On Sunday, February 10, 2013 1:55:15 PM UTC-5, John Clark wrote:
> On Sat, Feb 9, 2013 at 2:16 PM, Craig Weinberg
> > wrote:
>> > They [mathematicians] are just elaborating existing concepts of
>> geometry, not creating it from mathematical scratch.
> But 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.
> > It doesn't matter how many dimensions you make the machine, the tape is
>> still one dimensional
> Yes but it can make calculations in N dimensional space, and a Turing
> Machine might not even know that it is one dimensional, or even that it is
> a Turing Machine.
The N dimensions are figurative though. Literal geometric dimensions are
inaccessible to mathematics unless we correlate them ourselves. We have
access to multiple spatial dimensions of geometry through our sensory-motor
participation as a body in a universe of bodies, but mathematics has no
such access. Thus comp fails by overconfidence.
> John K Clark
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