This entry is to show that fractal demonstrates and shows us that Ceteris Paribus exists in reality, and has connections directly to our understanding of reality - even possibly at the atomic scale.
Ceteris Paribus is a central assumption behind economic models and analysis - and of course, unbeknown to others, all science itself. It assumes, or sets, all other factors equal or constant; this allows us to study the pattern of the object in question. Without it, the 'cause' and the 'effect' would not be discernible - or would be confused in the 'chaos'.
I often explain it to those outside economics that this is our way of achieving a controlled laboratory experiment. Alas it is also seen as the weakness of economics, as we don't (really) live in a 'ceteris paribus' world, we live in the chaos. I would strongly argue - again - that this is the weakness of all the 'sciences'. We may have theories, but we cannot predict precise outcomes with these theores.
With the Koch Snowflake (below) - or any fractal in isolation - there are no other factors (or, other fractals) in the image. The shape is a pure (sterile) function - it is pure Ceteris Paribus.
Scale and location are impossible to discern.
I am convinced that this problem of no scale or position is the problem we call (in reality) being lost. In reality, ceteris paribus locations or situations are ones where there are no reference points, no discernible scale, and where we literally feel lost. All we have is shape (or knowledge) and as the perimeter of the Koch frontier is infinite we have ourselves a reference point or positional problem.
Crators on the Moon
Ceteris Paribus exists in reality too - in monotonic 'landscapes'. Snow-scapes, dunes, waves and the like..
One example I picked up on - and demonstrates this observation - is the craters on the surface of the moon.
Watching the moon landings (yes they did land on the moon!) it must have been next to impossible for the pilots to discern height, there are no other reference points. It must be like this flying in cloud -without the aid of instruments. They relied on their radar - their instruments.
From (Fig. 1) below, ask yourself, what is the distance between these craters?
Is is 1:1?
We can't tell, can we. Height, scale, or location cannot be discerned - with the information provided.
Interestingly, one cannot even discern if it is the moon? It could be a tray of flour - with craters?
What we can discern though is that it at least looks like a crater landscape.
In Fig. 2 we now have references: 'eye at 104.10km - it is very large.
Is is the moon - I trust NASA and Google, and the two dominant craters in Fig 1. are at the top of the image. We actually get another reference, time: 11:27 2010-12-29, and it is in the age of Google.
This is how these craters looked around that time - and have done for many millions of years.
In Fig. 3 we can see that the large crater has a name, Lambert. This is another reference. Those 'in the know' will know Lambert, and of course the other craters in Fig 1.
Fractal Ceteris Paribus is where we can know, and understand the object in question; it is where we do 'science'; it is clean of all the 'chaos' of other influences.
And this leads me to believe that there is a direct relationship between the fractal and the atom - for this is the language I hear when the quantum world is described to me. Something I will pick up on soon.