-----------
The day after...sorry I swamped the list with all that wild ride math talk... I never get to talk or write like that anywhere except here. guess that's what they mean by loosing your inhibitions and judgement... Wish we were all in a bar in NYC about 1950... time warp.
There is a way to approach understanding the impact and development of set theory. Hilbert was a teacher, and Gottingen was something like a math giant professor production line.
There were twin developments in German mathematics that vastly condensed and systemized mathematical education proper. One theme of those developments followed the Geometry line. The other followed the Number line. In the former we have starting with Riemann, followed Felix Klein and his use and development of group theory and the re-axiomization of geometry which superceeded the controversy over the parallel postulate. The other or number line starts again with Riemann, then follows, Cantor, et al and sets. This line sort of has multiple branches that creat the `morass' of analysis. (There is also Riemann himself to look at, study, think about and how he generated such numerous and giant spawn: Cantor, Klein, Hilbert, Einstein in the next generation. Mostly his contributions go to what I think of as the join of number and space into analysis and physics, while the number-space line goes the re-organization of algebra, via the same core concepts used in sets and groups.
Together these two topics Set Theory and Group Theory, went through the entire spectrum of maths and re-codified, compacted, and `simplified' this entire branch of thought. They essentially created `modern' mathematics including many of its new branches like topology. Both Klein and Hilbert were also philosophers in their own way by doing what amounted to mathematical ontology. Both men and lines of thought --number, space-- have theoretical and concrete applications that were discovered along the way and that in turn re-fed back into these developments.Well like nuclear war and cyber space.
So when I read/study mathematics and/or about mathematics, I look for theme content in applications within mathematics and applications within the sciences, sometimes the arts.
I just ordered Number and Numbers to start. So we'll see maybe if I can get enough through it to figure out what's going on... In the meanwhile, I know I will use something like the above pov to think about Badiou. Will the math crew get anything out of these works. Will the sciences get something they don't know? Will I get insights I didn't have? Do I like the style, the method of approach, etc.? The thing is, I am into raw ideas, good, bad, true, false, great, small, and the lowest of the low, totally middle brow, whatever...
CG