The Reasonable Ineffectiveness of Mathematics

T he nature of the relationship between mathematics and the physical world has been a source of debate since the era of the Pythagoreans. A school of thought, reflecting the ideas of Plato, is that mathematics has its own existence. Flowing from this position is the notion that mathematical forms underpin the physical universe and are out there waiting to be discovered. The opposing viewpoint is that mathematical forms are objects of our human imagination and we make them up as we go along, tailoring them to describe reality. In 1921, this view led Einstein to wonder, "How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? " [1]. In 1959, Eugene Wigner coined the phrase "the unreasonable effectiveness of mathematics " to describe this "miracle, " conceding that it was something he could not fathom [2]. The mathematician Richard W. Hamming, whose work has been profoundly influential in the areas of computer science and electronic engineering, revisited this very question in 1980 [3]. Digital Object Identifier: 101109/JPROC.20132274907 Hamming raised four interestin