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Mathematics as the Science of Patterns - Mathematics as the Science of Patterns

Michael N. Fried

The characterization of mathematics as the “study of patterns” seems to have been first made by the British mathematician, G. H. Hardy.   Lamenting his waning mathematical powers, Hardy, perhaps as a curative for his despair, wrote a small book on his life as a mathematician.  Although the book was, indeed, an account of what it is to be a mathematician, it naturally could not escape also being an account of mathematics itself.  Thus, when Hardy wrote in A Mathematician’s Apology,

A mathematician, like a painter or a poet, is a maker of patterns.   If his patterns are more permanent than theirs, it is because they are made with ideas” (Hardy, 1992 , p. 84)

he gave us something like a definition of mathematics, and a beautiful one at that! 

Hardy may or may not have been the first to use the metaphor of patterns to describe the heart of mathematics, but he certainly was not the last.   In recent years the most well known and often quoted statement to this effect is that of Lynne Steen, who referred to mathematics as the ‘science of patterns’ (Steen, 1988 ).  Since then, the metaphor has become almost commonplace.  One finds it in key documents in mathematics education, such as the NCTM Principles and Standards (NCTM, 2000 ), in books such as K. Devlin’s Mathematics: The Science of Patterns (Devlin, 1994 ), and in the classroom as well.

That it has become commonplace to call mathematics a science of patterns is probably a sign that there is something right about it.   But what does it mean?  Certainly, patterns are often the explicit subject of mathematics—sometimes even in the perfectly ordinary sense of the word, as in the study of ‘tilings’ and ‘wall-paper’ symmetries.  Of course, the case may be made that the study of symmetry comprises a greater part of mathematics than might seem on first sight, but one hesitates to say that this is the reason it is right to call mathematics, in general, the science of patterns. 

Why does this word ‘pattern’ seem so apt?  No doubt because it connotes order, regularity, and lawfulness.  Moreover, as the pattern, say, for a shirt is not cloth but the plan, scheme, or idea for a shirt, the word ‘pattern’ calls up the fact that, as one writer puts it (in a book called again Mathematics as a Science of Patterns (Resnik 1999)!),  “…in mathematics the primary subject-matter is not the individual mathematical objects but rather the structures in which they are arranged” (Resnik, 1999 , p.201). 

Michael N. Fried, "Mathematics as the Science of Patterns - Mathematics as the Science of Patterns," Convergence (August 2010)