Quantity And Infinity

I've been thinking about possibilities. Given that information can only exist when relative, what would be the simplest way to store information?

First I concluded that energy and non-energy, and so energy and one dimension so that it could be "spread out" into somethingness and nothingness. It would be a line of black and white bits, in any random order like a row of infinite heads and tails on coins.

Energy is a term loaded with conceptions though. I immediately thought of light, electromagnetic energy, and in this place that's a wave and so would need extra dimensions to wobble in, and it moves, sort of, so might need extra dimensions like time. Thus, the term energy will be ignored.

Instead we'll turn to the heads and tails. If we had an infinite number of heads and tails, coins flipped at random then every possible combination of heads and tails could exist. One million heads in a row and one tail would exist; or a billion or a trillion heads in a row and then one tail. That's the wonder of infinity. But there is one possibility that couldn't exist, and that's all heads (or all tails). That's because, as previously stated, information can only exist relatively; the heads only exist relative to tails and neither is anything without the other.

This makes sense, but has implications for other infinities. What of numbers, for example? Numbers continue infinitely, up forever. With nothing to be relative to they cannot exist. So what do numbers exist relative to? An obvious answer must be zero. Numbers exist relative to zero, and so zero cannot be a number.

But is that true? If numbers measure quantity then surely they could exist relative to one as much as zero because one isn't a quantity, but two or more are? Let's examine this. What if there were just one thing in the universe, permeating everywhere? Well, a space around the one thing is essential to define it as one thing. Without that space, one thing (or any quantity of things) and nothing would be identical, so in this case zero, one or more are the same and can be discounted. Even with space around one object, one or two or an infinite number of identical things would be indistinguishable, and thus would not exist as a quantity.

It appears that with quantity though dimensionality is essential. Consider one circle that gradually becomes pinched, deformed into a figure eight shape and then pop, splits into two objects. One has become two. The split was a property of the topology and the malleability of the squidgy circle.

We need at least one dimension to establish separateness. Consider the data sequence "xox". "x" exists twice. That is a finite universe however, of just three things. Consider an infinite line of "x" and "o" is any and every combination. There would be an infinite number of each, so quantity couldn't be said to exist, as a whole. Perhaps the number of x's between two bracketing o's would count? That would still make space essential for establishing the concept of quantity.

Could there be a dimensionless example of quantity? What if two liquids were mixed in a flask, red and green. There might be more red than green, so establishing quantity. That could be visualised in many ways; in three dimensions in the flask, or in two as a line, but no matter how abstract, to measure a quantity a scale is essential because in effect the word quantity demands a result on a scale, and the simplest scale is one dimension.

Mark Sheeky, 11 January 2013