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Years ago, I studied undergrad mathematics at university. The understanding passed to us then was that the past centuries have had several phases of "programs", large-scale efforts that defined mathematical research of the time, e.g.

  • 18th century: Development of analysis
  • 19th century (first half): Foundations of modern algebra (separation of set and structure)
  • 19th century (second half) - ca. 1930: Rigorous foundations of mathematics (logics and set theory), further development of analysis
  • ca. 1920 - ca. 1970: Formalization of computation, metamathematics

Is there a "big picture" effort going on in mathematics at the moment also? Or is everything happening at the same time? I'd really like to know that, but not being in math myself, I thought I should just ask the pros directly.

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closed as primarily opinion-based by Bill Johnson, Yemon Choi, Stefan Waldmann, Steven Landsburg, Ryan Budney May 8 at 17:00

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise.If this question can be reworded to fit the rules in the help center, please edit the question.

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Starting in the mid 20th century mathematicians began reorganizing the foundations of algebra and geometry around the language of categories and functors. This program has accelerated in recent years with the introduction of higher categories in many different areas of mathematics. –  Paul Siegel May 8 at 14:34
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One should keep in mind that, while it is true that these were indeed the main concerns at the time, we are looking at them from hindsight. I doubt that mathematicians at the time thought of what they did as a "big program", this is a label we put on things with a perspective of some fifty years... –  Pablo Zadunaisky May 8 at 14:42
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Hmmm... when I hear "big program" I'm more inclined to think "Langlands program" than the type of program you listed. It may be the case that we only see things as you have put them in retrospect: for instance, Galois wasn't really trying to 'develop modern algebra' as much as solve a concrete problem about polynomials. It may well be that the concrete, but intractable problems are always what lead to what are seen (in hindsight) as fundamental methods. –  Sam Hopkins May 8 at 14:43
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Does "big program" mean something that is currently regarded as a single program, and that the majority of mathematicians are currently working on? If so, the answer is definitely "no." (And I suspect that the answer to this form of the question would also have been "no" in the time periods you listed; see Pablo Zadunaisky's comment about hindsight.) –  Trevor Wilson May 8 at 16:02

1 Answer 1

A rather large one (7m) is the category theory/homotopy theory framework being developed by Awodey, Voevodsky, and others; see http://www.cmu.edu/news/stories/archives/2014/april/april28_awodeygrant.html

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Don't you think it's a bit suspect to measure the importance of this program by the number of $$$ that the US government throws at it? –  David Loeffler May 8 at 16:04
    
I don't think it's fair to say that this program "define[s] mathematical research at the [present] time." (Although I suppose if it is successful enough then future commentators may forget about all the other unrelated mathematical activity of the early 2000s, but that would be a distortion.) –  Trevor Wilson May 8 at 16:06
    
@Trevor, I was responding to the PO's more specific question "Is there a "big picture" effort going on in mathematics at the moment also?" I agree that looking for a single program defining mathematical research as a whole would be futile. –  katz May 8 at 16:23
    
@David, my budget comment was tongue-in-cheek. It is the program (and the people involved) that's impressive, not the budget. –  katz May 8 at 16:25
    
On second thought, maybe your interpretation is correct, as the question's third example includes two separate programs. I think the OP should clarify the question. –  Trevor Wilson May 8 at 16:32

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