## Tuesday, July 15, 2014

### Numbers

I suppose humans don't really understand numbers.

You don't even have to get to crazy numbers like quaternions or complex numbers (which are understandably . . . complex).

Humans don't understand real numbers.

Hell, humans don't understand the natural numbers, the so-called counting numbers.

I suppose we can blame education systems for a large portion of it. I mean, how many people think of fractions as numbers?

If you ask a kid what a fraction is (or an adult probably), chances are likely they're going to say something along the lines of "it's a ratio" or "it's a division problem".

But well, you know, it's a bloody number. It's right there on the number line with all the rest of the numbers.  If I ask you to describe pi, amongst the top three most likely somewhat valid answers would be "a(n) (irrational) number", the other two being "3.14" and "circumference divided by diameter of a circle" and the mostly invalid but tasty answer of "food".

But one million is equally difficult to think about. One thousand thousand. For every one in a thousand, there's a thousand more.  Then there's a billion, trillion, quadrillion, pentillion and more. There's a googol. There's a googolplex.

And even though a googolplex is a number larger than the number of atoms in the known universe, there's still reason to use outrageously large numbers. If we had computers that could do that many operations in a second teleportation would be a snap.

I've read that if you took the known universe and shrunk it down to the size of an atom, the length of a small tree is THE length, as in, the elementariest of lengths, the smallest length, smaller than which is impossible with certainty, the length that if you try to break things up smaller you create virtual black holes, the length of strings.

Can we conceptualize? Well, we can anecdotalize I suppose and theorize and develop proofs and maths and more, but it takes processing.  It's not natural even if it's with natural numbers. The moon is 250 thousand miles away. One-quarter of a million. That's all. It's likely you'll eventually drive 250,000 miles in your lifetime, well, if you drove up that far, you could have driven to the moon.

Similarly, if I write 1/2 people cringe in fear even though if I ask them how much gas they have in their car they may response with the same number. Hell, if I ask them the chances of a fair shell game, they might respond with the dreaded 1/3.

People don't understand natural numbers or rational numbers, how could they understand real numbers which contain the irrational?

People still cling to the belief that numbers mean solidness. 3 is a number of apples. 1/2 is a portion of a pizza. Pi is geometrical, if not tangible.

What's the smallest positive real number?

On the one hand, there is no smallest positive real number.
Proof: Let x > 0 be the smallest positive real number.  Then x/2 > 0 and x/2 < x, which is absurd by our claim.  Therefore, our claim is absurd.  There is no smallest positive real number.

On the other hand, we assume that in some heretofore unproven type of  ≤ sort of way, there is a smallest positive real number.  There has to be by the well-ordering principle, which is equivalent to the axiom of choice, which is foundational to ZFC.  We just don't know the ≤ that makes a least (or a most).

For those that don't realize it, ≤ can be thought of as a type of function.  It takes two numbers and orders them.  But we can order numbers differently.  For instance, we could define ≤ to tell us the number of digits of a number.

If that was the case, then 2 = 5 since both 2 and 5 are 1, and a googol is 101. A million is 7. But 0.3 is 2? and 0.00000000001 is 12.  We could apply this to sets of numbers maybe more easily than digits, but then it would just be cardinality and that would be boring.

There is some way to order the real numbers, unless there isn't. But if we allow that we can pick any number, then we should be able to pick the smallest, yes? The next largest past nothing? The least is bigger than nothing? Right?

Err, well, if you've abandoned mathematics and gone back to primitive ways of observing, I suppose we could say that the least is bigger than nothing, and we can even show this to be true since the least must be contained within 0 < x < 1 and every x in that set are larger than 0, the nothing. But there is no 0.000...01 with an infinite number of zeros. There is no end to infinity.

Numbers, grounded in the most concrete of concrete, especially if you're counting statues, end up being insubstantial. Infinite in breadth and infinite in their skinniness.

. . . why don't more people like math?