“Butterworth provides an engaging description of the variety of tasks and tests by which a great variety of animals in their natural environments … have revealed impressive numerical abilities.”
Can fish count? Attempts to provide an answer to this question might have profound implications for our understanding of what a number is. Historically, the issue of whether nonhuman animals exhibit some understanding of numerousness (from now on, “numerosity” to conform to current neurocognitive jargon) was characterized by one false start and some early studies of largely neglected pioneers. The false start was, of course, the affaire of Clever Hans, the German wonder horse who was described as being capable of solving complicated mathematical problems, including cube roots. In 1911, however, psychologist Oskar Pfungst showed that Hans was, in fact, not using mathematics at all but rather subtle cues, provided unconsciously by his trainer, as signals to start and stop tapping with a leg . This was how the animal had been trained to provide the numerical response. While the experimental conditions devised by Pfungst to prove that Hans was in no way counting represented a triumph of the scientific method, they discredited for several years any attempt to demonstrate truly numerical capacities in nonhuman animals. This may also explain why the pioneering work of zoologist Otto Köhler on the ability of what he called “thinking unnamed numbers” in several species of birds and mammals was largely neglected  and . Köhler’s work, however, did not go unnoticed. It was actually very well known, respected for its accurate controls for possible Clever Hans effects, and cited in ethological circles. Still, it did not develop into an established research program, because no other scientists, particularly young scientists, showed interest in repeating and continuing the work.
Another example that intrigued me years ago—and that actually was a driver for my entry into the field of the comparative neurobiology of numerical cognition—is provided by the studies carried out by psychologist Géza Révész concerning the ability of chickens to grasp the serial aspects of numerical ordering. Révész’s chickens learned to peck only at every second grain in an evenly spaced line, even when spacing between the grains was manipulated [5, p. 73]. These findings anticipated the evidence for learning of ordinal (i.e., other than cardinal) aspects of number that has since been documented in a variety of creatures, from bees to zebrafish.
It is indeed quite a mystery—or perhaps a topic for the sociology of science—why some of these early contributions were neglected, in spite of being published in journals with a potentially wide readership. To give just one example, work on number cognition and other cognitive feats of canaries by Nicholas Pastore, a psychologist at Queens College in New York, was featured in Scientific American . Be that as it may, the wide variety of studies on numerical cognition in nonhuman animals is now summarized and elegantly described in the marvelous book Can Fish Count?, written by cognitive neuropsychologist Brain Butterworth, himself a pioneer in the field. In particular, Butterworth focuses on some of the turning points in research on number cognition.
He considers first the evidence for the existence of a magnitude-based estimation system, conceptualized by the accumulator mechanism proposed by psychologists Warren Meck and Russell Church in 1983. In that model, number is represented by a physical magnitude that is a function of the number of items to be counted. Imagine a pulse generator that operates at a constant rate, with a gate that allows the operation of counting to take place for a fixed amount of time. The outcome would be an analogous—rather than a symbolic—system in which the state of the accumulator (the total energy accumulated) is a direct linear function of number.
A second turning point came from the evidence provided by psychophysicists David Burr and John Ross. They showed that adaptation to visual numerosity can occur just like adaptation to color—such as the production of a complementary afterimage (after adaptation to a red square, people perceive a greenish illusory square)—or to movement, such as the illusion that a waterfall produces motion in the direction opposite to that of adaptation. Burr and Ross showed that perceived numerosity decreased following adaptation to a large number of visual items (small dots) and increased after adaptation to a small number of visual items . This has been an important result even from a philosophical standpoint, for it suggests that numerousness would be something that results from our immediate phenomenal experience, like color or motion, i.e., it is a “quale” in philosophical terms.
Of course, if something adapts, as shown by psychophysics techniques in visual behavior, there must be a population of neurons in which adaptation has been taking place. This represented another turning point directly associated with animal research, done mainly with monkeys (though evidence for homologous brain areas activated during numerical tasks abound in functional magnetic resonance imaging studies in humans). Jamie Roitman, Elizabeth Brannon, and Michael Platt discovered neurons in the lateral intraparietal cortex, a portion of the parietal lobe, whose frequency of discharge is monotonically related to the number of dots monkeys see in their visual receptive fields . This seems to represent the neurological instantiation of the idea of the accumulator mechanism hypothesized on the basis of behavioral evidence by Meck and Church .
Capitalizing on the idea of the accumulator model, Stanislas Dehaene and Jean-Pierre Changeux went on to hypothesize that the accumulator energy level is then mapped onto a mental number line in such a way that a specific neuron would respond more strongly to one specific numerosity . Of course, being an approximate analog number system, it would represent numerosities with a certain scalar variability. Thus, the neuron for, say, “four” would present a peak in discharge rate for four dots and weaker responses for neighboring smaller (three, two, …) and larger (five, six, …) numerosities. Moreover, variability in response would increase with magnitude. The neuroscientist Andreas Nieder provided evidence that in the parietal and frontal lobes of monkeys, there are indeed such “number neurons” (other neurons seem to respond to continuous quantities, such as the length of a line, and some also to both continuous quantity and numerosity).
These labeled-line number neurons appear to confirm a fascinating conjecture put forward by psychologist and neuroscientist Randy Gallistel and colleagues, namely, that it is the system of positive real numbers (and not that of the positive integers) that is the psychologically primitive system . Gallistel’s basic idea is that a system for arithmetic with real numbers evolved before language, a fact observed in nonhuman species and in infants; after language makes its appearance in our species, it picks out integers, making them the foundation of the cultural history of the number.
Butterworth provides an engaging description of the variety of tasks and tests by which a great variety of animals in their natural environments—as well as in more aseptic laboratory conditions—have revealed impressive numerical abilities. This supports the idea that possession of a sense of number, possibly by an accumulator mechanism and its neural instantiation in the neurons of number, is a common feature in the animal kingdom. Butterworth takes the reader on a tour through a multitude of animals and their brains. He starts with the impressive feats of our strict cousins—such as the chimp Ai, studied by primatologist Tetsuro Matsusawa, who is capable of associating collections of dots with Arabic numbers —and moves to numerical abilities of bees, ants, and beetles, which can count things like the number of landmarks, steps, rivals, and much more.
The evidence for quantity neurons (some selective for numerousness, others for continuous extent, and others for both) seems to support the idea that there could be a common system for representing both discrete (countable) and continuous quantity by means of mental magnitudes that would be, according to Gallistel, equivalent to real numbers. Other proposals have been made more recently, such as that rational numbers would be psychologically primitive . One open issue, however, is to specify better the role of continuous physical variables in the building of a more abstract (though nonetheless nonverbal and nonsymbolic) representation of numerousness. Recent studies suggest that areas in the brain devoted to early visual processing could already be selective for numerosities. In humans, early visual cortex responses to visual stimuli appear to increase monotonically with numerosity, regardless of size or density of the stimuli. Research with nonhuman animals also converges on a possible role of thalamic and tectal processing (subpallial regions) well before pallial (cortical) involvement [9, 12, 17].
The book also alludes to another crucial issue, namely, that possession of a number sense, implemented as an accumulator à la Meck and Church, is possibly innate in animals’ brains. Recent evidence for number neurons in the brain of very young chicks seems to be consistent with this . However, again, given that we are not dealing here with external symbols, as in exact formal arithmetic, an important issue is to specify the kind of visual (or from other senses) information that would represent the minimal condition for object individuation and thus numeracy. For instance, it has been suggested that at least in the early visual areas of the brain, selectivity follows aggregate Fourier power of the intensities of an image (at all orientations and spatial frequencies) more than numerosity, and that truly numerosity-tuned neural responses would emerge at a later processing stage in the cortex .
This wonderful book introduces the nonspecialist reader—but also the professional mathematician who wants to know about numbers and brains—to one of the most fascinating scientific enterprises. To paraphrase the famous question of neurophysiologist and cybernetician Warren McCulloch: What is a number that an animal may know it, and an animal that he may know a number?
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Vallortigara, G. Can Fish Count?. Math Intelligencer 45, 387–389 (2023). https://doi.org/10.1007/s00283-022-10221-x