Mandelbrot analogies

This is a bit of a digression on the Mandelbrot set as an example fractal universe, which lends itself to several useful analogies and thought experiments. It is a nice illustration of our strategy to work forward. The Mandelbrot set formula would be very hard to reverse engineer from the pretty pictures that it produces. Furthermore it is important to see that the computer does not create the Mandelbrot set, but merely explores it.  Finally it is a good example of a set that exhibits space and time. We will return to this a later.

Working out the formula from the data

Here is a magnification of a small area of the Mandelbrot set.


Note that there are characteristic swirls that could probably be described by laws, but they are a lot more complicated and less general than the simple formula that generated the entire set in its infinite complexity.

Even a world class mathematician of the pre-computer era (say Euler or Gauss) would have a tough job finding the defining equation (z2+c) when confronted with this picture.

This purpose of this section is to show that the backwards approach is hopeless. There are obviously a lot of regularities in the pattern, but they would require a lot of laws to describe them. A better approach would be the forward approach. Once the image has been identified as a fractal, a number of generating formulae can be tried until patterns with similar characteristics start to emerge. Without a computer to generate the images the task is hopeless.

Does the computer generate or recreate the Mandelbrot set?

Imagine that we look at some regions of the Mandelbrot set on our computer, then after some time we quit the program and switch off the computer.

“Does the Mandelbrot set still exist after the computer is switched off?”

It seems that it does, since the Mandelbrot set is a mathematical entity that exists independently of the computer. The computer only acts as a tool to visually explore it, not to generate it.

In fact it only generates an approximation appropriate for the resolution; the complete set would take an infinite time to calculate.

This then means that the Mandelbrot set exists completely independently of our universe.

Even if we imagine a universe where the physical laws, as we understand them today, were different, the Mandelbrot set would still exist, as it is a purely mathematically defined set.

Mathematics transcends physical reality

Some people say that we need objects to count something, so that mathematics is an artefact of our particular universe, but the author would disagree and argue that mathematics is universal. All possible mathematical sets exist in their own right. Our universe is one of them.

Although this is an important but philosophically contentious point and central to the theory presented here, we will not debate Platonism versus Empiricism, but assume that the set of the integers:
N={0,1,2,3,4,….} exists outside of our universe.

It may be argued that even simple and abstract maths is an artefact of our universe, and that the integers have been invented (and not discovered) for counting.

If this is the case then this theory will not make a lot of sense. Here we take it that mathematics creates our reality and not the other way round.

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