repetition of these words: is equal to: I will set as I do often in work use, a pair of parallels, or Gemowe lines of one length, thus: =, because no 2 things, can be more equal." (

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    $\begingroup$ I do love the fact that we know the name of the person who invented (or, at least, made popular) the equals sign. $\endgroup$ – David Richerby Jun 29 at 12:59
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    $\begingroup$ @David: Indeed, his name is a matter of recorde... $\endgroup$ – Rahul Jun 29 at 19:11
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    $\begingroup$ This also makes sense of using $\simeq$ for isomophism, since $\sim$ and $-$ are isomorphic spaces. Now we just have to convince people to use $\mathbin{\dot-}$ for equivalence.... $\endgroup$ – Oscar Cunningham Jun 29 at 21:08
  • $\begingroup$ @OscarCunningham I'm hooked! $\endgroup$ – mattecapu Jun 30 at 9:40
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    $\begingroup$ @OscarCunningham There's a variety of graph products whose symbols come from the way that product of two edges is drawn. See en.wikipedia.org/wiki/Graph_product#Mnemonic $\endgroup$ – Bob Krueger Jun 30 at 16:53
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Because not only do we have to learn the symbols, in order to understand it we have to say the real meaning in our heads.

That doesn't match my personal experience. For me, if someone speaks a formula aloud, I have to reconstruct in my head how it looks before I can begin to understand what it means. (Sometimes "in my head" doesn't work, and I need to use paper instead).

As for advantages, here is a bit I once wrote for another answer:

Also: cheating is allowed. Very often, large parts of a formula are identical to large parts of a previous formula -- and the only thing that really matters is how it differs from the previous formula. In those cases it is expected that you'll just think to yourself, "oh, this thing is the same as that thing over there", without bothering to understand or remember exactly what "that thing" was in detail. You can just compare the relevant parts symbol for symbol without thinking.

In fact, this last point is part of the reason why formulas are designed to be compact and dense with information. It increases the chance that you can keep the entire formula in your visual short-term memory as you move your eyes from one formula to the next, and thereby make it easy to spot that some parts of them look alike, without even being conscious of the individual symbols. (Who knew all those inane spot-the-differences problems you find in kids' magazines actually train a highly relevant mathematical skill? They do!)

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  • $\begingroup$ Thanks for writing in words the answer I wanted to write but was too lazy to. =) $\endgroup$ – user21820 Jun 30 at 16:22
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There are plenty of reasons for prefering symbols:

Brevity: Compare the two sentences you just wrote. Why would I want to scroll when reading an equation? Also, isn't it nicer to see it in just one line. Moreover, it takes much less time to write "*" than "multipliedBy". I just don't want to write "multipliedBy" a hundred times for doing basic calculations

Clarity and precision: While the meaning of words may change over time, mathematical symbols do stay the same. Also, they don't allow for any kind of ambiguity. For instance, we all agree that $-5 < -3$ but some people would think of negativeFive as a biggerNumber than negativeThree, and for good reason. Others would not.

Language variation: Here you have two choices: Either you force everyone who wants add two numbers together to learn English/Latin/whatever, or you will run into trouble with translations. You may want to translate circuloDeRadioDos as circleOfRadiusTwo. Congratulations! You've screwed up! You may argue that only a few words would have to be remembered for basic math, but while symbols have translated names that stick into one's mind, eventually, someone will scratch their head wondering "This plus thing, was it más or menos? Yeah! I'm pretty sure it was menos There was definitely a different keyword for más

Coding standards: If you are asking this question in these terms, I am 100% sure you are a software developer. I am also confident that you use Java. Well, first, camelCase is awful! Everyone know that! Deal with it! It also only applies to languages with upper/lowercase distinctions. Also, if not even a "small" community of R programmers can agree to how to write variables, how do you expect the entire world to do so?

As a final remark, I would like to point out that there are cases where replacing words by symbols is not advisealbe. While you should definitely use symbols rather than words for basic things like "+, -", and you should define sets with "{stuff that verifies X}" rather than

Set set = new Set();
set.append(stuff);
for(i in 1:length(set)) {
 if(set[i].verifies(X)) {
  set.remove(i)
 }
}

Wow! I am pretty sure you are really enjoying this last paragraph! The world should look like this in your eyes! OK, you probably thought of something more Let set be the SetContaining stuff suchThat stuff VerifiesX, but the point is clear. Math is not coding, and it should not look like coding. This comes as a new idea to many people in the Stack Exchange communities, but not everything in this world is code!

Finally, coming back to seriousness, there is a point in nor overusing symbols, as pointed out by previous answers (see Kezer). I would add that you should avoid symbols unless you are exactly sure of what they mean. Many students beginning their Math degrees are writing absolute nonsense for ignoring this point. At least as a beginner, you should always check that your mathematical symbols make sense in ordinary language terms. But please don't do it in CamelCase!

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  • $\begingroup$ Pedantic nitpick: you've actually pascal-cased "CamelCase". And yes, the world would be better off if we all moved to the strictly-superior snake case, but alas it will not. $\endgroup$ – Jared Smith Jul 1 at 16:01
  • $\begingroup$ You are right, but you have to admit that Set set = new Set(); deserves either an upvote or a flag $\endgroup$ – David Jul 1 at 16:05
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    $\begingroup$ It deserves a firing-squad. Which is of course your point. +1. $\endgroup$ – Jared Smith Jul 1 at 16:06
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There are lots of reasons.

A major one is of course brevity. Using mathematical notation is much shorter than writing things out in full, which will make a huge difference to a 100+ page proof.

What's more, in mathematical notation, synonyms don't exist. We can say the same thing in English in two different ways, but there is only one true way to express things in notation

The third, and most significant, problem with this is: not everyone speaks English! While we read this notation in our language, other countries will read it in theirs and interpret the same meaning.

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    $\begingroup$ I don't agree with 'synonyms don't exist'. In fact, the system of mathematical symbols have plenty of homonyms and synonyms: for example the 'disjoint union' has the following notations, among others: $\overset*\cup, \ \cup^*,\ \coprod,\ +$. $\endgroup$ – Berci Jun 28 at 22:42
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    $\begingroup$ Another source of synonyms is choice of symbols for unknowns. Then there are differing definitions of the same term, which lead to different people expressing the same quantity by different formulae, e.g. the quadratic form $ax^2+bxy+cy^2$'s `discriminant' is defined variously as $b^2-4ac$ and $ac-b^2/4$. Then there are several conventions for the formulae for the Fourier transform and its inverse -- a factor $\frac{1}{2\pi}$ must go somewhere, but where? $\endgroup$ – Rosie F Jun 29 at 6:47
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    $\begingroup$ @Berci And, conversely, homonyms exist: $A\subset B$ might mean strict subset or just subset; $f\circ g$ might mean $f(g(\cdot))$ or $g(f(\cdot))$ and so on. $\endgroup$ – David Richerby Jun 29 at 13:05
  • $\begingroup$ @DavidRicherby $f \circ g$ always means $f(g(\cdot))$ $\endgroup$ – David Jul 1 at 15:10
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An important point that the other answers haven't (I believe) mentioned: for many mathematical symbols, the natural language 'word-phrase' has a subtly different meaning.

A few examples:

  • $\exists$ is about whether or not an element with particular properties lies in our domain of discourse, whereas there exists, in natural language, is about metaphysical existence. For example, $\exists x: x=2$ is not controversial but there would be people who dispute that the number 2 exists (at least, in the same way that tables and chairs exist).
  • The exponentiation function (^) would presumably be rendered to the power of. But once we start writing expressions like $(2+i)^\wedge(3-2i)$ it's not clear that there is any useful intuition remaining behind to the power of.
  • $\vee$ is a logical connective that holds when at least one of the propositions either side is true. The most common natural language 'translation', or, is sometimes used in this way but often used for exclusive disjunction, depending on context (for example, someone who asks whether you want the vegetarian or the meat option in a restaurant would not expect 'both' to be a possible answer).
  • $\rightarrow$ (often also written $\supset$) is a logical connective that holds when the antecedent is false or the consequent is true. But there is plenty of literature about why this material conditional is a poor model for natural language connectives like if ... then.

If you wanted to use words for these, you either have the problem that the words you are using have different meaning to in usual life (in which case the words are a distraction), or you have to make up new words, in which case there is no particular cognitive saving.

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    $\begingroup$ "For example, $\exists x: x=2$ is controversial" - Did you mean to write that this is not controversial? $\endgroup$ – Tanner Swett Jul 1 at 17:43
  • $\begingroup$ @TannerSwett I did, thanks for the spot. $\endgroup$ – dbmag9 Jul 2 at 14:01
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Simply it is simpler to write it in symbols

If we are going your way the equation

$$\exists x,y,z,n \in \mathbb{N}: n>2 \land x^n+y^n=z^n$$

would become

ThereExists SomeNumber,AnotherNumber,AnotherAnotherNumber,AnotherAnotherAnotherNumber from theNaturalNumbers suchThat 
     AnotherAnotherAnotherNumber isGreaterThan Two and SomeNumber toThePower AnotherAnotherAnotherNumber plus AnotherNumber toThePower AnotherAnotherAnotherNumber equals AnotherAnotherNumber toThePower AnotherAnotherAnotherNumber

as you can see clearly instead of 2 lines where it is very hard to read and understand, it was written by half line that is can be read easily

in top of that math code is international language, people in Japan, France, Palestine, etc. they all can understand the math without speaking English

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    $\begingroup$ I'd say it's simpler in the sense that it's more concise and sometimes more readable. But often, it is simpler to use words instead of symbols - especially in the case of obscure notation. For your example, we could have started it with "There exists three natural numbers and a natural index, such that the index is greater than two and... which isn't too bad. In some contexts it might even be more readable! $\endgroup$ – Jam Jun 29 at 14:47
  • $\begingroup$ @Jam keep going, continue the sentence! "There exists three natural numbers and a natural index, such that the index is greater than two and the first number raised to the power of the index added to the second number raised to the power of the index equals the third raised to the power of the index." for me personally to understand this sentence will take much longer than simple coded sentence.. in top of that math code is international language, people in Japan, France, Palestine, etc. they will all understand what is the math without speaking English $\endgroup$ – asmgx Jun 30 at 21:52
  • $\begingroup$ My point isn't that natural language is always simpler than symbolic language, it's that it is sometimes simpler. Try coding your comment symbolically and you'll see what I mean. $\endgroup$ – Jam Jul 1 at 10:37
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I like the answers here so far, but I would just like to add:

In my personal experience, when I read mathematical symbols that I am familiar with, I don't necessarily say the meaning out loud in my head. I think that kind of thing comes with familiarity with the symbols you're using. For example, seeing $\frac{d}{d\theta}$ doesn't make me think, "the derivative with respect to theta." All that I hear/say in my head when I read it is, "d d$\theta$." The original meaning is still there, I'm aware of it, I just don't explicitly say it. Sometimes I don't even read symbols as words in my head at all - I might hear an equation as just a series of short noises in my head because I'm just acknowledging the symbols on the path to building up a picture of what the equation is saying. It makes for much quicker reading, because instead of having to slog through a bunch of words, the meaning is explicitly clear to me on sight.

I think if you keep doing math, you'll also develop a familiarity with the symbols and you won't need to explicitly state everything in your head.

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Yves Daoust's answer demonstrates how unreadable a formula with words instead of symbols tends to be, but doesn't really explain why it is so unreadable.

Certainly brevity is one thing: the more symbols there are in total, the more processing the brain needs to do to even lex the expression (to use the CS term), before we can even begin to parse it.

Reducing the length of the words intuitively reduces the total amount of information on the page – though actually this only works because if all parentheses etc. are written out as a word, there's lots and lots of duplication. Efficient coding avoids such duplication; the simplest way is to choose short indentifiers for everything that's used often. Well, mathematical notation drives this to an extreme by basically using only a single glyph for everything; that's not always viable, but in many formulas there are only few different variables and most of them are repeated quite a lot, so single-character symbols do in maths generally make more sense than whole words for everything.

But that's not all. Yves Daoust's example could be compressed with a coding

a: a
b: b
c: c
d: Derivative
f: From
g: IsGreater
i: Integral
l: LeftParenthesis
m: Times
n: Infinity
o: Of
p: Plus
q: Minus
r: RightParenthesis
s: squared
t: to
u: one
v: Over
x: x
y: Implies

to an extremely short form, that would nevertheless be almost completely unreadable:

ifaxspbpctnoatlxplxpcrvlxpurdxgifaxspbqctnoamlxplxpcrvlxqurdx
yifaxsqbpctnoatlxplxpcrvxpurdxgifaxspbqctnoamlxplxpcrolxqurdx

The real problem isn't solved:

it is completely unusable because it is lacking a geometric layout

Well, I wouldn't phrase it quite this way. Geometric layout is a helpful visual aid for humans, but really the point is that mathematical language is for the most part not “linear stream” like, as a prose story. Rather, it is organised as an abstract syntax tree, and that's where the maths notation shines: it uses geometry to make that structure evident much clearer than with open/close parentheses that need to be found. Note that this can also be done with the compressed string above, with the technique that programmers have for the purpose: indentation and whitespace.

  i f axs p b p c
    t n
    o a m l
           x p lxpcr
           v
           lxpur
          r
    dx
 g
  i f axs p b q c
    t n
    o a m l
           x p lxpcr
           v
           lxqur
          r
    dx
y
  i f axs p b p c
    t n
    o a m l
           x p lxpcr
           v
           x p u
          r
    dx
 g
  i f axs p b q c
    t n
    o a m l
           x p lxpcr
           v
           lxqur
          r
    dx

This would now actually be quite readable, with some practise. The main problem is that it takes up vastly more space than the standard notation, because of all that whitespace. Maths notation avoids this by just making the “top level separators” a bit larger or otherwise visible, instead of making them stand out in the indentation.

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1
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My favorite quote by Alfred North Whitehead, from An Introduction to Mathematics (1911):

By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and, in effect, increases the mental power of the race. Before the introduction of the Arabic notation, multiplication was difficult, and the division even of integers called into play the highest mathematical faculties. Probably nothing in the modern world would have more astonished a Greek mathematician than to learn that … a large proportion of the population of Western Europe could perform the operation of division for the largest numbers. This fact would have seemed to him a sheer impossibility … Our modern power of easy reckoning with decimal fractions is the almost miraculous result of the gradual discovery of a perfect notation. [...] By the aid of symbolism, we can make transitions in reasoning almost mechanically, by the eye, which otherwise would call into play the higher faculties of the brain. [...] It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilisation advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle — they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.

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  • $\begingroup$ Yes. The reason my brain imploded when I read Principia Mathematica was probably that the excellent notation relieved my brain of too much work at once. $\endgroup$ – leftaroundabout Jul 1 at 15:15
  • $\begingroup$ Mathematical advancements make hard things easy, and formerly impossible things merely very hard. $\endgroup$ – Daniel R. Collins Jul 2 at 0:56
  • $\begingroup$ True – before the Principia, humans couldn't have even though of doing something like 1+1, couldn't they? ... — Mocking aside, I actually really like this Whitehead quote and agree with most of the points. $\endgroup$ – leftaroundabout Jul 2 at 7:04

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