TITLE: Why are proofs not written as collections of logic symbols but are instead written in sentences?
QUESTION [12 upvotes]: 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?

REPLY [4 votes]: 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.