Justice Roberts Doesn’t Know What “Exponential” Means

From Winter v. NRDC, No. 07-1239 (Nov. 12, 2008):

There is an exponential relationship between radius length and surface area (Area = πr2). Increasing the radius of the shutdown zone from 200 to 2,200 yards would accordingly expand the surface area of the shutdown zone by a factor of over 100 (from 125,664 square yards to 15,205,308 square yards).

Once again, we see that even successful lawyers don’t know enough math. Just because there’s an exponent in the equation doesn’t make the relationship exponential. That there is a quadratic relationship.

This is an embarrassing mistake, or at least it ought to be, like referring to the plaintiff and defendant as “offender” and “defender.” The terms are valid, but not in this context.

Hat tip to Ken; the Volokh commenters are also on the case.

And/or his clerks don’t know what it means, I suppose.

(If he doesn’t understand math, good luck with marine biology.)

I don’t have a problem with this usage. I understand the meaning of “exponential” you’re referring to (an exponential relationship requires the variable to be in the exponent, like e^x — right?), but requiring non-mathematicians/scientists to distinguish between this technical definition of “exponential” and terms such as quadratic or cubic or whatever seems a bit much. I just don’t think that colloquial English has any other commonly used term to describe the kind of non-linear relationship here. And I think it would have done more harm than good if Roberts had pleased the specialists but confused his audience by using the technically appropriate term.

(Speaking of confusion — why is the word “quadratic” used for formulas that involve powers of 2?)


I disagree. This isn’t a colloquial usage of “exponential”; it’s simply an incorrect one. The colloquial English word covering both exponential and quadratic relationships is, as you use yourself, “non-linear.” (Though I would leave out the hyphen, I think.) He could have just said “non-linear,” or he could have reworded the sentence entirely. (“The area of the exclusion zone increases with the square of the radius.”)

The broader point, however, is that this type of innumeracy is problematic. This point may be considered by most lawyers to be hypertechnical—but it should not be. I doubt that many of the Justices would refer to the United Kingdom as England or confuse the Greeks and the Romans—and this point is one that, while unimportant to the case, should just as much be part of a Chief Justice’s liberal education.

I’m still not particularly troubled. I really do think that most people loosely use “exponential” to mean that one value grows at a rate that is some factor greater than another value; hence the common pairing of linear vs. exponential growth that Roberts was invoking here. This usage lumps x^2 with 2^x, but so what? If Roberts, say, had to compare those two functions, I’m sure he’d write something like, “Although both functions describe exponential growth, the rate of growth of 2^x far exceeds the rate of x^2” — leading to much gnashing of teeth, I’m sure, but also the right conclusion.

Full disclosure: maybe I’m just touchy because I also apparently didn’t know what “exponential” meant before this post! So I may just be working through a phase.