Date: Apr 23, 2013 4:25 PM
Author: fom
Subject: Re: Matheology § 246

On 4/23/2013 2:07 PM, WM wrote:
> On 23 Apr., 16:35, fom <> 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

This is the crap you engage in:

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.

This is WM's model of mathematics:

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:

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


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