Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Matheology � 258
Replies: 104   Last Post: May 5, 2013 2:26 PM

 Messages: [ Previous | Next ]
 fom Posts: 1,968 Registered: 12/4/12
Re: Matheology § 258
Posted: May 3, 2013 3:18 PM

On 5/3/2013 11:51 AM, Dan wrote:
> On May 3, 6:47 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>> On 3 Mai, 08:45, Dan <dan.ms.ch...@gmail.com> wrote:
>>

>>> 1/9 = 0.1111 ..... (infinity ones) it's ALSO a valid continuous
>>> magnitude .

>>
>> It is a limit, but it has no decimal representation.
>>
>> If it had a decimal representation, then it would be in the following
>> list which contains all decimal representations with the digit 1
>> having a finite index.
>>
>> 0.1
>> 0.11
>> 0.111
>> ...
>>
>> Or can you determine a finite index which is not occupied by a 1?
>>
>> Being infinite is a valid mathematical notion, but it is not a means
>> to distinguish decimal representations. If there is a sequence of 1
>> larger than all sequences of the list (not only larger than every),
>> then let me know the difference by naming the indices. Otherwise stop
>> asserting unmathematical properties.
>>
>> Regards, WM

>
> Every real number ,in decimal representation , is an infinite
> sequence . But those on your list are followed by an infinity of
> zeroes :
> 0.1 .............
> 0.110000000 .....
> 0.111000000 .....
>
> that we choose not to write , for convenience .

It is more than "convenience".

Let me qualify that with the provision that I hold
certain non-standard views.

There is an issue here with the nature of the identity
relation. One does not "know" that a terminated expansion
is exact except in relation to the algorithm that generates
the expansion.

However, in order for that algorithm to demarcate each terminated
expansion from all other terminating expansions, the algorithm
carries the presupposition of a completed infinity.

The difference between numerical identity (singular term) and
equivalence is understood with respect to bivalent logic and
mutually exclusive truth valuation.

self-identity:

x=x

Numerical identity, however, depends on something different:

(x=t_0) and (not(x=t_1)) and (not(x=t_2)) and (not(x=t_3)) and ...

or

(not(x=t_0)) and (x=t_1) and (not(x=t_2)) and (not(x=t_3)) and ...

or

(not(x=t_0)) and (not(x=t_1)) and (x=t_2) and (not(x=t_3)) and ...

or

(not(x=t_0)) and (not(x=t_1)) and (not(x=t_2)) and (x=t_3) and ...

or

...

where each t_i may be thought of as a canonical name. It is not
that there is some particular collection of names that is special.
Rather, it is that there is a presupposition of denotation by
singular terms. The paradigmatic singular term is a name. So,
while there need not be a particular canonical naming, the semiotics
of naming is presupposed prior to the presupposition of denotation.

Observe that when only the ontological self-identity is taken into
account, the list collapes to a Russellian enumeration,

(x=t_0) or (x=t_1) or (x=t_2) or (x=t_3) or ...

t_0 and t_1 and t_2 and t_3 and ...

This construct respects Bolzano's admonition against "doing violence
to the language" by adhering to Wittgensteinian identity,

not(i=j) -> not(t_i=t_j)

That is, there is no need of a sign of equality for this
presupposed enumeration of names. Each distinct inscription
purports to represent an individual and each purport of
representation excludes prior inscriptions from further use
to prevent representational ambiguity.

In general mathematicians consider the Peano axiom,

AmAn((m+1=n+1) -> (m=n))

as a statement fixing the successor function as a well-defined
function. Arguably, however, it expresses the fact that ordinal
series implicitly define the identity relation on domains.

Among other things, one of the changes that had been made in the
nineteenth century is an understanding of mathematics in terms of
systems. The sign of equality becomes relativized to the identity
relation of those systems.

If, for the moment, questions such as "What is a number?" or "Is a
completed infinity legitimate?" are ignored, then the question of
"How does the result of the Euclidean division algorithm designate
an individual in relation to the system identity?" can be seen for
the difficult problem that it is.

Of course, when one simply works within a system of axioms, the
problem is obscured by the presupposition of denotation. It merely
hides itself in set theory as the question of a well-ordering for
the reals.

Date Subject Author
4/29/13 Virgil
4/29/13 mueckenh@rz.fh-augsburg.de
4/29/13 Virgil
4/30/13 dan.ms.chaos@gmail.com
4/30/13 mueckenh@rz.fh-augsburg.de
4/30/13 dan.ms.chaos@gmail.com
4/30/13 mueckenh@rz.fh-augsburg.de
4/30/13 Virgil
5/1/13 dan.ms.chaos@gmail.com
5/1/13 mueckenh@rz.fh-augsburg.de
5/1/13 dan.ms.chaos@gmail.com
5/1/13 mueckenh@rz.fh-augsburg.de
5/1/13 dan.ms.chaos@gmail.com
5/1/13 mueckenh@rz.fh-augsburg.de
5/1/13 dan.ms.chaos@gmail.com
5/1/13 mueckenh@rz.fh-augsburg.de
5/1/13 Virgil
5/1/13 mueckenh@rz.fh-augsburg.de
5/1/13 dan.ms.chaos@gmail.com
5/1/13 mueckenh@rz.fh-augsburg.de
5/1/13 dan.ms.chaos@gmail.com
5/1/13 mueckenh@rz.fh-augsburg.de
5/1/13 Virgil
5/1/13 dan.ms.chaos@gmail.com
5/1/13 JT
5/2/13 mueckenh@rz.fh-augsburg.de
5/2/13 Ed Prochak
5/2/13 Virgil
5/2/13 mueckenh@rz.fh-augsburg.de
5/2/13 dan.ms.chaos@gmail.com
5/2/13 mueckenh@rz.fh-augsburg.de
5/2/13 Virgil
5/3/13 dan.ms.chaos@gmail.com
5/3/13 mueckenh@rz.fh-augsburg.de
5/3/13 dan.ms.chaos@gmail.com
5/3/13 Virgil
5/3/13 mueckenh@rz.fh-augsburg.de
5/3/13 dan.ms.chaos@gmail.com
5/3/13 mueckenh@rz.fh-augsburg.de
5/3/13 dan.ms.chaos@gmail.com
5/3/13 mueckenh@rz.fh-augsburg.de
5/3/13 dan.ms.chaos@gmail.com
5/4/13 mueckenh@rz.fh-augsburg.de
5/4/13 dan.ms.chaos@gmail.com
5/4/13 mueckenh@rz.fh-augsburg.de
5/4/13 dan.ms.chaos@gmail.com
5/4/13 mueckenh@rz.fh-augsburg.de
5/4/13 dan.ms.chaos@gmail.com
5/4/13 mueckenh@rz.fh-augsburg.de
5/4/13 dan.ms.chaos@gmail.com
5/4/13 mueckenh@rz.fh-augsburg.de
5/4/13 dan.ms.chaos@gmail.com
5/4/13 ross.finlayson@gmail.com
5/5/13 LudovicoVan
5/5/13 fom
5/5/13 ross.finlayson@gmail.com
5/5/13 ross.finlayson@gmail.com
5/5/13 mueckenh@rz.fh-augsburg.de
5/5/13 Virgil
5/5/13 dan.ms.chaos@gmail.com
5/5/13 mueckenh@rz.fh-augsburg.de
5/4/13 Virgil
5/5/13 mueckenh@rz.fh-augsburg.de
5/5/13 Virgil
5/4/13 Virgil
5/4/13 Virgil
5/5/13 mueckenh@rz.fh-augsburg.de
5/5/13 Virgil
5/5/13 mueckenh@rz.fh-augsburg.de
5/4/13 fom
5/4/13 Virgil
5/4/13 ross.finlayson@gmail.com
5/4/13 Virgil
5/4/13 Virgil
5/4/13 Virgil
5/4/13 trj
5/4/13 Virgil
5/3/13 Virgil
5/3/13 Virgil
5/3/13 fom
5/3/13 dan.ms.chaos@gmail.com
5/3/13 fom
5/3/13 gus gassmann
5/3/13 Virgil
5/2/13 Virgil
5/1/13 Virgil
5/1/13 JT
5/1/13 Virgil
5/1/13 JT
5/1/13 Bergholt Stuttley Johnson
5/1/13 rt servo
5/1/13 mueckenh@rz.fh-augsburg.de
5/1/13 Virgil
5/1/13 Virgil
5/1/13 Virgil
5/1/13 Virgil
5/1/13 Virgil
4/30/13 Virgil
4/30/13 mueckenh@rz.fh-augsburg.de
4/30/13 Virgil
4/29/13 ross.finlayson@gmail.com
4/29/13 Virgil