Math for the people

Below is from an e-mail for my teenage cousin in the Philippines, who wanted help for a "why is math important" assignment in school. Caveat: anally retentive types may have bones to pick with a few details below--I know I do--but the intended audience is thoroughly layperson, and rather than be pedantic I just wanted to get some general ideas across.

1. Why do you study mathematics?

I study mathematics because I enjoy it, and I enjoy it because it's beautiful. Math is interplay between logic (what you know is true) and intuition (what you think is true, but can't yet say exactly why). It's an art form, in some ways.

2.Why mathematics is very important in our everyday life?

With good reason, many people criticize math for being irrelevant to the real world. It's true there are hardcore math fanatics who spend a lot of time creating and contemplating wild abstractions with no clear connection to anything tangible or practical.

But a lot of math is tangible and/or practical; you could say it's the skeleton of the real world:
- Logic is math, and logic helps keep us from being fooled.
- Patterns are math, and patterns show up everywhere, e.g. the pokey points on the outside of a pineapple.
- Time is the topic of calculus: the derivative of where you are going (location at a certain time) equals how fast you are going (speed). The derivative of how fast you are going (speed) equals how fast you are getting faster (acceleration).
- Space is studied in geometry: how many oranges you can fit in a crate, etc.

Math also answers practical questions such as: can I afford to take out this loan? Is it better to buy one, get one at half price, or to get 30% off everything? If the survey says 52% of people prefer the incumbent, is that a meaningful lead or just chance? Will building this bridge reduce traffic? How can we build it so it won't fall down? Etc etc etc.

But when it comes down to it, however important math is practically, I think it's important in the way that music and poetry are important, and that's why I include the section below.

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You might not believe me that math is beautiful. In an attempt to illustrate, here is one of my favorite purely mathematical ideas:

There is more than one kind of infinity.

Which of these is biggest?
The number of words you could say if you could talk forever.
The number of days there would be if time went on forever.
The number of stars in the sky if the sky went on forever.
The number of ways you can put a stone in the palm of your hand.

The first three are the same kind of infinity. But the fourth, the infinity of your hand, is a different kind of infinity, and it's bigger! The first three are called "countable" infinity, which is as big as the set of whole numbers (1, 2, 3, etc). The infinity of your hand is an example of "uncountable" infinity, the same as the infinity of the plane or the line, i.e., the set of all real numbers including infinite decimals (1.234..., pi=3.14159256... etc). The number of ways you can put a stone in your hand is biggest, because, for example, you can put the stone right in the middle of your hand. Or you can put it a little bit to the left. Or you can put it a little bit less to the left but not quite the middle. And so on, forever, like the number of exact points in a line. This is bigger than the whole numbers 1,2, 3, etc, because those sit on the line, but so does a whole lot more.

Another illustration:
The number of musical notes you can sing (do re mi, etc.) is probably finite.
If you had a magical vocal range and could always sing higher and higher, the number of notes you could sing would be "countably" infinite.
But the number of -tones- you can sing just between "do" and "re" (go from one note to the next but gradually, in the way a flute can but a piano can't) is "uncountably" infinite like the points on the real line or in your hand. Again, the notes are like 1, 2, 3, etc. But the tones, like the points on the real line, include 1, 2, 3, and also all the points -between- 1, 2, 3, etc!