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[Off Topic] Re: 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 >>> > > ------------------------------------------ "There was a mighty king in the land of the Huns whose goodness and wisdom had no equal." Nibelungen-Lied