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MathGroup Archive 2005

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Re: Re: Types in Mathematica thread


Andrzej Kozlowski wrote:

> 
> On 8 Dec 2005, at 17:24, Andrzej Kozlowski wrote:
> 
>>>>
>>>> Exp[I Pi]==-1.
>>>
>>> Yes.
>>>
>>>> In that case would you say that you can also "derive" I form Pi and
>>>> E? What do you mean by deriving a number from another number?
>>>
>>> I meant to say that Pi can be defined in terms of E.  I am
>>> assuming the
>>> definition of complex numbers as a prerequisite.  It's something
>>> that's
>>> been in the back of my mind for quite some time.
>>
>>
>> But there is no reason at all to think that this would  help in
>> deducing that Pi is transcendental form the fact that E is. The
>> fact that neither Lindemann's nor Hilbert could do this using this
>> ancient formula of Euler, which they certainly new, would make most
>> people hesitate in claiming that it should be done once you have
>> got complex numbers "as a prerequisite".
>>
>> But if you really have an idea how to deduce that Pi is
>> transcendental from the fact that E is then perhaps you might wish
>> to prove that E+Pi is transcendental because somehow nobody has so
>> far been able to do it.
>>
>> Andrzej Kozlowski
> 
> 
> Although, I have to admit, that if you make use of the Lindemann-
> Weierstrass theorem (which was proved by Weierstrass much later than
> Lindemann's proof of the transcendentality of Pi) than indeed
> transcendentality of Pi does follow from the transcendentality of E
> and the relationship Exp[I Pi]= -1. But the Liendemann-Weierstrass
> theorem is far from easy to prove; and certainly requires more
> prerequisites than "complex numbers", see for example Alan Baker,
> "Transcendental Number Theory" (which I just have been looking at).
> 
> Andrzej Kozlowski

I offer an interesting observation about _defining_ Pi as Exp[I Pi]= -1. 
That seems to provide a means of distinguishing Euclidian geometry from
non-Euclidian 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.

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