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Re: Re: Re: Re: What is zero divided by zero?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg48623] Re: [mg48600] Re: [mg48585] Re: [mg48563] Re: What is zero divided by zero?
*From*: Murray Eisenberg <murray at math.umass.edu>
*Date*: Tue, 8 Jun 2004 00:48:14 -0400 (EDT)
*Organization*: Mathematics & Statistics, Univ. of Mass./Amherst
*References*: <4xm5ym42r3vg@legacy> <wzog6i63na4c@legacy> <c9k4bo$fi9$1@smc.vnet.net> <c9pfb1$s4l$1@smc.vnet.net> <200406051119.HAA11743@smc.vnet.net> <200406052358.TAA28968@smc.vnet.net> <200406070933.FAA10935@smc.vnet.net> <6190E72C-B898-11D8-B3A8-000A95B4967A@mimuw.edu.pl>
*Reply-to*: murray at math.umass.edu
*Sender*: owner-wri-mathgroup at wolfram.com
Names have great stipulative and connotative power.
Ceratinly the meanings of math names gets extended all the time, but one
can ask whether breaking certain constraints on use of a name would, for
purposes of communication and understanding, best result in a new name
-- or at least clear warning to the reader that an unvonventional use of
the term is being used.
With your intended broadening of the term, one would of course
immediately ask which established theorems about finite fields have to
be restated (if any), which definitions relaxed, etc.
That it might be useful to form this "field", under some name, allow the
multiplicative identiy to equal the additive identity, and thereby to
see that 0^(-1) = 0, so that 0/0 = 0 in this "field", still doesn't
provide a compelling argument to ME for saying 0/0 = 0 in other contexts!
Andrzej Kozlowski wrote:
> I find your argument strange. I am a mathematician and I have published
> papers where I have introduced new definitons and new terminology, as
> has practically every research mathematician. I am free to introduce any
> new concept and name it anyway I like (though of course I can't force
> others to use my terminology) if it is self-consistent, useful and I
> make my meaning clear. The fact that "folks don't ordinarily speak of
> it" is relevant only until sombody chooses to do otherwise.
> The set with one element with the obvious operations of addition and
> multiplication satisfies all the axioms of a field except the convention
> that 1 should be different form 0. It is perfectly well defined, it is
> useful for the purpose of this thread, and 1/0 =1 = 0 holds in it. I
> chose to call it a "field" though I could equally well have called it a
> "desert" but how does the name change anything?
>
> Andrzej
>
>
> On 7 Jun 2004, at 18:33, Murray Eisenberg wrote:
>
>> I'm not sure what Zen world you refer to, but so far as I have met the
>> term "field" in the actual mathematical world, the smallest field has 2
>> elements, not 1.
>>
>> Thus, from http://mathworld.wolfram.com/Field.html:
>>
>> "Because the identity condition must be different for addition and
>> multiplication, every field must have at least two elements."
>>
>> (I suppose you could say that, in the trivial ring consisting of just
>> the 0 element, 0 is its own multiplicative inverse, since 0 * 0 = 0 and
>> 0 is a multiplicative identity. But folks don't ordinarily speak of
>> multiplicative inverses, and hence don't speak of quotients, unless
>> there's a multiplicative identity 1 =/= 0.)
>>
>>
>> Andrzej Kozlowski wrote:
>>
>>> There is at least one mathematical context where it is perfectly well
>>> defined: the Zen-like world of the field with one element, where
>>> 0/0 = 0 = 1.
>>> Andrzej
--
Murray Eisenberg murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower phone 413 549-1020 (H)
University of Massachusetts 413 545-2859 (W)
710 North Pleasant Street fax 413 545-1801
Amherst, MA 01003-9305
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