Mathematician Tim Pennings watched his dog Elvis fetch balls thrown in the water and noticed that the dog consistently chose the quickest route. Running is faster than swimming, so the overall time the dog spends heading to the ball depends on how the dog decides to split his path into running and swimming parts.
Elvis could run directly into the water and swim a long way to the ball, which would mean traveling the shortest distance, but not getting there as fast as possible. Another possibility is to run on the sand until he is even with the ball, and then swim to it. A third option is to run part of the way along the shore and then finish traveling to the ball by swimming in the water. Elvis always chose this last option, which resulted in reaching the ball the fastest. Mathematicians describe his actions by saying that Elvis optimized his travel time.
With information about the position of the ball and the dog, and the dog’s running and swimming speeds, it is possible to use calculus to determine the exact place at which the dog should switch from running to swimming in order to minimize his travel time. Pennings has suggested that dogs do in fact know calculus , because their paths match what the mathematics of calculus predict.
I think it’s more accurate to say that dogs act as though they know calculus rather than to say that they actually know calculus. It’s a small, but important distinction. I agree that dogs act to optimize their travel time when fetching in the water—I’ve observed dogs doing this—but that does not mean they are making complex mathematical calculations. It’s more likely that their experience allows them to make choices that result in getting to the ball faster.
Watching dogs fetch in an optimal way is no less remarkable to me than if they were using calculus. Have you seen dogs performing the kind of behavior that led Pennings to suggest they know calculus?