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/ --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to [email protected] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~----------~----~----~----~------~----~------~--~---
