Le 26-mai-06, à 02:50, James N Rose a écrit :

>
> Bruno,
>
> You struck a personal nerve in me with your following remarks:
>
> Bruno Marchal wrote:
>>
>> They are degrees. The worst "unreasonableness" of a (platonist or
>> classical or even intuitionist) machine is when she believes some 
>> plain
>> falsity (like p & ~p, or 0 = 1). The false implies all propositions, 
>> so
>> that such machine believes everything, including everything about 
>> their
>> maximal consistent extensions or histories (which does not exist).
>> Those machines are just inconsistent.
>
> particularly ,
>
>     "some plain falsity (like p & ~p, or 0 = 1)".
>
> Rather than treat these as 'blatantly false' I have been
> exploring the notion for several years .. 'what conditions,
> situations, criteria or states would allow such statements
> to be 'true', and what would it mean in how we define and
> manipulate and operate the rest of mathematics?'.
>
> I have discovered that an unprecedentedly un-appreciated
> realm of mathematical relations has existed right before
> our minds.  The lack, having kept us trying to cope with
> 'anomalies' and math issues without the full toolkit of
> mathematical instruments.
>
> An example at the core of it is a most simplistic
> definition/equation.
>
>                 1^1 = 1^0
>
> [one to the exponent one  equals  one to the exponent zero]
>
> To all mathematicians, this is a toss-out absurdity, with
> no 'real meaning'.  n^0 is a convenience tool at best ;


n^0 = 1, because 1= (n^m)/(n^m) = n^(m-m) = n^0.
Or better n^0 = the number of functions from the empty set (cardinal 0) 
to the
set with cardinal n. This justifies also 0^0 = 1 (there is one (empty) 
function from
the empty set to the empty set).


> along
> with  'n/0 is 'undefined''.   We note the consistent/valid
> notation, but walk away from any active utility or application.
>
> My thesis is that doing so was a missed opportunity.
>
> To be hyper-consistent, the equation set-up
>
>                 1^1 = 1^0
>
> indicates that there -must- be some valid states/conditions
> (not just 'interpretation') when 0 and 1 are 'equal' in some
> real meaning/use of the word "equal".


Why? It is usual that a function (like y = 1^x) can have the same value 
for different argument.
 From (-5)^2 = 5^2 you will not infer that 5 = (-5), right?
 From sinus(x) = sinus(pi - x) you will not deduce that x = pi - x, 
right?


>  If they can be substituted
> in the above equation, without changing a resultant of
> calculations (they are embedded in), then they must somewhere
> somehow in fact be identical in some way or condition.


You talk like if all functions are bijections (one to one function).



>
> The entire ediface of physics is hamstrung because of this,
> because mathematical definitions and language compounded
> the error by applying - actually DIS-applying - a related
> concept .. the notion of 'extent' .. also known as 'dimension'.
>
> Physics and mathematics transform and wholly open up when
> we throw away the old concept of 'dimensionless' and instead
> reformulate -everything- as 'dimensional'.  Including zero;
> including numbers unassociated with variables.
>
> As musch as you are brilliant and mathematically inventive,
> your statement  "some plain falsity (like p & ~p, or 0 = 1)"
> shows you haven't quite awoken to everything yet.  I hope
> I'm in the process of stirring you from your slumber.



I am using the name 0, 1, ... for the usual numbers. 1 is different 
from 0 for the same reason that 1 cup of coffee is different from 0 cup 
of coffee, or that 1 joke is different from 0 joke ...


Bruno



http://iridia.ulb.ac.be/~marchal/


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