Why are proofs not written as collections of logic symbols but are instead written in sentences? Mathematical proofs are written as sentences and not as collections of logic symbols.
Through logical operations, it is much easier for me to visualize what the symbols are trying to tell us rather than English text filled with grammar. This is my personal opinion, others may have different opinions.
I just asked this question on another website to find out logical mistakes in my work which is entirely done in the language of propositional logic.
Some people suggested to write it down in sentences in English. Is there any kind of tragedy in writing proofs as collections of logic symbols?
 A: You have not translated the pages from Apostol's book into mathematical logic.
What you have done is to transcribe them into your own idiosyncratic shorthand, which may be useful to you but is less than meaningless to anyone else.
Let's start with the use of the symbol $\stackrel{\mathrm{def}}=.$
In normal mathematics, this tells us that the notation on the left is defined to represent the expression on the right in a general way.
For example, when we write
$$ \cosh x \stackrel{\mathrm{def}}= \frac{e^x + e^{-x}}{2}, \tag1$$
it is a definition of the $\cosh$ function.
In a definition of this sort, a symbol such as $x$ is a variable that can be substituted, so Definition $(1)$, above, tells us not only how to interpret
$\cosh x$; it also says how to interpret
$\cosh y,$ $\cosh t,$ $\cosh a,$ or $\cosh b.$
For example, Definition $(1)$ informs us that
$$ \cosh b = \frac{e^b + e^{-b}}{2}.$$
In your notes, you start with the definition
$$ [a, b] \stackrel{\mathrm{def}}= \text{closed interval in $x$-axis}. $$
Now, setting aside the fact that there are four English words on the right-hand side of that definition (what were you saying about using symbols rather than English text?), you have just defined a bracket notation for us,
"[" followed by a variable followed by "," followed by another variable
followed by "]" and you have informed us that this is a closed interval on the $x$-axis. Now it seems strange that your variable names do not occur on the right-hand side of this definition, and in fact this does make the definition relatively useless in strict logic: which closed interval is denoted by $[a,b]$?
But worse still, on the next line we find out that changing the variable names changes the definition to a closed interval on the $y$-axis, not the $x$-axis.
If you actually succeeded in translating the pages to pure logic, along the way you would realize that the labels "$x$-axis" and "$y$-axis" are hints to help you visualize things, not part of the strict logic of the mathematics itself.
You really need only define the closed-interval notation once.
I would say that some of your uses of $\stackrel{\mathrm{def}}=$ are actually logical definitions of symbols and notation. But many are not.
If you have a good definition of the product of two sets, it is not necessary to write out your interpretation of $P_x \times P_y$ as a "definition."
It would already be defined and (logically) unnecessary to write.
By the, way, symbols such as "$\ldots$" do not belong to the notation of mathematical logic; they are (again) merely hints to understanding.
You also seem to tend to use "$=$" to signify "is a" rather than the standard symmetric, transitive, and reflexive notion of equality.
For example:
$$ Ƃ:Q \to \mathbb R = \mathrm{SF} $$
would mean the same thing as
$$ \mathrm{SF} = Ƃ:Q \to \mathbb R $$
if you were writing in the language of mathematical logic;
and the meaning of the line in which it appears would still be ambiguous.
(Is SF a mathematical constant like $\pi$?)
If you actually were writing in mathematical logic you might have defined SF as a predicate, written in the form
$$ \mathrm{SF}(Ƃ:Q \to \mathbb R). $$
Later on that same line, however, you write $Ƃ:Q_{ij} \to \mathbb R,$
contradicting what you wrote earlier.
The domain of $Ƃ$ could be either $Q$ or $Q_{ij},$ but it cannot be both in the same definition. It seems you want to say that the restriction of $Ƃ$ to $Q_{ij}$ is a constant function, but you have neither the logical notation to describe a restriction of a function to a subdomain nor to say that a function is constant.
You end up defining $©_{ij}$ as a synonym for $Ƃ:Q_{ij} \to \mathbb R$ but not saying anything about new what the function does.
Frankly, without using Apostol's text as a Rosetta Stone for your work,  I think it would be very difficult for anyone else to guess what you mean by all your notations.
I see nothing wrong with making your own notes on a passage of text and equations in which you break everything out in a tabular format with displayed equations and no paragraphs of text.
Just don't expect anyone else to read it. It is for your own use in organizing your thoughts, and that is all.
If you really want to write things like this in mathematical logic, there are various computer-aided proof systems in which you can write your definitions and theorems in completely symbolic language and feed them into the software, which will check them for you.
But I don't know if you would actually find this easier to work with than the text in a book like Apostol's.
A: It seems like there's multiple different questions being asked here.
For the title question, the reason is simply that for most of us, words express our thoughts better than symbols. In fact, whenever you learn what a new symbol means, it's usually explained to you in terms of words, right? And sometimes those words are about as concise as the symbol, such as "for all" vs. $\forall$. When we see the symbol, we read it as "for all" and so all it's done is save space on the page. Useful for writing $\varepsilon$-$\delta$ proofs on the whiteboard quickly, but they don't encode a large enough amount of information to be worth it. Compare this to the notation $f: A \to B$, which encodes the idea of a function, domain, and codomain. This is a useful notation and you'll see it all the time in math papers and textbooks. So I guess the point is that there's a point at which a mathematical notation becomes more expressive than natural language, so that's generally when proof writers start to use them.
To answer the question in the body of your question: no, there's no "tragedy" in writing proofs using primary symbolic notation. But it really depends who/what you're writing for. If you're writing for yourself, you can write it down in the way that you best understand. This can help you study and review in the future. If instead you're writing for a test, a journal, or for a friend to review, there are going to be different expectations in each situation. The short version is "know your audience". When you are writing for anyone that's not yourself, you should be writing in the way that they want to read it. Otherwise your ideas might be misunderstood, which goes counter to your goal of writing. And, as noted previously, most people will prefer the wordy version over the symbolic one.
I should mention, as others also have, that what you've written isn't exactly propositional logic as you claim. There's a lot of English word definitions, non-standard notations, and appeal to geometry/visualization that is generally not encoded in logic. These ideas are part of the reason why words do better than symbols, because we can use words to paint pictures to our audience and help them "see" what we do. If this shorthand is helpful for you, then go for it. But no one else will find this writing nearly as intuitive as the text it was sourced from because we are not you.
Now, it seems to me that you're quite an outlier here, in that you find symbolic proofs (or at least your own version of them) easier to read than English ones. And again, that's okay; if you get something out of translating a textbook you're reading into your preferred form, I highly recommend it. But as for why you feel this way, it seems that English may not be your first language. Or at the very least, the English language is not your strong suit. It's obvious in your writing that the way you speak English is different from most native speakers, and that's likely also true about how you read and interpret it. The English used in most proofs isn't usually too difficult, but there is a certain grammar to it that can be especially confusing when symbols are interspersed in the text.
In the end, if you plan on reading proofs and textbooks in English, this is a skill you will have to pick up. If you plan on writing proofs for anyone but yourself, this is a skill you have to pick up. But for taking notes, whatever shorthand helps you understand is perfectly fine.
A: It might help to think of reading a proof as reading a story. When you read proofs in English (not purely written in logic symbols), you understand the motivation of the statement and the approach. The author could even emphasize on what is the key idea. I doubt you'll be able to convene these implicit information with pure logic symbols. That's why when an argument is complicated or gets too long, it's far easier to read a proof in English than pure logic symbols.
A: Advantages of natural language:

*

*Familiarity: people are taught to communicate with natural language starting when they are toddlers, and have the most experience hearing and using this medium.  On the other hand, they're introduced to abstract notation (if even they are ever are introduced) at a later age, probably 20's in most cases.  Nobody is ever going to come close in fluency with logical symbols as they are with their native tongue: neurological science says our language acquisition talent peaks when we are children.

*Audience: The audience does not need to know much about notation at all, if you are speaking language they understand. Why encumber the listener with having to train themselves to be good at parsing symbolic logic when you can just... tell them.

*Redundancy: this is a good thing, information theoretically. It means that when mistakes occur, the extra context can be used to decipher the correct meaning.  Claude Shannon, the father of information theory, wrote about this.

The only advantages I can see to maximizing use of symbols when expressing something are

*

*It might take less space, but any semantic or syntactic error is going to be a big deal.

*perhaps someone who cannot understand the language you would use, but who is comfortable with the same symbolic language, can read your meaning.  This, however, seems like a prize of little use. Academia has always valued being able to read multiple languages over being able to write flawless symbolic gobbledygook. (And, as user freakish also notes below, the written language of logic is not really much more universal than regular languages. There are still different conventions and notation used in different parts of the world.)

*(The content of this point was suggested by Joris Schellekens, whose contribution I do not want mistaken for my own) Having a symbolic version may be a useful thing to have if your content would be used in an automated proof assistant.  And this I agree with: there are probably some cases of complicated proofs in research literature where having an appendix with a symbolic version of your proof that is valid in some proof assistant is actually very useful.  However, it would be foolish to communicate it to your peers only using the symbolic version.

A: Most humans find it much easier to understand proofs written in a natural language (assuming, of course, it is a language that they are fluent in) with  logic symbols kept to a minimum. You may find it easy to deal with a proof of four or five lines written in logic symbols, but I suspect it would be quite a different matter with a $100$-page proof.  Natural language can be much better at telling you what is going on, while with logic symbols alone you would be lost in the details.
A: TLDR: Using sentences is necessary. The consensus about this in the mathematics community is overwhelming at least.
The basic idea is that you are not writing your work for yourself. Let me give a simpler example that I find fitting. I am not a native English speaker. It is much easier (for me) to write my thoughts in Greek. But right now I am trying to communicate with you - that means that I must write in a language that is easy for you to understand. That language is English. Most mathematicians will have serious problems understanding what you mean (I tried skimming through your work - it wasn't easy) if you insist on using logic symbols.
Assuming you are not famous and you haven't managed to prove something really "big" that means that your peers will not even bother reading your work.
I really enjoyed reading:
https://sites.math.washington.edu/~lee/Writing/writing-proofs.pdf
and laughed a lot with the last line in:
https://ocw.mit.edu/courses/mathematics/18-901-introduction-to-topology-fall-2004/assignments/commentsonstyle.pdf
A: In the beginning, there where no mathematical symbols just words!
So young mathematicians love to emulate the older mathematicians.
This why we still love to not use much symbols.
