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Topic: some amateurish opinions on CH
Replies: 57   Last Post: Apr 16, 2013 8:12 PM

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fom

Posts: 1,968
Registered: 12/4/12
Re: some amateurish opinions on CH
Posted: Apr 15, 2013 1:25 PM
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On 4/15/2013 11:02 AM, apoorv wrote:
> I have a few comments to make.
> It seems that the relation between sentential and predicate logic is not much discussed.
> Sentential logic can deal only with conjunction of a finite number of sentences. Predicate
> Logic,via the Universal Sentence allows us to make the leap from the finite to the infinite,
> ' to escape the prison of finitude' so to say . Yet, purely from the perspective of sentential
> Logic, the only wff or sentence that Implies an infinite number of independent sentences
> Is the Contradiction A&~A.In introducing the Universal Sentence ,we seem to be ignoring
> this.
> 'there is always a bigger number'representing ' potential infinity' actually represents infinite
> Information , and therefore carries with it the seeds of disconnect with reality.
> The point, as the actual limit of a sequence of segments of ever smaller lengths is
> The completed infinite, representing as it does, information to an infinite precision.
> But the fact remains that mathematics is amazingly successful in depicting and explaining
> Reality. That is its strength and the reason for its almost unquestioning acceptance by
> Everyone. Most other branches of knowledge have there share of diverse opinion and
> Theories, but peresent day Maths is almost unique in the sense of a monolithic edifice of knowledge.
> Yet , maybe it is worth to see if models of reality based on the assumption of the infinitely
> Divisible continuum and on the basis of a finite indivisible unit yield significantly different
> View of the world.
> Apoorv
>


Although my views are non-standard, the answer
will be no.

The infinitely divisible continuum relates to
a geometric notion whose reality is untestable.
But, mathematically -- more precisely, arithmetically --
it has to do with the more philosophical notions
of identity. It simply takes the presumption of
a completed infinity to interpret the use of the
Euclidean algorithm as providing exact names for
arithmetical manipulations.

In the history of first-order logic, there is a
period where description theories concerning the
nature of naming had been prevalent. Although
the names associated with infinitely long symbol
strings ideally output from a long division reflect
the complexity of the necessary identity criterion,
to say "the square root of 2" does not. But,
identity plays a role in the descriptivist theory
of names.

I have spent some time thinking about this matter.
I have no way to explain this in a way that is
even comprehensible, but this investigation has
led to certain significant finite geometries.
Moreover, the particular finite geometries involved
are closely related to certain information-theoretic
error correcting codes. In other words, a mathematical
theory of naming will lead one to information-theoretic
quantization.

Returning to the discussion of the continuum,
I have recently learned about synthetic differential
geometry. It appears that part of its axiomatization
is to admit arithmetical elements such that

x^2=0

does not permit one to conclude

x=0

What you are referring to as a "finite indivisible unit"
seems to be the necessary assumption in differential
geometry that coordinate charts can be constructed
from locally homeomorphic (do I mean diffeomorphic ?)
copies of real manifolds pasted together. The necessary
mathematical requirement involves differentiability.
In turn, that requirement involves neighborhoods about
points rather than points.

When one treats the cardinality of the continuum in
a purely arithmetical way as in set theory, one reason
for doing so is that it became clear how the continuity
assumptions relate to dimension for the geometric
continuum. Cantor and others showed that the cardinality
would be independent of dimension, and, later results
would show that there is a topological notion of
dimension -- hence, it is, in some sense, not possible
to have an infinitely divisible continuum that
explains the phenomena of the material physicalist
conception. But, it seems logically necessary to
have a notion of identity that can interweave
reference to points as boundary of parts with
reference to the coextensive parts whose coextensive
nature is required for that explanation.

Your observation about the "infinity" of the formula
'A&~A' is quite astute.

On the material side of things, there is a theory of
general relativity called scale relativity in which
differentiability is permitted to fail at small
scales. What appears is a duality from some sort
of broken temporal symmetry. So, when the differentiability
assumption is permitted to be relaxed, it manifests
itself as two separate equations with respect to the
directionality of time.

On the logical side of things, when Frege arrived at
his notion of a logical language, it is significant
that he viewed negation as "the mark of judgeable
content". Coming from the Kantian tradition, the
relation of temporality with logical structure would
have been implicit. This is made explicit with
Brouwer's rejection of formalism. And, in the latter
case, it is made explicit precisely in relation to
the use of negation and the perception of the continuum
as infinitely divisible in the sense of names
obtained by the Euclidean algorithm of long division.

So, what I really believe concerning the nature
of the argument over infinity is that it comes
down to an understanding of how the mathematics
relates to the various uses of mathematics rather
than a decision of which view of mathematics
is metaphysically correct.

It is, however, the nature of mathematics that
such a perspective can only be arrived at by
one individual at a time. And, in spite of the
fact that the study of completed infinities
does affect the relevancy of certain results in
applied mathematics, it is unlikely that this
fact will dissuade those who reject its notion
on metaphysical grounds.

Thank you for your thoughtful statements on
these matters.

One other thing. In Aristotle, there is the
admonition to not negate "substance". In
some ways, our ability to treat infinity as
a mathematical concept has been arrived at
precisely by ignoring that suggestion.






Date Subject Author
4/7/13
Read some amateurish opinions on CH
fom
4/7/13
Read Re: some amateurish opinions on CH
mueckenh@rz.fh-augsburg.de
4/7/13
Read Re: some amateurish opinions on CH
Bergholt Stuttley Johnson
4/7/13
Read Re: some amateurish opinions on CH
dan.ms.chaos@gmail.com
4/7/13
Read Re: some amateurish opinions on CH
mueckenh@rz.fh-augsburg.de
4/7/13
Read Re: some amateurish opinions on CH
dan.ms.chaos@gmail.com
4/7/13
Read Re: some amateurish opinions on CH
mueckenh@rz.fh-augsburg.de
4/7/13
Read Re: some amateurish opinions on CH
dan.ms.chaos@gmail.com
4/7/13
Read Re: some amateurish opinions on CH
mueckenh@rz.fh-augsburg.de
4/7/13
Read Re: some amateurish opinions on CH
Virgil
4/8/13
Read Re: some amateurish opinions on CH
dan.ms.chaos@gmail.com
4/8/13
Read Re: some amateurish opinions on CH
mueckenh@rz.fh-augsburg.de
4/8/13
Read Re: some amateurish opinions on CH
dan.ms.chaos@gmail.com
4/8/13
Read Re: some amateurish opinions on CH
mueckenh@rz.fh-augsburg.de
4/8/13
Read Re: some amateurish opinions on CH
dan.ms.chaos@gmail.com
4/8/13
Read Re: some amateurish opinions on CH
mueckenh@rz.fh-augsburg.de
4/8/13
Read Re: some amateurish opinions by WM
Virgil
4/8/13
Read Re: some amateurish opinions on CH
Virgil
4/9/13
Read Re: some amateurish opinions on CH
apoorv
4/8/13
Read Re: some amateurish opinions on CH
Virgil
4/7/13
Read Re: some amateurish opinions on CH
Virgil
4/9/13
Read Re: some amateurish opinions on CH
Guest
4/9/13
Read Re: some amateurish opinions on CH
dan.ms.chaos@gmail.com
4/9/13
Read Re: some amateurish opinions on CH
fom
4/10/13
Read Re: some amateurish opinions on CH
Guest
4/10/13
Read Re: some amateurish opinions on CH
dan.ms.chaos@gmail.com
4/10/13
Read Re: some amateurish opinions on CH
fom
4/10/13
Read Re: some amateurish opinions on CH
JT
4/11/13
Read Re: some amateurish opinions on CH
apoorv
4/11/13
Read Re: some amateurish opinions on CH
dan.ms.chaos@gmail.com
4/11/13
Read Re: some amateurish opinions on CH
apoorv
4/11/13
Read Re: some amateurish opinions on CH
fom
4/15/13
Read Re: some amateurish opinions on CH
apoorv
4/15/13
Read Re: some amateurish opinions on CH
fom
4/16/13
Read Re: some amateurish opinions on CH
Shmuel (Seymour J.) Metz
4/16/13
Read Re: some amateurish opinions on CH
fom
4/7/13
Read Re: some amateurish opinions on CH
Virgil
4/7/13
Read Re: some amateurish opinions on CH
William Elliot
4/7/13
Read Re: some amateurish opinions on CH
fom
4/7/13
Read Re: some amateurish opinions on CH
fom
4/8/13
Read Re: some amateurish opinions on CH
William Elliot
4/8/13
Read Re: some amateurish opinions on CH
fom
4/9/13
Read Re: some amateurish opinions on CH
William Elliot
4/9/13
Read Re: some amateurish opinions on CH
fom
4/9/13
Read Re: some amateurish opinions on CH
William Elliot
4/9/13
Read Re: some amateurish opinions on CH
fom
4/9/13
Read Re: some amateurish opinions on CH
dan.ms.chaos@gmail.com
4/9/13
Read Re: some amateurish opinions on CH
fom
4/9/13
Read Re: some amateurish opinions on CH
dan.ms.chaos@gmail.com
4/9/13
Read Re: some amateurish opinions on CH
fom
4/9/13
Read Re: some amateurish opinions on CH
dan.ms.chaos@gmail.com
4/9/13
Read Re: some amateurish opinions on CH
fom
4/10/13
Read Re: some amateurish opinions on CH
fom
4/11/13
Read Re: some amateurish opinions on CH
dan.ms.chaos@gmail.com
4/11/13
Read Re: some amateurish opinions on CH
fom
4/11/13
Read Re: some amateurish opinions on CH
dan.ms.chaos@gmail.com
4/11/13
Read Re: some amateurish opinions on CH
fom
4/9/13
Read Re: some amateurish opinions on CH
fom

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