David Knott (title later)
One of the most peculiar characteristics of mathematics is its seemingly limitless ability to accurately account for real world phenomena. The success of the discipline in providing a rigorous structure on which principles of physics and chemistry can be scaffolded is perhaps most strongly demonstrated by the justification frequently given for studying theoretical math: although some given piece of research may have no connection to the real world, there is a high probability that one will be found at a future time. The confidence that researchers have in the usefulness of mathematics, even before an actual use has been found, speaks volumes about the strong parallels between mathematical principles and the underlying architecture of reality. Mathematical research is a worthwhile endeavor because, among other reasons, so much of what is discovered in math corresponds to patterns found in nature, and analysis of the one often leads to understanding of the other. Even a passing familiarity with the history of science makes one feel the force of this argument, and so I will not spend time multiplying instances of mathematics playing an essential role in scientific and technological advancement, but will instead only refer the reader to the mathematics requirements of university physics and engineering programs.
Having made these observations as a sort of throat clearing exercise, I will get straight to the question on which the present essay seeks to shed light. Why might this ethereal (and in some sense arcane) discipline, rife with immensely abstract and unfamiliar definitions and arguments, cobbled together over the centuries by various eccentrics, and at first glance having no import outside the papers and computers of its practitioners, have so much to do with the world we live in? Why does mathematics play a role in the answer to virtually every serious scientific question? It is true that mathematical technique is often designed from the beginning to solve a particular problem, but there must be a basis on which our confidence in the field’s applicability rests. Why should the track record of mathematics be so good that we can safely bet on its continued relevance in new scientific problems? In short, why does mathematics work so well? Clearly a definitive answer to these questions is not forthcoming, but we can conduct a preliminary survey of the types of answers we might give. I will here discuss a few interesting possibilities, and let the reader conclude which of them (if any) makes sense.
We might tentatively define mathematics as a particular sort of logical system, often but not necessarily concerned with number and shape, that seeks to derive high level generalizations from simple axioms and definitions. That mathematics is very much like a system of logic results in a level of rigor unmatched by the natural sciences; there is rarely ambiguity of any sort, and its language is specific enough for a competent person to use mathematical principles without fear of misapplication. To tap into the strength of mathematics, we have to distill a phenomenon into its most essential elements, and at times transform a straightforward concrete event into an object of high abstraction. But having satisfied those requirements, we are given a framework within which profound moves toward understanding can be made. In this sense, mathematics is a tool that, properly wielded, bootstraps the human intellect into doing things beyond its ordinary powers. We start with a series of events throughout which some quantity is conserved, or a collection of objects and forces identified with some initial time, or a dynamic system evolving under known constraints, and we feed the data into our mathematical machinery. Before our eyes the information is transformed and mutated into a picture of the future, or if need be, the past. And the picture is something we could not have imagined, at least not with equal detail and coherence, without the mathematical aid. We could not have run the simulation on our own hardware. Mathematics allows us to offload the difficult computation into the cogwheels of previously established theorems.
When a student is doing calculus, she can use a formula to integrate functions without referencing any of the set theoretic underpinnings of real numbers, or even being concerned with knowing how the integral formula is derived. A good student who really wants to understand the material should have a strong familiarity with how integration formulas are proved, but it is not required to obtain the correct answer, and herein lies the clever move that mathematics performs. Once a certain theorem is proved, it is proved forever, and the laborious calculation required to reach the theorem can in some sense be forgotten. It is as though mathematicians use a ladder to reach a higher point, and then kick the ladder out from underneath. The effect of this phenomenon is that high level conceptual work can be done without going through the thousands of individual calculations and proofs that are implicit in even minor operations and steps. The mental lifting that is accomplished in using something like Stoke's theorem is, when viewed at the level of sets, stupendous, but the mathematician hardly lifts a finger. It would be premature to here decide the limits of human intellectual ability, but I think I am safe in saying that no person who has ever lived could reason directly from set theory to, say, research level topology, without first arduously learning the intermediate theorems. Through the mathematical paradigm of proof, then, we transcend our ordinary limitations and gain access to higher level conceptual manipulation.
The success of mathematics in accurately capturing the essence of physical phenomena is made less surprising when we consider what we have just observed: that it is effectively the greatest intellectual steroid to which we have access. But to make the point a little more forcefully, I wish to argue that this mental amplification is not only desirable but absolutely necessary to make any sense of physical reality. Transcending the set of mental tools we are born with is a crucial step in comprehending the universe for the simple reason that our minds are filled with conceptual and perceptual biases that lead to erroneous beliefs and expectations. The human brain is not, in any strict sense, a truth detection device; it is a survival device. Our minds were shaped by evolution to solve a certain set of problems faced in our ancestral environments. We successfully developed perceptual and analytical abilities for handling issues in social group dynamics, food finding and self preservation. And although the talents with which we have been endowed were, over the course of history, exapted to purposes never anticipated by natural selection, as we see in the blooming of science, philosophy and literature, there is absolutely no reason to suppose that the bag of tricks evolution has given us renders our minds sufficiently equipped to understand, say, the origin of physical law.
The argument is frequently espoused that our success thus far in understanding the cosmos is a trend that can be extrapolated indefinitely, and one might say that our mental deficiencies have already been conquered. But I would propose that the success we have had is not due to some enigmatic human resilience, but instead to a rejection of natural human intuition for the more robust conceptual framework provided by the scientific method and mathematics. The eminently quotable Richard Feynman once said, “I think I can safely say that nobody understands quantum mechanics.” Yet physicists daily use quantum theory to predict physical events to an impossible degree of accuracy, and students are introduced to the subject as undergraduates. Was the great physicist mistaken? No; he was presumably referring to the difficulty of grasping the subject in a truly intuitive way. Our Aristotelean intuition of physics urges us to frame all events in terms of billiard balls, but an honest reading of the experimental data forces us to reject such naive conceptions. The fact that we have come so far in our scientific understanding, then, does not negate my point, but instead strengthens it: mathematics is necessary to jump over erroneous human intuitions.
The reader may contend that I have exaggerated the limitations that human beings receive from their ancestors, but remember that evolution is frugal. Constructing highly specialized organs requires significant resource investment that a species may not be able to afford. It is for this reason that the human eye has a blind spot; it is more cost effective to build around the flaw than to backpedal millions of years and redesign the organ. Natural selection constantly imposes similar cost benefit analyses upon the different apparatuses and physiological systems that organisms possess, and the result is typically a hodgepodge of compromises and improvisations that serve a specific set of purposes within one environment at one time. Take a creature which has evolved in a moderate climate, stick it in the tropics, and its survival mechanisms will no longer adequately function. Take a mind that evolved to solve problems relevant to a social primate species in Africa, stick it in a physics class, and it will need scientifically derived empirical data to correct its naive intuitions, and mathematics to make its ideas rigorous, inscrutable and refined enough to test. There is zero survival value in having a brain that can truly, intuitively comprehend the oddities of quantum theory. Solid matter is almost entirely empty space, but our perception and interpretation apparatuses construe it as a continuous medium, because for all purposes related to food finding, it might as well be. Relativity theory strikes the unacquainted mind as unreal and ridiculous. The reader might think evolution would build the brain to understand these things intuitively, but it would simply be a waste of evolutionary resources to design a mind that understands Newtonian mechanics as a special case of relativistic principles. In the environment responsible for our evolutionary development, human beings never traveled at significant fractions of the speed of light or hovered near black holes. Therefore natural selection used cheap parts, and developed a mind that has no built in comprehension of relativity. This is why mathematics is necessary to understand the world around us.
Given this conclusion, we can readily understand the presence of mathematics in virtually every difficult scientific theory. A conceptual aid is clearly in order for a species as imperfect and new as our own. But this answer, although it is progress of a kind, prompts in turn an even more difficult question. Various techniques exist for enhancing the mind’s ability to comprehend difficult ideas, such as maps, graphs, analogies, computer simulations, repetitions and comparisons. There is no a priori reason for mathematics to be the one conceptual aid that overwhelms the competition to play a pivotal role in explaining the universe. What explanation, then, might we give for this peculiarity?
The great physicist John Wheeler said that black holes have no hair. He was not trying to create an innuendo, but rather was commenting on the fact that black holes can be ‘‘totally’’ described by just a handful of physical properties. They do not have any ill defined, messy characteristics like hair style or personality. As it happens, electrons are much the same way, and are in a significant sense all alike. There would be little point in keeping electrons as pets, for instance, because they have no individualistic traits by which we can differentiate them. We are far from a physical “theory of everything”, but the consensus amongst physicists seems to be that, whatever the fundamental building blocks of matter are, they will more or less be copies of each other. Mathematical objects are the same way. The function f(x)=2x is the same whether it is sitting in the middle of an equation, being analyzed in isolation, or acting on another function. It is the same whether it is drawn on paper, typed on a screen, or dreamily pondered while taking a bath. All mathematical entities are very much like this: they exist in a Platonic, idealized world, where if two things are “equal”, then they are really the same thing. If A=B, then what is true for A is true for B, and if you are talking about A, then you are simultaneously talking about B. This mirrors the interchangeability of fundamental physical objects. Our ability to make inferences with regard to distant galaxies, whose atoms we can never touch, and whose light reaches us epochs after departure, directly hinges upon this uniformity of matter. And mathematics captures that uniformity perfectly. Other conceptual aids do not.
The infinite flexibility of number, too, gives mathematics an edge over competing mental tools.
