What Is Good Mathematics?

A very interesting “opinion piece” by Terry Tao. I liked the list of five “hypothetical” ways in which a mathematical field can lose its ways. And the story of Szemeredi’s theorem, used as a case study of good mathematics, is so beautiful that it’s always good to hear it told.

10 thoughts on “What Is Good Mathematics?

  1. This sort of introspection study of the field is very common in mathematics. In computer science this is not so much the case. I’ve only heard Karp and Chazelle openly question if the field is moving in the right direction.

    We researchers in TCS could all benefit from hearing the perspective of some of the best and brightest as to where the think the most fruitful lines of research are.

    Last but not least, its amazing the clarity of Terry’s vision of mathematics. Clarity that I might add many famous mathematicians often achieved only until their older, wiser years.

  2. Can Anonymous please post a reference to where Chazelle and Karp question whether (theoretical?) CS is moving in the right direction?

    Thanks. –Yarden

  3. I think that anonymous 1 is confused. There were many questions about the direction of TCS a dozen years ago. Karp led the group raising the questions at that time. What transpired in the next few years demonstrated the extraordinary effectiveness of the field.

    More recently both Karp and Chazelle have been very effective at pointing out how valuable the work of TCS has been and how the insights of TCS are important to many other disciplines both within CS and outside.

  4. Sorry, I did not mean “question” in a critical way. Allow me to rephrase that as “I’ve only head Karp and Chazelle openly ponder if the field is moving in the right direction”.

    As anon 4 points out, in STOC 1995 Karp lead a discussion of where the field is going and where it ought to go. Chazelle held a similar session during SoCG 1994.

    Over the years I’ve seen several (sub)fields within CS blindly sailing towards the rocky shallows without much question as to the expected outcome. AI during the late 70s and early 80s comes to mind, yet anyone who dared openly question the direction AI was taking was seen as uncollegial. As we know, AI research and funding paid a heavy price during much of the 80s and early 90s from these mistakes.

    TCS is nowhere close where AI was. The field as a whole is very healthy. But this is part of my point: we shouldn’t need to be inches from the rocks to ponder what is the best course to take. Terry’s paper is very much in this spirit. He’s not claiming that math is in some state of crisis, he’s simply doing some healthy housekeeping and planning.

  5. The TCS community has a tradition of having the “best and brightest” expound on their view of where the field is going or ought to go. STOC, FOCS, and CCC have invited talks, and tutorial sessions that do just that. If you look at the original publication of Kapr’s NP-completeness paper, in the book “Complexity of Computer Computation” (record of a special conference at IBM Yorktown Heights), it concludes with the record of a public discussion on the meaning of these topics.

    Moreover, the challenges of Karp et al gave rise to a very spirited, and constructive exchange of ideas: see for example the early ACM Computing Surveys article (Dec 96).

    For a more recent version of what Theory should be, see

    The point that Tao emphasizes is that the vitality of an area depends on a continuous collaborative story, not on a single point of view, or a single definition of what is “good”.

    AI funding didn’t suffer as much from bad directions, as from exagerated, overoptimistic claims of what their techniques could achieve.

    I am not sure what anonymous (WHY ANONYMOUS?) saw as Theory sailing towards the rocky shallows–could we have some concrete examples? My impression is that we try to go in the directions where we can push the envelope of knowledge–which is not a bad heuristic.

  6. not on a single point of view, or a single definition of what is “good”.

    I do not expect nor seek unanimity in the voices. It’s the dialogue that we should seek: great thinkers pondering aloud “what’s the next big challenge? where can progress, as a field, be made?”.

    AI funding didn’t suffer as much from bad directions, as from exagerated, overoptimistic claims of what their techniques could achieve.

    Hype is only part of the story. The methods of AI were not conducive to growth. Today, progress in AI subfields is highly correlated to how long ago they adopted hard scientifically reproducible tests (e.g. machine learning, theorem proving).

    I am not sure what anonymous (WHY ANONYMOUS?)

    Answer by self-quote: “anyone who dare openly question is seen as uncollegial”.

    Theory sailing towards the rocky shallows–could we have some concrete examples?

    Outside theory: programming languages and software engineering, in which progress seems very weakly correlated to actual needs.

    Within theory and to use an example from Karp: PRAM research. The whole field came to screeching halt in the mid 90s, so much so, that SPAA was considering folding a couple of years back. IMHO all of this could have been avoided if PRAM research had been redirected in 1992-1993. By the time the LogP model came out, in an acknowledgment of this drift, it was too late. Today’s resurgence of SPAA proves that there was value in the field, what was needed was a different approach. Since the community could never muster this direction the field went into hibernation.

    A more modern example is CC which seems to be drifting a bit after the complexity-classes driven approach of the late 80s. I would love to hear a dialogue, say, between Lance, Christos and Lucca as to where CC ought to go.

  7. it’s very uncollegial of you to misspell my name.

    I read my son every night a story about a kid named “de Lucca” and no sooner had I clicked on “publish your comment” when the thought hit me: last name de Lucca is double c, first name Luca is single c. Sorry.

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