TITLE: A global mathematics library
QUESTION [23 upvotes]: Just for personal interest, I am not (yet) professionally involved in it. My question is about the state of arts in digitalization of mathematics and to what extent it is possible and reasonable.
There are different levels of digitalizations:

OCR scan all historical mathematical texts
organize metadata of references and authors (e.g. as graphs)
extract mathematical objects (like theorems, definitions, etc)
extract proofs and ideas
formalize mathematics such that it can be completely checked by theorem provers

The main effort I found is
https://imkt.org/
Steps 3/4 and 5 may be of independent interest and should be understood more parallel than chronological. Point 5 is more interesting in having (error-free) formalized math. It should also be allowed to choose different foundations of mathematics and the possibility to switch in between them. Point 3/4 is more interesting for a researcher that wants all references for a definition, a theorem, a keyword. It would be a wonderful source for doing data analysis of mathematical knowledge (historical, social, semantical, etc). In contrast to 5 it can contain errors and speculations. The main interest is in identifying and referencing mathematical objects over all the produced texts in history of math.
My question is:
The goal of https://imkt.org/ is huge but when looking at its first projects it looks (sry) a little bit disappointing. The main focus (also of other literature that I scanned through) lies on connecting the existing databases and languages, i.e. of theorem provers, computer algebra systems (and maybe wikis). I understand that different applications in math demand different systems (e.g. integer series http://oeis.org/ should also be part of it?) Can or shouldn't there be one system that contains everything that can be accessed (and is stored not just referenced!) through the same system? Are my dreams of such a system over the top?

One of the largest issues is the copyright of the big publishers.
More and more is going in direction open math. Until then it is
unclear to what extent a library can be complete (gaps are somehow
missing the point of this system).

The other issue is the efficiency in producing the content extraction
and to advance it by advertising the advantages of such a library to
mathematicians such that it will move itself at some point.


There have been many efforts in the past that were abandoned again or here for years (like Mizar) but far from being known and used in daily mathematics.

REPLY [3 votes]: There's a lot of work lately on formalizing large parts of classical mathematics using proof assistants, see for example the Lean prover and the accompanying library mathlib. You can see an overview of what has been formalized here; it includes a lot of classical abstract algebra and real analysis, although there's also a lot of undergraduate math they haven't formalized yet which they list here. If you think that's all "trivial", there's ongoing work to formalize the proof of the independence of the continuum hypothesis, the proof that it's possible to evert the sphere, etc.
An interesting facet of this project is that all the proofs are kept under a version control system and you can see the entire history of every proof. It's common for the initial proof of some result to be very long and verbose, only for people to find ways to shorten it or make it more elegant later -- for example, von Neumann's proof of the Radon-Nikodym theorem. Where before you might have had to trace through the citations of loads of historical papers that your library might not even have, now you can see exactly how this process played out by just doing git blame.