Canine Calculus

March 22, 2010 at 10:08 am 2 comments

Stories of “clever” animals that can spell, write and do math have amazed and delighted us for more than a century. These stories capture our attention because we have an odd tendency to be most impressed by animals when they seem to be able to mimic human activities.

A classic example of this phenomenon is Clever Hans, a horse who appeared to be able respond correctly to questions involving mathematical calculations and other advanced cognitive tasks by tapping his hoof.  Hans was a sensation.  People flocked to see the horse that could think like a man.

Of course now we know that Hans wasn’t spelling, doing mathematical calculations or telling time — he was responding to incredibly subtle physical cues picked up from his handler. And while some may be disappointed by this I think that the skills of clever animals like Hans are truly astonishing.  They’re just not astonishing in the way many of us hoped they would be.

The riddle of Clever Hans was solved by Oskar Pfungst who, unlike everyone else who studied the horse, focused on the handler instead of the horse. Pfungst didn’t assume that the key to the phenomenon was how Hans learned the answers to the questions – he wanted to know how the horse was able to give the right answers.

As Oskar Pfungst demonstrates, we learn a lot more about what goes on inside the minds of animals when we focus on seeing them as they are instead of imagining them as we’d like them to be.

And while stories of ‘clever’ animals still make the news, real insights on animal cognition are being discovered by mathematicians following in the footsteps of Oskar Pfungst.

Case in point: Dr.Timothy Pennings and his Corgie mix Elvis. Pennings noticed that when they played fetch together at the beach Elvis consistently chose a very efficient path to retrieve the ball. Pennings’ curiosity was piqued so he and his students developed an experiment to test whether Elvis was choosing an optimal path. Mathematically speaking, the optimal path is the route that allows Elvis to minimize his travel time through various media that affect his speed in different ways.

Following this work, Pennings published the article “Do Dogs Know Calculus“. In the article he discussed Elvis’ intuitive ability to solve a classic optimization problemwithout doing calculus. The world of mathematics is apparently almost as competitive as the world of dogs, and Penning’s article was soon rebutted by Peruchet and Gallego’s 2006 article “Do Dogs Know Related Rates Rather Than Optimization“. in which they demonstrated that a female Labrador named Salsa also seemed to choose an optimal path when retrieving in water.

Peruchet and Gallego postulated that instead of intuitively choosing an optimal path, Salsas’ performance was based on her ability to detect transient changes in the distance to an object (in this case, the ball) combined with an awareness of the relative speeds of running and swimming.

In 2007 Pennings and co-author Roland Minton responded in “Do Dogs Know Bifurcations?” Bifurcation theory is the mathematical study of how systems change as some parameter of the system is changed. It is typically applied to dynamical systems. A bifurcation happens when a small, incremental change made to the system causes a sudden change in its behavior. In biology it is used to study things like population dynamics and predator-prey relations.

According to the Roanoke College news page:

The experiment so interested Minton that he used it as a problem for his calculus textbook and soon e-mailed Dr. Tim Pennings, quickly forming a unique friendship sown entirely through electronic communication. Minton says they were “having a lot of fun topping each other” with various calculus equations that revolved around Elvis’ innate bifurcation point. Minton and Pennings became so enthralled with the subject that Dr. Pennings suggested Minton write his own article about their findings. Minton, however, felt that it wouldn’t have been right to take the credit, so they collaborated on the article, “Do Dogs Know Bifurcations?” instead, combining Minton’s ideas with Pennings’ and Elvis’ findings.

[...] Pennings’ findings show that when he mathematically calculated the optimal path of Elvis’ route, he arrived at an estimation that was very close to Elvis’ own bifurcation point. This indicates that Elvis may in fact have the ability to problem solve.

Minton says the experiment and the application of Elvis’ actions to calculus provide an excellent visualization for the teaching of calculus and is also an entry into finding out how dogs (and possibly humans) problem solve. Pennings continued the experiment, and instead of standing on the shore, he and Elvis began in the water, and he found once again that Elvis innately found the most efficient path to the stick. This experiment cemented his belief that Elvis has the ability to think ahead when solving a problem.

Pennings and Minton reported that Elvis had repudiated Peruchet and Gallego by introducing bifurcation into his strategy. They proposed that Elvis uses a small set of rules to find an optimal path to a ball thrown into the lake. If the ball is close, he just swims straight to it. If the ball is farther away then he gets out of the water and solves the shore to ball problem. Since this set of rules described Elvis’ behavior accurately, the remaining question was whether he knew how to bifurcate at the optimal point.

They returned to the lake to collect more data and discovered that Elvis’s bifurcation distance was consistently somewhat farther down the beach than the optimum point. From this they concluded that “Elvis knows bifurcations qualitatively, but not quantitatively” (a result that may be comforting to many frustrated calculus students).

Pennings, Minton, Peruchet and Gallego studied something dogs do well.They used a tongue-in-cheek approach (dogs do calculus!) to poke fun at each other and get attention for their work – but they also made of a point of stating that these clever dogs functioned as inspiration for the human mathematicians who did the real calculating.

In focusing on something dogs do well (choose an optimal path) instead of how they might mimic our behavior (doing calculus), these mathematicians were able to provide evidence that dogs use executive functions to solve complex problems much as we do. In a related bit of mathematical grooviness, they also discovered a nifty pictorial proof for the relationship between the geometric and arithmetic means of two numbers

Hope College has awarded Elvis an honorary degree. He is not, however, allowed to teach classes.

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2 Comments Add your own

  • 1. Rob McMillin  |  March 22, 2010 at 11:00 am

    This reminds me of the story of Sir Isaac Newton’s solution of Bernoulli’s two puzzles (one of which was the brachistochrone problem):

    In June 1696 Bernoulli addressed a letter to the mathematicians of Europe challenging them to solve two problems—(1) to determine the brachistochrone between two given points not in the same vertical line, (2) to determine a curve such that, if a straight line drawn through a fixed point A meet it in two points P1, P2, then AP1m+AP2m will be constant. This challenge was first made in the Ada Lipsiensia for June 1696.

    Six months were allowed by Bernoulli for the solution of the problem, and in the event of none being sent to him he promised to publish his own. The six months elapsed without any solution being produced; but he received a letter from Leibniz, stating that he had “cut the knot of the most beautiful of these problems,” and requesting that the period for their solution should be extended to Christmas next; that the French and Italian mathematicians might have no reason to complain of the shortness of the period. Bernoulli adopted the suggestion, and publicly announced the postponement for the information of those who might not see the Ada Lipsiensia.

    On 29 January 1697 Newton returned at 4pm from working at the Royal Mint and found in his post the problems that Bernoulli had sent to him directly; two copies of the printed paper containing the problems. Newton stayed up to 4am before arriving at the solutions; on the following day he sent a solution of them to Montague, then president of the Royal Society for anonymous publication. He announced that the curve required in the first problem must be a cycloid, and he gave a method of determining it. He also solved the second problem, and in so doing showed that by the same method other curves might be found which cut off three or more segments having similar properties. Solutions were also obtained from Leibniz and the Marquis de l’Hôpital; and, although Newton’s solution was anonymous, he was recognized by Bernoulli as its author; “tanquam,” says he, “ex ungue leonem” (we know the lion by his claw).

  • 2. Rob McMillin  |  March 22, 2010 at 7:51 pm

    Also, since you said the magic words “population dynamics”, here’s a fascinating mathematical take on vampire population ecology in Buffy the Vampire Slayer (PDF).

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