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.

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