The standard classical way to think about this would be to first consider what is a tree made from. We can pulp it all up into a block of wood pulp and weigh it and see how much "stuff" there is. Then we might consider how all this "stuff" was put together. We might isolate trunk, branches and twigs and show how they are stuck together end to end in a "branching" pattern - the very concept of branching here comes from familiarity with trees (an SRH problem).
But if we shift to thinking fractally we see that each part of the tree is identical with the whole tree. Cover up the rest of the tree to show just a branch and it is a copy of the whole thing. Obviously this only works with an infinite set, where removing a finite amount from the start of the set leaves the set still infinite. That is a branch of the tree really is identical (apart from scale) with the whole thing. Interesting that infinite regress is linked here with notions of self again, just as it is with paradoxes and non-halting programs. See the image to the left. So the real building blocks of the tree are actually the tree itself. And with just a simple geometric relationship defined between building blocks we can avoid having to define a substance at all and just say that it is made from itself!
Now this blows apart the SRH... but not quite.
The SRH says that things cannot be made from themselves because there is a contradiction in the idea of "substance" being built with itself. The proof is oneday supposed to show that things are always made from what they are not.
In a fractal however there is no substance either because things are made from the whole structure not a "part" or a "thing". The only substance is a relationship between parts, but parts with no substance.
It is a new way of thinking for me - that of total relationships. I need to "fix" this - it is what I'm searching for.
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The above was written in a hurry. Some more elaboration. 21/3/2011
Visualising fractals poses a problem because a definite point needs to be added to the system, or to put it another way a start needs to be made in what is an infinitely deep system. In the tree it is customary to start with a trunk somewhere (after our experience of real trees). This is really just a branch of an infinite fractal tree that could be projected into larger scales. We only focus on the smaller scales - the contracting affine transformations in an IFS.
So a fractal that really is infinite and doesn't have a start point is just a set of all the possible applications of the transformation rules, branching just like a tree but in logical space. This is a very familiar structure for example probability trees, or wave functions, mapping all the possible outcomes. But there is something unfractal about all these because they do "collapse" into discrete actualities, the substances of rendered fractals that we can sense and interact with. That said the point of this blog is the appreciation that the real structure of what we see does not lie in the inner "substance" but in the hierarchy of functions that relate the scales of a structure. That hierarchy is of the mind and not of substance and enables a thing to be made from itself which the SRH says is impossible.
However to recover the universality of the SRH it need only be pointed out that while the tree is made from copies of itself under transformation, the simple transformation rules that govern the shape of that hierarchy are not themselves built from the tree. There are three affine transformations needed to make the whole tree. These must be constructed "outside" the space in which the tree is constructed - e.g. in a symbol space for example. Thus the SRH is not really affected by this shift in understanding.
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