“Explanation is work for second-rate minds,” said the monster.

You want to drive me into a fit of blood-boiling rage? Here are some options:

  • Burn down a used bookstore.
  • Sponsor legislation banning the use of peanut butter in desserts.
  • Offhandedly mention that you think baby chimpanzees aren’t that cute.


  • Or, if you want to take the easy route, just bust out this quote from the great mathematician G.H. Hardy:


The blood…

The boiling…

Oh, the boiling of the blood…


Try as I might, I can’t hate Hardy. He wrote wonderful textbooks, proselytized on the beauty of mathematics, and did other good deeds.

But I can hate this view, this toxic meme, which I believe is latent in our stereotypes of mathematics: this belief that generating new mathematical ideas is man’s highest calling, while wallowing in old ideas is grunt-work fit only for mules, washouts, and the dim bulbs we call teachers.

Hardy didn’t invent this idea. He simply gave it voice.

So on behalf of the mules, the washouts, and the dim bulbs we call teachers, I wish to mount a defense of the art of explanation.


For my first witness, I call to the stand… Euclid of Alexandria!

Now, Euclid was not the first Greek scholar to tackle geometry. Far from it. Centuries of brilliant folks had left their signatures by the time he arrived on the scene. So why is he known today as the founder of mathematical proof? Why is traditional school geometry often known as “Euclidean”? Why am I name-dropping him 2500 years after his death?

It’s because he wrote the book.

It’s because he consolidated the ideas.

It’s because he explained.

Before Euclid, geometry lay scattered in pieces, like a collection of jewels thrown to the sands. The ideas were there. The proofs were there. But they lacked organization, structure, coherence.

Euclid unified geometry into a clear, logical system. He laid out simple assumptions (called axioms) and traced every geometric truth, no matter how remote or sophisticated, back to these axioms. His project brought together the work of diverse mathematicians into a single, coherent whole.


Euclid transformed mathematics not by creating new ideas, but by elucidating the connections between the existing ones.

That is: by explaining.

Not bad for a second-rate mind, right?


Euclid’s not the only one to change the world through mathematical explanation. Leonardo of Pisa (best known by his posthumous nickname “Fibonacci”) brought the modern numeral system to new lands.

Before Leonardo, Europe sweated it out with Roman numerals: clunky, slow, inefficient for computation. But in the shipyards of Algiers, the young Leonardo learned a better system. These numerals—1, 2, 3, 4, 5, 6, 7, 8, 9, and the curious 0—were born in India, perfected in Arabia, and now transmitted, through Leonardo, to Europe. They caught on among the merchant class. And eventually, they swept the entire world.

Leonardo re-explained arithmetic to a whole continent. In the process, he nudged history towards its modern, globalized language of number.

Score another one for the second-rate minds.


It was Albert Einstein—another dim bulb, clearly—who said, “If you can’t explain it simply, you don’t understand it well enough.” But it doesn’t require Einstein to see the wisdom here.

Every schoolchild knows that explaining something to a friend helps you master it yourself.

Every instructor (from primary school to grad school) has felt how teaching something can help snap into place the loose fragments of your own understanding.

And every researcher has witnessed how the effort to write or speak about your ideas winds up purifying and clarifying them—like boiling off excess water to make smoother, creamier sauce.

Explanation isn’t just good for other people. It’s good for the explainer, too.


Hardy was a proud, contributing member of the mathematical research community. So I find it curious that he so blindly and guilelessly separated the “research” from the “community.”

Now, to research mathematics is to develop new ideas. This adds to our library of collective wisdom. That’s obviously a very cool thing.

But when you teach new minds, or write accessible books—i.e., when you explain—you’re doing something equally cool.

You’re staffing that library.


If we all shared Hardy’s stated priorities, we’d simply be stuffing the shelves with new and largely unread books. No one would be editing or consolidating this knowledge. No one would be coordinating our efforts. No one would be recruiting new scholars into the library.

Hardy’s advice is self-defeating. It creates a research community that’s all research and no community.

It creates a library full of great books and with no readers to read them.


We ought to be glad, then, that Hardy saw past his own bad advice—that he wrote books like A Mathematician’s Apology and his other accessible texts, in spite of his grudging attitude towards such work.


Maybe you think I’m exaggerating about mathematicians’ antipathy towards explanation. Surely this pro-research, anti-teaching, anti-anything-that-smells-like-teaching prejudice can’t be all that pervasive and damaging. Can it?

In that case, I introduce my final exhibit: the ABC Conjecture.

First proposed in 1985, the ABC Conjecture is one of the great unproved statements in mathematics. It’s a powerful claim about number theory, and if it’s true, it has many deep repercussions.

And, in August 2012, it was proved.

Well… maybe.

We’re not sure.

The mathematician with the proof is a fellow named Shinichi Mochizuki. He spent decades developing his theory, which spans more than 500 dense pages, full of novel notation and new conceptual machinery. The work is so inventive that—unfortunately—no one has been able to check it.

His individual triumph of problem-solving has been swamped by a collective failure of explanation. Mochizuki’s proof sits there, unexplained, not budging, like an undigested meal caught in the throat of a snake.

We’d better get some second-rate minds over to help.

Right, Hardy?


Of course, my complaint isn’t really with Hardy.

My complaint isn’t even with the casual elitism of the academy. (Trying to purge the self-importance and sense of intellectual superiority from the research enterprise is like trying to purge the sex references from rock ‘n roll. It probably can’t be done, and you wouldn’t like the result if you did.)

My complaint is against the twisted belief that you should care deeply about ideas, but resent having to share them.

My complaint is against the self-defeating effort to separate research itself from the ecosystem of activities that support, sustain, and justify it.

My complaint is against anyone who thinks explanation is a lesser art.

And most of all, my complaint is against people who don’t think baby chimpanzees are cute.

I mean, come on, guys. Open your eyes!


14 thoughts on ““Explanation is work for second-rate minds,” said the monster.

  1. I recently mused that we’re driven to explain as a means of testing the truth of our ideas. An idea sitting in one’s own brain-box can go un-tested and ‘proven’ to the thinker our whole life. But there’s no value to that idea. It’s utterly useless, and could be very detrimental if it is false and lingers largely due to confirmation bias. So we reach out and ‘try it out’ on another brain – see whether that person sees flaws or openings it didn’t even dawn on you to consider. In doing so our ideas gain value both for ourselves and the community with which we share them.
    (by the way, that odious quotation made my blood boil too!).
    Thank you for a lovely post! 🙂

  2. Thank you for this! It’s one of your best! I shared this out liberally, with special attention to the parents of my students!

  3. Reminds me of one of my favorite Lockhart snippets:

    “Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity—to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs—you deny them mathematics itself.”

  4. Your description of Euclid made me think of Category theory and how it made rigorous the connections between different theories in mathematics. I wonder what can you draw about it! 😀

  5. I fully agree with you regarding those who explain, but I have a hard time considering ANY non-human animals to be cute, including baby chimpanzees. Sorry. 🙂

  6. Your point #3 reminds me of a hand-written motto that my college Biochemistry professor had taped up in his office, which I’ve replicated and taped up around my work desks a couple times. It’s easier to understand if you see it, because it’s not just text; specifically, the motto says, on the top line, “Clarity of Expression”, and a curved arrow points down to the bottom line, which says “Clarity of Thought”, and another curved arrow points back up to the top line. So I guess if you want to write it less fancily (and with math/logic notation!), you could write it as “Clarity of expression ⇔ Clarity of thought.” Which is pithy way of saying that if you understand something clearly, you can write about it clearly (and quickly—on a test!), and conversely, sometimes even more importantly, that only by having to explain a subject clearly and logically, in a succinct and organized way, do you develop a deep and coherent understanding of it.

    He also mentioned something like this to us once in class, probably before our first test, but I might have forgotten it, particularly the second half, if I hadn’t seen that cool sign in his office (the only time I went there!). That was 14 years ago now. So his hand-written/drawn motto drove home this very important message better than just hearing it or reading it somewhere would have. This just goes to show that not only teaching and explaining but also bad drawings are integral to learning and thereby improving ourselves and the world!

  7. My fairly short and inglorious venture into the world of research was punctuated by the terror of having to visit the yellow shelves in the library. You know, where they store all those scary Springer-Verlag graduate textbooks? They typically have impossibly banal titles like “Introduction to Arithmetic” by Professor Gustav von Brilliant of the Institut fur HardMathematik at the University of Leipzig.

    The introduction would be promising, something like: “This very basic introduction to a rapidly expanding field should be accessible to anyone who has mastered the elementary undergraduate courses in calculus and linear algebra [Yay! That’s me!] and has developed an interest in hyperplectic structures in paraboloid para-arithmetics.” Um, maybe…

    Any hope would be dashed upon reading line 1 of Chapter 1. “Consider a spasmoid ubervariety over an ideal of quasi-transfinite metablobs…”

    Time to find a new research area, I think, and I don’t remember any of that stuff from my elementary calculus or linear algebra courses.

    Most material written by successful academics seems to be primarily for the elevation of said academics to even greater success. The problem is, of course, that if you explain it well it might sound easy, and we couldn’t have that now, could we?

    I love the blog, by the way. I even like the drawings, but you need to work on your chimps.

  8. Is it possible that Hardy is the inspiration, if not the origin of “Those who can, do; those who can’t, teach.” Credited to George Bernard Shaw (Man and Superman)??

    It was always part of my mantra that “those who cannot teach cannot pass along their own brilliance.” And that might not always be such a bad thing. Humility does not always accompany genius. Fortunately, few new ideas die in the minds of solitary individuals. Radio was “invented” more than once, as a reasonable example. The shoulders of giants can support many.

    Thanks for the post.

  9. You mention Hardy’s famous apologia; as your blood boiled at his contempt for explaining, so did mine at his contempt for usefulness. A major part of his argument for mathematics being good consists in focusing on its lack of use and extolling this as a virtue. I reject lack of use as a virtue; and consider the useful bits of mathematics every bit as beautiful and worthy of study as the bits for which no-one has yet found a use. Even having no military use is not a virtue, though I tend to like other uses better.

    Still, he did pick his examples perfectly. Writing in about 1939, he chose as his two favourite examples of useless and beautiful (therefore the best) mathematics, those to whose virtues he devotes great space: Quantum Mechanics and Number Theory. Within six years, these two had each made a decisive contribution to winning a war, by their applications to code-breaking and the Manhattan project.

    Perhaps the enormity of his hubris will console your boiled blood.

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