P 0 I N T 0 F V I E W ) The Reasonable Ineffectiveness of Mathematics By DEREK ABBOTT School of Electrical and Electronic Engineering The University of Adelaide, Adelaide, SA. 5005, Australia T he nature of the relationship between mathema tics a nd the physical world has been a sour ce of debate since the era of the Pythagoreans. A school of thought, reflecting the ideas of Pla to, is that mathematics has its own existence. Flowing from this position is the notion that mathematical forms underpin the physical universe a nd are out there waiting to be discovered. The opposing viewpoi nt 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 in fluential in the areas of computer science and electronic engineering, revisi ted this very question in 1980 [3]. Digital Object Ident ifier: 101109/JPROC.2013227490 7 Hamming raised four interesting propositions that he beli eved fell short of providing a conclusive expl a- nation [3]. Thus, like Wi gner before him, Hamming resi gned himself to the idea that mathematics is unrea- sonably effective. These four poi nts are: 1) we see what we look for; 2) we select the kind of mathematics we look for; 3) science in fact answers comparatively few problems; and 4) the evolution of man provided the model. In this article, we will question the presupposition that mathematics is as effective as claimed and thus remove the quandary of Wi gner's "miracle," leading to a non-Platonist viewpoint. 1 We will also revisit Hamming's four propositions and show how they may indeed largely explain that there is no mira cle, given a reduced level of mathematical effectiveness. The reader will be asked for a mo- ment of indulgence, where we will push these ideas to the extreme, ex- tending them to all physical law and models. Are they all truly reified? We will question their absolute reality and ask the question: Have we, in some sense, generated a partly anthropo- centric physical and mathematical framework of the world around us? Why should we care? Among sci entists and engineers, there are those that worry about such questions and there are those that prefer to "shut up and calculate." We will at- tempt to explain why there mi ght be a 'This explains the in verted title of the present article, "The reasonable ineffectiveness of mathematics." 0018 9219 © 2013 IEEE VoL 101, No. 10, October 2013 1 PROC EE D INGS OF THE IEEE 2147