Language, Time, Love, Math, and Money.
Part 3: Math.
July 15, 2016
Are Dogs Smarter than Toddlers? You Do the Math!
Um ... E Equals What Now?
Canine Math Tests
In 2009 Dr. Stanley Coren did a study on dogs and their apparent math skills. This came at a time when dognitive science was making a big splash in the news: dogs had just become the “it” animal for scientific research into the nature of animal consciousness, and the question of whether animals have some of the same cognitive abilities as humans.
AOL: “The canine IQ test results are in: Even the average dog has the mental abilities of a 2-year-old child.” CNN: “Counting ability is tested in drills such as one in which treats are dropped, one at a time, behind a screen. When the researcher either sneaks away one of the treats or stealthily adds an extra before raising the screen, the dog will wait longer—appearing to puzzle over the bad math—before eating the treats.”
“Now we’re giving him the wrong equation,” Coren said. “The dog acts surprised and stares at it for a longer period, just like a human kid would.”
But why is the dog staring? Do we know it’s for the same reason a human kid would? Or is that pure supposition? And which is more likely, that dogs are able to feel an emotional attraction to things in their environment—toys, treats, their owners, other dogs—and can sense when some of those things have gone missing or have been added? Or that they engage in mental arithmetic and count out, by number, how many things were there initially and either do mental addition or subtraction in their heads to come up with a new number, reflecting whatever changes have taken place?
The truth is, all animals, including birds, have what’s called a “number sense.” It has nothing to do with math skills or even the ability to count. It’s a gut feeling, a visceral process, which, for instance, enables a robin to know how many eggs are in her nest. In dogs, this comes partially from the social nature of wolves. Like dogs, wolves can’t add, subtract, or do math, but they are able to keep track of the locations of their pack members.
Other questions remain. What if the study had been done with objects that don’t interest the dogs? Toddlers can be taught to count on their fingers and toes, or to count the number of cats in a drawing, or to count spoons or matchsticks or cracks in the sidewalk or other items that wouldn’t interest a dog in the slightest.
Plus, how do the researchers know the dogs were really surprised when the screen was removed? How can we know? Why can’t we just ask the dogs what they were thinking?
Oh, right. Dogs can’t talk. But toddlers can.
Counting to Five
Years ago I taught my dog Freddie to “count” to 5: I would give him a number between 1 and 5 and then give him a treat if he barked the correct number of times, but didn’t reward him if he barked out the “wrong” number. And after repeating this procedure over and over, many, many times, until we were both mentally exhausted, Fred learned to bark in accordance with the number given. People were always amazed when I would say to Freddie, “Four,” and he would bark 4 times in succession, etc.
The thing is, Freddie’s responses were based on a very simple process called pattern recognition: he was responding to how the patterns of certain verbal cues I gave him matched the pattern of his barking. Pattern recognition is an unconscious form of mental processing, an evolutionary precursor to logic, language and math. And it’s light years away from “having a basic understanding of arithmetic.”
Yet pattern recognition explains why dogs sometimes are “smarter” than toddlers. That’s because dogs pay close attention to dozens, if not hundreds of things we do on a daily basis. This means they actually have more of a feel for how the household operates than toddlers do. They’re constantly paying attention to our behavioral patterns.
This doesn’t mean that dogs have the capacity to do math, arithmetic, or even count. They just don’t. After all, numbers, which are the fundamental units of mathematics, are mental constructs, not real, physical things. You can't sniff, eat, bite, chase, play with, or pee on a number. As for the comparisons made between dogs and toddlers, three-year olds can do thousands of math-related things a dog could never do. They can speak the names and recognize the written shapes of numerals up to 9. Dogs can’t talk and they can’t read or interpret written symbols—you can write as many numbers as you want on a piece of paper, show it to a dog, and he’ll just wag his tail, not knowing what the heck is going on.
In 1930, mathematician Tobias Dantzig first proposed the idea that animals and humans have a kind of mental accumulator, giving them what he called a “number sense.” This is not the ability to count but a natural sense of knowing when something has changed in a small collection of items. In his book The Number Sense: How the Mind Creates Mathematics, modern French mathematician and cognitive scientist Stanislas Dehaene writes: “Whatever its exact neuronal implementation, if the accumulator model is correct, two conclusions must necessarily follow. First, animals can count, since they are able to increase an internal counter each time an external event occurs. Second, they do not count exactly as we do.”
Yet Coren insists, “These studies suggest dogs have a basic understanding of arithmetic, and they can count to four or five” the way toddlers do. In a way, perhaps that’s true. But it’s not a mental calculations, it’s a number sense. And while in Coren’s estimation dogs are capable of keeping track of 5 separate objects, a wolf pack can have up to 30 members, meaning their number sense is more developed than a dog’s.
So where does the ridiculous idea that dogs are better at math than toddlers come from?
Believers, Skeptics, and the Fallacy of Degree-Not-Kind
According to animal researcher Michael Tomasello, there are two kinds of dognitive scientists: boosters and scoffers. Believers tend to favor cognitively-rich explanations for animal behavior (operating under what I call the fallacy of degree-not-kind), while skeptics favor cognitively lean ones. Just as a quick guide to the players on both sides, Marc Bekoff would be the ultimate believer (with Stanley Coren running a close second), while Ray Coppinger would be the ultimate skeptic.
Believers see the differences in consciousness as a kind of sliding scale, a continuum, if you will. Skeptics feel that there are substantial and important differences of kind, not degree, where consciousness is seen as a spectrum, like a rainbow of different colors, not a continuum, with little or no differentiation. This is why Coren believes that dogs and other animals have a rudimentary ability to understand math and language.
The problem is, according to linguists, there’s no such thing as a simple or rudimentary language. It comes as a whole package. You either have the ability or you don’t. And if you don’t have it, you don’t have the ability to count either. You can’t add, subtract, do multiplication, etc. It’s impossible to do any of those things without the use of words and language.
Dogs Aren’t Good With Numbers, Wolves May Actually Be Better
So Coren’s magic trick with dogs and numbers made a big splash in 2009. But a newer study, done in 2013, shows that dogs do not, in fact, have the ability to count, but that wolves just might. At least that’s one interpretation.
From Discover Magazine’s website, science writer Elizabeth Preston tells us about a new study: “At the Wolf Science Center in Austria, Friederike Range and her colleagues raise both wolves and dogs by hand, then train them to take part in cognition research projects. Their interest in canine counting skill isn’t totally trivial.” She goes on, “If dogs have any grasp of numbers, they should be able to judge two sets of food items—say, three versus four Milk-Bones—and pick the bigger snack. Earlier research found that dogs are OK at this, but only if they can see both food piles. This means they might just be judging which pile takes up more space, not the actual amounts.”
To prevent any possible unconscious cues given to animals by the body language, etc., exhibited by the human operator, the researchers set up an apparatus (pictured below). Not only did the apparatus hide the researcher’s body, she also wore sunglasses to prevent the dogs and wolves from seeing her eyes.
Cheese cubes were dropped into an opaque tube, then a different number of cubes were dropped into another tube. The dog or wolf was taught to step on a buzzer to indicate which side had more cheese. When they chose correctly, the cubes dropped out of the tube so the animals could eat.
The wolves passed this test with flying colors. “They weren’t perfect,” writes Preston, “but they got the right answer more often than not.”
Meanwhile dogs were only able to pass the test when one tube had at least twice as much cheese as the other. In neither case was this clear, convincing evidence of an actual ability to count. What it does show is that a wolf’s number sense may be more finely-tuned than a dog’s. And this would, in all likelihood, be related to the way wolves hunt as a group.
Anyway, that’s how I see it.
“Life Is an Adventure—Where Will Your Dog Take You?”