There’s an adorable proverb that goes like this: The fox knows many things; the hedgehog knows one big thing.
(Cute, right? I love a good hedgehog.)
In his essay “The Hedgehog and the Fox,” Isaiah Berlin expands this into a playful metaphor for two kinds of writers. It makes for a great game.
Who’s a fox, and who’s a hedgehog?
The hedgehogs are writers with a master theory. They “relate everything to a single central vision… a single, universal organizing principle.” One classic example is Plato, with his Theory of Forms (and, more generally, his faith in pure reason to resolve any question or controversy).
Meanwhile, the foxes are rummagers, explorers:
those who pursue many ends, often unrelated and even contradictory, connected, if at all, only in some de facto way, for some psychological or physiological cause, related to no moral or aesthetic principle. These last lead lives, perform acts and entertain ideas that are centrifugal rather than centripetal; their thought is scattered or diffused, moving on many levels…
The classification game makes for good long-car-ride fun. If Plato was a hedgehog, then Aristotle was a fox. Dante was a hedgehog; Shakespeare, a total fox.
It’s also applicable to politicians, filmmakers, tech moguls, and whoever else you like. Steve Jobs and Henry Ford, hedgehogs; Elon Musk and Jeff Bezos, foxes. Nate Silver self-identifies as a fox (hence the foxy logo for FiveThirtyEight).
I’m curious about mathematicians. But I don’t feel I know enough mathematics or history to judge myself, so this is a bleg as much as a blog. Knowledgeable mathematicians, help me out! My questions:
- Who are the hedgehogs of mathematical history? Grothendieck? Lovelace? Euclid?
- Who are the foxes? Gauss? Newton? Erdos?
- What are the comparative merits of being a fox vs. a hedgehog in mathematics?
(Readers may notice a resemblance to Timothy Gowers’ “two cultures of mathematics,” theory-builders and problem-solvers. I suspect the two ways of carving up the world are not quite isomorphic – it seems to me that you could be a foxy theory-builder, or a hedgehog of a problem-solver – but feel free to argue otherwise!)
For more on the literary fox who dreamt of being a mathematical hedgehog, try my new book Change is the Only Constant: The Wisdom of Calculus in a Madcap World.
8 thoughts on “Who are the Foxes and Hedgehogs of Mathematics?”
You’ve left out Freeman Dyson’s division of mathematicians into “birds” and “frogs”!:
Is the bird vs. frog notion significantly different compared to the theory-builder vs problem-solver notion? My thought was that birds = theory-builders and frogs = problem-solver. But perhaps you can be down in the trenches with a theory and have a broad view of problem solving?
It occurred to me that David Mumford touched on this topic a few years back in a very stimulating piece (talking of math “tribes”):
Wiles is certainly a hedgehog, Tao certainly a fox.
Here are Szilard’s ten commandments to chew on:
1. Recognize the connections of things and laws of conduct of men, so that you may know what you are doing.
2. Let your acts be directed toward a worthy goal, but do not ask if they will reach it; they are to be models and examples, not means to an end.
3. Speak to all men as you do to yourself, with no concern for the effect you make, so that you do not shut them out from your world; lest in isolation the meaning of life slips out of sight and you lose the belief in the perfection of creation.
4. Do not destroy what you cannot create.
5. Touch no dish, except that you are hungry.
6. Do not covet what you cannot have.
7. Do not lie without need.
8. Honor children. Listen reverently to their words and speak to them with infinite love.
9. Do your work for six years; but in the seventh, go into solitude or among strangers, so that the memory of your friends does not hinder you from being what you have become.
10. Lead your life with a gentle hand and be ready to leave whenever you are called.
I was also gonna refer to “Birds v Frogs” by Freeman Dyson. One might also more crudely compare it to the saying, “jack of all trades, master of none”…!
Feynman once said that the trick of appearing to be a genius was to learn a handful of methods very well and to apply them to just about every problem one runs into. If they are any good and you understand them, you, like a blind squirrel, will find lots of nuts, and people will be convinced that you are brilliant.