There is a cultural problem in mathematical communication and pedagogy, which is keeping the discipline from having a wider accessibility and appeal.
I come from a background where I ended up studying mathematical logic without having any mathematical skills to speak of at all. You'd be surprised how far you can go in the former without any training in the latter. I'm now teaching myself maths.
The reason this is significant though is because of the notion of a proof as it is defined in mathematical logic is different to the proofs you see presented in mathematical papers and textbooks. In mathematical logic, what we care about is the precise mapping between the syntax and the semantics. The ideal (which for some systems is entirely met) is a perfect functional mapping between the two. And because we care about this - a proof is something which is defined carefully so as to ensure that no invalid WFF (well formed formula) is derivable from your axioms and rules of inference. Obviously maths doesn't want to prove falsehoods either, but the difference is in the level of explicitness. Every line in a proof of logic must have it's justification written beside it - whether it be by use of a rule of inference, or a previously derived theorem. You don't skip steps just because to you they seem 'obvious'. And my wonderful logic teacher at the time would take off marks whenever we did.
Reading proofs in mathematical logic - even long and complex ones - is more often than not straightforward. Even the meta-logical results about the systems I studied were presented with greater clarity than your average proof in normal mathematics.
Why is this? Well it seems to me that mathematicians un-apologetically skip steps in their proofs all the time. And it's extremely rare that they will state their justifications. To me this is as bad as un-commented, un-documented code with no unit tests. Any coder that inherited that sort of code would be rightfully annoyed!
And this leads to situations that require this blog post. I have a friend who is a maths PhD (who has been a great help in my mathematics studies) - who defends this aspects of mathematics. His main argument is that of simple convenience. The problem, he says, is not the presentation but my lack of mathematical experience. If every proof had to be written for those with little mathematical experience, then no maths would ever get done.
As I get more experienced with Maths - I can't disagree that reading proofs gets easier. But I feel his reply misses the point. These gappy proofs are endemic in pedagogical literature as well - not just research papers. And a pedagogical text shouldn't be about the convenience of the author. Different textbooks assume different levels of background knowledge and I find it to be a complete mess. Besides this, I generally have little sympathy for this reply because of my background in logic which trained me to be explicit. Whereas all these mathematicians are growing up thinking that it's normal to just assume everyone knows what you're on about.
And, besides the teaching context, in the research context I've heard things aren't exactly great - you get this absurd situation, described to me by my friend, where mathematicians gather at conferences - present papers, but generally have little understanding of what anyone else is talking about. Mathematicians seem fine with this, apparently. Why? I believe this to be a cultural reality that could be changed - not something essential to the nature of the discipline.
Don't get me wrong - there are some wonderful examples of mathematicians that are working really hard to change this. I found the Ohio State university calculus course on Coursera to really try to be innovative in their pedagogical approach (Their text book was a bit meh - but their online exercise application was brilliant). But in general, I just don't see a lot of this.
Agree? Disagree?