I have a penchant of writing mathematical pieces (proofs, theorems, axioms, etc.) in formal syntax, using as less English as possible. I don't know whether this is essential, I like doing this so I do this. I personally feel that this increases the precision of the mathematical statements that I am making. But there are various problems I encounter, like expressing non-logical strings like "we choose", "arbitrary", "let", "we define", "assume", "hence by contradiction" ,etc, thus making my writing dry for those who read them. I just want to know whether symbolism is essential in mathematics.
For some schools of the philosophy of mathematics, formality is what avoids the internal inconsistency of human logic, expressed in the various paradoxes of mathematics like Russell's, or Barry's.
Since we do not know exactly how far into mathematics these inconsistencies reach, it is safe to fall back on formality when there is a question. Officially, that formality is couched in set theory, with a specific kind of axiomatization to which the community has largely agreed.
Unless you adopt a firm philosophy of mathematics that somehow explains the right way to handle paradox, formalism is your only fallback position. Most practicing mathematicians are more interested in the work of mathematics than its philosophy, so they adopt this default fallback as their primary defense against paradox or other logical potholes.
But on a day-to-day basis, they accept a fully Platonic position: They assume that what can be visualized or captured in other ways compatible with human sense and senses is safely far away from foundational issues.
That leaves formalism as a necessary part of mathematics, but not necessarily as a productive part of most ongoing mathematical development.
At the same time, it is quite hard to do some kinds of mathematics with any speed unless you compress it into notation. That notation can usually be formalized, but does not need to be so. So most mathematicians find a balance between concise notation that allows them to capture thoughts in writing quickly enough that they do not evaporate or develop faster than they can be recorded, and the need to communicate things that are not necessarily that clear.
I would add that the level of formality is not related to the usage of symbols, or abbreviations - in other words, to the way things are "encoded" and/or expressed - but to what is missing in the conventional mathematical text, and missing on purpose, as much as possible: metaphors, non-technical explanations, merely evocative commentary.
There are at least two issues at stake in the way this conduct has been incorporated in the tradition of mathematical work:
1) First of all, it manifests the position that mathematical definitions and proofs are not made more cryptic than they could be, in the absence of these non-formal "appendages". The thing here is to try to understand, not only what is written, but why it has been written, namely, to convey a kind of knowledge that is (supposed to be), at least originally, immune to practical implications or worldly agendas. This is, of course, highly debatable.
2) There are usually, nonetheless, non-formal paths towards the understanding of a new mathematical object or problem, even for the initiated. That these paths are rarely, if at all, documented in any way, can be considered a problem for a field that remains extremely elitist, at a time when science desperately needs to be able to communicate better with the general public. There is, of course, remedy for that.
Euclid managed to be formal without being formalist; and when I was taught Newton's three laws of motion it was done in words and without equations; it's necessary to be precise but it's not necessary to be formal; the two, though related, are necessarily different.
The invention of formal languages is useful in allowing one to theorise about such languages - the first and standard instance where this has been important would be Gödel and his incompleteness theorems.
It's not normal practise to write mathematics in this manner, that is using a formal language; and in fact excessive formalism is usually frowned upon - a certain amount of judgement is called for.