TITLE: Writing "Semi-Formal" Proofs
QUESTION [23 upvotes]: I am very interested in proofs. I have taken an undergraduate course
called "Logic and Set Theory" which I found very interesting, but ultimately
unsatisfying.  My biggest disappointment has to do with the language in which
proofs are expressed. It seems to me that we have all of the symbols necessary
to express a proof in "pure math".  By which I mean, only using symbols and a
few specialized words (iff, let, ...). And yet most proofs that I have seen are
just walls of English text, interpolated with mathematical symbols.
When I read a complex proof, I find myself needing to transcribe it into pure
symbols before I have any chance at understanding it.  I have talked to a
professor about this, and he informed me that my "pure math" proofs were
actually considered informal, and not proper proofs at all! He seemed skeptical
that anyone would actually prefer symbols to English.
I have searched Wikipedia and Google for more information, and I see that there
is something called a "Formal Proof" (although I have heard this term used in
other situations, and so I am not quite sure it means what I think it means)
which uses a computer to verify a proof written in a special programing
language. As fascinating as that is, it seems to be a step further than what I
am looking for.
Is there a well known method for writing and sharing proofs of mathematical
statements that uses only mathematical symbols and is not a full blown
programming language?  And if not, why is this considered "taboo" or "informal"?
Thanks,
--jc
EDIT:
I guess this turned out to not be a real question?  Strange, I checked, it definitely ends in a question mark.  Thanks everyone for the help, advice, and links.  I appreciate your input.

REPLY [4 votes]: N. G. de Bruijn is known for, among many things, his work on the Automath project. Automath was a formal language for writing proofs that had a big influence on many of the languages used today for computer-aided mathematical proof (such as Coq and Mizar). However, de Bruijn also spent some time developing a system for "semi-formal" proof, which he called the "Mathematical Vernacular" (MV):

N.G. de Bruijn, The Mathematical Vernacular, A Language for Mathematics with Typed Sets

Here is some explanatory text from the introduction:

1.2. The word "vernacular" means the native language of the people, in contrast to the official, or the literary language (in older days in contrast to the latin of the church). In combination with the word "mathematical", the vernacular is taken to mean the very precise mixture of words and formulas used by mathematicians in their better moments, whereas the "official" mathematical language is taken to be some formal system that uses formulas only. [...]
1.4. Many people like to think that what really matters in mathematics is a formal system (usually embodying predicate calculus and Zermelo-Fraenkel set theory), and that everything else is loose informal talk about that system. Yet the current formal systems do not adequately describe how people actually think, and, moreover, do not quite match the goals we have in mathematical education. Therefore it is attractive to try to put a substantial part of the mathematical vernacular into the formal system. One can even try to discard the formal system altogether, making the vernacular so precise that its linguistic rules are sufficiently sound as a basis for mathematics. [...]
1.6. The idea to develop MV arose from the wish to have an intermediate
  stage between ordinary mathematical presentation on the one hand, and fully coded presentation in Automath-like systems on the other hand. One can think of the MV texts being written by a mathematician who fully understands the subject, and the translation into Automath by someone who just knows the languages that are involved. [...]
1.13. One might think of direct machine verification of books written in
  MV, but this will be by no means so "trivial" as in Automath.  Checking books in MV may require quite some amount of artificial intelligence.  In the first place MV allows us to omit parts of proofs, at least as long as no definitions are suppressed.

You can find an example of an "MV book" in section 18 (alas, it is a bit difficult to typeset here...).