The foundations of math or "Bunches of something different than nothing"

I'm taking a quick detour to explore the foundations of math. It seems that there is something that isn't settling right, and I have to make sure I understand it before I go on.

I'll just tell you this came from 0^0 debate. Is it 0, 1, undefined, indefinite, or what?

So I started to work through what exponents are. Seems easy enough; exponentiation is the multiplication of multiplication. So:

2^3 = 2 * 2 * 2 = (2 * 2) * 2 = 8

To sum it up better, x^y says multiply x against itself y times. Like in the example, you multiplied two against two and then again against two.

OK, but then what does multiplication mean? Well, multiplication is short hand notation for adding a number to itself so many times, right?

2 * 3 = 2 + 2 + 2 = 6

so we can expand our exponent example (above) to be:

2^3 = 2 * 2 * 2 = (2 * 2) * 2 = (2 + 2) + (2 + 2) = 8

Wow, That's so neat. But what is addition?

...

Um, OK, that's a little harder to explain. It's like making something more than it was, right?

2 + 2 = 4

So 4 is 2 more that 2. But 2 is pretty abstract. What is 2? Well, 2 is 2. This is why they teach kindergartners about apples. :(

But using apples, we can say that 2 apples is 2 more than no apples. That would insinuate that 1 apple is 1 more than no apples and no apples is just no apples. So, numbers, such as 2, are the change from nothing. (i.e. there was nothing and now there is 2 is more than nothing)

Now, what about -1 apples? Well, then that would mean you have less than nothing or a hole has formed in the universe and made a trans dimensional/temporal crack and the whole thing is going to collapse in on itself!

...

Or, it means that 0 isn't always really nothing. Maybe 0 can just be the status quo, and when there isn't anything to relate the status quo to, them 0 is actually nothing. So I guess that would mean that numbers are a change from 0, which is a change from the status quo, which may be nothing.

That would define numbers to have the following format:

0+2

That says 2 more than nothing and:

0+2 -1 = 0+1

says 2 more than nothing is the status quo, and there is 1 less than that.

Seems like boring stuff, but really, this is going some place. There is a raging debate about what 0^0 means, and if I want to know what the answer is, we have to understand the very fundamentals of math.

So now we have a good understanding of numbers, and numbers them selves are addition, as we saw in the last example. Next on the list is to understand multiplication:

2*2 = (0+2) * 2 = (0+2) + (0+2) = 4

What does that mean? The parenthesis makes them look like groups, kinda like what we learned in kindergarten. 2 apples and 2 more apples is a total of 4 apples. So that's saying 1 group of 2 apples was combined with another group of 2 apples to make 4 apples. so the multiplier is really just saying how many groups of the same size/content we have. Like this:

2*2 + 3*2 = (0+2) * 2 + (0+3) * 2 = (0+2) + (0+2) + (0+3) + (0+3) = 10

Imagine if that were 2 groups of 2 apples and 2 groups of 3 sticks. Well, you would have 10 things, but you would still have 4 apples and 6 sticks, so we have to group things that are the same, and bunched the same way. This leads to a new form for numbers:

1*(0+2)

which says "There is 1 group of 2 things". Addition works on changing the contents of a group, and multiplication works on grouping groups.

So let's look at the special cases:

1*(0+0) = 1 bunch of nothing:
How can you have a bunch of nothing? Let's do an experiment. Actually go and get me 1 bunch of apples that have no apples more than nothing in them. You can't, but conceptually, you brain is capable of conjuring into existence a bunch of something that has nothing in it. Your brain is capable of storing part of a fact and waiting for the rest to be filled in, and this skill has granted us the ability to abstract nothing into something.

Example: If I tell you you have a bunch of apples, you brain might draw up the mental image of a wicker basket full of apples. Now, when I tell you your bunch of apples has no apples in it, your brain suddenly empties the basket and you can see to the bottom of the basket. Your brain still holds onto the basket as the symbol for the bunch of apples that doesn't exist.

1*(0+1) = 1 bunches of 1 more than nothing:
One bunch of a thing is easy to physically manifest. You have 1 bunches of 1 more than 0 keyboards connected to your computer. That makes sense; you have 1 thing that actually exists as a single group.

0*(0+0) = 0 bunches of nothing:
This is also pretty easy; You have no groups of nothing. It simply doesn't exist in any fashion.

0*(0+1) = 0 bunches of 1 more than nothing:
This a bit odder again. Let's play that little mental game like we did before. So, I tell you to group the apples into groups of 1 (set them by themselves), but then I tell you there are zero groups. Well, how can you group nothing together? It's like the first special case. Your brain can abstract the idea, but it would be an impossible task to actually do.

So what do we do from here? Do we let our brains play these games, or do we stick to what we can physically accomplish? We could call these odd ball cases undefined, but the math community has decided to accept them as equal to nothing, making the answer to 3 out of the 4 cases 0. But that's OK, because it seems reasonable. 0 groups of any size still means the group never existed, which means there is nothing.

Additionally, multiple groups of nothing makes some sense. If I have nothing and you have nothing, then there are 2 nothings, right? That would make 2 bunches of nothing, and nothing is nothing more than nothing.

So what about exponential and 0^0?

2^3 = 2*(2*(0+2)) = 1*(1*(0+2) + 1*(0+2)) + 1*(1*(0+2) + 1*(0+2)) = 1*(1*(0+2) + 1*(0+2) + 1*(0+2) + 1*(0+2)) = 8

(I have to admit. I'm getting tired, so the writing may not be as good).

This is saying to do a lot of adding. Take a bunch of 2 more than nothing and double it, and then double it again. As the equation was expanded, parenthesis were added to show grouping.

Now imagine 1^3, That says you have 1 group of 1 bunch of 1 more than 0. How about 0^3? That says you have 0 groups of 0 groups of nothing more than nothing.

So here comes the million dollar question: What about 0^0? That would say, following the convention that multiplication is compounded addition and exponents are compounded multiplications, I don't know what to say here.

As far as this would suggest, 0^0 is simply 0, but we will keep exploring...