Re: Re: Types in Mathematica thread

*To*: mathgroup at smc.vnet.net*Subject*: [mg62947] Re: Re: Types in Mathematica thread*From*: "Steven T. Hatton" <hattons at globalsymmetry.com>*Date*: Fri, 9 Dec 2005 05:10:28 -0500 (EST)*References*: <dmp9na$hi2$1@smc.vnet.net> <roadnYOk3NcFDw7eRVn-jg@speakeasy.net> <200512050837.DAA08425@smc.vnet.net> <200512051840.NAA21063@smc.vnet.net> <200512060503.AAA02736@smc.vnet.net> <dn3jsl$8s0$1@smc.vnet.net> <5_ydnSmM8KqB-gjenZ2dnUVZ_v6dnZ2d@speakeasy.net> <dn5npi$nef$1@smc.vnet.net> <200512080504.AAA11638@smc.vnet.net> <7A0DFBF7-1606-40E2-995E-7AB809E68662@mimuw.edu.pl> <dn9581$kll$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

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. -- The Mathematica Wiki: http://www.mathematica-users.org/ Math for Comp Sci http://www.ifi.unizh.ch/math/bmwcs/master.html Math for the WWW: http://www.w3.org/Math/

**References**:**Re: Types in Mathematica thread***From:*Kristen W Carlson <carlsonkw@gmail.com>

**Re: Re: Types in Mathematica thread***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>

**Re: Re: Re: Types in Mathematica thread***From:*Kristen W Carlson <carlsonkw@Gmail.com>

**Re: Types in Mathematica thread***From:*"Steven T. Hatton" <hattons@globalsymmetry.com>