The problem has nothing to do with formalism vs intuitionism. The problem is that mathematical proofs are written by humans and read by other humans, yet are supposed to adhere to an inhuman level of consistency. It is human to err, and as such errors compound, purported absolute proofs become..somewhat less so.

I made the analogy to programming because this audience is one that has experienced first-hand the chasm between what you think that your program should do, and what it will do when put into a computer. I drew the analogy to a program that has only been specified because, at the research level, most proofs have significant gaps and steps that the reader is expected to be able to fill in. (Sometimes, of course, these gaps can't be filled in. At least not easily.) This is part of why a professional mathematician is not surprised to find that they can only read research papers at a rate of a page or two a day.