Dknott-Philosophyofmath - Rhea

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 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? 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.

Mathematics as a Conceptual Aid

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 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. 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 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?

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’’. Almost.

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 Path

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.

We are beginning to see an explanation for the success of mathematics, and I now wish to talk about the discipline’s predictive power. If a mathematical equation is seen to account for observations in some interval of circumstances (say, in a certain temperature range), then we can be confident of its accuracy within that interval. But there is typically nothing preventing the insertion of numbers representing circumstances outside the initial interval of observations, and so a predictive model built to account for one set of data often has something to say about an entirely different set of data. A model which accurately reproduces data obtained from a low pressure experiment Once Einstein had formulated his equations of general relativity, other people began finding solutions to the equations corresponding to phenomena Once a piece of mathematics has been properly anchored to the world, experimentation can be done without ever going to the lab.

When modeling physical processes, often it is easier for the intuition to come up with a differential equation than an explicit function to describe what is happening. That the differential equation and the function derived from it refer to entities that are essentially the same is what allows the

Intuition supplies the differential equation, mathematics transforms the equation into (say) an explicit function of time, and then the mind resumes control