Sunday, October 31, 2021

When did Math Get So Hard?

I have been on many Math PhD thesis defense's  as the Dean's Representative. This means I don't have to understand the work, just make sure the rules are followed. I've done this for a while and I used to understand some of it but now there are times I understand literally none of it. As a result, when the student leaves the room and we talk among ourselves I ask

When did Math get so hard?

I mean it as a statement and maybe a joke, but I decided to email various people and ask for a serious answer. Here are some thoughts of mine and others

1) When you get older math got harder. Lance blogged on this here

2) When math got more abstract it got harder. Blame Grothendieck.

3) When math stopped being tied to the real work it got harder. Blame Hardy. 

4) Math has always been hard. We NOW understand some of the older math better so it seems easy to us, but it wasn't at the time. 

5) With the web and more people working in math, new results come out faster so its harder to keep up.

6) All fields of math have a period of time when they are easy, at the beginning, and then as the low-hanging fruit gets picked it gets harder and harder.  So if a NEW branch was started it might initially be easy. Counterthought- even a new branch might be hard now since it can draw on so much prior math. Also, the low hanging fruit may be picked rather quickly. 


  1. Are math doctorates at UMD about creating something so obscure and complex that it will be seen as a new contribution?

    1. 1) If so then I doubt UMD is unique in this.
      2) Another reason that I didn't list is that Math has been around so long that it requires so much prereq knowledge to understand things. So it may not be obscurse, just beyond my knowledge base.
      3) To directly answer your question- might depend on what `obscure' means

  2. I suspect that mathematicians looking at the solution of the Connes embedding problem may be wondering when computer science got so hard.

  3. Along the same lines as what J said, often there are no incentives to make your research intelligible, and there can even be perverse incentives to make your research unintelligible. Jared Diamond wrote an interesting article on this topic back in 1997.

  4. 5) With the web and more people working in math, new results come out faster so its harder to keep up.

    This seems to be the most relevant to me. Together with the fact that math and CS seem to have a culture that allows them to really thrive in that environment. I wonder more how math and CS could help adjacent fields to thrive similarly.

  5. I would refuse to serve on a committee if the candidate could not spend the first 5 minutes motivating their work in a way I could understand

  6. That might be a good way of getting out of being on committees!

    But to be fair, as Dean's Rep its not my job to understand the material, though I would like to. Has math gotten so hard and abstract that this is impossible? NO- I often DO ask the student

    ``Tell me what problems, perhaps from a long time ago, and perhaps no longer as connected to your research as you would like, motivated work in your field.''

  7. If you think that P!=NP (which is my guess, and seems like the conventional wisdom), then proofs are generally going to be easier to check than find. (And presumably to "understand", although that's not exactly the same as "checking", I guess.)

    In which case, even if we rely on future math AIs (or smarter people) to find proofs, at least there is a window in which we'll understand the proofs, even if we can't find them ourselves.

    I think that's (kind of) reason for optimism. (Although I guess at some point, even checking the proofs becomes difficult also.)

  8. How do you sit through a foreign language movie if you do not understand anything while the movie is about three to five people sitting in a room and standing in front of them is the leading actor doing most of the dialogue with some slides? Coffee?

    1. I Do listen for as long as I am not totally lost, but when I get totally lost I might proofread a paper, or work on a problem set, or make up a HW, or ... And the odd thing is that NOBODY in math things this is rude or inappropriate. They do the same at their Monday Colloquiums where very few people understand the talks.

  9. @Bill: My question is whether von Neumann would agree with this;
    supposedly, he was one of the last giants to understand a large
    portion of mathematics. I keep forgetting the famous quote that
    states the percentage (20%?).
    Even Terence Tao seems to have a threshold when dealing and posting about certain topics in modern mathematics -- there's one area particularly, that he has not touched on. So understanding where the threshold is and why it indeed represents such a challenge maybe not a bad approach.

  10. I understand 90%--- of the work done on The Muffin Problem :-)

    One measure of how hard math has gotten is to have two parameters x,y

    If x% of the people can understand y% of mathematics, then when
    x is small and y is small, math has gotten hard.

  11. "Mathematicians are programmers ... the only problem is that they are using the wrong programming language" :-)