Math argot

There’s a lovely picture on the cover of the Proceedings of the National Academy of Sciences this week:

And a description of what it depicts, which is intriguing:

Cover image: Pictured is a modern version of the Borromean rings, a topological arrangement of three interlocked symmetric rings that owes its name to the Borromeo family of Italy on whose coat of arms the rings appear. Although the three rings cannot be pulled apart, no two of them are linked—a fact that becomes apparent when one of the rings is hidden from view. Jim Conant, Rob Schneiderman, and Peter Teichner derived this particular realization of the link from their theory of Whitney towers, where it represents the Jacobi identity, or IHX-relation. See the article by Conant et al. on pages 8131–8138, which is part of the Special Feature on Low Dimensional Geometry and Topology. Image courtesy of Jim Conant, Rob Schneiderman, and Peter Teichner.

But when you go to the paper, you’ll see that its abstract is so opaque to a non-mathematician that it might as well be written in Martian:

We show how to measure the failure of the Whitney move in dimension 4 by constructing higher-order intersection invariants of Whitney towers built from iterated Whitney disks on immersed surfaces in 4-manifolds. For Whitney towers on immersed disks in the 4-ball, we identify some of these new invariants with previously known link invariants such as Milnor, Sato-Levine, and Arf invariants. We also define higher-order Sato-Levine and Arf invariants and show that these invariants detect the obstructions to framing a twisted Whitney tower. Together with Milnor invariants, these higher-order invariants are shown to classify the existence of (twisted) Whitney towers of increasing order in the 4-ball. A conjecture regarding the nontriviality of the higher-order Arf invariants is formulated, and related implications for filtrations of string links and 3-dimensional homology cylinders are described.

(Presumably “Arf invariants” don’t refer to the unchanging vocalizations of a dog. )

This shows how far removed mathematics is from even other scientists.  Or are our own biology abstracts just as opaque to mathematicians?


Conant, J., R. Schneiderman, and P. Teichner. 2011.  Higher order dimensions in low level topology.  Proc. Nat. Acad. Sci. USA 108:8131-8138.

h/t: Matthew Cobb


  1. FloM
    Posted May 19, 2011 at 8:44 am | Permalink

    I would say: pretty much. Just a random pick here:

    B cell-specific coactivator OCA-B, together with Oct-1/2, binds to octamer sites in promoters and enhancers to activate transcription of immunoglobulin (Ig) genes, although the mechanisms underlying their roles in enhancer-promoter communication are unknown. Here, we demonstrate a direct interaction of OCA-B with transcription factor TFII-I, which binds to DICE elements in Igh promoters, that affects transcription at two levels. First, OCA-B relieves HDAC3-mediated Igh promoter repression by competing with HDAC3 for binding to promoter-bound TFII-I. Second, and most importantly, Igh 3′ enhancer-bound OCA-B and promoter-bound TFII-I mediate promoter-enhancer interactions, in both cis and trans, that are important for Igh transcription. These and other results reveal an important function for OCA-B in Igh 3′ enhancer function in vivo and strongly favor an enhancer mechanism involving looping and facilitated factor recruitment rather than a tracking mechanism.

    • Tim
      Posted May 19, 2011 at 8:56 am | Permalink

      This abstract sounds appropriate for a specialist’s conference, but should not appear in a journal like PNAS.

      Context and audience-appropriateness is everything.

    • Buzz
      Posted May 19, 2011 at 11:30 am | Permalink

      I went to graduate school in mathematics, and yet I understand this biology abstract at least as well as the math one.

    • gillt
      Posted May 19, 2011 at 7:20 pm | Permalink

      Come on! Enhancers and silencers do exactly what they say to promoters which initiate or promote transcription. Transcription is a high school term. Cis and trans and in vivo are Latin.

      I’m sure most people, if they actually tried, could fake their way through most of this bio-med stuff.

      • Alex, adv. diab.
        Posted May 20, 2011 at 2:02 am | Permalink

        4-Balls are also exactly what they say. ;)

    • Posted May 20, 2011 at 2:23 am | Permalink

      Oh well, i don’t know… i studied both maths and biology and, while i can undestand, at least, what you’re writing about (even if i don’t know what OCA-B and TFII-I are), the mathematical abstract is martian, for me :)

      • gillt
        Posted May 20, 2011 at 10:00 am | Permalink

        OCA-B: Bone marrow-cell (backronym!) specific transcription coactivator.

        These things are only abbreviations.

  2. Posted May 19, 2011 at 8:46 am | Permalink

    That math abstract is already opaque to many mathematicians.

    • Diane G.
      Posted May 19, 2011 at 1:01 pm | Permalink

      That’s comforting!

      • Saikat Biswas
        Posted May 19, 2011 at 2:42 pm | Permalink

        The math abstract would certainly appear opaque to anyone who has not taken graduate-level courses in differential topology and knot theory. I have taken both and it’s not entirely transparent to me either.

  3. Tim
    Posted May 19, 2011 at 8:53 am | Permalink

    I feel like maybe one or two sentences in most of my abstracts might be incomprehensible to someone with no knowledge of biology whatsoever. But, I think there’s an obligation to make most of it (at the very least the first few introductory sentences) clear to any educated reader, especially for such a widely-read journal as PNAS. These guys dive right into Sokal-level jargon in the first sentence and never look back. I don’t know what a single sentence in that abstract means.

  4. Posted May 19, 2011 at 8:56 am | Permalink

    I do wish more scientific papers would trumpet cool visuals. And have more cool visuals.

    …And pay for more cool visuals. I gotta eat.

    • Diane G.
      Posted May 19, 2011 at 1:10 pm | Permalink

      Which caused me to check out your website. Nice!

    • leslie
      Posted May 19, 2011 at 1:21 pm | Permalink

      You might consider joining the Guild of Natursl Science Illustrators –

      • Posted May 20, 2011 at 11:35 am | Permalink

        Sorry to Jerry – just realizing how trolly that came off. And thanks Diane and Leslie – I’m a fan of the GNSI already though not a member.

        My intention wasn’t to troll for hits, but just to agree that cool visuals can enhance understanding and enthusiasm in science. And right now, there are more and more science-related artists and image-makers doing tremendously cool stuff. I maintain the ScienceArtists Feed for, and there’s awesomeness there daily.

  5. MadScientist
    Posted May 19, 2011 at 9:04 am | Permalink

    Any field of research within mathematics is likely to be unintelligible to many people from other fields of math. Topics like the Borromean Rings come up in the mathematics of topology and manifolds and would likely be incomprehensible to, say, a statistician. I’m sure you can take a paper from some field of biology and show it to a person in another field and find that they don’t understand much at all.

    • darkgently
      Posted May 19, 2011 at 10:51 am | Permalink

      Yeah, gone are the days when someone like Euler could be at the cutting edge of many different fields at the same time. I’ve gotten used to it over the course of my PhD; now I don’t expect to understand any paper or seminar unless it’s closely related to my work.

      I’m sure there was an xkcd comic about this recently…

  6. Murray
    Posted May 19, 2011 at 9:08 am | Permalink

    “Arf invariant” sounds like it could be the name of a Frank Zappa song.

    • sasqwatch
      Posted May 19, 2011 at 11:46 am | Permalink


      I envisioned an “Arf invariant” as the state of Evelyn, who was only able to go “Arf” while pondering the significance of short person behavior, in pedal-depressed and other highly ambient domains.

      Except, the above sentence actually makes more sense than the math article, especially if you have experience stepping on a piano’s sustain pedal while making all kinds of noise above the strings.

      • sasqwatch
        Posted May 19, 2011 at 12:07 pm | Permalink

        (pedal-depressed, pan-chromatic, and other highly ambient domains… excuse me)

    • Ken Kukec
      Posted May 19, 2011 at 11:48 am | Permalink

      Flip side of “Poodle Lecture”

  7. Teapot
    Posted May 19, 2011 at 9:24 am | Permalink

    By Pure Maths standards the paper is extremely readable.

  8. stvs
    Posted May 19, 2011 at 11:17 am | Permalink

    This shows how far removed mathematics is from even other scientists. Or are our own biology abstracts just as opaque to mathematicians?

    Wow. No. Here, let me google that for you:

    What’s worse, Andy Rooney scooped you by over two decades:

    The Miami-Dallas game was over, and the football theme music segued into the dreaded ticking timepiece of “Sixty Minutes.” I was too exhausted from the game to change channels or turn off the television. Instead I drowsed off, feet propped up on the coffee table, head nestled awkwardly in the corner of the sofa, a Xeroxed article on abelian semi-rings resting, unread, on my lap. …

    I awoke to the nasal twang of Andy Rooney giving the damnedest editorial I’d ever heard.

    “I was in the library last week, looking through the current journals. I couldn’t read a thing in any of them! Not the Annals. Not
    Inventiones. Heck, I couldn’t even make sense of the stuff in the Math Monthly!”

    Andy held up a copy of some nebulous journal with a pale gray cover.

    “The first problem is the titles. You might think the table of contents would help you find an article you can understand. But listen to these titles: Root systems of hyperbolic typeA note on GPIs alld their coefficientsSome remarks on nilpotent
    . Here’s one I like: Dynamical systems on dragon

    “Now tell me, is this algebra, topology, or what?

    “The titles I don’t like are the ones that only refer to people’s
    problems: On a problem of MagnusDarbeaux’ problemOn Waring’s problemHilbert’s 22d problem. Just how mixed up was this guy Hilbert? I don’t know about you, but I do mathematics to get away from people and their problems.

    “And even when you do find a title you can understand, the article itself is likely to be like something the legal department drew up. How’s this for an opening line: ‘Let K be an algebra defined over Q, let pi be the relative Galois group of K, let sigma be an element of pi, let T be a fixed topology on K, and let C—that’s script C, folks—super sigma be the collection of continuous functions on K invariant under sigma.’

    “Can you imagine a mathematician writing Moby Dick? ‘Let my name be Ishmael, let the captain’s name be Ahab, let the boat’s name be Pequod, and let the whale’s name be as given in the title.’

    “I think there ought to be a new rule for authors: Don’t let ‘let’ appear until at least the second paragraph.

    “And don’t you love articles that use weird terms without defining them? It’s hard enough for me to remember what a group is—but what in the heck is a ‘quasi-perfect semi-monoid’? At least tell me which reference I can look it up in!”

    Andy folded his hands on top of the stack of journals. He maintained his characteristic quizzical gaze into the camera, as if the viewer were a trusted conspirator in his iconoclastic theories.

    “I’ll tell you a secret: sometimes I peek at the end of the article, like in a mystery novel, to see who done it. But it usually doesn’t help.

    “The excuse for all these articles is that we’re not supposed to be able to read them. They’re only written for the other four or five experts in the field. I don’t buy it, do you? Is it really necessary to publish? Haven’t these guys ever heard of chain letters?”

    Andy’s bushy eyebrows danced, his pudgy hands pushed the journals

    “But articles are a godsend compared to lectures. Have you ever sat through a one-hour talk on Boolean networks in metaplectic sheaves? Do you find yourself wondering why the speaker can’t tuck his shirt all the way in? I do. Or just what it is on his tie that he had for breakfast? Or why he mumbles? I went to one talk where the speaker actually used his coat sleeve to erase the blackboard.

    “Are you like me? I always take something to work on when I go to these talks. Even if it’s one of these journal articles that I can’t make heads or tails of.”

    Tick tick tick tick. …

    The commercial—for a pain reliever—woke me up for real. I managed to get the set switched off before the show returned. It was time to get back to work. I found myself looking at the article in my lap, wondering why it had been written, and why I was trying to read it.

    My God, I thought, what if “Sixty Minutes” decides to do an exposé of the tenure system? (Barry A. Cipra, “Andy Rooney, Ph.D.“, Mathematical Intelligencer, 10(2),

    • Posted May 26, 2011 at 4:26 am | Permalink

      haha I loved it! and I am going to use that all the time now.

  9. Posted May 19, 2011 at 11:35 am | Permalink

    “Arf” is actually a proper name (of a Turkish mathematician) who discovered this algebraic object; to understand it you need a rigorous course in linear algebra.

    The paper you talked about is about 4-dimensional topology; mathematicians who are non-specialists (even other topologists) would have difficulty understanding the statement of the abstract.

    If you want to understand it, I’d suggest treating Shmel Weinburger to lunch; he is one of the world’s best topologists and he is at your university. He doesn’t know me, but I’ve seen him give invited lectures several times.

    And yes, biology papers are, from my point of view, filled with technical language. I fully understand why; when you and your colleagues write papers, you are communicating to experts. You can’t start from scratch every time. :)

    Speaking of the Borromean Rings, I might recommend this reference? :)

  10. sasqwatch
    Posted May 19, 2011 at 12:01 pm | Permalink

    I work in social networks analysis, which has its own highly specialized set of jargon. It’s pretty abysmal, actually.

    I work more on the practical applications side of it, but it gets really opaque the more you get towards the theoretical, modeling end of it.

    An example…

    “If there is evidence of Markov (and social circuit) dependence in y we may rule out “long-range” dependencies in the data generating process. The action
    of removing an actor i does however induce dependencies among the tie-variables that are not of the type, Markov (and social circuit) dependence, that were assumed for y. Loosely speaking, the MD approach is able to pick out interdependencies between tie variables, that should be conditionally independent according to a model defined on the induced subgraph, as stemming from unobserved potential ties, the AC approach is unable to cope with this since it does assume that there are no unobserved tie

    from: “Extreme Actors – Outliers and Influential Observations in exponential random graph (p-star)

    I’m in the field, and that takes me too long to figure out what has been said, if I’m able to figure it out at all.

  11. Posted May 19, 2011 at 12:11 pm | Permalink

    Indeed biological abstracts are opaque to mathematicians. Having a degree in mathematics and working in a biological lab for the last few years, I can attest that most mathematicians are absolutely unable to follow any biological text. However, it should be said that the mathematical abstract cited here is also not understood by most mathematicians. Currently mathematical disciplines such as topology and, say, statistics are so removed from each other that people working in one field have no possibility of exchanging even the abstract description of their results… How is it between biologists? Is it imaginable for a biologist to not understand a PNAS from a different sub-discipline of biology?

  12. Kevin
    Posted May 19, 2011 at 12:55 pm | Permalink

    The answer to your question is “yes”.

    I just came from a conference where all the talk was about SMADs and MIRs (miRNA) and HIF1-alpha, and all the rest.

    Every priesthood needs its language.

  13. Torbjörn Larsson, OM
    Posted May 19, 2011 at 1:47 pm | Permalink

    I see several commenters noted already how specialist this math is, and how inappropriate for PNAS.

    Unfortunately this is the type of abstract that got a strong push when knot theory became important for the latest fad in theoretical physics, string theory. IIRC Whitney moves tells you how to classify and unravel knots with respect to their topology.

    I assume obstructions to moves tells you which knots persist. And I had to google filtrations: apparently they connect the topology to geometry, which takes you even further towards some nebulous use in theoretical physics.

    • Posted May 19, 2011 at 5:18 pm | Permalink

      I think you’re thinking of Reidemeister moves. “Whitney moves” here refers to what’s also called the “Whitney trick”, a way of getting rid of (self-)intersections of lower-dimensional manifolds inside higher ones. It only works when the dimension is high enough, and the fact that it fails in dimension 4 is (so I’m told by people who know this stuff much better than I do) connected with the fact that lots of topological things are easy in dimensions 5 and up or 4 and up, but go wrong in dimension 3 or 4 or thereabouts.

      Talk of “measuring the obstructions” usually means that there’s some mathematical object that somehow classifies the ways in which something fails. It often seems to have something to do with cohomology. That appears to be the case here, but I regret that I don’t understand nearly enough of the paper to say anything more coherent.

      • Posted May 19, 2011 at 6:40 pm | Permalink

        Yes. I am very familiar with the Reidemeister moves, and the above is about the trick of removing self-intersections.

        Dimension 5 and up are easier because we have the h-coboardism theorem, knots make stuff in dimension 3 hard, and 4 is just plain weird for a whole host of reasons.

        • Alex, adv. diab.
          Posted May 20, 2011 at 1:59 am | Permalink

          Yey WEIT!

          Learn something new every day here! And there seems no subject to remote to not at least attract two or three experts ;)

      • dale
        Posted May 20, 2011 at 9:17 am | Permalink

        the fact that lots of topological things are easy in dimensions 5 and up or 4 and up, but go wrong in dimension 3 or 4 or thereabouts

        Is the key the understanding the Pointcare conjecture.

        Which essentially deals with how can you tell if something looks like the surface of a four dimensional ball.

        In Mathmateese

        Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

        The problem was solved for all dimensions except four and finally solved after a hundred or so years of work just a couple of years ago.

        • Posted May 24, 2011 at 6:00 am | Permalink

          I know that by “closed” you mean “compact, without boundary” but that should be emphasized.

      • dale
        Posted May 20, 2011 at 9:20 am | Permalink

        “Whitney trick”, a way of getting rid of (self-) intersections of lower-dimensional manifolds inside higher ones

        In other words Whitney came up with a trick to “Hide the decline” in dimensions. These scientists can not be trusted.

  14. Diane G.
    Posted May 19, 2011 at 2:01 pm | Permalink

    How much of a problem is caused by journals’ pressure to save space? There’s technical language, and then there’s dense technical language…

  15. Sawdust Sam
    Posted May 19, 2011 at 2:54 pm | Permalink

    Engineers do it too . . .

  16. Posted May 19, 2011 at 9:43 pm | Permalink

    I’m taking cytology right now at university, and some parts of my textbook might as well be written in Klingon. Tough class. Class average on 1st 2 midterms was below 55%.

  17. Posted May 20, 2011 at 12:19 am | Permalink

    Or are our own biology abstracts just as opaque to mathematicians?


  18. Alex, adv. diab.
    Posted May 20, 2011 at 2:27 am | Permalink

    You might enjoy the High Energy Physics paper generator:

    We’ve actually played the game

    and it’s not always easy even for an experienced theorist to tell which is the snarkticle.

    • Matthew Cobb
      Posted May 20, 2011 at 4:47 am | Permalink

      That is a hoot! I only got 50%, which is of course chance.

  19. Posted May 20, 2011 at 7:06 am | Permalink

    As commented by many other people, math is an extreme case of specialization. Differential geometry/topology/knot theory doubly so. However, maybe it is worthy of nothing that there are still subfields of math that are accessible to normal people. Before I switched to biology, I was a math major. I used to walk past the office of one of the world’s leading experts in combinatorics. Every time the professor became impatient in solving some problem, he posted a note on his office door, promising a certain amount of monetary reward. The problems that he worked on tend to be explainable in less a page. Interestingly, some of those problems were solved by random people who walked by. The professor was very passionate about engaging lay person into solving original math problems.

  20. CanadianChick
    Posted May 20, 2011 at 11:15 pm | Permalink

    I barely comprehend math beyond fairly basic algebra, but I LIVE the graphic. I soooooooooo want to create an embroidery piece using that graphic…

  21. CanadianChick
    Posted May 20, 2011 at 11:16 pm | Permalink

    Argh – LOVE, not live. Damn fat fingers on the iPhone…

  22. Peter Beattie
    Posted May 26, 2011 at 8:36 am | Permalink

    Of course, there is a rather famous short essay by Jared Diamond, bemoaning the same trend in biology papers. He gives a couple more reasons why all science should go to considerably longer lengths to be understood by non-experts. Highly recommended!

    • Diane G.
      Posted June 2, 2011 at 1:04 am | Permalink

      You’re right, that is wonderful! Thanks for the link!

  23. Jim Conant
    Posted May 31, 2011 at 5:15 am | Permalink

    As an author of the paper, just let me comment that we tried to make the introduction readable and understandable to non-specialists and non-mathematicians, but we followed the standard mathematical convention of having the abstract describe the contents of the paper in technical detail. Seeing as how this was perceived, I now feel we should have also made the abstract less technical. Pure math papers in PNAS are a relative rarity, so we are treading on unexplored ground. We learn from our mistakes, and in I hope to see many more pure math papers appear in PNAS in the future.

    • Diane G.
      Posted June 2, 2011 at 1:22 am | Permalink

      It’s always nice when the subject of a post shows up for the conversation!

      Much as I would love to be able to understand everything in every journal (HA!), there simply have to be subjects with specialized vocab so arcane as to seem like a second language. Which means some papers will indeed be opaque to most unless one footnotes every discipline-specific term with a lengthy, reader-friendly definition. I’m sure that would go over well. :rolls eyes:

  24. Jim Conant
    Posted May 31, 2011 at 5:19 am | Permalink

    BTW, we’re happy everyone liked the cover graphic. For me, it conveys the spirit of our discipline (low-dimensional topology) more effectively than words anyway.

4 Trackbacks/Pingbacks

  1. [...] Jerry Coyne’s post is rather funny: But when you go to the paper, you’ll see that its abstract is so opaque to a non-mathematician [...]

  2. [...] Math Argot Math Argot [...]

  3. [...] Coyne calls our attention to this abstract, from a recent issue of Proceedings of the National Academy of Sciences: We show [...]

  4. [...] as his starting point my “rant” about the impenetrability of scientific papers in mathematics, Jason Rosenhouse has written a nice essay on what it’s like to be a mathematician, and to [...]

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