Last month, a handful of science news outlets slightly bungled their reporting of a new study discussing the mathematics behind dolphin echolocation, leaving the public believing that dolphins can “do nonlinear mathematics.”

The opening sentence from the Discovery News article that seems to have been Patient Zero from which the other “dolphins are math geniuses” reports sprang is as follows:

“Dolphins may use complex nonlinear mathematics when hunting, according to a new study that suggests these brainy marine mammals could be far more skilled at math than was ever thought possible before.”

This is, however, a rather misleading overview of the situation. The scientific research in question has nothing to do with dolphins’ math skills in the way readers normally think of math skills. Here’s a quick overview of how the science reporting went wonky.

The original press release upon which the news articles are based can be found at this link. The study (abstract at this link) showed that a nonlinear mathematical model was the best way of describing how signal processing works for dolphin echolocation. Dolphins are able to gain useful information about objects (like fish) in water that is filled with bubbles, which usually flummoxes human-made sonar. By changing the amplitude (loudness) of their echolocation clicks, the click echo information that dolphins receive helps them find objects in cluttered environments. But this feat can only be explained by nonlinear sonar processing models – standard mathematical models simply can’t explain how dolphin echolocation works in a cluttered environment.

In trying to put this study in context, Discovery News related these findings to work on dolphin cognition:

“In terms of dolphin math skills, prior studies conducted by the Dolphin Research Cetner (sic) in Florida have already determined that dolphins grasp various numerical concepts, such as recognizing and representing numerical values on an ordinal scale.”

But the kind of “math” the echolocation study is talking about is completely unrelated to the kind of “math” these cognition studies are talking about. And both the cognition and echolocation studies are discussing kinds of “math” that are unrelated to the “math” you or I do when we sit down to solve a nonlinear algebraic equation. Here are the three kinds of math that are being conflated:

**MATH 1** (perception)

Brains are able to process incoming stimuli (e.g., sound waves that tickle ear hairs, photons that activate retinal cells) using mental calculations that results in the brain perceiving the world around it (i.e., hearing, seeing). The mathematical process of turning sensory information in the form of nerve impulses into mental representations (i.e., perception) is entirely unavailable to the part of the brain that either consciously or unconsciously works with these mental representations.

**MATH 2** (numerosity)

Brains of many animals are able to categorize objects in the world in ways that allow them to make judgments about relative quantity. This basic form of numerical cognition might be ubiquitous in the animal kingdom, and allows animals to understand that, for example, five items are more than three items. Dolphins, pigeons, and even insects are capable of relative quantity judgments.

**MATH 3** (symbolic math)

The human brain is able to represent numerical concepts symbolically. This then allows us to consciously perform advanced mathematical operations (e.g., addition, subtraction, algebra) involving these symbols. It does not appear that many non-human animals can perform these kinds of calculations – with the possible exception of a handful of subjects (e.g., Alex the parrot, Sheba the chimpanzee) that can perform very basic addition.

The echolocation study in question is discussing the kind of subconscious mathematical processing described in MATH 1 that happens in the dolphin brain (i.e., perception). Dolphins are entirely unaware that their brains are performing these dazzling feats. But the news reports seem to be suggesting that dolphins are “doing math” like we see in MATH 3 – as if dolphins are whipping out pen and paper to calculate echolocation click amplitude levels in order to find mackerel. And of course the MATH 2 skills that dolphins have shown in the lab (see here or here) are entirely unrelated to MATH 1, so there’s no real reason to bring them up at all when trying to make sense of the echolocation study.

Here’s an example from the human world to better understand the nature of the confusion. The human brain – like the brain of many species – is able to determine the location of a sound (i.e., sound localization) by (among other methods) calculating the different arrival times of the sound at each of our two ears. If a dog is positioned a few meters away directly to my right, the sound of the dog’s bark will reach my right ear before it reaches my left ear. My brain makes a calculation using some sort of brain-math that can be represented with this nonlinear calculation (taken from Reid & Milios, 2003):

The result of this brain-math is my perception of the dog bark as being “on my right.” But it is not the case that I make this calculation by sitting down at the table and plugging numbers into the above formula. I couldn’t even solve this formula right now (or ever) if I tried. It all happens subconsciously and instantly; I am completely unaware of the kind of math underpinning my perception of sound. It would be madness to claim that I am doing “nonlinear mathematics” in order to localize sound. I – the conscious me that thinks about mental representations and symbols – am not “doing math” at all. And, contrary to media reports, neither are dolphins when they echolocate.

Related references

Leighton, T., Chua, G., & White, P. (2012). Do dolphins benefit from nonlinear mathematics when processing their sonar returns? Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 468 (2147), 3517-3532 DOI: 10.1098/rspa.2012.0247

Reid GL, & Milios E (2003). Active stereo sound localization. The Journal of the Acoustical Society of America, 113 (1), 185-93 PMID: 12558259

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