On 4/23/2013 2:07 PM, WM wrote: > On 23 Apr., 16:35, fom <fomJ...@nyms.net> wrote: >> On 4/23/2013 3:59 AM, WM wrote: >> >> >> >>> 1 >>> 2, 1 >>> 3, 2, 1 >>> ... >> >> WM is an unabashed ultrafinitist > > No.
Sorry, but yes.
Equating Kroenecker's version of the natural numbers with Moses's ten commandments is nonsense.
"Knowing" what mathematics is without the burden to explain it is voodoo.
Mathematics is respected for its exactness, its correctness, and its efficiency of representation in application.
It gets tarnished by debates where one party or another wishes to use mathematics to justify a belief. The same holds when someone attempts to use mathematics to win arguments when the participants are not clear of what is presupposed in a given application.
Sadly, metamathematics had arisen from a period when scientific pursuits hoped to use mathematics for those kind of justifications. So, it is taught without a careful discussion of how to use the words "true" and "false" in a metamathematical context. I had been impressed by Kleene's book on the subject when I ran across a candid explanation of that fact. Unfortunately, the book from which I had been taught made no such distinctions.
Unfortunately, I let myself be dragged into the same.
The fact that n=n is true of the natural numbers is not an account of natural numbers. So defend your ideas properly instead of blathering one piece of rhetoric after another and reinterpreting your own statements in whatever manner is convenient.
You are an ultrafinitist until you can produce a philosophy of mathematics that can stand on its own account rather than merely criticize the existing paradigm. And, one tires of your version of Moses.
Our Kronecker, who art in Heaven hallowed be thy name...
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.
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:
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:
If a is an element of D, then a>=a
If a, b, c are elements of D such that a>=b and b>=c, then a>=c
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.