Formal proofs do nothing to help understand the mathematics, all they do is help to check for errors in logic -- have you ever read a formal proof? They are leagues more obtuse and difficult to understand than any piece of mathematics from any field.
A program is technically truth-value equivalent to doing a proof, but the resulting document/piece of text are MILES apart. The proof is an explanation in plain language to others who understand the topic why something is true. The formal proof is a sequence of opaque inference rules that the computer promises is equivalent to a real proof.
It's the difference between understanding a formula and plugging it into your calculator. The calculator is more likely to compute correctly, but it gives you no help when it comes to actually understanding mathematics.
Mathematicians as a community completely reject the notion of using formal proofs (while sometimes begrudgingly accepting they may be useful to check our work) because they completely miss the point of what we are actually doing.
Only in the sense that writing in brainfuck is equivalent to writing in Python...
Doing formal proofs manually is like writing turing machine instructions. Computer-assisted proving gives you higher level tools, but apparently it is still not powerful enough to be accepted in mainstream. And unlike programming where you get stuck if you haven't invented higher level languages, mathematicians have the widely accepted tool of "it trivially follows". Bourbaki might disagree, though.
Users of high-level languages, typically do not think about what the compiler is doing, and if in fact they do not know the compiler action--often the only way to find out is by reading the compilers source code.
This is akin to looking a math paper seeing "it trivially follows..." and your only recourse to find out why it so trivially follows is to get a mathematics degree.
With programming the information is always there. If you don't understand a higher level language but do understand the language in which the compiler is implemented, you can always read the compiler.
The "it trivially follows", the information is simply missing. Gaining a mathematics degree will often give you the intellectual power of finding the missing pieces yourself, but the only sure way is to ask the authors themselves.
Getting a math degree just to be able to fill in the gaps created where 'it trivially follows' is equivalent to being required to memorize the API for a framework because clearly defined documentation is hidden behind a paywall.
so called 'intellectual power' has little to do with it. What you're describing is familiarity with navigating a minefield of poorly structured information and tapping a academia (ie at a very high cost of entry) for 'special access' to resources that bridge the gaps.
Of software development suffered from the same informational constraints and lack of innovation, we'd still be playing pong.
To be cynical about it.
I guess that's not much different from most fields' use of specialized vocabulary, etc. There are many informal methods of dissuading people from competing in your own little corner of the labor market.
Hmm you are from US aren't you? :)
> Only in the sense that writing in brainfuck is equivalent to writing in Python...
Actually, in the very concrete sense that a program is a proof for the forumala that is equivalent to its type[0].
[0] https://en.wikipedia.org/wiki/Curry–Howard_correspondence
Mathematics with a solid logic foundation has also existed only for the past century, and writing programs is actually equivalent to doing a mathematical proof as some have already pointed out.
The actual problem is that when programming you are talking to computers, so you have to lay out each step, or the computer will not know what to do. Sure you can use libraries to do very complex things all at once, but then that's because those libraries have already laid out the steps.
But when doing mathematics you are talking to humans and you often skip steps in a very liberal way. There is indeed a library of theorems that is written down, but there is also an unwritten library of "it trivially follows", which, if a mathematician is asked to actually write it down, might feel humiliated.
When you think that there is something so trivial that computers must be able to do, you still have to find or invent a library for that. When you think there is something in mathematics so trivial that it must be true, you just need to convince your audience.
One day all mathematical papers will come with a formal proof, but that day has not yet arrived.