Do theoreticians check proof of all lemmas and theorems proposed by others while being used in their own proof?

Question

I am trying to figure out how theoreticians work. In my understanding, theoreticians work to find new theorems from previously established theoretical systems. To prove new theorems, they have to read a lot of previous papers to find paths for their own proofs. I am wondering, in the process, do they have to read and understand all proofs of theorems that build their own construction carefully? Simply using an existing lemma or theorem is convenient, but the intricate construction process will be ignored, which may contain valuable hints for the problem of the theoretician himself/herself.

Although not working as a pure theoretician, from my research experience in parallel algorithms of computational physics, I can imagine the difficulty of fully digesting theoretical papers, especially when a systematic understanding of a whole theoretical system is needed to proceed. The general structure of a system is more important than the path to build it, for most scholars. But for theoreticians, the building path may be of more interest.

P.S. I purposely avoided using the word "mathematician". The reason is, I'd like to incorporate researchers not only in pure math, but also other areas, e.g., theoretical computer science, theoretical neuroscience, theoretical physics, etc. I guess anyone who works on establishing, or expanding a theoretical system (in the sense of formal logic) should be counted as working in theory. These researchers may not directly contribute to the mainstream of mathematics, but they also rely on constructing proofs in their career.

My question is mainly from my observation of theoretical computer science, especially from many theoretical works for understanding deep learning. Many excellent young scholars publish several (around five) theoretical papers on top venues every year, full of proofs for lemmas and theorems. I was astonished by their publication speed, and was wondering if they really completed understanding all previous relevant papers to write their own. For me (who is not working directly in theory, but has to read some theoretical papers for implementations), understanding each of these papers thoroughly could take at least weeks (considering reading reference papers). Honestly, I am not a theoretician, but this publication speed really exceeded my imagination, and I am figuring out if I misunderstood theoreticians' work.

Answer 1

It depends, but I'd say: to a large extent yes.

There are certainly some theorems which are simultaneously extremely useful but have very difficult proofs, which get used as blackboxes at least some of the time. At the same time I can confidently say that every result I've used is one whose proof I've understood completely, at least at some point in time (I may have forgotten it since then), and at present at least I have no interest in using a result without fully understanding its proof first. Certainly this is made easier by the nature of my subfield (mathematical logic is relatively young, and reverse math and computable structure theory in particular don't really have "big theorems" as such), but I don't really think it's particularly rare.

There's certainly no rule saying that you can't use existing results without knowing their proofs, and in principle one could build a career based entirely on combining known theorems in new ways (although I don't think this is actually possible in practice). However, I'd strongly warn against that, partly because of the lack of insight, partly because of the drastically increased possibility for error (referees aren't perfect and there are plenty of results which are snappy-sounding but have delicate hypotheses), and partly because to be honest taking too much on faith will make it harder, not easier, to see how to apply or generalize the results you already know.

Answer 2

If you're talking about theoretical mathematics, it depends on the subject. There are some areas of mathematics that have advanced so quickly and build on such a huge foundation that most of the people working in them do not understand much of the context or proofs for many results they make use of.

To understand every single detail of every lemma or theorem you make use of would require you to read tens of thousands of pages of technical details. Instead of doing that, many mathematicians find it easier to keep a collection of working examples in their minds. The examples are not themselves proofs, but in understanding the examples, one is sufficiently convinced of the truth of many theorems which in full generality are difficult to understand why they are true.

If you work in the Langlands program of algebraic number theory, for example, there is a good chance you are using a lot of algebraic geometry, and that is a huge timesink to try and work out the details. Stacksproject, which includes detailed proofs of a lot of foundational algebraic geometry, is 7000 pages long. Even the foundational papers in the Langlands program themselves are long and incredibly difficult to read. Langlands' paper on Eisenstein series is notoriously difficult, and many people who work with Eisenstein series just take those results for granted.