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

4. On Circles and Giving Up

4.4. The Mixture Separates

As you might have guessed, the (surprisingly equal) numbers we called # and have another name. This number usually goes by the name π, and its numerical value is slightly more than 3. We don’t know its numerical value yet. Whether we call it or π or anything else, at this point it’s simply a name we invented for the unknown answer to a problem we got tired of trying to solve. Eventually, we will discover mathematical sentences where mysteriously shows up again — sentences that will let us figure out exactly which number it is. However unimportant and uninteresting its particular numerical value might be, these sentences will let us discover that is indeed roughly 3.14159.

Let’s keep our ignorance in plain sight until we’ve conquered it. In order to constantly remind ourselves of what we don’t know, we will continue to use the symbol instead of π, for now. We will only start calling it π once we figure out how to calculate this number for ourselves, to any accuracy that we desire.

Having gone through everything above, I called this example a “conceptual centrifuge” because it helps to separate out different ideas that are usually presented all at once. When we’re simply told that the area of a circle is πr², we’re being served a weird omelette of necessary truths, definitions, and historical accidents all scrambled together. Unscrambling the mixture, we get a list like this:

Necessary truth: is always the same number, no matter how big the picture is.

Definition: The symbol π is defined to be that number. That is, .

Historical accident: Calling it π rather than # or or ♣ or anything else.

Necessary truth that is not made obvious by any of this: π = 3.14159 . . .

Actually, the symbol π is usually defined in the same way that our friend Mathematics defined its give-up number in the above dialogue. However, we saw that # and turned out to be the same number. Because of this, the fact that π is usually defined in terms of lengths rather than areas is a historical fact, but it’s not a logically necessary fact. As such, we can add another item to our list:

Historical accident: The fact that π is typically defined in terms of lengths rather than areas. That is, that π is defined by the behavior (circumference) = π(diameter), rather than by the behavior .

In summary, being able to separate the historical accidents from the logically necessary truths is one of the most important skills to hone in developing a deep understanding of mathematics. Indeed, much of this book’s non-standard approach was chosen toward the goal of separating out the necessary from the accidental. Much about mathematics — its notation, its terminology, the level of formality with which it is explored, and the social conventions for communicating its content in textbooks — these things can all be changed. But even after changing every one of these things, certain fundamental truths remain. Those fundamental truths, however one chooses to express them, constitute the true essence of mathematics, and it is only by stripping away and turbulently varying everything accidental that we can finally come to see the invariant truths underneath.