How Not to Be Wrong: The Power of Mathematical Thinking

by Jordan Ellenberg

of vision.

Galton called his measure correlation, the term we still use today.

If Galton’s

ellipse is almost round, the correlation is near 0; when the ellipse is skinny, lined up along the northeast-southwest ax...

1.

the association

two variables,

richer states tend to skew more Democratic, and the ellipse points northwest to southeast.

if you have a problem whose solution admits a simple mathematical description, there are only a few possibilities for the solution.

next simplest curves are those cut out by quadratic equations,*

Thomas Pynchon’s

Harry Reid,

The “majority rules”

system is simple and elegant and feels fair, but it’s at its best when deciding between just two options.

Any more than two, and contradictions start to seep into the maj...

“independence of irrelevant alternatives.”

Each one of us is a little nation-state, doing our best to settle disputes and broker compromises between the squabbling voices that drive us.

Pascal’s wager,

Riemann’s Lines, which are circles, have finite length but are boundless; one can travel along them forever

one can prove that Riemann’s geometry of Points and Lines on a sphere is the same

as that of the projective plane.

once you understand that the first four axioms apply to many different geometries, then any theorem Euclid proves from only

those axioms must be true, not only in Euclid’s geometry, but in all the geometries where those axioms hold.

non-Euclidean geometry is not just a game; like it or not, it’s the way space-time actually looks.

Each one of these little local uncertainties percolated through the simulation, and the uncertainties about different parameters of the model fed back into each other, and by 2050, the noise had engulfed the signal.

It is not the critic who

counts; not the man who points out how the strong man stumbles, or where the doer of deeds could have done them better. The credit belongs to the man who is actually in the arena, whose face is marred by dust and sweat and blood; who strives valiantly; who errs, who comes short again and again,

because there is no effort without error and shortcoming; but who does actually strive to do the deeds; who knows great enthusiasms, the great devotions; who spends himself in a worthy cause; who at the best knows in the end the t...

at least fails while dari...

survival of civilization depends on the triumph of the bold, commonsensical, and virile against the soft, intellectual, and infertile.*

John Ashbery, whose poem “Soonest Mended” is

uncertainty and revelation can mingle, without dissolving

When you reason correctly, as Silver does, you find that you always think you’re right, but you don’t think you’re always right. As the

drop one tiny contradiction anywhere into a formal system and the whole thing goes to hell.

Bertrand Russell did to Gottlob Frege’s set theory what Kirk did to uppity robots.

Proving by day and disproving by night is not just for mathematics. I find it’s a good habit to put pressure on all your beliefs, social, political, scientific, and philosophical.

To the greatest extent possible you have to think as though you believe what you don’t believe. And

Scott Fitzgerald

“The Crack-Up,” his 1936 essay about his own irreparable brokenness.

Bolyai created an entirely

The lessons of mathematics are simple ones and there are no numbers in them: that there is structure in the world; that we can hope

to understand some of it and not just gape at what our senses present to us; that our intuition is stronger with a formal exoskeleton than without one. And that mathematical certainty is one thing, the softer convictions we find attached to us in everyday life another,

and we should keep track of the differe...

let go of the idea that the beliefs of groups should be subject to the same rules as beliefs of individuals;