Burn Math Class: And Reinvent Mathematics for Yourself (2016)
Act I
3. As If Summoned from the Void
Interlude 3: A Backωαrd Look to the Future
There will come a time when you believe everything is finished. That will be the beginning.
—Louis L’Amour, Lonely on the Mountain
Part I: The End
From the beginning, our stated end has been (i) to invent mathematics for ourselves, and (ii) to occasionally tie what we’ve created back to the mathematics you’ll see in other textbooks. It’s worth pausing and asking ourselves where we are. Let’s remind ourselves of everything we know and everything we don’t. We’ll use both our terms and the standard terms to give you some idea of what you know in the language of the standard textbooks.
First, we have already invented and seen the surprising simplicity of quite a few of the things from high school that may have seemed utterly mysterious at the time.
Things We’ve Invented That You Were Probably “Taught” in High School, at Which Time They May Have Seemed Utterly Mysterious
1.The concept of functions (machines)
2.The concept of area (our first example of how to invent a mathematical concept)
3.The distributive law (the obvious law of tearing things)
4.Goofy acronyms like “FOIL” that don’t deserve their own names
5.The concept of slope (our second example of how to invent a mathematical concept)
6.Lines can be described by functions of the form f(x) ≡ ax + b (this was a consequence of how we invented the concept of slope)
7.The Pythagorean theorem (formula for shortcut distances)
8.Polynomials (plus-times machines)
9.Exponentiation, nth roots, negative nth powers, arbitrary fractional powers (all of this follows from our generalization in Interlude 2)
Second, we invented calculus, and spent a bit of time exploring the new world we created.
Things We’ve Invented That You Usually Hear About in Calculus
1.The concept of local linearity (the infinite magnifying glass)
2.The concept of the derivative (defining steepness for curvy things by zooming in)
3.The concept of infinitesimals (infinitely small numbers)
4.The concept of limits (a contraption that lets us avoid infinitely small numbers, if we ever decide we don’t like that idea)
5.How to find the derivative of any polynomial
6.The sum rule, its derivation, and its generalization to n functions (the hammer for addition)
7.The product rule, its derivation, and its generalization to n functions (the hammer for multiplication)
8.The chain rule and its derivation (the hammer for reabbreviation)
9.The power rule and its derivations for positive integer powers, negative integer powers, inverse integer powers, and arbitrary rational powers (note: the power rule is the textbook name for the pattern (x#)′ = #x#−1, which we kept rediscovering in various contexts)
10.Any number can be approximated to arbitrary accuracy using rational numbers (“rational number” is the standard name for numbers that can be written as one whole number divided by another whole number)
11.Mathematical induction (ladder-style reasoning, which we used to build more general versions of our hammers)
Third, we’ve learned some things that you rarely hear in most textbooks or classrooms. Ironically, this third group of things is arguably the most important.
Things You Rarely Hear
1.Equations are just sentences.
2.All of the symbols in mathematics are just abbreviations for things that we could be saying verbally, except we’re too lazy (the good kind of lazy).
3.How to invent good abbreviations: they should remind us of the ideas we’re abbreviating.
4.How to invent a mathematical concept (we’ll do more of this later).
5.More than anyone likes to admit, mathematical definitions are influenced by aesthetic preferences, such as what we think is most elegant.
6.Mathematics has two parts. The first half is anarchic creation. The other half is trying to figure out what exactly we created.
7.How to derive the formula for time-dilation in special relativity, using some pretty simple mathematics.
8.Sometimes, a meaningless change of abbreviations can have a large effect on our ability to understand the meaning of an expression (e.g., the two ways of writing the hammer for reabbreviation, a.k.a. the chain rule).
9.In mathematics, nothing should ever be memorized. . . unless you want to.
On the other hand, there are some things we still don’t know which are generally considered to be very basic.
Some “Simple” Things We Still Don’t Know, Perhaps Because They’re Not As Simple As We’ve Been Led To Believe.
1.We still don’t know that the area of a circle is πr2.
2.We still don’t know that the distance around a circle is 2πr.
3.The symbol π is completely meaningless to us at this point.
4.The machines sin(x) and cos(x) are not part of our current vocabulary, and we certainly won’t be calling them by these silly names except to make fun of them.
5.We don’t know that logb (xy) = logb (x) + logb (y). We don’t know how to “switch bases.” We don’t know a single property of logarithms.
6.We have no idea what a “logarithm” is, but it reminds us of depression.
7.We have no reason to expect that ex has a special relationship with our infinite magnifying glass.
8.The symbol e is completely meaningless to us at this point, except as the first letter of the word “except”. . . and a few others. . .
Part II: The Beginning
We’ve already invented the mathematical concepts in the first three lists above, but what about the final list? In the next two chapters, we will find that the vast majority of that unrelated bag of facts called high school mathematics (the “prerequisites” to calculus that we have not yet discussed) can easily be invented with the tools we already have, or with simple cousins of these tools that we’ll invent by accident along the way. In every case, we’ll find that these so-called prerequisites are in fact postrequisites, requiring calculus before we can fully understand them. So! Having summarized the “advanced” ideas we’ve invented, and the “basic” ideas we do not yet know, let’s begin our backward look to the future. Ready? Here we go.
(Nothing happens. . . Almost as if this spot were already reserved. . .)