TITLE: How many words (i.e. not "math" symbols") should I use in my proofs? ${}{}$
QUESTION [19 upvotes]: I must once again resort to the advice of this great community.
As I was reading about the pigeonhole principle something about its proof struck me as odd. Allow me to explain:
After reading the "The Foundations: Logic and Proofs" chapter in Rosen's "Discrete mathematics and its applications" book I was left with the feeling/notion that I can (and I ought to) describe all my proof's statements in symbols.
Yet, as you can see in the pigeonhole principle's proof:

We use a proof by contraposition. Suppose  none of the k boxes has
  more than one object. Then the total number of objects would be at
  most k. This contradicts the statement that we have k + 1 objects.

Without a doubt it has more English words than symbols. Yet the proof is actually without flaws. I struggle with the fact that I can't convert it into the A -> B format (I hope you understand what I'm trying to convey).
However, is there a way to symbolically represent what it states (as I'm trying to put it in my mind)? 
Or the only way to argue for this proof, is by using words?
And if so, is there any guide or principle that should tell us when to use words or symbols in our proofs?

REPLY [8 votes]: Natural language contains more redundancy and is therefore more robust against errors.
When you write proofs, chances are that you will sometimes make mistakes.
This is not a feature of mathematics, but of life in general.
If you mistype a character in a symbolic proof, it can become utterly undecipherable — and wrong, but that's not the biggest issue.
If people understand your story, they can reconstruct a symbolic proof from your story if they want.
If you write your proof in plain English and repeat important things, it is easier for the reader to recall past things when needed and fill in the holes when you leave them.
Natural language is redundant, and therefore messages go through better.
If you ever grade a math exam, you will learn the hard way that people will forget all kinds of silly things and write things unclearly.
A fully symbolic proof is readable only if it's completely correct, a language-oriented proof is readable if it's at least somewhat correct.
Symbolic proofs only contain the proof.
A good proof in words contains more: it describes what is going on and motivates maneuvers instead of only executing them.
This makes a difference to whoever reads your text, including yourself.