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.
This is a concept we are accustomed to. Few and unfortunate are the students who walk into a physics course not expecting a heavily mathematical treatment of the subject matter. And virtually no one believes that the much sought after physical "theory of everything" will be properly described without reference to math. But there is no a priori reason for this to be the case. Why does mathematics so elegantly reveal the nature of our world? This is a question that I cannot dream of answering, but it is sometimes useful to ineffectually bash one's head against a problem in the hope that something insightful pops out. I will begin the head bashing, then, by talking a little about what mathematics is.
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. The power of the discipline can be found both in its generality and its rigor: a theorem which has been proved usually says something about an infinite number of objects, and says it with unrivaled specificity and clarity. To utilize the computational 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 mental feats can be accomplished. In this sense, mathematics is a tool that, properly wielded, bootstraps the human intellect into doing things beyond its ordinary powers. Impressive as human minds are, they can only juggle so many concepts simultaneously, can only perform so many operations per second, and can only maintain focus for so long. Mathematics allows us to offload difficult computation into the machinery of previously established theorems, whose if-then statements churn out desired information from previously known, less useful values.
When a student is doing calculus, she can use a formula to integrate functions without referencing any of the technical 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 a mathematician uses a ladder to reach a higher point, and then kicks the ladder out from underneath him. 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 with extreme magnification, stupendous. 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.
Our question concerning why mathematics accurately captures the essence of physical phenomena might be informed by this argument. Adopting an engineering standpoint, we can reasonably say that human 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 generally. 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. This repurposing of our biologically inherited abilities has been an unequivocal success; some people are so impressed that they imagine even the deepest mysteries of nature being brought under the umbrella of human comprehension. I would contend that much of the understanding we supposedly have is actually offloaded conceptual acrobatics being handled by mathematical machinery.
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, but I urge caution on this front. Feynman is famously quoted as saying, “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. Unless contemporary students have more insight than Feynman did, it seems probable that a truly intuitive understanding of quantum mechanics is not essential to successfully employ the theory’s techniques. In ordinary situations, the student does not have to know how or why the theory works; they have, as I have stressed repeatedly, offloaded the difficult conceptual work into the established body of theory and mathematics. And this is why I urge caution when predicting limitless growth of genuine human comprehension: much of the understanding we have obtained is only had in a superficial sense. Mathematics, it would seem, is more adept at processing the strange truths of the natural world than human minds are.
Evolution is frugal. Constructing highly specialized organs requires significant resource investment that a species may not be able to afford. That is why our eye has a blind spot: it is more cost effective to leave the design flaw alone than to backpedal untold millions of years and redesign the organ, and even if redesign were truly worth the sacrifice, natural selection cannot look into the future to make the necessary cost benefit analysis. We are lucky in having minds that are capable of performing more than the functions for which they were originally developed, but the conservatism of evolution tells us not to expect too much out of them. From this, we can understand why people have so much difficulty understanding quantum mechanics: there is zero survival value in having a brain that can truly, intuitively comprehend the subject. It is well known that our evolutionarily bestowed intuitions for physics and probability are frequently dead wrong, and this is exactly what we should expect. 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 idea that time is relative strikes the untrained mind as ridiculous, and relativity theory as a whole seems unreal and bizarre to an unacquainted student. Again, this is in line with our expectations: in the environment responsible for our evolutionary development, human beings never traveled at significant fractions of the speed of light, or hovered near a black hole. Consequently our intuition for physics reflects a Newtonian understanding of the world, but mathematics knows no such disabling prejudice; it was not shaped by concerns of usefulness and survival. Therefore mathematics is the superior comprehension device.
