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 <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.

This is the crap you engage in:

http://en.wikipedia.org/wiki/Argument_from_ignorance

http://en.wikipedia.org/wiki/Eristic

http://en.wikipedia.org/wiki/The_Art_of_Being_Right#Synopsis

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:

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.