# Pigeons, rats, monkeys and real numbers

I’d like today to stay on the topic of mathematics from the cognitive science perspective, and in particular, to make available another set of interesting studies summarized by C. R. Gallistel, Rochel Gelman and Sara Cordes. The studies are described in their contribution to the book Evolution and Culture (edited by Stephen C. Levinson and Pierre Jaisson and published by MIT press) and entitled: The Cultural and Evolutionary History of the Real Numbers. A pdf of this selection can be found here. These are provocative ideas that don’t seem to be getting a lot of attention yet.

Their premise:

that a system for arithmetic reasoning with real numbers evolved before language evolved. When language evolved, it picked out from the real numbers only the integers, thereby making the integers the foundation of the cultural history of the number.

Observations of the conceptual need for the real numbers, as well as their sometimes unwelcome presence, is peppered throughout the history of mathematics. But they were only formerly defined in the 19th century. The authors clarify that this real number system – a continuous, uncountable set of rational and irrational numbers

is used by modern humans to represent many distinct systems of continuous quantity–duration, length, area, volume, density, rate, intensity, and so on. Because the system of real numbers is isomorphic to a system of magnitudes, the terms real number and magnitude are used interchangeably. Thus, when we refer to “mental magnitudes” we are referring to a real number system in the brain. Like the culturally specified real number system, the real number system in the brain is used to represent both continuous quantity and numerosity.

I liked their summary of the observed weakness of the rationals.

The geometric failing of the integers and their offspring the rational numbers arises when we attempt to use proportions between integers to represent proportions between continuous quantities, as, for example when we say that one person is half again as tall as another, or one farmer has only a tenth as much land as another. These locutions show the seemingly natural expansion of the integers to the rational numbers, numbers that represent proportions. This expansion seemed so natural and unproblematic to the Pythagoreans that they believed that the natural numbers and the proportions between them (the rational numbers) were the inner essence of reality, the carriers of all the deep truths about the world. They were, therefore, greatly unsettled when they discovered that there were geometric proportions that could not be represented by a rational number, for example, the proportion between the diagonal and the side of a square. The Greeks proved that no matter how fine you made your unit of length, it would never go an integer number of times into both the side and the diagonal. Put another way, they proved that the square root of two is an irrational number, an entity that cannot be constructed from the natural numbers by a finite procedure.

And this is the heart of my interest:

Our thesis is that this cultural creation of the real numbers was a Platonic rediscovery of the underlying non-verbal system of arithmetic reasoning. The cultural history of the number concept is the history of our learning to talk coherently about a system of reasoning with real numbers that predates our ability to talk, both phylogenetically and ontogenetically.

What I find provocative about the history of mathematics is while it may look like mathematics is just the conscious organization of practical symbols, over time it is inevitably discovered that these symbols contain more than was put into them. They grow deeper, become more entwined and produce unanticipated new possibilities. This has always suggested to me that every formalized idea emerges from a well-spring of possibilities to which the mathematician keeps gaining proximity. This alone is full of implications about the nature of abstract ideas, what they accomplish, and what moves the development of human culture. Recent papers, like this one on the evolutionary history of the real numbers, consistently encourage me to keep thinking along these lines.

The way these investigators identify the presence of this primitive use of continuous mental magnitudes is interesting. Some of the first studies cited involve pigeons, rats and monkeys, where their memory of ‘duration’ is observed by exploiting one of the difficulties with continuous measurements. The difference between nearby numbers is difficult to discern, for example, numerosities are represented by voltage levels, because of the noise in voltage levels. This is contrasted with numerosities represented by digital computers. Experiments were designed to identify one of these creature’s subjective judgment of durations, by using the behavior of the animals, as the indicator of their memory of duration. The variability in these judgments (called scalar variability) increases as the remembered durations get longer. It is believed that this is because the noise in a magnitude is proportional to the size of the magnitude. Their observations are fairly precise, and even extended to allow the observation of non-verbal animals doing arithmetic with these continuous magnitudes. Other studies designed to produce non-verbal counting in humans produced the same results. These mental magnitudes were also seen mediating judgments of the numerical ordering of symbolically presented integers.

I expect this kind of evidence will continue to grow.

For now I’ll leave you with their summaries:

In summary, research with vertebrates, some of which have not shared a common ancestor with man since before the rise of the dinosaurs, implies that they represent both countable and uncountable quantity by means of mental magnitudes (real numbers). The system of arithmetic reasoning with these mental magnitudes is closed under the basic operations of arithmetic, that is, mental magnitudes may be mentally added, subtracted, multiplied, divided and ordered without restriction.

In short, we suggest that the integers are picked out by language because they are the magnitudes that represent countable quantity. Countable quantity is the only kind of quantity that can readily be represented by a system founded on discrete symbols, as language is. It is language that makes us think that God made the integers, because the learning of the integers is the beginning of linguistically mediated mathematical thinking about both countable and uncountable quantity.