Friday, March 28, 2014

... on Coastlines and Fractals being No Challenge to Thomistic Concept of Mathematics

1) ... on Honesty of Numberphile / Cambridge University, Mathematics, 2) ... on Coastlines and Fractals being No Challenge to Thomistic Concept of Mathematics

Video commented on:
Measuring Coastline - Numberphile
[They have a logo which is a π, which is not a number ...]
Speaker and Calculator:
Steve Mould

Coastline per se
Sinuosity of rivers earlier mentioned may suffer from a similar problem?

A ruler getting into millimetres would be getting hundreds and millions of times greater than the ordinary scale?

I would say rather that the lesser the scale of the ruler, true, the longer you get, but the less is proportionally added. Meaning that even the first measure is a fair approximation.

[I was wrong, see further down.]

Besides, getting millimetres can never be a true answer of a coastline, since the water gets up and down on a scale of decimetres or even a metre or two, excepting tidal regions with even greater variation.

The variation of waves needs to be evened out by an average for a fixed as opposed to fluctuating coastline to exist at all. In tidal regions this can of course mean two coastlines. Inner and outer. But even those are not fixed, since along the month tidal variations vary in intensity. There you get an even greater area of averaging out.

You might answer that the actual length of a coastline without averaging exists and fluctuates.

In that sense, man cannot measure it. Man has not the resources to pick out a moment and measure the smallest details of the coastline at that precise moment. It has a length, and a finite length though. And what it is is known - for each moment - to God.

And the second answer for a coastline length was actually including islands - which are off the coastline. Might be part of reason why it is more than double.

Actually I tested by using Koch sequence as each step meaning *4/3 and got from 3 to 39 ... I see your point.

Let us add that God knows the length at each moment as measured in each possible scale (English feet, French feet, English inches, French inches, English lines, French lines AND French points, not to mention any un-known to man of which there are infinitely more - which may give an idea how Divine Omniscience means "infinita scire"). But man who can only measure with one unit at a time does not.

On fractals, metioned as a parallel
The parallel you make with what looks like a side of a Koch snowflake (yes, I had to look Koch up) ... two points.

The total is NEVER at once three and four times the detail.

The detail is a third of total length previous to adding it and a quarter of total length simultaneously to adding it.

Any fresh addition is an addition which may be made ad infinitum, in the sense there is no limit predetermined on how much you can add, except the practical ones.

And if we look at the computer simulation of a Mandelbrot sequence ... well the smaller details do not physically exist in the pixels until you zoom in.

Saying that the length of the circumferences of a Mandelbrot sequence are infinite is treating it as it never ever exists in reality.

Hardly a solid basis for redefining certitudes about more practical realities, as some Pyrrhonists would abuse it!

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