fom
Posts:
1,035
Registered:
12/4/12
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Re: Matheology § 203
Posted:
Feb 2, 2013 3:57 AM
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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?
How do you prove this assertion?
To even have the discussion presupposes that you ascribe
meaning to the phrase "actually infinite sequence."
This is an old problem. Nevertheless, you are
maintaining your assertions relative to the fact
that one cannot ostensively prove the existence
of an infinite sequence.
But, one cannot prove that a sequence is not
infinite unless one can get to an end. So,
since your position entails a claim concerning
all sequences, the handful of finitary sequences
that you accept do not constitute a proof
of your own claim.
To assume some finite sequence of symbols
represents an exact real number presupposes an
infinite terminal sequence of constant 0's
in so far as that number is understood
only in relation to a system of names (to be
contrasted with understanding through the act
of manipulating strings by a long division).
So, you are arguing that mathematics is
only as exact as the limitations associated
with some given acts of measurement.
Every real world situation is understood
relative to some normative ideal. Jurisprudence
should be just, politics should be virtuous,
etc.
Why is a normative ideal with respect
to which acts of measurement are given
a science so objectionable?
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