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Re: Types in Mathematica thread

On 9 Dec 2005, at 01:01, Steven T. Hatton wrote:

> I offer an interesting observation about _defining_ Pi as Exp[I Pi] 
> = -1.
> That seems to provide a means of distinguishing Euclidean geometry  
> from
> non-Euclidean geometries using purely symbolic manipulations.  That  
> is, and
> I offer no proof, the value Pi is unique to Euclidian geometry.  I  
> contrast
> this observation with Einstein's assertion in _The Meaning of  
> Relativity_
> that "?physicists have been obliged by the facts to bring down from  
> the
> Olympus of the a priori ?(the concepts of time and space)?in order to
> adjust them and put them in a serviceable condition?"
> I contend that Kant is somewhat vindicated by my observation.  This  
> is a
> deeply philosophical topic which certainly transcends the  
> objectives of
> this newsgroup.  So I will not persue it further here, other than  
> to say
> that my observation would almost certainly be welcomed by Einstein  
> as food
> for though, and not as a refutation of the remainder of his work.

I confess I do not understand much of the above, and in particular,  
the phrase: "the value of the value Pi is unique to Euclidian  
geometry". I have always believed that Pi is

4*Sum[(-1)^i/(2*i + 1), {i, 0, Infinity}]

What is Euclidean about that?

If what you have in mind is the fact that the 5th postulate of Euclid  
is equivalent to  the sum of the angles in a triangle being equal to  
two right angles  (180 degrees or Pi in radians)  then this is true  
but the profound significance of this for number theory escapes me.  
Why is not 180 unique to Euclidean geometry? Pi, being a number,  
lives just as comfortably in the Bolyai-Gauss geometry or  
Lobachevski's geometry  as in Euclidean geometry.

I do agree however that you have not refuted Einstein's work.

Andrzej Kozlowski

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