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So my understanding is that a while back a group of mostly French mathematicians, under the pseudonym Bourbaki, wrote a somewhat austerely written series titled "Elements of Mathematic(s)" covering a large swath of the discipline, starting from the very ground up with an axiomatic set theory. The idea was that the books be rigorous, and entirely self contained, and to a large degree they were throughout the series.

My question is, is Bourbaki unique? Has anyone else, or any other group, collaboratively written a similarly comprehensive series before or since? I don't necessarily mean simply a series of textbooks on a few topics, I mean a directed series of volumes which specially rely on each other and are built on top of one another, starting from the very axiomatics. If so, has it been as influential? If not, why not? Why do I not see any other such series' lining the library bookshelves today?

If such a series exists, could you recommend it to me?

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  • $\begingroup$ The foundations of mathematics as we know them today only really began to emerge in the first half of the twentieth century. So it was the right time to write them out. If we were to write them down again today, they wouldn't be very different from those in Bourbaki. So there is no point in redoing what Bourbaki did. It is possible that in the future, the foundations used for mathematics will change, and at that point a new Bourbaki will emerge. However, that period was the first time mathematics had been entirely formalized, to the extent that proofs could, if written formally, be checked... $\endgroup$ – Keith Jun 8 '15 at 8:58
  • $\begingroup$ in an entirely mechanical way. That kind of qualitative leap is unlikely to happen again. Instead of going from limited formalization to total formalization, we'd be going from one kind of formalization to another. $\endgroup$ – Keith Jun 8 '15 at 9:00
  • $\begingroup$ To some extent, the ssreflect ( ssr.msr-inria.inria.fr ) project can be viewed as a modern Bourbaki, however focusing only on formalization as opposed to writing a reference text. I suspect more projects like this -- serving both machine and man -- will crop up once constructive foundations are better understood and once proof languages have become more mature. $\endgroup$ – darij grinberg Jun 8 '15 at 18:12
  • $\begingroup$ Yeah, that appears to get the formalisation part, at least. $\endgroup$ – Nethesis Jun 8 '15 at 18:13
  • $\begingroup$ Also, Bourbaki never covered more than half of existing mathematics (and even half is a rather imprecise upper bound), so there is a lot of room for improvement! $\endgroup$ – darij grinberg Jun 8 '15 at 18:13

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