In article <KdGdnfqxb_A8gu7MnZ2dnUVZ_rqdnZ2d@giganews.com>, fom <fomJUNK@nyms.net> wrote:
> On 4/20/2013 12:03 PM, WM wrote: > > > > All atoms of the accessible universe and all positions they can take > > belong to a finite set. > > > > Well, that is a strange statement given > that they are forming Bose-Einstein > condensates with atoms now.
Not to mention: "As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.
Albert Einstein > > Anyway, what scientifically falsifiable > natural number does WM assert to be the > number of positions available to the > atoms of the accessible universe? > > > WM is an unabashed ultrafinitist who refuses to fix > a largest finite number. Each "n" in his description > depends on the subsequence of triangular numbers. > > > F(n)=Sum_i(1..n)(i) > > > > 1 :=> 1 > > 2 :=> 3 > > 3 :=> 6 > > 4 :=> 10 > > > > and so on > > According to Brouwerian intuitionistic reasoning, > when WM's construction reaches the point where > the sequence of triangular numbers exceeds the > ultrafinitist limit, the contradiction nullifies > the construction. > > This is WM's model of mathematics: > > http://en.wikipedia.org/wiki/Finite_model_property > > until he reaches his contradiction and > it vanishes. > > ===================================== > > The triangular numbers correspond with > the number of 'marks' representing numerals > or significant denotations occurring in any > of WM' representations of the form: > > 1 > 2, 1 > 3, 2, 1 > ... > n, ..., 3, 2, 1 > ... > > ------------------------------------- > > This number of 'marks' satisfies a structural > feature of the natural numbers called a > directed set: > > Defintion > > A binary relation >= in a set D is said > to direct D if and only if D is nonempty > and the following three conditions are > satisfied: > > DS1) > > If a is an element of D, then a>=a > > DS2) > > If a, b, c are elements of D such > that a>=b and b>=c, then a>=c > > DS3) > > If a and b are elements of D, then there > exists an element c of D such that c>=a > and c>=b > > > So, WM's geometric reasoning for any given > n obtains a finite model domain with its > cardinality given by the associated > triangular number. The triangular number > is the "element c" of condition DS3 from > the definition. > > ------------------------------------- > > Finally, Brouwer's explanation for finitary > reasoning is used because WM refuses to > commit to any mathematical statement with > coherent consistent usage. > > Brouwer distinguishes between results with > regard to 'endless', 'halted' and > 'contradictory' in his explanations > > "A set is a law on the basis of > which, if repeated choices of > arbitrary natural numbers are made, > each of these choices either > generates a definite sign series, > with or without termination of the > process, or brings about the > inhibition of the process together > with the definitive annihilation > of its result." > > WM cannot be an ultrafinitist and > expect others to not hold him to > task for it. In constrast to > Brouwer, he repeatedly states > that there is absolutely no > completed infinity. Therefore, > there must be a maximal natural > number for his model of > mathematics. Beyond that > number, there is no mathematics. > > That is WM's belief as surmised > from his statements and reasonings > as opposed to what he says with > rhetoric. --