On Sun, Feb 10, 2013 Craig Weinberg <whatsons...@gmail.com> wrote:
> all those concepts of geometry, like the trigonometric functions, can be
>> derived from one dimensional numerical sequences with no pictures or
>> diagrams involved and if told that a particle with N degrees of freedom
>> changes in a certain way and then changed again in a different way but one
>> that is still consistent with those functions a one dimensional geometer
>> could still specify what the coordinates of that particle will now have in
>> N space.
>>
>
> >> That's my point. There is never any need to have more than one
> dimension. All there need be is numerical sequences.
>

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Then why can't a one dimensional Turing machine do geometry, after all the
number of points in a one dimensional line is the same as the number of
points in a square or in a 3D box (or 4D or 5D or 6D or...). Obviously if
you insist he remain one dimensional the poor machine can't draw a 2
dimensional triangle, but ask a question about triangles and he will give
you a good answer, or ask what a tesseract will look like when that 4
dimensional cube intersects with 3 dimensional space at a particular angle
and it will figure out a accurate description.
> We have access to multiple spatial dimensions of geometry through our
> sensory-motor participation
>
There is no reason a machine couldn't do the same, and we might only have
direct sensory access to 2 spatial dimensions not 3. Recent theoretical
arguments indicate that the maximum amount of information that can be
contained inside a volume of space is proportional to the area of a sphere
enclosing that space not of its volume as you might expect. So our 3D world
could be a sort of holographic projection of a flat 2D surface.
John K Clark
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