Why has evolution repeatedly made sure that animals can understand not just that four is less than five but that “four squares” is in some way conceptually the same as “four circles”?
SOME SCIENTISTS ARE SO MULISH THAT THEY ACT LIKE IDIOTS BY NOT ACCEPTING A GUIDING FORCE IN EVOLUTION CALLED BY PEOPLE AS GOD
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cognitive science
Animals Can Count and Use Zero. How Far Does Their Number Sense Go?
The intelligence of corvids like ravens and crows is well known. Recently, crows were even shown to have a numerical ability seen in few other species so far: a grasp of the concept of the empty set — the numerosity zero.
An understanding of numbers is often viewed as a distinctly human faculty — a hallmark of our intelligence that, along with language, sets us apart from all other animals.
But that couldn’t be further from the truth. Honeybees count landmarks when navigating toward sources of nectar. Lionesses tally the number of roars they hear from an intruding pride before deciding whether to attack or retreat. Some ants keep track of their steps; some spiders keep track of how many prey are caught in their web. One species of frog bases its entire mating ritual on number: If a male calls out — a whining pew followed by a brief pulsing note called a chuck — his rival responds by placing two chucks at the end of his own call. The first frog then responds with three, the other with four, and so on up to around six, when they run out of breath.
Practically every animal that scientists have studied — insects and cephalopods, amphibians and reptiles, birds and mammals — can distinguish between different numbers of objects in a set or sounds in a sequence. They don’t just have a sense of “greater than” or “less than,” but an approximate sense of quantity: that two is distinct from three, that 15 is distinct from 20. This mental representation of set size, called numerosity, seems to be “a general ability,” and an ancient one, said Giorgio Vallortigara, a neuroscientist at the University of Trento in Italy.
Now, researchers are uncovering increasingly more complex numerical abilities in their animal subjects. Many species have displayed a capacity for abstraction that extends to performing simple arithmetic, while a select few have even demonstrated a grasp of the quantitative concept of “zero” — an idea so paradoxical that very young children sometimes struggle with it. In fact, experiments have shown that both monkeys and honeybees know how to treat zero as a numerosity, placing it on a mental number line much as they would numerosity one or two. And in a paper published in the Journal of Neuroscience in June, researchers reported that crows can do it, too.
The fact that those three species are from diverse taxonomic groups — primates, insects and birds — suggests that certain numerical abilities have evolved over and over again throughout the animal kingdom. Scientists are puzzling over why nature has gifted so many animals with at least a rudimentary knack for numbers, and what if anything that might tell us about the deep origins of human mathematics. There are still more questions than answers, but neuroscientists and other experts have learned enough to amend and broaden perspectives on animal cognition. Even in “tiny brains like those in bees or even ants,” said Brian Butterworth, a cognitive neuroscientist at University College London and author of the forthcoming book Can Fish Count?, “there is a mechanism that enables the creature to read the language of the universe.”
A Competence for “Number”
Nearly 120 years ago in Berlin, a horse named Clever Hans attained celebrity status: He could seemingly do arithmetic, tapping out the solutions to addition, subtraction, multiplication and division problems with his hoof. But a psychology graduate student soon realized that the animal was really just paying very close attention to subtle behavioral cues from his trainer or audience members who knew the answers.
The incident entrenched a skepticism about animals’ numerical capabilities that persists today. Some researchers, for example, propose that while humans have a “true” understanding of numerical concepts, animals only appear to be discriminating between groups of objects based on quantity when they’re instead relying on less abstract characteristics, like size or color.
Crows notice the quantities of items that they see, and have neurons that are tuned to those quantities, allowing them to distinguish a set of four items from a set of three or five.
Courtesy of Andreas Nieder
But rigorous experiments during the past two decades have shown that even animals with very small brains can perform incredible feats of numerical cognition. One mechanism common to all of them seems to be a system for approximating numerosity that’s correct most of the time but is sometimes imprecise in specific ways. Animals are most effective, for instance, at distinguishing numerosities far apart in magnitude — so comparing a group of six dots to three dots is easier than comparing six to five. When the difference between two numerosities is the same, it’s easier to deal with smaller quantities than larger ones: Discriminating 34 items from 38 is much more difficult than discriminating four from eight.
Those strengths and weaknesses were reflected in animals’ neural activity. In the prefrontal cortex of monkeys, researchers found neurons that were selectively tuned to different numerosities. Neurons that responded to three dots on a screen also responded weakly to two and four, but not at all to more distant values, such as one or five. (Humans demonstrate this approximate sense of quantity, too. But they also associate numerosities with specific number symbols, and a different population of neurons represents those exact quantities.)
That observation seems to imply that a “sense” of number is innate and deeply rooted in the brains of animals, including humans. “Underlying the sense of number, there is a very ancient, fundamental psychophysical law,” Vallortigara said.
Once “you realize that almost every animal, or maybe even every animal, has some ability to do a numerical task, then you start wanting to know … what’s the threshold? What’s the limit?” said Scarlett Howard, a postdoctoral research fellow at Deakin University in Australia who studies numerical cognition in honeybees. If animals had this natural, hard-wired ability for telling quantities apart, scientists wanted to determine what other abilities might emerge with it.
First up was arithmetic. Several species have demonstrated that they can essentially add and subtract. In 2009, researchers led by Rosa Rugani, a psychologist and Marie Skłodowska-Curie Actions global fellow at the University of Padova in Italy, found that when newly hatched chicks were presented with two groups of items on which they had imprinted, the days-old birds tended to approach the larger group. Then the team obscured the groups of objects with screens, and moved some of the items from behind one screen to the other while the chicks watched. No matter how many items were moved, the chicks consistently chose the screen that hid more of them. They seemed to be performing computations akin to addition or subtraction to keep track of each hidden group’s changing numerosity. No training was required for them to do this. “They deal spontaneously with these kinds of numerosities,” Rugani said.
Newly hatched chicks imprint on objects presented to them. The psychologist Rosa Rugani of the University of Padova (right) has shown that imprinted chicks seem to be able to use arithmetic to keep track of the numbers of these objects.
Rosa Rugani
Wild monkeys can do something similar. While monkeys watched, scientists placed several pieces of bread in a closed box, then periodically removed one or more of them. The monkeys could not see how many pieces remained, but they continued to approach the box until the last piece was removed — which suggested that they performed subtraction to inform their foraging.
Honeybees, meanwhile, can be taught simple arithmetic. In 2019, Howard and her colleagues trained the insects to note the colors and numbers of objects they saw, and then to add one to numbers of blue objects or subtract one from numbers of yellow objects. For example, if the bees flew through a maze that contained three blue shapes, and they were then presented with a choice between two or four items, they consistently chose the group of four.
“They’re able to do these tasks because in their natural environments, they have to learn so much,” Howard said. No one knows whether the bees add or subtract in the wild without training — such behavior has never been observed, but scientists also haven’t had reason to look for it until now. Still, the bees already have all the building blocks for doing arithmetic at their disposal. And “their environment can be its own sort of training ground,” Howard added.
In behavioral studies, honeybees have demonstrated an understanding of the numerosity zero. They have also been trained to perform simple acts of arithmetic, although it is not known whether they use this ability in the wild.
Anne Moffat for Quanta Magazine
These kinds of findings motivated researchers to probe for even more abstract forms of numerical representation in animals. In 2015, a few years after their arithmetic study in chicks, Rugani and her colleagues found that the animals associated smaller numerosities with the left and larger ones with the right — much as humans spatially represent ascending values on a number line. “That was thought to be our human invention,” said Adrian Dyer, a vision scientist at the Royal Melbourne Institute of Technology who works with honeybees and was Howard’s doctoral adviser. But it may “just be something which is within some brains, [part] of how we process information.” (Dyer is now testing whether bees also use such a number line representation.)
Insects, birds and primates have also been trained to link symbols to numbers of elements. “We took the bees and taught them as if they were in primary school: this symbol represents this number,” Dyer said. “And they got the association.” Chimpanzees that have been trained to link numerosities to number symbols could also learn to touch the digits in ascending order.
Now researchers are exploring other kinds of numerical tasks. Rugani and her team are studying whether monkeys can bisect a quantity to identify the concept of “middle,” which requires them to count and compare the number of elements from both the right and left of a lineup. So far, she said, “the results are kind of impressive.”
Over and over again, she and others are finding evidence not just for a relatively simple, ubiquitous sense of numerosity in animals, but also for a growing inventory of much more abstract and complex forms of numerical cognition. That’s why for some neurobiologists, the current great frontier is in learning whether some animals’ grasp of numerical abstractions extends to the slippery concept of “nothing.”
A Special Quantity
All numerosities are abstractions. The numerosity “three” can refer to a group of three dots, or three chairs or three people. “Having a sense of number at all means to be able to assess or evaluate the size of the set, irrespective of its members” and minor differences between them, Butterworth said. “Even when you’ve got bees counting petals, each flower is different from the other flowers in some respects — in its location, the exact conformation of its petals.”
But one numerosity is different from the rest. “Zero is quite particular and peculiar,” Rugani said. “It’s not just an abstraction of perceiving something, but also perceiving its absence.”
Even humans struggle with zero: Very young children, for instance, don’t seem to regard the empty set as a numerical quantity at first. Instead, they consider it an absence, a category of its own, unrelated to other values. While children usually grasp the counting numbers by age 4, it often takes another two years for them to gain an understanding of zero as a number.
That’s because using zero in this way “requires some transcending of the empirical world,” said Andreas Nieder, a neurobiologist at the University of Tübingen in Germany — a recognition that the empty set can be considered a quantity, and that “nothing” can be represented as something. After all, he said, “we do not go out to buy zero fish.”
Moreover, he added, “When you look at the history of mathematics, it turns out that [zero] is an extreme latecomer in our culture as well.” Historical research finds that human societies didn’t begin to use zero as a number in their mathematical calculations until around the seventh century.
“From this human perspective,” said Aurore Avarguès-Weber, a cognitive ethologist at the University of Toulouse in France who works with Howard and Dyer on honeybees, “zero seems not to be biological but much more cultural.”
But Nieder suspected otherwise. Some animals, he thought, might be able to regard zero as a quantity, even if they didn’t have a symbolic sense of it in the way that humans did. Sure enough, his group demonstrated in 2016 that monkeys have neurons in their prefrontal cortex tuned to a preference for zero rather than other numerosities. The animals also made a revealing mistake when using zero: They mixed up the empty set more often with numerosity one than with numerosity two. “They are perceiving the empty set, or nothing, as a quantity that is next to one on this number line,” Nieder said.
In 2018, Howard, Avarguès-Weber, Dyer and their colleagues found behavioral evidence of this in honeybees as well. To Howard, these findings suggested that what she called “this numerical cognition, this high level of understanding abstract numerical concepts” is innate. An understanding of zero could be a more general trait across the animal kingdom than had been thought.
Honeybees can count landmarks — in this study, a series of yellow tents — to help them navigate toward a food source.
Lars Chittka
That honeybee study raised eyebrows, not just because it showed that an animal with fewer than one million neurons in its brain (compared with the human brain’s 86 billion) could treat zero as a quantity, but because bees and mammals diverged in evolution 600 million years ago. Their last common ancestor “was [barely] able to perceive anything,” Avarguès-Weber said, much less count. According to Nieder, who was not involved with the insect work, this implied that the ability to grasp the empty set and other numerosities evolved independently in the two lineages.
“A whole different neural substrate … produced such high-level cognitive capacity,” said HaDi MaBouDi, a cognitive scientist at the University of Sheffield in England. Unfortunately, researchers have so far been unable to study the neural activity of honeybees as they perform numerical tasks, making it difficult to compare their representations of zero with those of monkeys. To get answers about how and why the ability to quantify “nothing” evolved more than once, scientists realized they would have to explore the brain of another animal.
A Parallel History
And so Nieder and his team turned to crows, which haven’t had ancestors in common with primates for more than 300 million years, and which evolved to have very different brains. Birds do not have a prefrontal cortex; instead, they have their own “intelligence brain centers,” Nieder said, with a distinct structure, wiring and developmental trajectory.
Yet despite these differences, the researchers uncovered a familiar numerical understanding of zero: The crows mixed up a blank screen more often with images of a single dot than they did with images of two, three or four dots. Recordings of the crows’ brain activity during these tasks revealed that neurons in a region of their brain called the pallium represent zero as a quantity alongside other numerosities, just as is found in the primate prefrontal cortex. “From a physiological point of view, this fits in beautifully,” Nieder said. “We see exactly the same responses, the same type of code, represented in the crow brain as in the monkey brain.”
One explanation for the same neural framework evolving in such different brains is simply that it’s an efficient solution to a common computational problem. “It’s actually exciting, because it suggests that it’s just the best way,” Avarguès-Weber said. Maybe there are physical or other internal constraints on how the brain can process zero and other numerosities. “There could be … a very limited number of ways in which you can build up a mechanism to encode numbers,” Vallortigara said.
Still, just because crows and monkeys seem to be encoding an abstract concept like zero in the same way does not mean that it’s the only way. “It could be that different solutions have been invented during natural history, during biological evolution, to perform similar computations,” Vallortigara said. Researchers will have to study other animals to find out. In a paper just published in Cerebral Cortex, for instance, Vallortigara and his colleagues identified a brain region in zebra fish that seems to correlate with numerosity, although they haven’t yet tested the animals’ ability to assess zero.
Bees might also hold some surprises as the foundation for their numerosity becomes better understood. In a study published last year, MaBouDi and his colleagues “showed that the bumblebee counts by a fundamentally different strategy” when presented with up to four objects, he said. He thinks their findings hint that the mechanisms underlying honeybees’ grasp of numerosities, including zero, might indeed be quite different from what’s been observed so far.
But perhaps the more fundamental question about numerical abstraction in the brains of diverse animals isn’t how the ability works but why it exists. Why should animals have to recognize specific quantities at all? Why has evolution repeatedly made sure that animals can understand not just that four is less than five but that “four squares” is in some way conceptually the same as “four circles”?
According to Vallortigara, one reason might be because arithmetic ends up being so important. “Animals continuously have to do arithmetic. Even simple animals,” he said. “If you have an abstract representation of numerosity, this is very easy to do.” Abstracting numerical information allows the brain to perform additional computations much more efficiently.
That’s perhaps where zero fits in as well. If two predators enter an environment and only one leaves, the area remains dangerous. Rugani speculates that an animal needs not only to be able to subtract in this situation, but also to interpret zero as “the result of previously performed numerical or proto-numerical subtraction” — which the animal can then associate with particular environmental conditions. In this case, “whenever you reach the lowest value, which is zero, the environment is safe,” Rugani said. When foraging for food, zero can map onto a need to search in a different location.
Nieder, however, isn’t convinced. He doesn’t see a pressing need for animals to understand zero as a numerosity, since viewing it as an absence should usually suffice. “I don’t think that animals use numerosity zero as a quantity in their day-to-day living,” he said.
An alternative possibility is that an understanding of zero — and numerosity more broadly — might simply have emerged from the brain’s need to recognize visual objects in the environment. In 2019, when Nieder and his colleagues trained an artificial network to recognize objects in images, the ability to discriminate numbers of items arose spontaneously, seemingly as a byproduct of that more general task.
A Glimpse of Math’s Building Blocks
To Nieder, the presence of talents for numerical abstraction in animals indicates “that there is something already laid out in the brains of these animals … that may constitute an evolutionary basis for what in us humans can develop into a full-blown understanding of the number zero.”
But impressive as the animals’ accomplishments are, he emphasized that there are critical differences between how animals have been shown to conceptualize numerosity and how humans do it. We don’t just understand quantities; we link them to arbitrary numeric symbols. A set of five objects is not the same as the number 5, Nieder said, and the empty set is not the same as 0.
Even when animals can be trained to associate two items with the symbol 2 and three items with 3, “that does not mean they could put those symbols together to [get that] 2 + 3 = 5,” Dyer said. “Now, that’s a trivial mathematical problem for an elementary school student.” But experiments designed to test for that kind of symbolic reasoning in animals, he noted, have yet to be performed.
By taking this step beyond numerosity and building a symbolic system of enumeration, humans have been able to develop a more precise and discrete concept of number, manipulate quantities according to specific rules, and establish an entire science around their abstract use — what we would call mathematics.
Nieder hopes that his work on zero can help demonstrate how an abstract sense of number might emerge from a more approximate and practical one. He is currently conducting studies in humans to explore the relationship between non-symbolic numerical representations and symbolic ones more precisely.
Vallortigara, Butterworth and some of their colleagues are now collaborating with Caroline Brennan, a molecular geneticist at Queen Mary University of London, to pin down the genetic mechanisms underlying numerical ability. They have already identified genes that seem to be associated with a math learning disability in humans called dyscalculia, and are manipulating the equivalent genes in zebra fish. “I think that the genetic part of this story is, in a sense, the future of this field,” Vallortigara said. “Identifying genes for number would really be a breakthrough.”
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