Lets take the simplest of equations. 1 + 1. Most people who have gone to school would have been taught that the answer to that equation is 2. When in fact if we look at the real world it is rarely the case, I will explain. Lets take 2 raindrops, if you have 2 raindrops that are separate, one drop and then another next to it. 1 + 1 According to the popular belief we would have 2 rain drops, but what happens when they touch? They become one. So it is possible that 1 + 1 can equal 2 and it is also very possible that 1 +1 can equal 1 as when they join up they become one raindrop. This is of course one of the simplest equations that is taught in school and I have pointed out what is wrong with it. Now if I can show you want is wrong with this equation, you must assume that I can do so with more complex. This is why I have a hard time accepting mathematics as is taught in the educational system.

Wow. You know, I was almost starting to have respect. No more: sorry.

Raindrops are not numbers; numbers are not raindrops. Number is an abstraction of the property of discreteness that we observe in the world around us. Instead of particular instances of three sheep, buildings, tables, people, pens, keyboards, flutes, erasers, blah, blah, we just say, "three" and use that idea to talk about all instances of the number three.

(Dude: if you'd read anything, you might know that there is even evidence that babies, even a few months old, can do very limited counting—they can tell the difference between one and two; later on they can distinguish one, two, and three, all without knowing what those are "mathematically".)

Mathematics begins, essentially, as a way of doing precisely this kind of abstraction. When examples like the raindrop defy the idea of discreteness, we can use concepts like, oh, I don't know, volume, to conclude that the

**volume **of the two raindrops combined is the same as the sum (see, more math) of the volumes of the two raindrops individually.

I guess, however, that you trust mathematics enough so that the money you have in your bank account doesn't spontaneously become one dollar because the numbers all rolled together to make "one".

Man, you need to go to school, or something. One raindrop plus one raindrop doesn't make one raindrop: it makes one

**bigger **raindrop. And 1 + 1 is still equal to 2. If this argument weren't so fatuous I'd have more time to make the point.