As long time readers of this blog know, one of the things that drive me crazy – in fact, one of the things that led me to start this blog – is the rampant innumeracy of our society. The vast majority of

Americans have no real knowledge or comprehension of numbers or mathematics, and what makes that even worse is that most really, truly, fundamentally *don’t care*.

A vivid example of that is demonstrated in a recent Supreme Court ruling in a case dealing with the use of sonar in submarine training

by the US navy in waters inhabited by whales.

The basic idea behind the case is that environmental groups had sued to prevent the navy from doing Sonar tests and training in waters where they were likely to harm marine mammals.

Before getting to the innumeracy, it’s interesting to just

take a look at the basic idea of the case, and how some basic numbers fit into it.

Sonar is not an innocuous technology. Many of us have intuitions

about it taken from movies and television shows, where it’s just a “ping” sound. But the point of sonar is to create a sound wave in the water that is powerful enough to produce accurately measurable reflections off of bodies at great distances away. Because of the way that sound propagates, there’s an inverse square relationship between signal strength and distance. So if you’ve got a target 100 meters

away, you’ve first got an inverse square reduction in the strength of the signal before it even hits the target. Then *part* of the

signal is absorbed by the target, and part is reflected. The reflected part again diminishes by inverse square. The point here is that

it takes a *very* powerful sound pulse to get good readings.

A typical sonar pulse originates at a volume of around 235 decibels.

That is *loud*. That is amazingly loud. In fact, that’s louder

that the *loudest possible sound* in air by roughly 16 times! (*I
originally wrote 4 times, when it fact it’s four doublingsin volume.*)

To put that into context, it’s worth looking at a couple of examples to give you a sense of how much energy we’re talking about in a sonar pulse.

- If you’re standing on a platform on the NYC subway, a

train arriving in a station is around 100 decibels on the platform. - If you’re standing 100 feet away from the jet engine on a Boeing

737 when it’s pushing the plane up the taxiway, the sound of the

engine will be roughly 140 decibels. - Standing immediately in front of the speaker towers at a rock concert is about 150 decibels.
- 180 decibels will instantaneously rupture your eardrums.
- In air, the maximum possible loudness of sound carried through the atmosphere at typical air pressure is about 195 decibels.

Decibels are logarithmic – each 10 decibel increase corresponds roughly to a *doubling* of volume, and volume pretty much corresponds to the amount of energy packed into the sound wave.

So when we’re talking about sonar, we’re talking about a bloody hell of a lot of energy being pumped into the water. It’s a concussive wave of immense force. There are well documented instances of sonar pings causing bleeding around the eyes and ears of whales; there’s also some poorly understood data showing high correlation between high-energy sonar use and whale beachings.

The Navy policy is to not do Sonar tests if marine mammals were sighted within 200 yards. The Natural Resources Defense Fund sued the Navy to try to force them to not do Sonar tests if marine mammals were sighted within 2000 yards.

It’s a damn big difference. Given the inverse-square relationship between the power from the sonar ping at a point, and the distance away from the source of that point, you’re talking about reducing the maximum

exposure to Sonar waves by more than 30 decibels – a very significant

difference!

Now, on to the innumeracy: In his decision, Chief Justice Roberts

wrote:

The District Court’s injunction does not include a graduated power-down,

instead requiring a total shutdown of MFA sonar if a marine mammal is

detected within 2,200 yards of a sonar-emitting vessel. There is an

exponential relationship between radius length and surface area (Area =

πr^{2}). 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).”

Where’s the problem? The surface area affected by a given sighting range is, as he points out, a circle with area πr^{2}. But that’s a *polynomial* expression, *not* an exponential one. We’re not talking about a trivial distinction here.

So the problem is the word “exponential”. It doesn’t mean what he thinks it means. You might say that I’m just being pedantic here – so what if he got a word wrong?

But he’s a *judge*. His career is in studying laws, legal

judgements, and carefully piecing apart the precise meanings of those

words. If you’ve ever known an appeals judge, you’ll know that they’re

incredibly pedantic about precise meanings of terms – they have to be to

do the job. So we’re talking about a guy who’s supposedly an expert on

language semantics, at least where legal issues are involved. If anyone were to make a legal filing where they used the word “libel” where they meant “liable”, he’d throw the case out on it’s ass.

And he *clearly* thinks that he’s showing off, by including

an *equation* and fancy mathematical words in his judgement. But he can’t be bothered to actually understand the meanings

of the words that he’s using.

And those words mean something *very* different. Again, it helps to look at a simple example to compare. If we use 2 as our base, in the polynomial expression πr^{2}, the

change from a 200 yard radius to a 2200 yard radius is different

by a factor of 121 times. If we used an exponential expression based on 2, π2^{r}, we’d get a difference factor of, roughly,

1.15×10^{604}.

That’s the difference between a number that my five year-old son can count to in about three minutes, and a number that’s absolutely unimaginable – something so large that saying that it’s more than millions of times larger than the number of particles in the entire universe doesn’t *begin* to approach it.

Hell, replace the “2” with something smaller in the exponential;

say, 1.01. That’s something that’s going to increase very slowly

according to an exponential curve, right? It’s barely more than 1.

Raising it to the fifth power only gives you 1.05 – it’s a *very*

shallow exponential curve. But take the 200 versus 2200 from the

lawsuit, and you’ll find that the ratio of the exponential, π1.01^{r} compared to the quadratic polynomial πr^{2} is, roughly, 439 million.

Big difference. And the fact that that difference, between *an equation with an exponent*, and a *exponential equation* is totally lost on the chief justice of the U. S. Supreme Court is just pathetic.

*(On the other hand, the person who originally sent this to me – and several folks in comments in the newspapers that carried this story – claimed that πr ^{2} is the wrong equation, because you should be considering the surface of a sphere. Alas, the relevant regulations are for sighting marine mammals within a given radius on the surface That’s a bit of a silly confusion too – a nuclear submarine typically never goes below about 1600 feet; their hulls will collapse from pressure at around 2400 feet; there’s just no way that a spherical space makes sense. Submarines go beneath the water for stealth reasons, but they can’t go very deep at all; one half mile, and they’ll be crushed like a tin can. No one would really talk about a spherical area around a sub; the way they operate, it’s got no value.)*