Burn Math Class: And Reinvent Mathematics for Yourself (2016)
Act II
5. Aesthetics and the Immovable Object
How, in fact, does one decide which things in mathematics are important and which are not? Ultimately, the criteria have to be aesthetic ones. There are other values in mathematics, such as depth, generality, and utility. But these are not so much ends in themselves. Their significance would seem to rest on the values of the other things to which they relate. The ultimate values seem simply to be aesthetic; that is, artistic values such as one has in music or painting or any other art form.
—Roger Penrose, quoted in S. Chandrasekhar, Selected Papers Vol. 7
But the unmoved mover, as has been said, since it remains permanently simple and unvarying and in the same state, will cause motion that is one and simple.
—Aristotle, Physics
5.1. Reaching into the Void
5.1.1The Topic Without a Home
In this chapter we will encounter more evidence that the so-called “prerequisites” to calculus require a sizable amount of calculus before we can properly understand them. The focus of this chapter is a topic without a home; a topic that when taught in the standard manner possesses more sleep-inducing power than morphine; a topic whose name strikes fear and boredom into the hearts of students everywhere. This topic goes by the name “logarithms and exponentials.”
The standard manner of explaining — or rather, failing to explain — these concepts obscures both their underlying aesthetic elegance and their importance in application. They are typically introduced in a confusing course called “Algebra 2” (a course containing various topics assumed to be prerequisites to calculus), though they manifestly do not belong there. It is as if every history class, unable to find an appropriate time to discuss the French Revolution, simply inserted said discussion in the middle of a section on the Roman Empire, without ever telling the students that these two events happened at quite different periods and have very little to do with each other. There are indeed many ways in which one could introduce these topics, many of them involving applications such as population growth. But however important these applications may be, they are not an honest representation of the underlying motivation for the ideas themselves. This chapter will introduce these ideas in an admittedly unusual way that attempts to clarify where they come from. As a consequence, we will also gain a better understanding of how they relate to other ideas and how they might be generalized to wilder and weirder contexts. Here we go.
5.1.2Starting from Things We Know
Despite everything we’ve done, it is still not too much of a stretch to say that we only know about addition and multiplication. After all, as we’ve discussed before, “powers” began their lives in our universe as a meaningless abbreviation for repeated multiplication, which made sense only for positive whole-number powers. Later, we generalized this meaningless abbreviation to a genuine concept. How did we do this? Well, you’ll remember that we simply declared that whenever we write powers that are not positive whole numbers, we will mean whatever we have to mean in order to preserve the truth of the sentence (stuff)a(stuff)b = (stuff)a+b. While this equation looks like a statement about things called “powers” or “exponents,” a more honest description would be to say that it is a statement about our own ignorance. Recall that when we were in the process of generalizing the idea of powers, we found that there were an infinite number of possible ways in which we could perform the generalization. If all we wanted was a generalization that agreed with our definition for positive whole numbers, then we could have defined (stuff)# to be # copies of (stuff) multiplied together whenever # is a positive whole number, and fifty-two (or anything else) whenever # is a fraction or a negative number. The reason we didn’t generalize the concept of powers in this way was not that it was illegal, but that it was uninteresting. Though such a generalization would not be incorrect, we would immediately find that there was nothing to know, nothing to discover, and nothing to say. It would have been a completely sterile generalization.
In a sense, it was our ignorance of anything other than addition and multiplication that led us to define the concept of powers in the way we did. It is not an accident that the two equations sasb = sa+b and (sa)b = sab are only really talking about addition and multiplication, though it is addition and multiplication “upstairs.” We made it that way because, at bottom, that’s all we know. If our generalization didn’t behave in a way that allowed us to understand it by doing things we know, then we would have no use for it.
When we invent a mathematical concept, we reach into the Void and pull something out that does what we want. Our goal might be (i) to describe the real world, or (ii) to invent a mathematical concept that corresponds to and generalizes an everyday concept, or (iii) to invent a mathematical concept that generalizes another mathematical concept with which we are more familiar. In each of these cases, we perform a similar dance. We always tailor our idea to our goal and force it to behave like the thing we are attempting to describe. We never reach into the Void blindly.