Got it. Been working on this one for a while, I've seen it in glimpses over the years, but it just fell into place. Infinity is the backside of zero. Just like a quarter has a heads and a tails, this thing I've been calling "the origin" for a long time has a heads and a tails.
This idea is connected to the idea of the number ring I've contemplated many times but never was I able to see it so clearly as now. I think there might be two rings, not one.
Let's look more closely at this.
1. Normal counting
We normally think of a number line, starting at zero, incrementing one by one, off toward infinity. And then when we include the negative numbers it does the opposite, starting at zero, incrementing negative by negative, off toward "negative infinity" as if there are multiple infinities.
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
<-----------------------------|------------------------------>
2. Unite the infinities
But if you use Occam's razor to simplify things and combine the two infinities into a single infinity by making a ring of the numbers, you end up with a zero marker on a ring, opposite an infinity marker. Imagine it like this: grab the two arrowed ends above, and pull them upward to meet in the middle, making a straight line into a circle like so:
COMBINED
INFINITY
. 8 .
. * * .
-5 * * 5
NEGATIVE -4 * * 4 POSITIVE
-3 * * 3
-2 * * 2
-1 * * 1
0
ZERO
This is where I got stuck for years. I've definitely tried combining the zero and the infinity but although that solves some problems, it creates others. I've even thought the phrase "infinity is the backside of zero" before, but I never could get a visual on it, til now.
3. Now combine zero and infinity
Now I see "infinity is the backside of zero," like a quarter with a heads and a tails.
* *
* *
* *
* *
* *
* | *
ZERO
INFINITY
That line above ZERO and INFINITY is the "quarter," seen from the edge. The heads side of Zerofinity is zero, and the tails side of Zerofinity is infinity. Both of them are different ways of seeing nothing and everything.
4. Uh oh, close inspection reveals something unexpected
Thinking about this further, it appears that there are TWO rings, one positive and one negative (at this point, the ASCII art gets a little weird, but use your imagination and you can see it):
-6 -7 7 6
-5 * * -8 8 * * 5
* * * *
-4 * NEG * -9 9 * POS * 4
-3 * * -10 10 * * 3
-2 * * . 11 * * 2
-1 * * . . * * 1
ZEROFINITY ZEROFINITY
If you're puzzling over this step, give it a minute. This took awhile for me to see. It may hurt the brain to see two rings where, moments ago, we only had one, nice and neat. Why would Occam's razor develop something more complex, rather than something simple?
Two rings instead of one nice number line?
But think about it: in fact, it is more simple, because now the underlying duality is completely obvious.
5. How to envision that last step
Try to see it this way: the two rings are two opposite ways of counting one underlying structure. We're not used to separating the act of counting from the underlying thing being counted, but in fact "counting" and "the thing being counted" are two separate things.
Look carefully at just the positive side, and you'll see how it works:
Start at zerofinity's zero side, counting to the right -- positive -- and as you count, the ring grows larger, one increment at a time. From this model, zerofinity's infinity side would be conceptually always one increment after the largest number. As soon as you stop counting, you're one increment away from infinity, no matter when you stop counting.
7 6
8 * * 5
* *
9 * * 4 (counting counterclockwise)
4* *3 10 * * 3
5* *2 11 * * 2
.* *1 . * * 1
ZEROFINITY ZEROFINITY
6. Let's do that again, but this time clockwise
Now try it from the other direction: Start at zero counting to the left, and likewise, the ring grows larger, negative, as you count, with zerofinity always being whatever is just beyond the largest you've counted.
7. Now consider multiple levels of abstraction
In other thought experiments, I've previously tinkered with the idea of multiple levels of abstraction happening within basic mathematics. To me, this is a clean way of arranging a subtlety within counting that needs to be sorted out more precisely.
I've observed before that negative numbers behave like a level of abstraction above positive numbers. Note that history matches this arrangement. Negative numbers historically only came into existence after we came to understand zero ... which is also a level of abstraction above counting numbers.
- positives came first: 1, 2, 3
- then we discovered zero: 0
- then we discovered negatives: -1, -2, -3
8. Finally it begins to make sense
Combining these three abstractions into a single thought experiment allowed me to see the two rings in a properly layered perspective. In the illustration below, each layer represents an abstraction above the one beneath.
First, counting, so ancient that even animals can count to two or three. Then zero slowly evolved over centuries. Then negative numbers arrived -- as soon as we accepted the zero, we could see and start using negatives. (A similar thing happened with complex numbers as soon as we accepted the square root of negative one, but let's keep things simple for now.)
Take a look:
-9 -8 -7 -6 -5 -4 -3 -2 -1
0
1 2 3 4 5 6 7 8 9
<-----------------------------|------------------------------>
9. We should be careful in how we use these implicit layers
We think of a SINGLE number line because our numerical thoughts in mathematics are compressed into a single layer, but really we should imagine it as two (or all three), in order to keep the layers of abstraction in proper order (also important because there are other layers. Fractions are in another layer, for example, appearing historically after counting but before zero).
We should not move between layers randomly, which we do when we compress them into a single layer; as with all of math there are rigorous rules about how to move between the layers. (The way we think of imaginary numbers, for example, is breaking some of those rules.) We need to systemetize how to move between layers.
But that's a thought for another day. Just wanted to annotate this one as it came up.
10. Hmmm... is this describing a sphere?
[Update] a little later. Still thinking about this, and I just realized this is more like a sphere than I was seeing before. When it is a single ring, it's 2-D; just a circle, no sphere implied. But when toying around with two rings, it's easier to see an implicit sphere may be involved (spin a quarter on edge and you'll see the implicit sphere of rings). In this world, of many rings, negative numbers are only ONE way of mirroring positive numbers. Maybe there could be colored numbers, or audible numbers...?
Hm.