textured number line

4 05 2010

For a while now, a thought has been compiling in the back of my head regarding the number line. After reading Brian Rotman’s book, which I shall write a detailed post about soon, and some work by Chaitin, I am assured that this level of question has practical benefits, philosophically as well as computationally. Namely, the texture of the number line, which is a metaphoric mapping of the real numbers to a graphic straight line.

If you think about the integers, 1, 2, 3…, as being top level, and numbers with one decimal place as being one level deeper, etc, you get a textured line. The peaks are the integers, and the bottomless depths are eg root 2, or π.

We could think about 0.2 as being 0.20000… but let’s ignore this for now, since that would flatten the texture.

One simple observation, is to think of decimals as being numbers that tend to a limit. Some decimals tend to a constant, while others cycle (cycloids), and others still don’t appear to have a pattern, eg π.

The received way of thinking about this is computationally. Turing came up with the notion that all numbers are effectively approximate in 1936. And Rotman is talking about discrete numbers. And this ties into my idea that nothing, nothing is precise, and the illusion of precision is what the whole edifice of our society and knowledge and self and computing and science is based on. And if we shift that to everything is approximate, then there is no pretense that even our numbers are based on reality out there, but are applications to our mind’s projection. And thus, it is all self-referential. Hence, we derive a maths that is not based on founding axioms; and we take a step further than Godel and the notion of a relative and useful axiom set contextual to a given exploration. That is, newtonian physics and the rule set that derives it. We finally might actually have a system where the axiom links back to a higher derived result. That is, loops in scale, or time — as the essential (rather than fundamental) process or structure (even element if we push it…).

That is, if the integers are applications of terms to our mind’s capacity to imagine discreteness, how does maths shape up?

When walking around Arthur’s Seat with Phil, I tested out the XQ premise and tried to lead him to the insight on counting. He came up with an observation, that counting was based on there being a thing that is in process, and the ability for it to distinguish discrete parts. That is, consciousness and moments. We understand 1 is different from 2 because we make it so. Our minds provide a difference in terms of our senses. Counting is the minimal acknowledgement of duality in time.

What that means in terms of computing…. or animals… I don’t know.

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