On 17 Sep 2009, at 18:17, John Mikes wrote:

> Dear Bruno,
> it is not very convincing when you dissect my sentences and  
> interject assuring remarks on statements to come later in the  
> sentence, negating such remarks  in advance, on a different basis.
> I argued that - upon what you (and the rest of the multimillion  
> mathematicians past and present) live with  - the applied  
> nomenclature is incomplete. It is not a counter-argument that "it is  
> used by many" or "for so many purposes". Of course it is in use,  
> that was my point.
> I am not basing my position on opinions from "within" the argued  
> position.
> (May the "-2 level" point to a 'total senselessness' of my opinion?  
> I did not understand it, nor did I the (N + *) structure, which  
> therefore I find irrelevant in the question what "I" raised. (I, not  
> Rieman, Cantor, etc.).
> There is the idea of including 'quantities' in our worldview (excuse  
> my naive reference, but you illustrated earlier "2" as "II" and "3"  
> as "III" etc. and THIS in my mind means sort of a quantity) and such  
> 'system' would be qualitatively
> infinite if we try to include quantities from all directions (math  
> is the level of handling such quantities that 'came up' in the past  
> - gradually - and we may expect more to come, new discoveries,  
> extending the qualitative inventory)
> although in your words 'everything' can be expressed by (many many?)  
> of your natural numbers (except  square root 2?) - what is exactly  
> my point.
> I did not want to open a scientific argument - I am no match for  
> you, or any other 'mathematically educated' person. I scribbled a  
> 'qualitative' idea of thinking in 'wider' terms than the defined  
> 'natural numbers' in a worldview of a (qualitative) "totality" -  
> what I pursue, but do not understand in my sci.fic agnosticism.
> I am sorry if I bored you with my remark.

I apologize if I gave that impression, but I try sometimes to be not  
too much long in the mails, and being short can have given that  
impression. Sorry. My point was just that there is a sense where  
natural numbers are not enough in math, and that is why mathematicians  
have extended the set N. N, then Z, then Q, then R, then the complex  
numbers, then the quaternions, octonions, etc.
But, once we assume comp, N is "ontologically" enough, all other sort  
of numbers do necessarily appear as unavoidable epistemological  
constructions, if only to understand the (additive-multiplicative)  
behavior of the natural numbers, a bit like Riemann use complex  
numbers to provide information on the prime (natural) numbers.

Without digging a bit more on the technical issue, I can hardly say  
more than my usual: there is only natural number, together with the  
additive and multiplicative law. This, assuming comp, already defines  
a "matrix" of number's dreams, and those cannot avoid the internal  
phenomenological appearance of richer structures, like the "other"  
numbers, and indeed like the whole physical appearances.

Does this help you a little bit?




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