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Topic: Matheology § 253
Replies: 30   Last Post: Apr 22, 2013 2:44 PM

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fom

Posts: 1,968
Registered: 12/4/12
Re: Matheology § 253
Posted: Apr 19, 2013 8:28 AM
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On 4/19/2013 7:04 AM, WM wrote:
> On 19 Apr., 12:05, fom <fomJ...@nyms.net> wrote:
>> On 4/19/2013 3:41 AM, WM wrote:
>>
>>

>>>> If numbers existed only as names then different names would necessarily
>>>> represent different numbers,

>>
>>> No. Look, the unicorn does not exist other than as its name. But there
>>> are different written names and many different pictures describing it.

>>
>> So, you now turn to an argument about negative existential
>> statements.

>
> No, I only explained that Virgil's argument was mistaken. There can be
> many names for an idea that has no material counterpart.


Those are called negative existential statements.

>>
>> How do you justify your analysis of the
>> statement
>>
>> "The unicorn does not exist"

>
> Why should I? The unicorn exists in several forms as ideas of many,
> contrary to individuals that cannot even exist as ideas.
>


So, you do not identify existence with material forms
because you can distinguish between a negative existential
that refers in the imagination and a negative existential
such as

"The round square does not exist"

that cannot refer because of self-contradictory
description.


=========================================================


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 statements and reasoning
as opposed to what he says with
rhetoric.









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