Numerosity is the concept of someone being able to look at a group of items and determine the rough size of each. Presented with two pictures, one with five dots and one with five hundred, for example, they should be able to comprehend the five hundred dot image as more numerous than the five dot image.
This is a quality possessed by many animals, from apes to invertebrates. It makes sense for this to be the case: if you’re a sardine, you’re going to want to be able to differentiate between a group of five other sardines, and an entire shoal. This has certain evolutionary advantages, not least because many animals group together for safety.
Mathematics is frequently conceived to be a supervenience of numerosity. That is to say, it is a widely-held belief that mathematics is a sort of higher-level version of numerosity; an extension of basic numerical skills, such as adding up and grouping together, which places it higher in the hierarchy, but nonetheless places it in the same category.
Simon Conway Morris from the University of Cambridge, however, argues differently in a paper recently released in Studies in History and Philosophy of Science, and I am inclined to agree with him.
Morris points out that “no chimpanzee knows what a square root is, let alone a complex number.” The upper echelons of mathematical knowledge – and even the more prosaic square roots – are concepts that cannot be grasped by any animal other than humans, even animals with signifcant numerical abilities.
Numerosity requires sensory processes that follow psychophysical principles. You look at a shoal of fish and you instinctively know there are more than five, not through some abstract process, but because by seeing yourself as one, and relating yourself to a group, you gain an impression of addition and subtraction. One plus one is two. Two is more than one. Twelve is more than two. Eight hundred and fifty-six is a comparatively large group. And so on.
These are the kinds of concepts that are useful to animals in an evolutionary sense, as discussed earlier.
But mathematics is more than that. Irrational numbers, perfect numbers, square roots… the systems that make up human mathematical education are examples of abstract processes. While sensory mediation may be required, ultimately the process of doing mathematics involves a predisposition for abstract reasoning. With this requirement comes a need for language; generally fairly complex language, including the idea of substitution of some values for others, as in algebra, and graphical representation such as is used in geometry.
Morris argues that a true understanding of the nature of mathematics depends on a proper explanation of language, and by implication, of consciousness. “In this light,” he continues, “concepts of purpose are not intellectual mirages but legitimate descriptions of the worlds in which we are embedded.”
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