naturals, integers, and rationals

There is quite a difference between math and science.  If you had time enough, and were smart enough, and probably you would need a pencil and paper, you could sit in a room and replicate all of mathematics.  But in order to replicate science you would have to go out into the world and measure stuff and do experiments.  A lot of nature is just arbitrary.  Why do subatomic particles have the mass and the charge that they do?  Nobody knows.  Before Galileo nobody knew that when you dropped a cannonball off a tower it accelerated at a regular rate that could be measured.  Once they discovered that you could describe things with an equation, things took off.  Subatomic particles can not be described as like anything in the world we know, they are just equations.

But do numbers exist, in the way that a cannonball exists?  We can imagine a universe where the subatomic particles have different masses and charges, but not a universe where two plus two sometimes equals four and sometimes equals three,  There is a whole world of mathematical philosophy with varying schools, and one school believes they exist like Plato said, and another that thinks it is just a way we do things, something we invented.  Some schools embrace infinity and others eschew it.

Well  you know you start with the natural numbers: 1, 2, 3. and so on.  You take the operations addition and multiplication and every time you add or multiply them you get another natural.  All is peaceful in the kingdom.  But then you take up subtraction, seems simple enough, just the opposite of addition, a little odd because whereas 3 + 5 equals 5 + 3, 5 - 3 does not equal 3 - 5.  And now we have what are called the integers, a whole mirror image of the naturals only negative, but they seem well behaved enough.  It's odd how when we multiply two negative numbers we get a positive number, but it sort of makes sense, the way a double negative is logically a positive statement.

But in addition to the negative numbers we now also have zero, which doesn't seem all that odd, after all we all know that sometimes we have some candy bars and sometimes we have none.

But now let's add division to our operations, the opposite of multiplication, kind of like subtraction in that 7x8 equals 8x7, but 8/7 does not equal 7/8.  And how about that 7/8?  Hardly an integer, an arrested development, we put a 8 under the 7, and that's all we can do.  When we add all those fractions we get the rational numbers.  And suddenly that affable, but mysterious, zero is giving us a big headache when we try to divide anything by it .  Some say that gives you infinity, and others say there is no such thing as infinity, just that no matter what number you think of there is always one bigger than that, and bigger than that one, and scooby dooby do on, which sounds a lot like infinity to me, but they claim it isn't, and they simply say you just can't divide anything by zero, and don't even think about it either.

There are more surprises in the kingdom of numbers, namely the reals and complex numbers.  I'll get around to them tomorrow,