Reunion - Ex Nihilo - Burn Math Class: And Reinvent Mathematics for Yourself (2016)

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

Act I

1. Ex Nihilo

1.3. Reunion

Okay, in this chapter, we forgot everything we knew about mathematics except addition and multiplication, and started building our own mathematical universe. Occasionally, we left the privacy of that universe to compare the things we’d created to the corresponding concepts in the outside world. We learned:

1.Why textbooks write x as an abbreviation for stuff (because you can’t say “sh” in Spanish, and because humans are slow to change things that are no longer helpful).

2.Textbooks call our machines “functions.” It’s not clear why.

3.Textbooks call abbreviations “symbols,” and they call sentences “equations” or “formulas.” These terms are a bit fancier than the ideas they stand for, but we’ll occasionally use them anyway, just to get ourselves used to them.

4.The standard way of describing a machine is to use super-abbreviated sentences that look like f(x) = 5x + 3. This sentence contains three different abbreviations just on the left side of the equals sign: (i) the name of the machine is f, (ii) the name of the stuff we’re feeding it is x, and (iii) the name of the stuff that f spits out when we feed it x is f(x). So the left side is three names rolled into one. The right side of the sentence describes how the machine works.

5.By using different-looking equals symbols like ≡, , and , we can say that two things are equal, and also remind ourselves why. If this ever confuses you, just pretend I used =.

6.The obvious law of tearing things is obvious, and it keeps us from having to remember silly acronyms like “FOIL” or any of its more complicated friends. We can invent them all whenever we need to.

7.Just by using the idea that stands for “whatever number turns into 1 when we multiply it by stuff,” we can invent many of the so-called “laws of algebra” for ourselves, thus forever avoiding the need to memorize them.

8.A mathematical concept is invented like this: We start with an everyday concept that we want to make more precise or more general. There’s no single method for doing this. We usually have a few behaviors in mind that we want our mathematical definitions to have, but usually there are lots of candidate definitions to choose from. If our qualitative ideas aren’t enough to determine one and only one definition, mathematicians usually just pick the one they think is prettiest or most elegant. Students rarely hear this, so they often end up blaming themselves for not understanding the definitions better.

9.If we invent the concept of steepness like we did in Section 1.2, and if we assume that an unspecified machine has the same steepness everywhere, then we can discover the fact that this machine looks like M(x) = ax + b. This non-obvious expression is therefore a straightforward consequence of our intuitive, everyday concepts.