fom
Posts:
1,969
Registered:
12/4/12
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Re: Matheology § 224
Posted:
Mar 17, 2013 7:17 PM
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On 3/17/2013 3:48 PM, Virgil wrote:
> In article
> <ded07b82-5853-4a1e-af1d-a68c3ef695aa@j9g2000vbz.googlegroups.com>,
> WM <mueckenh@rz.fh-augsburg.de> wrote:
>
>> On 17 Mrz., 08:18, fom <fomJ...@nyms.net> wrote:
>>> On 3/16/2013 4:37 PM, WM wrote:
>>>
>>>> On 16 Mrz., 21:19, Virgil <vir...@ligriv.com> wrote:
>>>
>>>>>> In potential infinity there is no necessary line except the last one.
>>>>>> We know that with certainty from induction. Every found and fixed line
>>>>>> n cannot be necessary, because the next line contains it.
>>>
>>>>> AS soon as something is identifies as a natural or a FIS of the set of
>>>>> naturals, it has a successor. It cannot be either a natural nor a FIS of
>>>>> the naturals without a successor. at least by any standard definition of
>>>>> naturals.
>>>
>>>> As soon as a second becomes presence, it has a successor.
>>>
>>> And what fantasy is this?
>>>
>>> The successor to the present has existential form but
>>> has not yet happened.
>>>
>>> That is not the Kantian aprioriticity of time.
>>>
>>> That is not the Hegelian becoming of the present.
>>>
>>> It is the unfounded object of unjustifiable belief.
>>
>> It is the well known and established natural way how time passes and
>> how the system of human actions in time goes off.
>
> Mathematical truth is independent of time.
Well that depends on how a philosophy which makes
such a statement addresses the issue.
Frege specifically addresses the sense expressed
by WM:
"Next there may be those who will
prefer some other definition as
being more natural, as for example
the following:
if starting from x we transfer our
attention continually from one object
to another to which it stands in
relation phi, and if by this procedure
we can finally reach y, then we say
that y follows in the phi-series after
x.
"Now this describes a way of discovering
that y follows, it does not define what is
meant by y's following. Whether as our
attention shifts, we reach y may depend
on all sorts of subjective contributory
factors, for example, on the amount of
time at our disposal or on the extent of
our familiarity with the things concerned.
Whether y follows in the phi-series of
x has nothing to do with our attention
and the circumstances in which we
transfer it; on the contrary, it is a
question of fact, just as much as it is
a fact that a green leaf reflects light
rays of certain wavelengths, whether or
not these fall into my eye and give rise
to a sensation, and a fact that a
grain of salt is soluble in water whether
or not I drop it into water and observe
the result, and a further fact that
it remains still soluble even when it
is utterly impossible for me to make
any experiment with it.
"My definition lifts the matter into
a new plane; it is no longer a question
of what is subjectively possible but
of what is objectively definite. For
in literal fact, that one proposition
follows from certain others is something
objective, something independent of the
laws that govern the movements of our
attention, something to which it is
immaterial whether we actually draw the
conclusion or not. What I have provided
is a criterion which decides in every case
the question "Does it follow after?"
wherever it can be put; and however much
in particular cases we may prevented by
extraneous difficulties from actually
reaching a decision, that is irrelevant
to the fact itself.
Although Frege eventually retracted
his own definition, what he is saying
here is that defined relations relative
to a defined logic constitute the matter
of an objective mathematics.
To understand the distinction, one may
contrast Frege with Weyl. The latter
is, at least, tentatively willing to
admit a set theoretic ground that does
not yield the transfinite. He says this
in reference to Dedekind:
"A set-theoretic treatment of the natural
numbers such as that offered by Dedekind
may indeed contribute to the systematization
of mathematics; but it must not be
allowed to obscure the fact that our
grasp of the basic concepts of set theory
depends on a prior intuition of iteration
and of the sequence of natural numbers."
Let me give credit to WM for rejecting
Dedekind. He has shown enough consistency
to realize that a Dedekindian ground is
a ground that fixes the successor relation
with respect to a system that constitutes
a completed infinity. Weyl apparently
misses the inconsistency of his position
out of desire to reject transfinite
arithmetic.
But, a few pages earlier, Weyl makes an
interesting statement concerning the nature
of "objective" fact.
Note the explicit rejection of logic
and definition in his statement,
"Therefore, how two sets (in contrast to
properties) are defined (on the basis of
the primitive properties and relations
and individual objects exhibited by means
of the principles of section 2) does not
determine their identity. Rather, an
objective fact which is not decipherable
from the definition in a purely logical
way is decisive; namely, whether each
element of the one set is an element
of the other, and conversely. [...]"
So, as a reader of this statement, I
am first expected to reject prior
definitions and to reject logical
relations. Then, I am expected to
understand the discursive assertion
explaining what it is that cannot
be explained.
However, I am to understand that this
is sensible with respect to some
other prior principles explained
elsewhere. And, I am to understand
that what cannot be explained to
me can sensibly be expressed as
a rule.
The statement goes on to say,
"Moreover, we see that the description
of a finite set in individual terms
is, considered formally, just a special
case of that based on a rule. For
example, if a,b,c are three objects
of our category, then
P(x)=J(xa)+J(xb)+J(xc)
is the judgement scheme of the derived
property 'being a or b or c'; and the
set having just those three objects
as its elements correspond to this
property."
What is relevant from section 2 that
I am expected to not ignore while
being told to ignore is the following:
"By simple or primitive judgment scheme
we mean those which correspond to the
individual immediately given properties
and relations. To these we add the
identity scheme J(xy) (meaning 'x is
identical to y' i.e., 'x=y')"
So, once again, the situation resolves
to the concept of "immediately given
individual properties" or the objective
fact that the purport of singular
reference suffices as an establishment
of singular reference.
And, once again, searching through these
philosophies and the definitions leads
to the fact that presentations of
Leibniz law such as
http://plato.stanford.edu/entries/identity-relative/#1
misrepresents what, in fact, Leibniz
actually wrote:
"What St. Thomas affirms on this point
about angels or intelligences ('that
here every individual is a lowest
species') is true of all substances,
provided one takes the specific
difference in the way that geometers
take it with regard to their figures."
Leibniz
Returning to how time might inform
mathematics in relation to arithmetical
progressions, there is Kant:
1)
Time is not an empirical concept
that is derived from experience.
[...]
2)
Time is a necessary representation
that underlies all intuitions
[...]
3)
The possibility of apodeictic principles
concerning the relations of time, or
of axioms of time in general is
grounded upon this a priori necessity.
[...] We should only be able to
say that common experience teaches
that this is so; not that it must be
so. These principles are valid as
rules under which alone experiences
are possible; and they instruct us
in regard to experiences, not be
means of them.
4)
Time is not a discursive, or what is
called a general concept, but a form
of pure sensible intuition.
5)
The infinitude of time signifies
nothing more than that every determinate
magnitude of time is possible only
through limitations of one single
time that underlies it."
And, should anyone who wishes to reject
the mathematical aspect of Kant's
philosophy in relation to his remarks
here on the basis of nineteenth and
twentieth century "progress" one need
only consider the author to whom George
Greene directed me to learn about why
Kant had become outdated. Boolos
writes:
"[Crispin] Wright regard's Hume's
principle as a statement whose role
is to fix the character of a certain
concept. We need not read any
contemporary theories of the a priori
into the debate between Frege and Kant.
But Frege can be thought to have carried
the day against Kant only if it has
been shown that Hume's principle is
analytic, or a truth of logic. This
has not been done. [...]
"Well. Neither Frege nor Dedekind showed
arithmetic to be a part of logic. Nor
did Russell. Nor did Zermelo or von Neumann.
Nor did the author of Tractatus 6.02 or
his follower Church. They merely shed
light on it."
And, George's recommendation seemed
particularly odd when I found that
Boolos quoted Hao Wang's remark,
"The reduction, however, cuts both
ways. It is not easy to see how
Frege can avoid the seeming frivolous
argument that if his reduction is
successful, one who believes firmly
in the synthetic character of
arithmetic can conclude that Frege's
logic is thus proved to be synthetic
rather than that arithmetic is
proved to be analytic."
So, one may clearly hold the contrary
to Virgil's statement depending on
how one views time and its relationship
to mathematical thought without,
apparently, being in too much
mathematical jeopardy.
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