Why are symbols not written in words? We could have written = as "equals", + as "plus", $\exists$ as "thereExists" and so on. Supplemented with some brackets everything would be just as precise. 

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

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.
"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.
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.