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From the circle to the square and onwards

G J Boris Allan

The history of how people count numbers in different societies is not without some controversy, particularly with respect to the origin of counting. Two diametrically opposed views are:

  1. Counting developed independently in different societies, each society having its own spontaneous origin of the facility.
  2. Counting was invented in one society, and knowledge of how to count then spread to other societies by some process of cultural diffusion.

Probably the truth is somewhere in between these two extremes, in that there were multiple seed societies but in many societies knowing how to count (and why it was useful) was learned from other societies.

Looking at these two approaches to the development of counting, underlying each is a hypotheses about how we know and what we know: we have an “innate” hypothesis that predominates in the first, and a “learning” hypothesis that predominates in the second – that is:

  1. Innate hypothesis – there is something inherent in humans and their social situation that means the appearance of counting is inevitable in any society.
  2. Learning hypothesis – once counting had started (who knows how?) knowledge of counting spread by a social process in which the members of one society learned counting from members of another.

This distinction has parallels with work in language, thought, and mind.

In most societies, triangles, squares, and other many-sided shapes are well-known, and they appear in many different situations from (say) architecture to art. Though it is difficult to check now, because of the widespread diffusion of ideas, it has been argued that (at one time) there were cultures without squares – and knowledge of squares only appeared along with other knowledge obtained from (usually) the conquerors. It has even been argued that in early societies the circle came first, and the square came later (a result of ritual needs), after which societies without squares learned about squares from societies with squares (Abraham Seidenberg took this view).

If we go back to early Greek philosophers, there is an interesting distinction between philosophical idealism and philosophical realism.

Plato was a philosophical idealist in that he thought any empirical square (say, written, or engraved) was only a square to the extent that it approximated a “real” square, an ideal, an abstraction which was the Platonic form of a square. G H Hardy was a mathematical idealist when he thought mathematical objects (idealizations) had almost an objective existence – and the job of a mathematician was to investigate this ideal realm, almost like an explorer of the cosmos.

Aristotle was a philosophical realist who did not think there was a world of forms (abstractions) to investigate, rather he wanted to investigate how things are, and how we know what they are (the essence of things). This meant that he was interested in how we construct the notion of a square from our empirical experience of shapes, and what we meant by “square-ness”. Do not confuse terms (as Aristotle would tell us): a philosophical idealist is not necessarily a dreamer (say, a political idealist), and a philosophical realist is not necessarily a “pragmatist” (say, a political realist).

Applying these philosophical notions to the example of a square, Plato thought the concept of a square – the form of the ideal square – came before our observations, and Aristotle thought the observation of (exposure to) shapes – and what we thought were connections between shapes – came before our concept of a square. In this sense Noam Chomsky is in the Platonic camp, because he thinks linguistic categories and grammar come before language acquisition (they are innate), and he might even think the notion of a square is built into our mind – there as a computational primitive. George Orwell was more Aristotelean in that he thought that by exposure to language we create linguistic categories and grammars, and thus by manipulating language (changing categories and grammars) we can change what people think about situations they encounter. For philosophical idealist Chomsky, thought structures language (our innate ability to think determines our use of language), and, for philosophical realist Orwell, language structures thought (the language we use determines our thinking processes).

Suppose we have a machine, and the purpose of the machine is to recognise shapes. Forgetting about all kinds of considerations such as how it could see, we show the machine a shape, and the shape is almost a square.

A philosophical idealist would assume that a human being has an innate idea of a square, and so the problem becomes how to program the machine to recognise an almost-square – given that the machine has an internal definition of a square as part of its program. It is a problem in inference: given a empirical observation, is the shape close to any of the ideal shapes (hypotheses such as line, triangle, quadrilateral, ...) and then one of the more specific hypotheses (such as square).

A philosophical realist would not assume any innate notion of a square, but would see the first task as a different type of inference. Given a variety of empirical observations, can the machine construct any hypotheses about ideal shapes? – an area in which Turing was interested. Only when the machine has made a variety of such hypotheses can we start seeing what the machine makes of new shapes. This is a much more difficult task than taking a notion of a square and seeing if the machine thinks a shape is a square.

The more encompassing philosophical realist approach, which does not assume ideal forms, seems the only legitimate approach, unless we assume that people do have an innate concept of square, and do not have to learn what we mean by a square.

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