Some stuff I found interesting about number theory research

I do have a possibly relevant joke: A math student asks his professor how to visualize a result in 7-dimensional space. The professor responds: "It's easy. I visualize the result in n-dimensional space, then simply take n = 7."

A typical research paper is written at a very high level; often steps in the argument will assume the reader is also a skilled mathematician, and invite them to fill in the lower-level details themselves. This reasoning is as much intuitive as it is formal; I suspect it will be a while before proof assistants are as intelligent as the typical reader of a mathematics paper.

I think there's an outside perspective that if math isn't 100% logically verified, it's worthless, which doesn't really match up with my own experience. Most results rely more on the intuition of the authors than on the precise logic they write down; thus the surprising result that papers with logical gaps are, very often, still correct.

It's telling how you can tell the article was written by a mathematician, apart from the obvious fact that it's about mathematics. I'm talking about the structure. For example:

"How do mathematicians prove theorems? This question introduces an interesting topic, but to start with it would be to project two hidden assumptions:

(1) that there is uniform, objective and firmly established theory and practice of mathematical proof, and

(2) that progress made by mathematicians consists of proving theorems"

It reminds me of a joke I was told by my math teacher when I was at school about how mathematics shapes your thought (I may do a poor translation):

An air balloon traveller encounters very dense mist and gets lost and so is forced to descend to the ground, where he finds a guy taking a walk. The traveller asks the guy "Excuse me sir, could you tell me where am I?" The guy, after a very long pause answers "You are on a balloon". The traveller smiles and says "You are obviously a mathematician". "How do you know?" asks the guy to which the traveller replies. "Easy, there are three very good reasons. The first one is that you took a long time to answer, obviously considering just the facts and proceeding with strict logic. The second one is because the precision of your answer. And last but not least because your answer is useless"

In top-level mathematics its that you have no choice but to treat results as black-boxes because the complexity otherwise is just too high. Its the only way forward, and mathematicians would prefer not to take this approach.

In software development, black-boxing is a way of accelerating development. It allows people to abstract away and disregard complexities. And it allows re-using big works. It comes at the cost of some integration losses and performance losses. But, since they are not at the complexity limit, software developers can dive into their black boxes if required for performance or other reasons.

There's 'understand', and 'believe you understand', and in other fields (say, software engineering...) you meet lots of people who think they understand, when they don't. I also think that in the SE case, because it's more 'relatable' ('look at my website') , people generally assume more easily that they understand when they don't.

Also if you look at theoretical comp sci results (calculability theory, etc) , or even complex software engineering systems (actual working distributed systems, etc) , I very much doubt that many people (apart from the people who designed and wrote them, and who may even be wrong about them! ) can _actually_ claim to understand them fully. You will understand, to some extent, the 'theorem' (ie the system), but the implementation will remain a black box.