## Mathematics as the Ultimate Conceptual Aid, Part One

David Knott

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 introductory observations, 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 in which we live? 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? A definitive answer to this question is clearly not forthcoming, but I will here provide an hypothesis, and subsequently compile a laundry list of arguments for thinking the hypothesis is true.

## How Mathematics Enhances Comprehension

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 theory.

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 should show 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. 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. 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 cheaper 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 necessary for a species as imperfect and new as our own to make sense of the cosmos. 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?

## Generality, Interchangeability and Modeling

One of the fundamental working assumptions of the scientific method might roughly be called the principle of uniformity. No person has ever travelled to a distant galaxy, but we have full confidence that the laws of nature behave similarly whether we are talking about our local neighborhood of the solar system or the edges of the known universe. In a similar vein, many of the building blocks that comprise known matter are essentially interchangeable: it would be silly to have an electron as a pet, because there is no permanent, core difference between the pet electron and any other electron in the universe. And as the physicist John Wheeler put it, “Black holes have no hair,” meaning that black holes have no complex, ambiguously defined characteristics (like hair style) on which one might base principles of individualism; rather, their properties are few and well defined, and two black holes which are alike in these traits (by having equal mass, for instance) are more or less indistinguishable. The assumption that physics is ultimately the same in all places and at all times has proved essential in making inferences about objects with which we cannot directly interact, and the interchangeability of particular physical entities allows us to make, from particular observations, general statements about entire classes of objects without actually seeing them. Now, if we were to construct a framework for understanding the universe, it would be necessary to somehow capture this uniformity and interchangeability. Mathematics does this in spades, and consequently has a key advantage over other conceptual models.

Suppose that I propose two mathematical models: model A, which successfully predicts the behavior of planet Alpha, and model B, which successfully predicts the behavior of planet Beta. Are the behaviors of the planets governed by a unified general principle, or are they subject to distinct and contradictory rules? If model A can, through valid mathematical manipulation, be transformed into model B, then we have shown that the two planets are governed by a single principle. To demonstrate that a general rule explains a multitude of behaviors, the equivalence of two descriptions of two different objects must be demonstrated; it must be shown that model A, particular to planet Alpha, is the same as model B, which is particular to planet Beta. The structure of mathematics allows for exactly this sort of thing to take place. It is possible to express any given function in an infinite number of ways (how many different ways, for instance, can you express the sine function?), and yet each description, no matter how different superficially, is thought to have the same referent. This immense diversity of expression increases the probability that mathematics can properly describe two different phenomena. And the equivalence of the expressions allows us to equate not only two functions, but the physical objects to which they refer. Mathematical objects inhabit a Platonic world of idealized entities. Regardless of whether I write f(x)=sin(x) on a piece of paper, or type up its Taylor expansion on a computer, or verbally utter “the imaginary part of Euler’s formula”, I am referring to a unique conceptual object. If we can identify two distinct physical objects with two equivalent expressions of a mathematical idea, then we have perfectly represented the interchangeability of the two physical objects. And, in an eery way, mathematics almost steps beyond its ordinary domain of influence and proves something about the real world.

A convincing argument for the generality of a model is crucial if one wishes to have any kind of coherence amongst physical explanations (imagine there being a different theory of motion for every single celestial object). And if there exists a non-mathematical way of proving equivalence between descriptions, I have not heard of it. The Lorentz transformations of special relativity reduce to the familiar Galilean transformations as one lets the velocity approach zero; in this way, the generality of special relativity is proved. Without mathematics, we would not have a proof, but instead a mere suggestion, and arriving at it would be a significantly longer and more taxing exercise. So we see that, in addition to being a conceptual aid necessary in overcoming erroneous and unrefined human intuition, mathematics has the crucial ability to demonstrate equivalence between descriptions of objects and behaviors, allowing for generality and interchangeability to be achieved, which are in turn required to bring nontrivial numbers of objects under our understanding.

## The Parallel Pathway

Imagine a densely populated city whose complicated streets are badly choked with traffic. If I am driving in the city, and wish to travel from point A to point B, then I will be forced down one way streets, pushed around by aggressive drivers, frequently stuck in traffic, and lost in a disorienting maze of dead ends and wrong turns. But suppose that, near to point A is point A’, where I can merge onto a straightforward, low traffic, unidirectional highway that straddles the entire city. And suppose further that, near to point B is point B’, where an off ramp allows me to reconnect to the labyrinth underneath. The highway functions as a “parallel pathway”: the work we wish to accomplish is travel between two points, and the highway does just this. But in order to utilize the shortcut, we have to anchor the pathway to the tangle of city streets. The on and off ramps serve this purpose, and provide points of conversion between the ground level and the highway. Mathematics functions in the same way with respect to intuition. When modeling physical processes, it is often easier for the intuition to come up with a differential equation than an explicit function to describe what is happening. The reader is probably familiar with tank problems, where a certain concentration of saltwater is entering a vessel at a particular rate, and water with a different concentration of salt is leaving the tank at another rate. That the change in saltwater over time is equal to the rate of salt entering the tank minus the rate of salt leaving the tank is relatively obvious. Consequently the differential equation modeling the situation makes sense intuitively. But the explicit function that describes the amount of salt present in the tank at time t, depending upon the particulars of the tank problem, is often something you would have never imagined on your own.

In tank problems, we first use our intuitions to create a differential equation. This is like setting up an anchor point, or merging onto the highway I mentioned a moment ago. After that, the mathematical machinery kicks in: we employ the known techniques of solving differential equations. This part of the process, I wish to strongly emphasize, has nothing to do with tanks, or water, or salt. Solving the equation is conceptual work, performed in an abstract space of mathematics. The navigation that is done in this space is a pathway to the desired solution, and it runs parallel to the physical reasoning one would have to perform if access to differential equations was not available. At the end of the pathway is a second anchor point, the explicit function describing the amount of salt for a given time. This anchor point allows us to get off the highway in mathematical space, and return to the actual problem involving real physical objects (the tank, the water, and the salt).

This process of switching between conceptual worlds is utterly impossible with other mental aids. If we were to use normal, non-mathematical analogical reasoning, then the analogy’s imperfections would be grossly amplified every time we got on and off the highway. Take, for instance, the analogy that physicist Brian Greene uses to describe relativistic conceptions of gravity: a star’s gravitational influence is rather like the effect of placing a bowling ball in the middle of a trampoline. This analogy is instructive enough, but trying to perform any serious work in the analogical space results in nonsense. Suppose the bowling ball is so heavy that the indentation of the rubber surface becomes effectively vertical, corresponding to the effects of a black hole. If we roll a tennis ball across the trampoline, then its path is changed by the curvature of the surface. The analogy is useful so far. But suppose we roll the tennis ball very quickly, directly towards the center of the trampoline. The ball simply flies over the hole and continues onward. This is not what would happen with a black hole. The analogy breaks down, as every analogy does when pushed too far. Mathematical models, themselves special kinds of analogies, similarly explode when given data sufficiently different from the observations that motivated them. But when this happens, we do not discard mathematics; we merely say that the formulas are incomplete, are missing a term, have incorrect coefficients, or are the limiting case of a better mathematical description.

## Fine Tuning

Another argument for mathematics being the best conceptual aid can be found in this last observation. When a normal analogy breaks down, we discard it and look for a different way of understanding the situation. On the other hand, when a mathematical model is effective in predicting events within one range of values, but fails in some other range, we modify it. We add another term, or we fiddle with the exponent, or we play with the signs of the coefficients. If we are clever enough, the resulting predictions will asymptotically approach data obtained through experiment. The power of this technique, unavailable in other analogical spaces, is in part derived from our ability to alter mathematical descriptions by arbitrarily small degrees. If a model’s prediction differs from real world observation by one tenth of a unit, then we can adjust some parameter of our equation by a small degree until the error is on the magnitude of hundredths. But at this point, further reducing the error of our predictions will require minute changes; any large adjustment is likely to widen the gap between prediction and observation. Infinitesimal adjustment is impossible and useless in other conceptual aids (how would you adjust the trampoline analogy on the order of thousandths?), but is readily permitted in mathematics. This flexibility is built into the nature of numbers themselves, as can be observed in studying the completeness of the real number system.

Varying coefficients to scale the relative strengths of different terms is one powerful way to adjust a mathematical description. Using different families of functions is another. Functions are wildly diverse in their behavior, and all have varying applicability and usefulness in describing different phenomena. There are exponential functions, logarithmic functions, integral functions, differential operators, matrix functions, hyperbolic functions, and many more besides. Each type is able to represent certain behaviors better than others. Therefore the mathematician has a highly specialized toolset, a deep pool of descriptions he can draw upon in sculpting a mathematical model, and if one type of function fails, a kaleidoscopic variety of others offers potentially viable alternatives. And the list is expanded all the time. No other conceptual aid possesses such a robust set of toys.

## Notation

Beginning mathematics students are instructed in converting verbal descriptions of problems into purely notational descriptions. The necessity of this instruction is readily observed by considering the following statement:

Find the number such that, when squared, and added to one half of itself, results in the number five.

For as many words as the preceding sentence consumes, it refers to the modest equation of

$x^2+\frac{1}{2}x=5$

Notation allows for a tremendous amount of compression and simplification. Obviously this reduces the occurrence of carpal tunnel syndrome amongst mathematicians, but something more significant is simultaneously achieved. I have at times described mathematics as being like a machine that transforms one form of data into another, and at other times likened it to a path running parallel to normal human comprehension, in which one can get to a destination more quickly than is normally permitted. And at still other times, I have referred to it as a kind of analogy that represents concepts in the actual world. What I wish to emphasize here is that the same sort of idea can be readapted, twisted around, and applied to mathematics itself. Good notation does for mathematics what mathematics does for minds. Roughly speaking, each alphanumeric character in an equation refers to a specific concept, and out of many such references comes a representational quality whose utility we have seen before. Learning introductory algebra in secondary school, students are taught to solve simple polynomial equations by doing things like:

• Multiplying by the reciprocal of a fraction
• Combining like terms
• Cross Multiplying
• FOIL

When first learning these techniques, students are typically oblivious to the underlying mathematical principles that justify their use. Instead of being understood as derivatives of more fundamental properties of algebraic objects, these methods are conceived as entities in and of themselves. The notion of canceling, for instance, might be described by students as “the thing where you cross out opposite corners of the fractions”. As much as such a description might make mathematicians cringe, it is not entirely erroneous: crossing out opposite corners when the fractions are reciprocals of one another produces correct results, and that is why the method is taught. Manipulating visual characters on a piece of paper by moving them around is more amenable to a child’s intuition than rigorously applying abstract notions of transitivity and commutativity. To the child, objects are being moved from one side of the equation to another, or flipped upside down, or being connected visually with arrows and switching places. These transformations are easy for a species as visually gifted as ours, but they would not be possible without notation.

Just as mathematics provides an analogical space within which one can do difficult conceptual work, without knowing the corresponding “ground level” details, notation provides a still higher space inside of which we can do real problem solving without worrying about the more ugly axiomatic underpinnings. And although most readily seen in the work of children, this advantage of notation continues to hold throughout more serious math. Scalars can be “pulled out” of the integral; the inverse of a function can be found by “switching” x and y; separable differential equations can be solved putting the t’s and dt’s on one side of the equation; etc. These techniques are visual, notation level operations that do not require the user to understand what he is doing.

What we see when we incorporate notation into our comprehensive picture of mathematics is a stratified system of work reducing analogical techniques. The individual takes what she knows about the physical system in question and quantifies the essential values involved, exploiting the flexibility of numbers themselves. Behaviors, patterns and influences are represented by functions whose characteristics are similar to the behaviors of the system. At this point, the system has been represented mathematically, and work can be done on it in the analogical space of mathematics. To perform that work efficiently, the functions and numbers chosen to represent the system are themselves represented by alphanumeric characters. Metamathematical operations can then be performed by utilizing the representational fidelity of the notation. After sufficient manipulation has been performed, a certain notational expression is arrived at, and the individual reverses the analogical process. The notation is decompressed into a statement about mathematics, and the statement about mathematics is in turn decompressed into a claim about the physical system.

## Alumni Liaison

Ph.D. on Applied Mathematics in Aug 2007. Involved on applications of image super-resolution to electron microscopy

Francisco Blanco-Silva