The surreal numbers are an interesting construction, but they don’t really have applications outside of combinatorial game theory. That is the sense in which I meant that practically nobody uses them!

I mean, the surreal numbers are an extremely technical subject that practically nobody uses!

The “Wow! Zowee!” factor of surreal numbers is much higher than their “you need to know this to work in applied mathematics” index.

Beyond vocabulary, one aspect of talking about mathematics that I often see overlooked has to do with the value of ideas to different individuals in different fields. Often that value doesn’t carry over from one field to the next, or a general audience (and the change can go both ways, lest you think math is always less valuable outside of mathematics!). That matters a lot to how things are framed and described.

It’s been my experience that many mathematicians have only a foggy notion of the potential value of their work outside of mathematics, and that can greatly hinder their ability to give a good “cocktail party” summary of what it is they do. Furthermore, the (many) mathematicians that do work on something of value in other areas are often poorly equipped to translate what it is that they do and why that has value to an audience of non-mathematicians.

Being able to eloquently interpret mathematical questions and results into some other context with different values, objects, processes and questions is often a very difficult thing to do. In most mathematics programs, nobody is taught how to translate “mathematical question and it’s mathematical answer” into “science question and science answer” as this is a substantially difficult thing to do in many cases — especially when you also need to explain the science Q&A to a general audience.

So it seems then that writing about mathematics requires three levels of insight: 1) knowing the relevant mathematics, 2) understanding how to interpret the results into some non-math context, and 3) knowing why any of that matters to your audience.

[PS: Just so people know where I'm coming from here, I'm currently finishing up an applied mathematics PhD, with a heavy biology/ecology background.]

It’s been my experience that many mathematicians have only a foggy notion of the potential value of their work outside of mathematics, and that can greatly hinder their ability to give a good “cocktail party” summary of what it is they do. Furthermore, the (many) mathematicians that do work on something of value in other areas are often poorly equipped to translate what it is that they do and why that has value to an audience of non-mathematicians.

Being able to eloquently interpret mathematical questions and results into some other context with different values, objects, processes and questions is often a very difficult thing to do. In most mathematics programs, nobody is taught how to translate “mathematical question and it’s mathematical answer” into “science question and science answer” as this is a substantially difficult thing to do in many cases — especially when you also need to explain the science Q&A to a general audience.

[PS: Just so people know where I'm coming from here, I'm currently finishing up an applied mathematics PhD, with a heavy biology/ecology background.]

Your most humble of servants,

fpqc

For instance, here is Greg Friedman’s illustrated introduction to simplicial sets (which is excellent, by the way) http://faculty.tcu.edu/gfriedman/papers/simp.pdf , but it’s written for math students, and even still, it’s 50 pages long!

I think that calculus, some probability theory, some number theory, and some combinatorics might be accessible to laypeople given the right presentation, but there is a certain boundary in mathematics, which delineates “how far you can get doing mathematics on the side” and “how far you can get doing mathematics as your main area of study”.

Yes, when I try to explain things to actual laypeople, I start with the notion of triangulating polyhedra, then non-triangulable spaces, singular complexes, etc., but to even get them that far is a huge challenge. When people ask me what it’s good for, though, I say a little bit about 3d modeling and animation, but I feel like I’m not being completely genuine when I cite that example.

Dear Jennifer,

I think that if I expanded out each and every one of my jargon terms, I’d be forced past any sort of page limit. My problem is that not only do people not share a reference frame with me, but they often don’t know enough to make anything out of anything defined algebraically. I could draw things on a blackboard and maybe get the audience to understand what I mean, but not in any sort of useful way.

I admit that it’s frustrating to not be able to communicate precisely what we do, but there is some satisfaction to be had in walking out to where the audience is and bringing them at least a little bit of the way. When people ask me what topology is, I usually start by reminding them that in geometry they learned about squares and circles and triangles and that size and length mattered. Then I tell them that in topology we don’t worry about size and length and angles and that to us those are all equivalent things. Then to give them an example of what I mean by that, I talk a little about knots and how a topologist only really cares about how the knot is knotted and not how long the piece of string is. Most people appreciate these concepts. When they ask me what that’s good for, I say a little about protein knotting. Does all of this have anything to do with what I do? Not really. But telling “lay people” that I do work on the intersection homology of high-dimensional stratified spaces is a quick way to end a conversation. If you really want to tell people what it is you do, you’re going to have to build a bridge with analogy.

So maybe the short answer to your question about whether or not to give up on talking to lay people is yes and no: you shouldn’t give up talking to lay people about mathematics, but perhaps getting lay people up to speed on your research work is too big a task and an unnecessary one. Don’t give them what you do – give them an idea of what you do. If it’s a longterm relationship, build up to the details.

He can do this because Bisi has thought deeply and worked very, very hard to come up with non-jargon-filled, metaphor- and analogy-rich descriptions of his work. He takes the time to introduce and carefully define, in laymen’s terms, those concepts/jargon that must be part of the explanation, and gently walks his listeners through each step, answering any questions along the way. As Blake Stacey points out, Feynman was a master at this. But even he was hardly infallible; not everything he wrote was for a lay audience, or precocious 6th graders like Blake. (Which is fine, BTW. Not everything has to be. But we’re talking about whether it’s even possible. And I say that it is.)

fpqc’s comment at #27 is a prime example of what NOT to do — and I recognize he wasn’t even attempting to put things in a way a lay person could understand. But that’s the problem, isn’t it? Imagine how little would have been accomplished in mathematics if those same folks had made the assumption that certain things just couldn’t be done and that’s that.

I don’t buy this “Well, you can maybe do it for calculus and probability but there’s no way you could do it for the hugely important and fancy math that _I_ do for a living” argument. It’s not easy, I’ll grant you that. Someone like me would need to study for years before even attempting it. But… really, fpqc? You can;t think of _any other phrases_ to use besides “discretized homotopy theory”? There is absolutely no other option than to talk about “concatenations of edges”? Would it kill you to carefully define what is meant by a combinatorial device, category theory, and simplicial sets, to give your listeners a framework for the discussion? Is there honestly absolutely nothing in anyone’s life experience than can be used as a rough analogy for your advanced concepts? I highly doubt it. You don’t notice the jargon because it MEANS something to you, a specialist’s shorthand. The trick is to bring your audience to the point where those words now mean something to THEM. It can be done. That part is just a vocabulary problem.

Good communication is very, very hard, particularly when it comes to advanced math and science (although really, any discipline’s jargon is just as daunting — checked out literary criticism lately?). It’s a demanding discipline in its own right, and giving up too soon and concluding something can’t be done is — well, a bit like my former math-phobic self giving up too soon on math and concluding it just wasn’t my thing. We “laypeople” [and that is a loaded term in its own right, implying a sort of divine priesthood of pure mathematics] are not all stupid. Some of us are actually quite tenacious and may WANT to understand. We really appreciate it when we meet folks like Bisi who are equally tenacious in finding ways to improve our understanding.

My explanation requires at least basic knowledge of two areas that are considered extremely abstract and technical: category theory and homotopy theory. These are subjects about which most laypeople have never even heard, let alone studied. It’s so far removed to the ordinary experience, and involves the infinite even in its definition.

As CZ said, basics first. Instead of explaining your own current research, can you explain what category theory is and why one should care about it? Instead of explaining category theory for people browsing the science shelf at Barnes-and-Borders-A-Million, can you explain it to students who’ve had a term of calculus? Science and mathematics could benefit from better exposition at all audience levels.

As a high school teacher turned full-time dad, I have tremendous appreciation for the ease of producing bad writing and the difficulty of training good writing, especially when the conventions differ from one discipline to the next.

Thanks for the prompt and engaging response! I really appreciate it! However, I feel that I would be remiss if I didn’t bring up an objection about the book that you linked: Probably eighty to ninenty percent of pure mathematics done today does not feature calculus or probability theoy in any sort of central role (although the techniques from calculus are often used with impunity).

For instance, I do research on the homotopy theory of simplicial sets. Here is how I’ve tried to describe what I do in the past (to little success): A simplicial set (originally called a CSS complex) is a combinatorial device discovered in the 1940s and 1950s by Saunders Mac Lane and Sammy Eilenberg that allows us to give a sort of discretized homotopy theory (think of covering up a space by all possible configurations of simplices, and take this collection of simplices (along with the data of their orientations) to be a combinatorial object (this concept is extremely difficult to grok, but it’s something that we mathematicians use all the time)).

However, work of Boardman and Vogt realized not only that such objects could model spaces, but also could be used to give a “homotopy-coherent” version of the diagram calculus of category theory. The way they achieved this was to raise the bar for what it meant for two simplicial sets to be equivalent.

Their solution was to realize the classical homotopy theory of simplicial sets as a “localization” of their new homotopy theory….

et cetera…

My explanation requires at least basic knowledge of two areas that are considered extremely abstract and technical: category theory and homotopy theory. These are subjects about which most laypeople have never even heard, let alone studied. It’s so far removed to the ordinary experience, and involves the infinite even in its definition. For instance, any Kan complex, which is a simplicial set that has certain properties that make it extremely useful for computing the homotopy theory of spaces, has an infinite number of nondegenerate (nontrivial) higher simplices, which track all possible concatenations of edges, as well as which concatenations of edges admit a filler.

Should I just give up trying to talk to laypeople?

Just as sentences are not words casually linked together, paragraphs are not just a random package of sentences. When we start a paragraph, we should know what we’re in for, and the paragraph should live up to that promise. It should not meander from subject to subject. And the link from one paragraph to another must be obvious and inescapable.

Interestingly, McPhee’s writing often doesn’t follow this advice. He doesn’t connect the dots for you: the connection from sentence to sentence and paragraph to paragraph isn’t always immediately apparent. Nor, especially in his longer pieces, from section to section.

His technique is almost pointillistic. But the points aren’t all there simultaneously. They appear one at a time; first over here, then one up over there, a third down in the corner, another one somewhere else, until gradually the picture starts to fill in.

[CZ: It's not wise to try to become McPhee in the first few months of trying to be a writer. First come the basics. Think of Picasso--masterful figurative art came before the Cubism. ]

However, it seems the discussion hasn’t really addressed the practical value in which good jargon can be used to express an idea much more succinctly and precisely than a jargon free explanation. An equation is perhaps the penultimate form of jargon, but once familiar it becomes like a good picture — better than 1000 words. Actually many good scientific figures are like this too: jargon that is powerful to express an idea only to the trained eye, but look like nonsense to the untrained eye.

So black diamond science writing is when you must teach the jargon?

Another observation along these lines:

The trouble is that physical theories are not like novels: rather, they consist of an intricate logical structure in which technical terms have precise meanings (which differ subtly but crucially from the everyday English senses of the same words); and it is hopeless to try to isolate the philosophical “themes” of a theory that you understand only at the level of metaphor. (For example, one non-scientist friend asked me, quite reasonably: Isn’t it contradictory for quantum mechanics to exhibit both discontinuity and interconnectedness? Aren’t these opposites? The brief answer is: quantum mechanics exhibits “discontinuity” and “interconnectedness” in very specific senses — which require a mathematical understanding of the theory to make precise — and these senses are in no way logically contradictory.)

— Alan Sokal, Beyond the Hoax (2008), p. 12.

When unchecked, this problem leads to what I call Gell-Mann’s Law.

It seems to be characteristic of the impact of scientific discovery on the literary world and on popular culture that certain items of vocabulary, interpreted vaguely or incorrectly, are often the principal survivors of the journey from the technical publication to the popular magazine or paperback. The important qualifications and distinctions, and sometimes the actual ideas themselves, tend to get lost along the way. Witness the popular uses of “ecology” and “quantum jump,” to say nothing of the New Age expression “energy field.” Of course, one can argue that words like “chaos” and “energy” antedate their use as technical terms, but it is the technical meanings that are being distorted in the process of vulgarization, not the original senses of the words.

— Murray Gell-Mann, The Quark and the Jaguar (1994)

The layman searches for book after book in the hope that he will avoid the complexities which ultimately set in, even with the best expositor of this type. He finds as he reads a generally increasing confusion: one complicated statement after another, one difficult-to-understand thing after another, all apparently disconnected from one another. It becomes obscure, and he hopes that maybe in some other book there is some explanation. . . . The author almost made it — maybe another fellow will make it right.

— Richard Feynman, The Character of Physical Law (1964)

My current favorite example of this Black Diamond science writing is The Calculus Diaries: How Math Can Help You Lose Weight, Win in Vegas, and Survive a Zombie Apocalypse.