Burn Math Class: And Reinvent Mathematics for Yourself (2016)

The Infinite Beauty of the Infinite Wilderness

1. The Surprising Unity of the Void

—Richard Feynman, Nobel Prize Lecture

Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere.

—W. S. Anglin, Mathematics and History

.1.1Taking the Analogy Seriously

At the end of the previous interlude, we noticed a direct relationship between vectors and machines. In this chapter, we will take this relationship seriously and develop calculus in an infinite-dimensional space — a subject commonly known as the “calculus of variations.” As we saw in Chapter N, multivariable calculus is constructed from exactly the same simple concepts as single-variable calculus. That remains true in this case — the symbolic operations of calculus of variations basically behave like they always have — but there is a sense in which it feels very different from single- or multivariable calculus. This different feel is the result of the unification of vectors and machines that we arrived at in the last interlude. That conceptual unification will allow us to switch back and forth between two different interpretations of any given formula, and thus any formula in the calculus of variations can be thought of as saying two different things. This gives us two layers of intuition with which we can proceed.

Up to this point in the book, our approach has been rather unconventional. We have largely avoided limits in favor of infinitesimals; we have frequently invented our own terminology; we have taught the subject “backwards,” beginning with calculus and using it to invent the topics commonly assumed to be its “prerequisites”; we have sworn off the standard practice of presenting our mathematical arguments in a doctored and polished form, which — though elegant — often obscures the processes of thought that led us to discover them, and thus

Mathematics: WE HAVE ATTEMPTED TO ILLUSTRATE HOW PARTICULAR

Author: discoveries can be made only by first wandering down particular

Mathematics: BLIND ALLEYS AND ARRIVING AT PARTICULAR CONFUSIONS.

Author: Thus, to leave out the confusions is to leave

Mathematics: OUT THE UNDERSTANDING. HOWEVER,

Author: despite the many ways in which this book has been unconventional to this point, this final chapter may be the most unconventional of all. Calculus of variations, despite its beauty and the simplicity of its underlying ideas, is virtually never taught in a way that makes this simple beauty apparent. The standard method of teaching calculus of variations in mathematics courses and textbooks is so cautious and so formal (even in comparatively informal courses on applied mathematics) that the exactness of the analogy between calculus of variations and more familiar forms of calculus is nearly always unclear. The standard presentation of these ideas, though logically correct and understandably cautious, is a pedagogical nightmare, getting itself all tangled up in contraptions called “test functions,” “distributions,” “generalized functions,” “linear functionals,” “weak derivatives,” “variations,” and so on. All of these contraptions are beautiful ideas in their own right, but they’re much more complicated than one needs to grasp the underlying ideas. I will attempt to explain the calculus of variations in as simple a manner as I can, at every stage making clear the direct analogy with things we already know. As always, this similarity of the new and the old is not an accident, but a direct result of the fact that old familiar concepts provide the raw materials from which the innovators construct the new.

.1.2Fortune Favors the Bold!

At this point, we have no idea whether all the familiar operations of calculus will continue to make sense once we attempt to apply them in infinitely many dimensions. Perhaps they will not. But after all, that’s the same position we’ve been in throughout the book. The test of whether a generalization makes sense is, and has always been, whether it does what we want it to do. In most cases, “what we want it to do” is to agree with our intuitions in simple cases where our intuition can readily perceive what the answer should be. As such, we can confidently proclaim that, although we don’t yet know whether what we’re about to do makes sense, it’s well worth the risk. We choose to simply roll the dice and forge ahead. Alea iacta est.