Islamic Patterns and Symmetry Groups

There have been many attempts to describe what mathematics is, and in the classic book by Courant and Robbins (1941/1996) the authors intend, through examples from many areas of mathematics, to give an impression of this. They state that they do not intend to go into a detailed philosophical analysis of mathematics but rather they want to paint a picture as broad as possible by visiting many fields without going into great details in any of these fields. However, they warn against placing too much emphasis on the deductive-postulational character of mathematics. Later Hersh (1997), who in the title of his book is alluding to the book by Courant and Robbins, takes the stance that “mathematics must be understood as a human activity, a social phenomenon, part of human culture, historically evolved, and intelligible only in a social context” (p. xi). Devlin (1994) tries to define mathematics by looking at some common features characterising the activities that mathematicians are occupied with. His conclusion, which he claims to be commonly accepted, is that “mathematics is the science of patterns” (p. 3, emphasis in original). He goes on to say that the patterns that mathematicians study could be numerical patterns, patterns of shape, patterns of motion, to name some examples. G.H. Hardy, in his famous book from 1940 (Hardy, 1940/1992, p. 84), writes that the patterns of mathematics are made with ideas. He contrasts the mathematician’s patterns with the patterns of a painter which are made with shapes and colours, and those of a poet which are made with words. He recognises that also painters and poets, to some extent, work with ideas but “[a] mathematician, on the other hand, has no material to work with but ideas” (pp. 84-85). It is exactly at this place that Hardy writes what probably is the most famous statement from this book.