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
With each mathematical conjecture we prove, we add another floor to an enormous skyscraper of knowledge.
