fom
Posts:
1,969
Registered:
12/4/12
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Re: Matheology § 238
Posted:
Apr 11, 2013 4:12 PM
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On 4/11/2013 9:33 AM, WM wrote:
> On 11 Apr., 13:52, fom <fomJ...@nyms.net> wrote:
>
>> {{d_1, ..., d_n} | n in |n}
>>
>> {d_n | n in |n}
>>
>> They still look like different sets to me.
>>
> d_n are the elements of a sequence.
> (d_1, ..., d_n) are the elements of a series, namely its partial sums.
Once again WM seems to be using mathematical jargon
badly.
The elements of an arithmetical series are connected
by arithmetical signs.
The elements of a sequence of partial sums are the
elements of a sequence, connected set-theoretically
by nested ordered pairs. More generally, they are
connected by the successor operation on the natural
numbers when they are viewed as images of a function
with the natural numbers as a domain.
It seems to me that the statement above had been
my revision after having been told by WM that he
had not been talking about sequences.
> They look different, but the latter include the former.
>
> In a Cantor list the argument can be written:
WM has yet to discern what a Cantor argument is.
> For every n: (d_1, ..., d_n) differs from every entry (qk1, ..., qkn)
> with k =< n, i.e., i.e., every entry of the first n lines. That is
> exactly Cantor's argument, not more and not less.
>
No. That is not Cantor's argument.
Cantor's argument is more like:
Given any fixed listing of infinite
representations purported to be the
names for all real numbers, each
element of the list contributes information
to construct a name for a real number
not on the list.
At no point does Cantor attempt to make any
nonsensical claim comparable to
"Every dog is every animal"
That is, of course, what WM does regularly.
What WM meant to say above would be more like:
For every n: the sequence (d_1, ..., d_n)
differs from the sequence (q_1_1, q_2_2 ..., q_n_n)
at every corresponding term.
> In a list containing all rational numbers, the counter-argument can be
> written:
> For every n: (d_1, ..., d_n) does not differ from infinitely many
> entries (qk1, ..., qkn) with k > n.
As stated, this is true.
As stated, this is irrelevant.
>
> Why should the "for all n" only in one case be exhaustive?
Universal quantification is always arbitrary.
Universal quantification always applies to any given case.
It is when one thinks of universal quantification as "exhaustible"
relative to a "course-of-values" definition that this meaning
for the term is changed. This redefinition is a consequence
of the debates going back to Newton's refusal to explain the
mathematics of his theories. If the mathematics is expected
to explain the theories, then the mathematics should be changed
to serve that purpose.
This is at the heart of modern predicative instantiations of
logic.
It is in classical logic as well. Aristotle does not deal with
individuals. He claims that genera are prior to species. So,
that aspect of the logic is intensional. However, to address
the problem of vacuous statements, he introduces "essence"
(compare it with a fraction in lowest terms) as the distinguishing
feature between a definition and a distinctive property. Then,
he attaches the notion of "substance" to "essence" and endows
the individuals with primary substance.
In the modern logic, Frege and Russell developed description
theories for names to deal precisely with negative existential
statements for non-instantiable "objects". For example,
"The round square does not exist"
This is where your complaints about the directionality of
quantifiers comes from. Universal quantification is a
"top-down" quantifier and existential quantification is
a "bottom-up" quantifier. The "essence/substance" and
"truth/existence" connections introduce the circularity
of "analysis/synthesis" in the practice of mathematics.
One analyzes a problem by including a presupposition
that the problem has been solved. This identifies
necessary connections from intermediate facts that
can lead to the solution. Once the problem is
understood, the synthetic solution is constructed to
demonstrate the connection between the antecedent
conditions and the consequent.
Issues arise when one needs the mathematics to be
the philosophy, as had been the case with Newton.
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