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MathGroup Archive 2004

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RE: [Off Topic] Re: Re: What is zero divided by zero?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg48722] RE: [mg48678] [Off Topic] Re: [mg48652] Re: What is zero divided by zero?
  • From: David.Cousens at csiro.au
  • Date: Fri, 11 Jun 2004 03:53:06 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Janos,

Are you talking about a discrete form of algebra which has no zero. Zero is in the axioms of our mathematics system, it is not a construct of it. If we have the concept of unity, then zero is the concept associated with the absence of that quantity. In this sense it is an abstract limit quantity by definition and has no association with physical realization of the concept. The concept of "nothing" is I think different from zero as this has the implication to me of the inverse set of the set of all sets. 

The question of a  quantum limit is just a matter of what we define as unity in our system. As we have explored the quantum aspects of nature in physics we have had to redefine our units. We may think we have found an absolute fundamental set of units, but we can never be really sure that there is not some finer scale at which we are unable to, at least with our present knowledge and understanding, make observations.

Is the concept of a continuous variable an abstraction? Our experience in physics would indicate that it is. In mathamatics however it is a definable construct at least in terms of being able to infinitely divide any interval into a finer structure. 
I feel we can only consider the question of 0/0 as a limit process. If you have nothing in a set then how can you divide it. 

Dr David Cousens                      Phone 61-(0)7-33274564
CSIRO Exploration and Mining  Fax     61-(0)7-33274455
QCAT, Technology Court,         Email:David.Cousens at csiro.au
Pullenvale 4069, Brisbane, Qld., 
Australia



-----Original Message-----
From: János [mailto:janos.lobb at yale.edu] 
To: mathgroup at smc.vnet.net
Subject: [mg48722] [mg48678] [Off Topic] Re: [mg48652] Re: What is zero divided by zero?


This discussion brought back long forgotten memories.  In the early 
70's I had some discussion with my analysis professor at that time Dr. 
Daroczy Zoltán of KLTE.  I asked him why do we have to go with the dx 
to 0, and why not to stop somewhere at the Planck length or between the 
Planck length and zero.  He then told me something about some continuum 
hypothesis  and selection out axiom which I long forgotten and 
explained that without the ability to let dx go to zero we would not 
have higher mathematics , or it would be very "ugly".  I did not buy 
totally his argument at that time, but because I just forgot what I 
knew, I am not in better situation today either.  However I have a deep 
suspicion that 0 is very much overloaded.  I think that there are 
different kind of 0s.  For example  the having none is different in my 
mind than dx->0.   The abstract objects of mathematics are coming to us 
from the abstraction of macroscopic objects of nature.  It is easy to 
imagine having no turkey for diner, or loosing all soldiers on the 
battlefield - having none - , than having no free electron around an 
oxygen molecule.  First of all we do not really know what an electron 
is, second we have just vague impressions how an oxygen molecule looks 
like.  The boundary between the macroscopic and the microscopic is not 
well mapped yet.  The entanglement between two photons "one meter 
aside" is measurable, but I am not convinced that they are really one 
meter apart.  The metric - coming from the macroscopic abstraction - 
applied might be totally wrong and in REALITY the distance might be 
much closer to 0.

I have the deep conviction when the nature of the quantum will be 
explored as much as the nature of the macroscopic and the human mind 
will be able to create abstractions from the quantum world as naturally 
as it done in the macroscopic, then a new world of mathematics will 
come alive where 0/0 will be better defined - or undefined - than it is 
now.

János
On Jun 9, 2004, at 4:17 AM, Bobby R. Treat wrote:

> And yet again, it remains undefined. Let's leave it that way.
>
> Bobby
>
> Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote in message
> news:<ca3hin$s0f$1 at smc.vnet.net>...
>>> 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!
>>
>> But then nobody ever said that. In fact it was obvioulsy a joke,
>> though
>> I guess it needs a certain kind of sense of humour to appreciate it. 
>> As
>> for Zen ,  well  ... never mind.
>>
>> Andrzej
>>
>>
>>
>>
>>
>> On 8 Jun 2004, at 01:19, Murray Eisenberg wrote:
>>
>>> 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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wisdom had no equal."
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