Innumeracy and the U. S. Supreme Court

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

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

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 =
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).”

Where’s the problem? The surface area affected by a given sighting range is, as he points out, a circle with area πr2. 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 πr2, 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, π2r, we’d get a difference factor of, roughly,

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, π1.01r compared to the quadratic polynomial πr2 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 πr2 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.)

0 thoughts on “Innumeracy and the U. S. Supreme Court

  1. 6EQUJ5

    “The Navy policy … within 200 yards … within 2000 yards.”
    An isotropic radiator’s intensity at 2000 yards, referred to 200 yards, would be 20 dB down, not 30.
    The judge is thinking in terms of plane geometry, not spherical geometry, the existence of which he probably is unaware. Clearly he got buffaloed by some lawyer who could outsmart him.
    That’s the problem with legal arguments: they are only lawyerly arguments created with the intent to snow gullible judges. (Rule: all judges are gullible.)

  2. Mark C. Chu-Carroll

    I’m willing to cut him slack on that; they’re talking about area on the surface of the ocean; calling it surface area could be correct terminology according to the Navy.

  3. micah

    In fact, that’s louder that the loudest possible sound in air by roughly four times!
    Shouldn’t that be “four orders of magnitude”?

  4. Daniel Martin

    Jason: I think we can give Chief Justice Roberts a bit of slack on that term, since he was concerned about the area on the surface of the water that would now be excluded. So “surface area” means a section of the earth’s surface area, and for that pi r^2 is a close enough approximation given the distances involved.
    Marc: Unfortunately, the kind of innumeracy judges are subject to is the kind that gets coded into dictionary definitions. Justice Robert’s mangling would almost certainly be consistent with definition (1) given at that link.
    It used to drive me nuts when I was employed tutoring calculus students and one of them insisted on using the verb “derive” when he meant “take the derivative of”. As in: “So then I derive x-squared and get two x”.

  5. Flooey

    Slight correction: you say that the maximum sound pressure in air is 195 dB, and sonar pings are generated at 235 dB. That should be 10000 times as powerful, I believe, not a mere 4 times, since decibels are base-10 logarithmic.
    (Since you always say “loud”, I’m not sure if you’re silently accounting for the fact that increases in sound pressure are translated to increases in subjective loudness non-linearly. If so, it might be a good idea to mention that.)

  6. Sam C

    OK, so let’s apply your logic with micah’s comment: on this evidence, you apparently don’t know the difference between “N times greater than” and “N order of magnitudes greater than”. (Of course, we know you do really and this was just a slip!)
    A judge refers to a correct formula with an exponent in as exponential, you say “nyaa nyaa it’s polynomial”. Who is more wrong? The judge with the correct formula and a mildly misused term or the professional mathematician who apparently doesn’t understant some basic concepts?
    Unfair? A pedant hoist on his own petard?

  7. Robert Thille

    If the energy originates at a point and spreads out equally in all directions in a spherical wavefront, then the energy absorbed by a given area (ear drum) at a given distance (r) would vary by 1/(r^2), correct? Logically this is because the energy isn’t distributed evenly through the volume it’s traversed, but is rather ‘focused’ at the leading edge (the shell of the sphere).
    However, that assumes that the energy spreads equally in all directions and none is reflected (by the air/water interface, or the bottom or other structures). Also I was assume that for many applications of sonar, focusing/directionalizing the sound would be done, if possible to allow a larger range with a lower total power input. Were these considerations taken into account?

  8. micah

    Sam C: I have this radical theory that we should hold published court opinions to a higher standard of accuracy than random blog posts.

  9. Mark C. Chu-Carroll

    Re Micah (#4):
    Actually, you and I are both wrong.
    Each 10 decibels corresponds to roughly doubling the perceived volume of the sound.
    A sonar at 235 decibels is roughly 40 decibels louder than the loudest possible sound in air. That’s 4 doublings – so 16 times louder. 4 orders of magnitude would be 10,000 times.

  10. micah

    Are you sure? All the internet sources I can find use base 10 logarithms (or, to be more precise, base 101/10 logarithms) in their definition of decibels. I don’t have a physics textbook handy to confirm, but…

  11. Uncephalized

    Well that depends on if you’re talking about decimal orders of magnitude or binary orders of magnitude, in which case 4 doublings and 4 orders of magnitude are the same thing. 🙂

  12. Ktesibios

    Mark, there are some additional errors here, which have less to do with innumeracy than with ignorance of acoustics.
    SPL expressed in decibels is not directly comparable between the air and water media.
    For sound propagating in air, the reference 0 dB SPL pressure is 20 uPa RMS, which corresponds to an intensity (power density) of 10-12W/m2. For sound in water, the O dB reference pressure is defined as 1 uPa, which corresponds to an intensity of approximately 6 x 10-19W/m2, due to the fact that the characteristic acoustic impedance of water is approximately 3400 times that of air.
    So, one has first to deal with the different 0 dB reference levels, and also to ask which is the truly relevant quantity- pressure or power density.
    It might also be worth mentioning that the inverse square law applies only to a spherically diverging wave. If the divergence is constrained to 0 in both the horizontal and vertical dimensions, as in a plane-wave tube, the SPL remains unchanged with distance save for losses due to viscous effects in the medium. If it’s constrained in only one dimension, e.g., by the surface and bottom of the ocean, the divergence becomes cylindrical at a distance comparable to the depth of the water, in which case the power/distance relationship is inverse linear.

  13. Philip Roberts

    I see Ktesibios has beat me to it but I’ll post this anyway.

    Actually there is several more things that may need to be considered.
    First, as I suspect you know, decibel are meaning just a ratio, so it’s meaning less unless the reference level is either specified or infer able from context.
    Sound pressure level in water and air use different reference pressures, in air 20uPa is used while in water 1 uPa is commonly used, giving a dB SPL number 26 dB greater in water relative to air. I’ve also seen mPa used, a 34 dB difference, which means for the same pressure add 34 dB to the SPL level in air.
    Second, even accounting for this the threshold of human hearing in water is 67 dB (ref 1uPa, see wikipedia “Sound Pressure”).
    The pain threshold may also be different relative to the threshold of audibility, see EXPOSURE OF DIVERS TO UNDERWATER SOUND IN THE FREQUENCY RANGE FROM 800 TO 2250 Hz where “Bareheaded diver subjects experienced dizziness, vibration, and movement of the visual field at underwater sound levels above 170 dB re.1mPa.”, note the 1mPA reference level, this absolute pressure corresponds to 204 dB SPL in air.
    And one last this to consider is the overpressure, deltaP/P, a wave at 200 dB SPL ref 1uPa is 10,000 Pa. Taking 100 m as the depth we are interested in the pressure is 980,665 Pa giving an overpressure of just over 1%. And is the equivalent of a 1m wave passing over you on the surface.
    I’m still not sure what the sum total effect of all these additional complications are. I’ll still agree that 235 db SPL re ??? in water from most any reference is crazy loud and not anywhere I want to be even remotely close to.

  14. Kmeson

    Ktesibos strikes a key point here in the difference between 0 dBA and 0 dB re 1uPa. In those units it is then also relevant that casual whale conversation uses signal levels that are ~180 dB re 1uPa @ 1 m from the source (these are at similar mid-frequencies) and uses pulses to probe the near environment (higher frequency) that can be as loud as 220 dB. Does MF sonar cause whales harm? Perhaps but it is not analogous to being acoustically blasted by a jet engine. If anything it is analogous to being scared by distant air raid siren and darting into traffic.

  15. Jonathan Vos Post

    Stephen Hawking’s Metalaw of Equations is that each
    equation in a book cuts in half the number of copies
    Mark CC is exatly correct here. Chief Justice John G. “Right Wing Catholic Bush-Appointee” Roberts is horribly wrong.
    Why do lawyers (and judges) and diplomats and doctors and pharmacists use Latin terminology?
    Because Latin is a dead language. Hence these words do NOT change their basic meaning over time, the way that words in English, French, German, Mandarin, Arabic, Spanish, Portuguese, and the like do.
    Appeals (State or Federal Distict) are much more expensive and procedurally exacting than Superior Court cases. Supreme Court (State or US) are much more expensive and procedurally exacting than Appellate cases.
    Theoretically, this is due to the hierarchy associated
    with Stare Decisis.
    Economically, it is the iron law of Supply and Demand.
    There is a smaller supply of lawyers who have experience in Appeals. Still smaller is the fraction who’ve argued in Supreme Court.
    The demand on their time is higher.
    Thus, the price that they charge is higher.
    Now, more theory.
    [Disclaimer: IANAL: I Am Not A Lawyer]
    [Disclaimer: TINLA: This Is Not Legal Advice]
    What we call “The Law” (State and Federal courts) is a chaotic attractor in the space of all possible laws. Law, being precedent-based rather than axiom-based (as in Math), or Empirical basis (Physics experiments) have a sensitivity to initial conditions (i.e. precise language of legislation) and feedback mediated by Stare decisis which [United States Internal Revenue Serv. v. Osborne (In re Osborne), 76 F.3d 306, 96-1 U.S. Tax Cas. (CCH) paragr. 50,185 (9th Cir. 1996)] is the policy of the court to stand by precedent; the term is but an abbreviation of stare decisis et quieta non movere = “to stand by and adhere to decisions and not disturb what is settled.” Consider the word “decisis.” The word means, literally and legally, the decision. Nor is the doctrine stare dictis; it is not “to stand by or keep to what was said.” Nor is the doctrine stare rationibus decidendi = “to keep to the rationes decidendi of past cases.” Rather, under the doctrine of stare decisis a case is important only for what it decides — for the “what,” not for the “why,” and not for the “how.” Insofar as precedent is concerned, stare decisis is important only for the decision, for the detailed legal consequence following a detailed set of facts.
    As wikipedia begins:
    John Glover Roberts, Jr. (born January 27, 1955) is the seventeenth and current Chief Justice of the United States. Appointed by President George W. Bush, Roberts is generally considered a member of the more conservative wing of the court.
    Before joining the Supreme Court on September 29, 2005, Roberts was a judge on the United States Court of Appeals for the District of Columbia Circuit for two years.
    Previously, he spent 14 years in private law practice and held positions in Republican administrations in the U.S. Department of Justice and Office of the White House Counsel.
    Thank you very much President George W. “Most Anti-Science President Ever” Bush. Your legacy of ignorance and superstition continues unabated.
    Good thing that President Obama will replace some Supreme Court Justices. The sooner the better.
    Litmus Test: must believe that this is the real world, which is bound by Physical Laws that trump man-made laws, and which follow Mathematical equations.

  16. Gray Gaffer

    Nobody caught this bit – as an audio engineer at times, I had to know this: 3db is a doubling of the signal, not 10db. 10 db is doubling three times (approx) therefore a factor of 8+, not 2. The nonlinear compressive response of the ear makes for an un-reliable measuring device.
    So it is >4 times worse than discussed.


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