Effectiveness of Mathematics
A reply to Eugene Wigner’s paper, "The unreasonable effectiveness of mathematics in the Natural Sciences" and Hamming’s essay "The unreasonable effectiveness of mathematics."
Jef Raskin 1998 [edit of 19 Jan 2001]
In physics we often describe phenomena in terms of mathematical relationships between quantities that represent observable attributes of the natural world: Double the tension on a spring and the amount it extends doubles; the intensity of light from a point source changes precisely in inverse proportion to the square of the distance from the source. Even quite abstract mathematical constructs -- which were created without any reference to physics or the physical world -- turn out later (sometimes much later) to be excellent representations of newly-discovered experimental data. For example, group theory proved to be eminently useful in crystallography and in understanding the organization of elementary particles.
Why should one physical parameter be a mathematical function of some others? This is the philosophical problem that Nobel physicist Eugene Wigner called "the unreasonable effectiveness of mathematics in the natural science" in his 1960 paper of that name in Communications of Pure and Applied Mathematics. Albert Einstein put the problem this way in his Sidelights on Relativity, "How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?"
While working on my book, The Humane Interface, I found it necessary to try to understand why we had certain mental abilities and not others. Plausible hypotheses arose not from cognitive psychology, as I had expected, but from the biology of evolution. The present question, the fit of "our" mathematics to the external world has generally been approached as a question of philosophy and the nature of the physical world. But considering, in retrospect, that understanding of the physical world in terms of mathematics is a human trait, and that humans are biological creatures, I should not have been surprised that, once again, biology, and indeed one of the most powerful theories of biology, evolution, should provide a measure of understanding.
Discovering the fit of mathematics to the physical world is not typically an easy task. It may require a considerable amount of effort to determine which math to use where and, as with Newton and some of his contributions to the development of the calculus, the mathematics may be invented to meet the needs of science at the moment. But this does not explain why physical phenomena can be represented mathematically in the first place. Another proposed explanation is this: Given an independent and a dependent pair of physical variables, their relationship can be either (1) random, in which case we can use the mathematics of statistics to describe it, or (2) lawful — or possibly a mixture of the two. Therefore nothing escapes the scope of mathematics. However, this argument demonstrates only that if a physical process can be represented as a pair of variables, then there will be a mathematical relationship between them. It still begs the question of why many (if not all) phenomena are lawful and why there is a mathematical treatment of randomness.
Physical chemist R. Stephen Berry of the University of Chicago, wrote, in an email, that "Bill McNeill, the historian, told me he'd intended to go into physics until he encountered thermodynamics, where, he decided, he came to understand physical science as a field in which everything worked out because you defined the variables so that they would make things come out. At the time, I did not have the wit to recognize how close he was— not quite to the point, but near it. Here's what I now believe is the resolution of the Wigner question: The marvel of science is not that mathematics describes it, but that the human creature has been able to recognize that it is possible to make constructs related to observables for which precise, quantitative relations can be stated."
Certainly it is true that we abstract from nature — by a process that is often not easily accomplished and that is certainly not well understood — observables that quantify nicely. And no doubt that is part of the reason that math and nature correspond; we search for (or create) measures that have the necessary attributes. Nonetheless, the Berry/McNeill explanation avoids the question Wigner was addressing and asks instead the question of how we proceed to find and form laws (an amazing process in itself). In particular, it does not tell us why the variables can be disentangled and the laws can be found (or created) in the first place. It is conceivable that our universe and our mathematics would have no points of contact, but, as has often been observed, even our most abstruse mathematics seems to find applications in physics or technology.
Where does mathematics itself come from? Its historical underpinnings seem to rise from counting and land measure, but from at least the days of Euclid it was recognized that another property made it unique among human studies, and that was its purely deductive nature; that is, mathematics is based on using logic to move from axioms (originally taken to be too obviously true to be questioned) to theorems — and from proven theorems to new theorems. In other words, mathematics is built on a foundation of logic. This observation eventually led to an attempt to build all of mathematics from a finite set of axioms (the Hilbert programme), an attempt that was derailed when Kurt Gödel proved that for any system of sufficient richness (those at least complex enough to generate arithmetic, for example) there were true theorems that could not be proved from any finite set of axioms (or, to be technical, axioms and axiom schemas).
But Gödel’s work, and that which followed it, did not displace logic as the basis for rational thought. Even Gödel’s proofs are reasoned solely with traditional logic, and such logic has sufficed to create the mathematics and physics which gave rise to the question this essay addresses.
Some philosophers and scientists have thought that the brain’s structure inherently includes some logical principles, just as some have concluded that a propensity for the structures of language are built in. Sir James Jeans wrote that the laws of mathematics came from mathematicians’ own inner consciousness and without drawing to any appreciable extent on their experience of the outer world.
Even if true, Jeans leaves the question of how our brains became so inclined unanswered, and so merely pushes the question down one level. A different approach is expressed by Morris Kline in his book, Mathematics: The loss of certainty. Klein noted that the answer to the source of the effectiveness of mathematics might be "that mathematical concepts and axioms are suggested by experience. Even the laws of logic have been acknowledged to be suggested by and therefore in accord with experience." But he goes on to reject the idea as being "far too simplistic. It may suffice to explain why fifty cows and fifty cows make one hundred cows... But human beings have created mathematical concepts... that are not suggested by experience." It may well be that the axioms for particular mathematical structures (e.g. geometry) are suggested by experience, but Klein’s thesis cannot explain how logic came to be embedded in our brains. In Steven Weinberg’s Dreams of a Final Theory, while he suggests that a possible explanation for the effectiveness of mathematics is that "the universe itself acts on us as a ... teaching machine", his following discussion is framed in terms of a "natural selection of ideas". However, we need something stronger than experience and learning to build the strictures of logic into our inherited brain structure.
I suggest as an explanation, instead of Weinberg’s Lamarckian evolution of ideas (which may be how science and civilization proceed but which does not affect the genome) that Darwinian evolution has selected, in its aeons of experimentation, those individual animals whose minds happen to work in accord with the way the world works. It seems natural to most of us that we can assert
p or not p
for any proposition p that has a truth value. However, if there were a creature c that, due to its neurological wiring, acted as if it could assert
p and not p
then c was probably eaten by a predator that c concluded was not there even though c saw that it was. In other words, any creature whose logic did not accord with that of the physical world is unlikely to survive. Innate logic certainly predates mankind. Of course, this does not imply that animals apply logic consciously.
It is because we have evolved so as to have brains that work the way the world does, that part of what has evolved are the logical (to us) processes of deduction. As we build mathematics we build it in conformity with the physical world because the foundations of logic, the very nature of what makes sense to us, was dictated by the physical world. The inherent abilities of our brains were established, and those abilities reinforced, by natural selection. If we have been schooled by the physical world, should we be surprised that our works reflect its teachings? From this point of view, we should be surprised only if mathematics, built on a logic derived from the way the world behaves, was not able to describe the world. We do not need to resort to Penrose’s mystical explanation, which is based on a "belief in the profound mathematical harmony of Nature" as he proclaimed in his book, The Emperor’s New Mind. In an appropriately skeptical book, The Demon-Haunted World, Carl Sagan came close to my position, saying that, "our notions, both hereditary and learned, of how Nature works were forged in the millions of years our ancestors were hunters and gatherers. I think the roots of logic are perhaps deeper than our sentient ancestors." I move from "perhaps" to "must have been".
R. W. Hamming, in his essay "The Unreasonable Effectiveness of Mathematics" (in The American Mathematical Monthly, Vol 97 No 2, Feb 1980) anticipated much of my argument in this essay by a few decades, although he concluded that because "if you pick 4,000 years for the age of science, generally, then you get an upper bound of 200 generations. Considering the effects of evolution we are looking for via selection of small chance variations, it does not seem to me that evolution can explain more than a small part of the unreasonable effectiveness of mathematics." He finds himself "forced to conclude" that "mathematics is unreasonably effective." This is too narrow a view of evolution. If Eldredge’s and Gould’s theory of punctuated equilibrium is true, we might even find that there has been no relevant change in our genomes in the full span of human history. But no matter, as the groundwork for mathematics had been laid down long before in our ancestors, probably over millions of generations. Adding to this foundation language, curiosity and possibly (as Hamming points out) the development of an aesthetic sense, may be all that is required to eventually progress from an inherent logic to mathematics.
Consider a rainbow trout, and how closely the view of its body from above approximates the form needed to minimize drag, a form that looks much like the cross section of a modern symmetrical airfoil. It has taken scientists a great deal of work, and much mathematics, to find such shapes, but the trout has achieved it by evolution. A lower-drag trout is a faster trout, a faster trout is a less-eaten trout, and only an uneaten trout can have offspring. As evolution shaped the trout’s form for survival, so did evolution build mental abilities on the logical shape of the world. It is no objection to point out that our brains go beyond logic, sometimes for better, sometimes for worse. We can do what a Turing machine (a purely algorithmic computer) cannot. Real computers, such as the one on which I draft this essay, are not subject to the limitations of Turing machines and much less so are we; as a consequence, Gödel’s work does not apply to them or to us.
That modern physics has come upon events whose explanations seem to stretch our logic is also no objection. Animals do not move at relativistic speeds, and our bodies are not equipped to deal directly with the sub-microscopic world where the rules of quantum mechanics predominate. It therefore seems possible that our inborn logic might break down or have to be extended to cover phenomena at extremes of speed, size, or other physical parameters outside of the range that could have affected the development of our cognition. It could be that we are merely lucky in that the world on which we evolved is so typical of the rest of the universe that the logic we acquired here has permitted our knowledge to range so far into the cosmos.
Perhaps there are amathematical, alogical phenomena that will be forever beyond our understanding because they are based on schemas for which we have not inherited the mental wherewithal needed to comprehend them — perhaps, but there is currently no evidence that this is the case, or perhaps it is that we could not comprehend such evidence. Hamming, whose article I had not seen when I first wrote this essay, was driven to the same thought, and put it this way, "Perhaps there are thoughts we cannot think." Mystics may find God or paranormal phenomena implied by this speculation, but I mean nothing of the sort. Psychic or religious claims can be made freely and without fear of contradiction because they cannot be physically tested. As the Tractatus Logico-Philosophicus of Ludwig Wittgenstein noted, "what can be said at all can be said clearly, and what we cannot talk about we must consign to silence". But rather than that being solely a consequence of the structure of logic or language as in the Tractatus, it is here seen as a consequence of natural selection.
But those concerns are outside of the question with which we started, a question the science of biology answers: Consideration of how we evolved to have the thought patterns we do solves Wigner’s puzzle, completes Hamming’s search, and answers Einstein’s question of why mathematics is so effective at describing the material world. Human logic was forced on us by the physical world and is therefore consistent with it. Mathematics derives from logic. This is why mathematics is consistent with the physical world.. There is no mystery here -- though we should not lose our sense of wonder and amazement at the nature of things even as we come to understand them better.
|
