Celebrating misunderstanding

stacksdiagramMike Shulman has asked another question that is forcing me to think more carefully about my motivations in writing and publishing the book to which this blog is dedicated; even to remember some ulterior motives that I had forgotten long before I started writing at all.  He writes

I do not personally find it pleasurable to be forced to figure out what I’m reading about when the author could instead have made it clear.


while such writing may not be “wrong”, in the sense that it’s a choice that you as an author have the right to make, I would say it’s also a choice that excludes a substantial number of people from your audience.

Shortly after I moved to France in 1994 I found French expressions and syntactic constructions creeping across the poorly policed borders of my spoken as well as written language.  I began to write in part as a form of self-defense, or defense of my intellectual coherence, and I explicitly set out to write sentences that could not easily be translated, no so much by using an unfamiliar vocabulary but rather by combining stylistic levels and juxtaposing material in a way that (so I thought) would lead to fruitful misunderstanding if translated literally.

Now that I’ve written a book I realize this is a self-defeating strategy; I have no right to complain if no one wants to translate it.  Fortunately it turns out that I underestimated the ingenuity of translators.  Just this month I had the honor of checking a lively and startlingly idiomatic rendition of one of my book reviews by the translator responsible for making this book available to Italian readers.  And Nathalie Sinclair prepared a wonderful French translation of the Afterword to MWA for the Quinzaine Littéraire (unfortunately behind a paywall).

I’ll have more to say about these later.  Once I got into the habit of writing in English I stopped worrying about threats to my identity and now I am rooting for the translators.  “Kein Mensch ist illegal,” as (some) Germans say.  What worries me now is a more insidious form of border crossing, and I am peppering my prose with (apparent) non sequiturs less in the spirit of the dérive celebrated by the Lettrists and Situationists and recorded in Greil Marcus’s Lipstick Traces than to confound the automated trackers in a probably vain attempt to stave off the kind of cyberpunk apocalypse to which I allude in a series of monstrous run-on sentences at the climax of Chapter 8 (which also happens to be where all the loose ends that have accumulated up to that point are tied together in an attempt to prove that the non sequiturs were only apparent after all):

Like the (often inadvertently) anarchist heroes of Pynchon’s novels, who struggle to maintain their values, their personal identities, even their lives in the face of the implacably deterministic projects of more or less precisely identified Powerful Beings, cyberpunk stages a confrontation between an artificial intelligence programmed to impose its version of certifiable seriousness and a (Neu-)romantic trickster hero whose quest is to preserve what remains of our common humanity.

Or, in plain English, I want to do my part for human beings by writing prose that they can understand (with a little effort) but machines cannot (no matter how hard they try).  This is only halfway facetious.  Having lived twice (at least) for extended periods of time in certifiable police states, where I could be sure my telephone communications were monitored, I didn’t want to make them so easy to understand that they could be handled in a routine manner by automated language processors.  At least not by language processors that were not only automated but also lacking a sense of humor.

Some readers will recognize the sample text at the beginning of this post as an excerpt of a recurrent neural network‘s contribution to the Stacks project that has blossomed under the guidance of my neighbor Johan de Jong.  The results so far may be unstructured gibberish but the automaters are persistent and it would be unwise to count on having the last laugh.  But let me get back to Mike Shulman’s question.  My overriding concern in writing MWA  was that authors of books about mathematics have a tendency to “make things clear” by relying on what I call “unexamined preconceptions that can stand to be examined.”  Clarity is achieved in much popular writing about mathematics at the cost of smoothing over the rough edges.  It’s the rough edges, I claim, that make the whole enterprise worthwhile, and I have worked on developing a writing style that makes them as visible as possible, in the spirit of (fruitful) emotional and conceptual disorientation.*

But some potential readers may be looking for books that reassure them that the rough spots are temporary blemishes that in any case don’t deserve to be mentioned.  That’s why I can say a lousy review like the one by Mark Hunacek may be a blessing in disguise, even though the American Mathematical Monthly advertises itself as “the most widely read journal of collegiate mathematics in the world.”  Most of the people who will be discouraged from buying the book after reading such a review probably would have felt “excluded … from [my] audience,” as Shulman puts it, if they had bought it.  I suspect the business model of Princeton University Press (if they have such a thing) is not predicated on extracting profits from unwary readers who dislike the books they buy, but even if it is, a pure cost-benefit analysis shows the model is likely not to be sustainable.  (Readers who buy the book expecting it to be something other than what it is may well take their revenge by writing one-star reviews on Amazon, with a negative net effect on PUP’s bottom line.)

I’m not entirely satisfied that I’ve made myself clear in this post and I may revise it, more than once, over the next few weeks.

*”The spatial field of a dérive may be precisely delimited or vague, depending on whether the goal is to study a terrain or to emotionally disorient oneself.”  (Guy Debord, from Internationale Situationniste #2, December 1958, translated by Ken Knabb)


22 thoughts on “Celebrating misunderstanding

  1. Jon Awbrey

    A decade and a half on the web has taught me something that I did not know before, that some entities desire to learn what they did not know before and some entities prefer to preserve the illusion that what they don’t know is not worth knowing.


  2. sntx

    I always find talk of (or call to) clarity unsettling, as if there is a thing called clarity that is independent of the reader’s background hum of beliefs, motivations and receptivity at the moment of reading something. I have painful memories from school of staring at Milnor’s (!!) characteristic classes book trying to figure out something I dont remember now and at the same period being able to easily consume mouthfulls of, say, Foucault’s Les mots et les choses (it was a strange experience of understanding what he was trying to say without knowing exactly the corresponding collection of sentences with which he was achieving that). A few months later this situation would reverse before finally evening out.


  3. Martin Krieger

    I do not believe MWA is particularly difficult to understand, whatever your goals were. Namely, there is a major theme or two or three, and there is lots of detail. Often, the detail may be above my pay grade, but I can get the major themes.

    If this were a mathematics proof, at the blackboard, that there would be stuff above my pay grade makes me incapable of discerning the adequacy of the proof. But I then ask you questions, again at the blackboard or in the seminar, and maybe you can make it clear to me that I do understand the above-pay-grade stuff. Eric Livingston, in his writing about what happens in proving focuses on those questions. (In his books, and currently in the Journal of Humanistic Mathematics) Archetypically, I keep thinking of Wiles presenting his ‘Fermat” proof (building on Ribet and Langlands and…) and having someone in the audience asking that question that took him another year, with Richard Taylor, to answer.

    Sometimes such a question is devastating to the claims being made. “Devastating” since you thought you had filled in all the details, and it is now clear that the work is perhaps fruitless. Of course, you may well have developed enough mathematical techniques and lemmas, so that the contribution is less strong but still valuable.


    1. mathematicswithoutapologies Post author

      Some of the points I thought were straightforward were not always picked up by human readers (not to mention the hypothetical AI reader). With regard to utility, for example, I made two points that I thought would not be controversial. First: however useful pure research may turn out to be, pure mathematicians are generally not motivated by the (remote) prospects of useful applications. Second: there is not general agreement on what applications are or are not useful. Two negative reviews argued, roughly, that this is not the sort of thing you should say within earshot of elected officials. I don’t know whether this is because they misunderstood the points or because they understood them correctly.


  4. Mike Shulman

    It’s curious that this post is phrased as a response to a question that I asked, since I wasn’t aware of asking any question, only of making a comment. (-: So I’d be interested to know what the question was.

    Anyway, you have some interesting thoughts here. I do reject the (perhaps unintended) suggestion that I, at least, am “looking for books that reassure them that the rough spots are temporary blemishes that in any case don’t deserve to be mentioned”. In fact, one of the things I found frustrating about MWA was that I had a sense that you had interesting things to say that went beyond the usual generalities, and I wanted to understand and talk about them, but because of the writing style I was unable to figure out exactly what they were.

    I think the main point is that I don’t understand how a confusing writing style has anything to do with making the rough edges of the subject matter visible. If I’m teaching a class on real analysis, one of whose goals is to bring out all the rough edges of nowhere differentiable functions and epsilons and deltas that get glossed over in calculus, I still organize my lectures logically and do my best to explain clearly to help my students understand. In the words of Sam Brown, “Never offend people with style when you can offend them with substance.”


    1. sntx

      @Mike Shullman

      Would you agree that in many cases the manner of telling is itself an aspect of what is being communicated that cannot be reduced to what is being said (like the warm sincerity of a heartfelt ‘thank you’ over and above its illocutionary force and content)?

      Also, is it not true the manner of presentation of, say, a proof in a textbook fails to convey the inside-out, convoluted, backtracking path actually followed by the mind when arriving at the proof. I think the mind is for the most part pretty lazy so while these paths-to-proof may not look optimal in some sort of ‘logical space’ perhaps they are in some kind of ‘brain space’. In which case sometimes (when its not too time-consuming) it may actually be better to explain a result by establishing it in the approximate manner it was found (a sort of mental ontogeny recapitulating mental phylogeny, as it were).

      Personally, none of the things I understand well sits in my head like they were presented to me in speech or writing. To take your example, the anything-goes character of continuity of real-valued functions was not conveyed to me by the clear formulations of theorems but by the process of doing many problems during which my mind was mostly in a state of confusion.


      1. mathematicswithoutapologies Post author

        I agree with you, though I’ve pretty much worked through my confusion about real analysis and I feel I would not be doing my students a favor if I insisted on keeping them in a state of confusion to the end of the course. I do think they benefit greatly when I alert them to the practically inevitable confusion that precedes (relative) clarity, by pointing out some of the more common misconceptions.

        Questions about philosophy of mathematics are — interestingly — quite different. The more I examine them, the more I realize how confused I am, so that I believe my confusion would expand exponentially were the margins of my mind not too small to contain so much confusion. The most honest authors, I believe, are hardly less confused than I am, and they do their readers no favors by pretending otherwise.


      2. Mike Shulman

        @sntx, I agree with all of that, but I don’t think any of it contradicts what I said. Clearly explaining the origins of an idea can be part of clearly explaining that idea, and is different from being intentionally confusing. And it’s true that a learner often needs to pass through a period of confusion before understanding something, but that sort of productive confusion is not created by the teacher being intentionally disorganized or obscure about what is going on. Even if the subject matter to be conveyed includes the honest confusion of the author, that confusion can itself be communicated either clearly or obscurely, and I prefer clarity.

        If I may, allow me to quote from a book about quite a different genre of writing: How to write science fiction and fantasy by Orson Scott Card:

        …it’s terrible when it’s the readers, not the characters, who are doing the struggling. The mystery in these cases is not a single question — Who killed this man? Why does this large planet have such low gravity? Instead the questions are more basic. What’s going on? Why am I reading this?


      3. sntx

        (I hope this comment comes at the right place in the tree)

        I don’t want to deny that clarity (assuming for the moment we can give it some sort of meaning: surely you’ve had students who find verbose textbooks clearer than terse textbooks and vice-versa) is important. Perhaps mathematics was a wrong choice of subject to outline my position; since in our mode of presentation we value the propositional content of our mathematical statements above everything else and it seems unassailable that if we want to convey the propositional content of a statement then we want to make the manner of conveying it as invisible as possible – why compound the difficulty of ‘parsing’ the content of what is being said by adding an extra layer to be ‘parsed.’

        I am claiming something slightly different and I think it applies to mathematics as well as it does to philosophy and I believe it is similar to what (one of) the author(s) of MWA holds in context of this discussion.

        As a simple example, lets suppose I want to convey to my non-mathematician friend the ingenuity of a solution (over and above its content) to a puzzle. Now I’m willing to bet that simply stating the solution wont be as effective as adding an extra bit of communication inbetween the stating of the problem and the stating of its solution wherein I (lets assume successfuly) urge him to try and solve it first by himself. I have clearly made things more difficult for him than before when I was simply stating the solution by adding this extra performative but were he to accept it he’d come out the other end with a better appreciation of the ingenuity than before.


      4. Mike Shulman

        Yes — but you clearly stated the problem, and clearly urged him to solve it for himself, so that he understood what you were proposing and why. Imagine if instead you rambled on at length around the edges of the problem, without ever actually stating it or suggesting an activity for him, expecting him to figure out for himself that there was a puzzle, what the puzzle was, and that you were hoping he would try to figure out the solution. I would say that in that case you would have made things even more difficult for him, to no advantage.


      5. mathematicswithoutapologies Post author

        That’s a rather unkind parallel to what happens in MWA, and it’s not really accurate either. Chapter 7, which deals with the most complicated matters in the book, starts by stating a precise question: “How can we talk to one another, or to ourselves, about the mathematics we were born too soon to understand?” It then immediately branches out (like a graph, one of the two main visual images of the chapter) with a question about “mind-altering drugs,” which can be read as a (more or less pleasurable) non sequitur, or as the introduction of one of the chapter’s recurrent themes, but which can also be seen as a literal answer to the question, in the spirit of Jon Awbrey’s comment this morning. And so on. I would apologize for writing in such a way that the form and the content reflect one another but (a) this blog, like the book, is “without apologies” and (b) anyway, yesterday I heard the playwright David Hare make the comment (among many other wonderfully insightful comments) that it makes no sense to replace a line in one of his plays by a paraphrase with the same meaning, because he wrote it to sound a certain way (and this is a paraphrase; unfortunately I didn’t take notes).

        I often use a kind of Socratic dialogue with a student to help uncover misconceptions that interfere with the understanding of a concept I’m trying to clarify. Real analysis classes are a good place to look for such misconceptions, and I don’t think it’s a bad idea to guide the student to find the problem as well as the solution. Similarly, I’m wondering what problem HoTT is designed to solve.


  5. sntx

    I heard the playwright David Hare make the comment (among many other wonderfully insightful comments) that it makes no sense to replace a line in one of his plays by a paraphrase with the same meaning, because he wrote it to sound a certain way…

    Right. ‘Being clear’ makes absolutely no sense in the case of literature. A good writer is generally adept at warping the relevant portion of his text around the consciousness of his characters (I think this technique is called free indirect style). One of the great joys of reading Gogol is to notice these alterations of his voice – the madder Poprishchin becomes, the madder the text becomes.


    1. mathematicswithoutapologies Post author

      It would be pretentious for me to claim any literary merit for my book, but it is intended to be read by people for whom a clear text is not necessarily one that can easily be parsed by a machine. Readers of literary texts do not see clarity and ambiguity as opposites.


      1. Mike Shulman

        Maybe that’s why I’ve never really enjoyed fiction that styles itself as “literary” — and even if I did, I would be surprised to find a nonfiction book written in such a style. Which brings us back to the point of my original comment, namely that choosing to write in a certain style may exclude certain people from your audience — and not necessarily, as you seem to imply in this post, due to their dislike for the “rough edges” of the subject matter.


      2. mathematicswithoutapologies Post author

        Surprised? I guess you don’t have much use for Wittgenstein, not to mention Nietzsche or Walter Benjamin, or any of a thousand French authors. That’s one reason I made a point of devoting the Afterword to Hausdorff, and to his literary alter ego Paul Mongré.


      3. Mike Shulman

        No, I’ve never tried reading any of those authors. My surprise is based on the many nonfiction books I’ve read that were written in English in the past few decades (a category to which MWA also belongs, so it is a reasonable comparison).


      4. sntx

        > or any of a thousand French authors.

        Speaking of the french I am quite fond of Vincent Descombes thesis that mental vocabularies are deeply historical. This (unaddressed and dare I say unclear) notion of ‘clarity’ strikes me as quintessentially no-nonsense, we-drill-for-semantic-oil-hither Anglo-americana.

        Also, to add to your Hausdorff, I’d say, more recently, Weyl used to write wonderfully as well.


  6. sntx

    Oops. I got the name wrong, it should be Akakievich, the protagonist from The Overcoat not the guy from the Diary of the Madman (of course).


  7. Pingback: Working the red carpet, part 2 | Mathematics without Apologies, by Michael Harris

  8. Pingback: Problems of a problematic vocation | Mathematics without Apologies, by Michael Harris

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