To any system of Order-Principle^x the actions of the same system after another iteration of the Ordering-Principle will seem like a conspiracy.

MAthematics - its basics and their consequences

And other thoughts, of course.

Arising out of the basic building blocks of denkern: I do not need to know the reality for me to create a worklihood. Every worklihood I create from reality has consequences for which I carry the responseability.
Much like a teenager has to set himself apart from his parental  home to develop his independence, so must any system developing go through this phase. It starts with the di-vision, the setting apart and this gets refined over time. It is a naturally occurring necessity.
But as time progresses, so does the realisation that the roots were not so wrong after all, so does the convergence and eventual integration into the current view of the world. This was also so with mathematics, it is now an established discipline - first seed, then growth and development and differentiation, setting apart from the world, and then claim to be the sole saviour of the world.
Well, now is the time  where this teenager realizes that there is indeed more than himself to the world. Will this maturing process be a healthy one? We shall be the ones forming it.When mathematics realises that it is not the end- and be-all, that it is not alone in the world, then yes, it will be.

On this page I discuss, within the framework of denkern, some of my insights into the logic rule-set of mathematiics and what this pulls out of reality, i.e. its Ordering-Principle, as well as the consequences of this.
These are not always complete, as it does take effort and time to word it, yet I put them here for your perusal.

1. Mathematics is a definition

Mathematics cannot prove itself. And it cannot prove anything else. It is a tool to use to sculpt our experience of reality. A tool, what any definition is, no more, no less, albeit a very simple one and thus very robust. Much like a knife. You would not know mathematics if you did not have a knife to divide what is whole into two, where it is only this division that allows you to create/define a counting system.

2. The Ordering-Principle of mathematics is :: +1=

It is this simple form (algorithm), which you can apply as often as you like, which makes mathematics into such a powerful tool. Why? Because it is such a simple format to follow you can check, you can very every manipulation that you do in the field of mathematics by simply counting +1. Every! manipulation can be cross-checked and verified by stoically applying +1=.

3. Applying the Ordering-Principle of mathematics repeatedly generates a string (of particulates, here: numbers).

A string (of numbers,) the first definition of a linear dimension. Folding that string creates the definition of a planar dimension. Repeating the folding logic creates the definition of a cubic dimension. For more on this topic see the blog: DnA and the fourth Dimension. 

This is also the underlying reason, to me, why Quantum Mechanics  thinks the way it does, smart brains entangling themselves and then postulating that they observe that in reality.

4. Any formula or algorithm generated on this basis is simply a shortcut.

Any formula generated, like 2+2=, is an endeavour to shorten the path, to evade having to go the long way: +1 +1 +1 +1 =.
Folding the long way thus generated into a D2 or D3 or D4 are all simpler or more elegant ways to reach a position in this string.
It would be akin to inventing the aeroplane to get more quickly from London to New York, instead of having to walk step by step. Though I do sometimes question whether the one or the other actually does use mor energy than the other - inventing all the parts for the aeroplane is quite a few steps more than just walking...

5. The bound of mathematics is the = sign

Much as +1= makes mathematics a robust tool, simple to verify, easy to proof, it is, by defintion, bound by the = sign. This means that mathematics is a powerful tool within its bound, but cannot be applied outside of it (or if you do, with spurious results).

It cannot postulate or discover what has become termed dark energy or dark matter.

The = sign does trains our brain to not be able to grasp exponential growth. That is not a failing of our brain, it is the failing of the training imposed by the = sign.
This is an area where the concept of OP_n^x makes visualising exponential "growth" more amenable to the functioning of our brain: Understanding, for. example, the growth of an algae that doubles each day: On day 9 it has covered half of a lake, how much will it cover on day 10? as a further iteration of the logic rule set OP_n^x -> (developing into) OP_n^(x+1) gives a conceptual handle to more easily manipulate this type of information - that at least is my experience, it develops a more intuitive grasp.

6. We need a developmental mathematics: eMaths

A rule set for a developmental mathematics would have to be devised, agreed upon and implemented.
It could read thus: +1 -> (develops into).
Biological systems show that it can be done. 

7. Any number can be viewed as an address or information container, identified by that number

Any manipulation on the number would "move" the information container, leaving the contents unaffected.
Entering a car to travel (move (information container) human ) from (information container) Berlin to (information container) London leaves the contents of (information container) human mostly (he does eat, adding bits of information, and pees, discarding some others) unaltered.

8. Dimensions

I consider the concept of three axes at vertical angles to each other the most inhibiting in the view of dimensions. Trying to iterate a fourth of fifth vertical axis into visual perceptions ties the brain into a knot.
I suggest the following: Any formula can be viewed as an addressing algorithm, formulated to more easily access an information container in the string  (D1) created by OP_mathematics :: +1=. .
"Folding" this string (into D2) and "folding" it again (to give D3) is more like a conceptual brain hack. That we are able to see 3-D is because we "leant to see" it this way.
Therefore - to see 4-D, and 5-D and others we need to take a step back, examine the logic that brought us 3-D and discard that 90deg view-point.
Rather view 4-D and 5-D... as formulae sets that set access paths to information in a data set.
Libraries have a D4 in the Dewey-Decimal System, directing you quickly to a string of letters in a book,in a department, in a library.
See the blog post DnA and the 4th Dimension for more detail on this.

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