I mentioned Jenson’s Paradox. it really is no paradox at all, when you think of it in terms of real numbers.

just to clarify, I am talking of connecting 2 points with a zig-zag path.
this path has infinitely many rectangular angles and a total length of 2.
that path looks like a diagonal line though.
also mathematically it can’t be distinguished from the diagonal line.

for every point below the diagonal, at some moment in the construction the zig-zag line will be above that point.
for example the point C, or any point between C and D.
already in the 2nd step they all are way below the constructed zig-zag shape.

however, the concept of infinity makes it clear that this isn’t a diagonal line.
the problem is that there are uncountably many points not being put onto the diagonal line.

in my construction the corners get mirrored onto the diagonal between A and B
this process is repeated only countably infinite many times
this leaves uncountably many parts that aren’t part of the diagonal.
what’s worse, you don’t even need to believe in uncountable quantities.
is \sqrt 3 an x-coordinate for a point of the zig-zag line lying on the straight line connecting  A and B?
mathematically it is. it converges there. but at which step does that point get added?
unfortunately mathematics doesn’t come up with tools for talking about such cases. the notion of “derivative” comes close though…

therefore it isn’t surprising that the length is 2, as the constructed line is below the diagonal most of the time. wouldn’t even be surprising if the length of a similar line was infinite!

and this is what fractals are about: a discrepancy between size and actual objects of that dimension.
thanks to a point of view that takes infinities into account we can investigate the fractal’s shape.
of course our actual world doesn’t contain true fractals. and the uncertainty-principles prevent that in small sizes too.
however, abstract processes can be of fractal nature, our thoughts are such a beast of infinite repetition.

of course we don’t live forever, and so also infinite thinking makes no sense.
eventually there’s a  level where thinking would become coarse-grained.
but that’s beside the point! to estimate where your finite thinking will go, you need an approximation in terms of infinities.

additionally there’s the theory that part of our thinking is happening in the way of a quantum-mechanical computer.
if that’s true, then infinities would play an important role in studying how we think.
then our thoughts would be sort of fractals created by simple rules.
and just like fractals, our thoughts most likely will never encompass everything. there always will be holes, just like in above fractal…

as a side-note, is above construction really creating a fractal?
when you scale it up evenly, the length will be doubled, just like the straight line.
neither we have self-similarities nor “rough shape”, it’s merely lacking derivatives.
however, stretch it in the x-axis, and it will be size 3. a line stretched that way would be size \sqrt 5

 

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