Long Time No See - or - Much Ado About Nothing

I once theorized that, since a video screen has a finite number of pixels, that it would be possible to write a computer program that would progressively update the screen one pixel at a time until every single possible image was created. Through this process, every conceivable image (within the screen dimensions) could be realized - even a rendition of all the great artist's paintings. All by chance. Or, rather, all within the progression of the imaging sequence.

I also once wrote a program to do this on a very small scale - like with a 30x30 black and white image. I let it run for a few hours, but very little ever changed. For one thing, that was because it was running on a slow (1MHz) Apple II+ computer at the time. I also didn't realize just how long it would take.

Last night, the thought crossed my mind again, and I performed a quick calculation. Based on a tiny 10x10 pixel image in black and white, running at a rate of making 100,000 updates per second, it would take nearly 402 billion trillion years to generate every conceivable black and white image that would fit in a 10x10 square. 402 billion trillion years! I did the math several times to be sure, as I simply could not believe it.

The element I was initially missing is the fact that a 10x10 pixel black and white image requires 100 bits of data to describe. This doesn't sound like much, but what that represents is 2 to the 100th power. This works out to be an extraordinary number of possible images that can be displayed there.
1,267,650,600,228,229,401,496,703,205,376 to be exact. That's 1.2 thousand trillion trillion trillion images!

That's a lot.

So, forget about a 1024x768 full-color image. Assuming 32-bit color information, that would require 25,165,824 bits to describe the image. That means there are 2 to the power of 25,165,824 possible images.

That's inconceivable.

All this math got me thinking in my sleep, apparently. I woke up this morning thinking about zero exponents and how (or why) anything to the power of 0 is considered 1. I think that is merely a definition to satisfy equations - sort of a special case to accomodate 0, which, conceptually, often represents nothing.

They say anything divided by zero is undefined. I think what division by zero really means is that the division operation, itself, is nullified, and the dividend remains unchanged. After all, how do you divide something by nothing? You don't. If there is nothing to divide, no division occurs. Right? But if you have an expression that represents x^y in a divisor, the expression would fail if y was 0 and x^0 returned 0, because then you would be attempting to divide something by nothing, and that would take forever since no division would ever occur no matter how many times you tried.

Same with multiplication. z*x^y would return zero if y was zero and x^y returned zero, because anything times zero is zero. But, since x^0 is defined to be 1 and not 0, then z*x^y would return z. This is the same as if x^y was removed from the expression entirely - it effectively nullifies the factor from the equation.

But is this really what is desired in all cases? It seems to me that multiplication conceptually represents how many instances of a particular quantity exist. For example, if you have 9 boxes that are all full, and each box can hold 4 objects, then you have 9x4, or 36 objects. But if you have no boxes, then you have 0x4, or 0 objects. No boxes, no objects.

But what if there was some quantity that occurred exponentially? Like a group of cells that only existed in powers of 2 (each dividing at the same time). And what if each box, instead of containing 4 objects, contained 2^n cells? Well, if you have 9 boxes and each box has cells that have divided 3 times, then you have 9*2^3, or 72 cells. Well, what happens if you have 9 empty boxes - perhaps the cells all died? Well, then you would have 9*2^0, or 9 cells. But wait - that's not right! There are no cells! We have 9 boxes, but they're all empty. So, then, wouldn't it be better in this case if anything raised to the power of zero was zero? Then 9*2^0 would equal zero, which is the correct number of cells.

I would have to conclude that it depends on what your math represents as to what the quantity raised to the power of zero (x^0) should be.

This leads me down another path of thought about the concept of nothing. What is nothing? Is it really nothing? I mean, it has to be something, otherwise why would it have a name? What is nothing? If you take an arbitrary region of completely empty space (no cosmic rays, no light, no energy, no mass of any kind), we might think there is nothing there. But there is something there - it's called space. And we are conceiving it, whether we think of it as "nothing" or as "empty space".

What is space made of? It seems the true notion, or idea, of "nothing" can only hold true if there is truly nothing in existence - even space - even the absense of "nothing", itself, or any concept of it. But if even "nothing" doesn't exist, then what is there? Where is "there"? "There" wouldn't even exist. But if there is no place and no space, how can that be?

My contention is that nothing is, in fact, something. So, when someone asks me what is wrong, and I say, "nothing" - you can rest assured that I mean what I say, and that this whole business about "nothing" is really bothering me.

Have a nice day.