Date: Feb 2, 2013 5:17 AM
Author: fom
Subject: Re: Matheology § 203
On 2/2/2013 4:02 AM, WM wrote:

> On 2 Feb., 09:57, fom <fomJ...@nyms.net> wrote:

>> On 2/1/2013 9:48 AM, WM wrote:

>>

>>

>>

>>

>>

>>> On 1 Feb., 16:35, William Hughes <wpihug...@gmail.com> wrote:

>>>> Let P(n) be

>>>> 0.111... is not the nth line

>>>> of

>>

>>>> 0.1000...

>>>> 0.11000...

>>>> 0.111000...

>>>> ...

>>

>>>> Clearly for every natural number n

>>>> P(n) is true.

>>

>>>> This means there is no natural

>>>> number m for which P(m) is true.

>>

>>>> It is not simply that we cannot find m,

>>>> we know that m does not exist.

>>

>>> More. We know that P(n) = 0.111... = 1/0 does not exist as an

>>> actually infinite sequence of 1's.

>>

>> Hmm....

>>

>> As I watch you make these arguments, it occurs to me...

>>

>> What proof do you have that some sequence is not infinitely

>> long?

>

> Even with no regard to Tristram Shandy who disproves actual infinite,

> we can say: The sequence for 1/9 = 0.111... cannot have indices that

> differ from all indices of its finite approximations. So you cannot

> distinguish 0.111... by looking at digits from its finite

> approximations. And you cannot use it in any discourse because every

> message is finite and needs an endoffile signal to be meaningful.

>

> Concluding: The property that a sequence of digits does not end cannot

> be obtained from its digits. (Remember, 0.111... is not a sequence of

> digits but is only a rule to construct a sequence of digits. The rule

> yields the sequence, but the sequence does not yield the rule.)

Ok. So, you are introducing the kind of arguments

used by Wittgenstein. Of course, Wittgenstein never

gave a coherent explanation for classical mathematics.

His criticisms, however, are easily seen to be forebears

of much of the discrete mathematics that has become

so important with the advent of computing technology.

>

> That means in mathematics, understood as discourse, there is no

> decimal representation of 1/9. What God may use for his mathematics is

> not my problem.

And, this points to modern pragmatics. The tradition of Russell,

Carnap, Tarski, Quine, ... is classified as "ideal language

theory" in contrast to "natural language theory." This is one

of the disservices done by mathematics departments that have

classes on foundational topics without a coherent curriculum

surrounding those topics.