benjayk wrote:
>
>
> Bruno Marchal wrote:
>>
>>
>> On 08 Oct 2011, at 20:51, benjayk wrote:
>>
>>>
>>>
>>> Bruno Marchal wrote:
>>>>
>>>>
>>>> On 08 Oct 2011, at 13:14, benjayk wrote:
>>>>
>>>>>
>>>>>
>>>>> Bruno Marchal wrote:
>>>>>>
>>>>>>
>>>>>> On 04 Oct 2011, at 21:59, benjayk wrote:
>>>>>>
>>>>>>>
>>>>>>>
>>>>>>> Bruno Marchal wrote:
>>>>>>>>
>>>>>>>>
>>>>>>>> On 03 Oct 2011, at 21:00, benjayk wrote:
>>>>>>>>>
>>>>>>>>
>>>>>>>> I don't see why.
>>>>>>>> Concrete objects can be helpful to grasp elementary ideas about
>>>>>>>> numbers for *some* people, but they might be embarrassing for
>>>>>>>> others.
>>>>>>> Well, we don't need concrete *physical* objects, necessarily, but
>>>>>>> concrete
>>>>>>> "mental" objects, for example measurement. What do numbers mean
>>>>>>> without any
>>>>>>> concrete object, or measurement? What does 1+1=2 mean if there
>>>>>>> nothing to
>>>>>>> measure or count about the object in question?
>>>>>>
>>>>>> It means that when you add the successor of zero with itself you
>>>>>> get
>>>>>> the successor of one, or the successor of the successor of zero.
>>>>> But that does this *mean*? These are just a bunch of words. You
>>>>> could as
>>>>> well write
>>>>> "It means that when you colmüd the pööl of ämpod with itself you
>>>>> get
>>>>> the pööl of trübda, or the pööl of the pööl of ämpod.".
>>>>
>>>> Exactly! That is the point of axiomatization.
>>>> Hilbert said this to explain what his axiomatic geometry means: "you
>>>> can replace the terms 'points', 'lines', and 'planes', by the term
>>>> 'elephant', 'table' and 'glass of bear'.
>>>> Now, doing this would not be pedagogical, and we use the most
>>>> commonly
>>>> used symbols. That is "+" for colmüd, "s" for pööl, and the symbol
>>>> "0" for your ämpod. We already have some axioms for logic and
>>>> equality, and all you need consists in agreeing or not with the
>>>> following principles:
>>>>
>>>> 0 ≠ s(x)
>>>> s(x) = s(y) -> x = y
>>>> x+0 = x
>>>> x+s(y) = s(x+y)
>>>> x*0=0
>>>> x*s(y)=(x*y)+x
>>>>
>>>> The intended meaning being 0 is not a successor of any number, etc.
>>>> You can say "the ämpod is different from all pööls". No problem,
>>>> but
>>>> it is obviously quite unpedagogical, I think.
>>> You don't get the point. Of course I can agree with these principles
>>> concerning countable and measureable things.
>>
>> But then, unless you see a flaw in the reasoning, you should know that
>> at the obtic level, we don't need more, nor can we use more than the
>> countable collection of finite things, once we assume mechanism.
> For the flaw in the reasoning, see my post above.
>
Sorry, I mean below!
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