[Extracts from The Pythagorean Perspective]
Literature, music, mathematics, art, are constituents of culture and each of them has its separate history. But each of them can also be seen as a manifestation of a human biological drive, a drive towards exploration, experimentation, the analysis of human perception.
Is mathematics an art? Can it reasonably be treated along with music, poetry and the visual arts? Can anything be said about its evolution? ... To answer these questions we perhaps need to have some clearer idea of what mathematics currently is, as a body of knowledge and as an activity. Mathematics can be defined, its content described and the way mathematicians approach their work examined. The objects with which mathematics deals can be stated, the emphasis on proof as characteristic of the mathematical process considered and then larger questions dealt with about the foundations and evolution of mathematics.
A contemporary definition is that mathematics is the science of pattern and deductive structure (replacing an older definition of mathematics as the science of quantity and space). The extent of mathematics is indicated by Davis and Hersh`s estimates that mathematics is contained in 100,000 volumes; that a skilled mathematician might know 10%, that there are 3000 categories of mathematical writing and that some 200,000 theorems are published every year. Mathematics is generally thought to place major emphasis on proof and to be concerned with a variety of mathematical objects such as circles, polyhedrons etc. Some professional mathematicians think the emphasis on strict proof is a mystification. G.H. Hardy, one of the most eminent pure mathematicians, commented: "There is strictly no such thing as mathematical proof; proofs are what Littlewood and I call gas, rhetorical flourishes, devices to stimulate the imaginations of pupils". How mathematicians see their activity is often equally surprising. Quoting G.H. Hardy again: "I have never done anything useful. The highest aspiration in mathematics is to achieve a lasting work of art". Poincaré also emphasised the aesthetic rather than the logical aspect of mathematics. Davis and Hersh (from whom the Hardy quotations are drawn 1983: 29, 85, 173). suggest that mathematics is a great art form distinguished from other humanities only by its science-like quality, its conclusions not subject to permanent disagreement like ideas of literary criticism. The meaning of mathematics is to be found in the shared understanding of human beings, not in any external nonhuman reality. The object of mathematics is to extract structure and invariance from apparent disorder. A mathematician, like a painter or a poet, is "a master of pattern" (Hardy). In their creative work, mathematicians rely, not on language or logic but on a semi-conscious stream of analogic thought, visual, or sometimes even musical (Hadamard). Views about the nature and foundations of mathematics are in process of change. Since the Greeks, mathematics has been seen as the `Queen of the Sciences', the only sure source of truth. Recent books such as those of Davis and Hersh, Lakatos,and Ormell contend that formalistic accounts of the nature of mathematics have had their day. Instead, mathematics is to be thought of as human-made, connected with the rest of knowledge, and just as much a cultural product as literature and music. With this change it starts to make sense to consider the evolution of mathematics in parallel with the evolution of other aspects of culture such as music, the visual arts and poetry.
There have long been speculations, both in philosophy and psychology, about the origin of mathematical knowledge and numerical concepts. The evolutionary view is that mathematics began with the study of real things, and was a product of the evolution of the brain, along with language and other aspects of culture. Mathematics is a form of perception which draws its unique power from the narrowness of the range of phenomena with which it is concerned, number, shape, pattern. At first sight, one might ask, with Davis and Hersh: How can man impose his mathematical will on the great cosmic processes? This is parallel to the problem considered by Lorenz how Kant`s categories could relate to the real world, to which he answered that our cognitive apparatus evolved over time precisely in such a way as to make its operations valid, effective in relation to the real world. Mathematics has evolved as a symbolic counterpart of the universe. Certain mathematical concepts and procedures have survived and others have been abandoned, survival of the fittest symbols, models, processes, constructs.
Mathematics began when the perception of three apples was freed from apples and became the integer three. There seems something specially natural about arithmetic. There are no preliminary axioms; arithmetic furnishes us with experimentally verifiable facts about the world. Quite young children can learn or discover basic arithmetic and even animals have some awareness, as experiments have shown with rats, canaries, parrots, raccoons, pigeons, and chimpanzees. Infants as young as 2.5 months old showed elementary addition abilities (1 + 1 = 2) as did rats and chimpanzees. Infants were able to recognise a group of three objects as such without counting. Wynn (1992), from whom the material in this paragraph is drawn, concludes that such abilities in a wide range of species and at a very early age in human infancy suggest that the knowledge is innate. This fits rather neatly with the long-disputed ideas of Brouwer on intuition as the foundation of mathematics. Brouwer's position was that the natural numbers are given to us by a fundamental intuition which is the starting point for all mathematics. Wynn comments moderately that it seems reasonable that our initial numerical knowledge somehow serves as a basis for the development of mathematics. For Kant, the truths of geometry and arithmetic are forced on us by the way our minds work but, following Lorenz, the way our minds work is the product of evolution of our cognitive structures to match the real structures of the environment in which our ancestors had to survive. Mathematical perception is a specialised segment of perception, which, along with other forms of perception, evolved in the service of survival. Davis and Hersh quote (1993: 319) Rene Thom: "I don't see why we should have any less confidence in this kind of perception, i.e. in mathematical intuition, than in sense perception"; both may represent an aspect of objective reality.
In summary, one can understand The Unreasonable Effectiveness of Mathematics in the Natural Sciences (title of an article by Wigner) if one accepts that there must be a structure in the universe and this structure is what is studied by every science, including mathematics, and determines the shape of mathematics. (Ormell) And one can add that mathematics as a specialised form of perception can legitimately be considered alongside music, the visual arts and poetry which are forms of representation of the contents of perception of rather more complicated, many-factored, aspects of the world and of the individual human being`s existence. Mathematics is another form, besides these other arts, which exploration of human presence in the universe can take. Like other arts, the prerequisites for mathematical exploration are to be found in the structure of the brain.
All the arts can be seen as a manifestation of one and the same impulse. ... The basic biological drive, impelling composers to compose, artists to paint, poets to write and even scientists to do their science is towards exploration of the perceived world (the outer world and the inner world) and the attempt to replicate it in some durable form. In a sense artists and musicians are engaged in exploration of the properties of the eye and exploration of the properties of the ear. ... We are perceiving creatures set in a multi-sensory world who have acquired an awareness of our perception and even perception of our awareness of our perception. We have a drive to the externalisation of our perception, of our awareness. Externalisation is transduction - transfer from one (neurological, physiological, cerebral) system to another - conversion of simultaneous patterning into time-patterning or extended experience into immediate unextended structure. The ability to transduce in this way must depend on cross-modal connectivity in the brain. On this view the production of art is a process of cross-modal transfer, for example, from neural emotional patterning to music, painting, the words of a poem.
Mathematics is no exception. It is not exceptional in the reliability of the information it provides about the world - visual and other forms of perception also produce reliable forms of information (verified by action). Mathematics is exceptional only in the narrowness of the category of information with which it deals. The mathematician's art materials are number and shape and the interrelation of number and shape. In the case of other forms of art, other forms of perception, the information is many-faceted, dependent upon the point-of-view, not susceptible to instrumental measurement. Between artistic and scientific perception there is no essential difference. As Hayek (1952: 165) suggests: "The apparatus by means of which we learn about the external world is itself the product of a kind of experience. We cannot regard the phenomenal world in any sense as `more real' than the constructions of science" - or, I would add, any less real.