List, Jerry & al.
An interesting discussion seems to be going on.
Jerry wrote:
"CSP's notion of the continuum effectively separates the concept of the continuum from the concept of counting. The difference that makes a difference in his CSP philosophy is the meaning of the term "count" in relation to the term "continuum". Most of the pragmatic scientific community ignore the subtle distinction as it has little practical significance in terms of measurements, which seldom exceed 10 decimal points."
Yes, I agree. But as far as I have followed this discussion (& ealier ones in
the list) there is a distinction never properly discussed. That is the distinction
between continuity and continuums. CSP's synechism is about the principle of
continuity. Which is a philosophical statement (having to do with mathematics, but
not a mathematical statement).
There seems to be a fuzzy use of the concepts "continuum" and "continuity". A kind of vaccillation between the two. BUT in an asymmetrical way:
If and when continuity is taken up, people - without duly noticing - change the topic and start to talk on continuums. - After that the common topic stays with continuums. - Never to return to the topic of continuity.
Well, continuums are easier to grasp. - And to all my knowledge, Peirce himself did not get the idea of the (subtle?) difference between these two very clear untill very late in his life.
The problems of continuums and counting kind of belong together. But with continuity counting just loses its meaning. Even the meaning which emerges with the problem of continuum.
If anyone writes "continuity" and then changes into talking about continuum,
that is something to be duly noticed. That is what I wish to say. There is a difference.
Best,
Kirsti
Jerry LR Chandler [[email protected]] kirjoitti:
List, Michael, Ben
Michael:
Your logical construct is sound from the perspective of Cantor's set theory and its numerous extensions, such as the category of sets. No disagreements with your conclusion if one assumes the propositions of Cantor.
(For an overview of Cantor's view, see:
http://en.wikipedia.org/wiki/Cantor's_theorem#A_detailed_explanation_of_the_proof_when_X_is_countably_infinite
)
I presume you are aware that CSP never accepted Cantorian set theory or Russelian logic. Consequently, your line of explanation could hardly be the basis for his definition. Further, CSP's definition does not distinguish between "infinite sets" or "uncountably infinte sets", which is one of the most important conclusions from Cantorian set theory for mathematical analysis.
My remark was restricted to the rhetoric of CSP's definition as given in Ben's post in comparison to CSP's other definition's of the continuum.
My rhetoric was based on graph theory, not the diagrammatic logic of lattice
theory. Lattice theory, which can be inferred from the power set of Cantorial
counting, can also be represented as a special form graph theory. CSP's
definition does not invoke Cantor's propositions of sets, subsets of sets, and
subsets of subsets of subsets and so forth as can be constructed within lattice
theory. See the wikipedia article for a simple example of inference by the
diagrammatic logic of power sets, lattice theory and the growth of the graphs
representing lattices.
Further more, my rhetoric does not depend on the science of chemistry, either.
It does depend on concept of the potential for addition of graphic node to
create multiple nodes, which is synomyous with the concept of electrical
relations or of chemical valence.
For example, consider a connected graph with nodes of valence five. If the graph is connected, then a connection must use one node from each to form the connection. To concatenate the graph into a chain, another pair of nodes must be used. Thus, for every object with valence five, at least two nodes are necessary to concatenate the links of the chain. What are the possibilities for the other nodes? Each of three nodes may initiate a new branch in the graph. Thus, 3^n new branches can be created where n is the count of the number of branches. Although the algebra is clear, the diagrams rapidly become to perplex to draw.
Now, if the power set is defined as 2^n counts of the possible subsets, then
3^n grows faster than 2^n.
The difference that makes a difference (a la Bateson) is that my rhetoric is
based on the conceptualization of what is being counted.
Cantorian set theory counts members of a set.
My logic counts relations among relatives. (A rough sketch of this form of
counting was published in Discrete Applied Mathematics in 2009.)
Physically, Cantorian set theory can be viewed as each point/object
representing a point mass.
Physically, my way of counting can be viewed as each object as a graph composed
from unbounded numbers of electrical valences at each node.
In other words, two clearly separate and distinctive methods exist for counting physical particles.
In my post, I pointed to the weakness of the concept of exhaustion with respect
to the definition of the continuum. You apparently agree, basing your
assertions on Cantorian set theory.
Here, I am simply pointing out that the methods of counting of graphic relations among relatives, (homologous with some aspects of chemical logic, biochemical logic and cellular logic) increase faster than the power set. Although I use the example of a valence of five, it should be apparent to all readers that higher valences (6,7,8,& (infinity - 1)) will grow relations among graphs even faster, faster by any desired multitude (of the power set) less than infinity.
Michael, I agree that
BTW, do you consider the Cantorian propositions to be natural or unnatural?
Also, do you consider the arithmetic propositions to be natural or unnatural?
Thanks for the provocative reply.
Cheers
Jerry
On Nov 10, 2014, at 11:26 AM, Michael DeLaurentis wrote:
> Jerry That you can endlessly, and apparently recursively, add any number of
elements means only that you have a potentially countably infinite collection.
Cantors power-set operation allows increasing magnitude beyond the limit of
countability. CSP is effectively saying no operation can move beyond the
cardinality of any rank, no matter how constructed, that arrives at his conception
of a true continuum. Certainly, the m-value branching of any number N of elements
doesnt get you there.
>
> From: Jerry LR Chandler [mailto:[email protected]]
> Sent: Monday, November 10, 2014 11:59 AM
> To: Benjamin Udell
> Cc: [email protected]
> Subject: Re: [PEIRCE-L] Re: Continuity, Generality, Infinity, Law, Synechism,
etc.
>
> Ben, list:
>
> On Nov 10, 2014, at 9:33 AM, Benjamin Udell wrote, quoting CSP:
>
>
> A true CONTINUUM (q. v.) is something whose possibilities of determination no multitude of individuals can exhaust.
>
>
> A minor comment with respect to this definition of a continuum.
>
> The concept of "can exhaust" is a weak concept of continuity relative to the notion of a mark on a line or the welding of points together.
>
> A simple counterexample of this definition arises in chemical logic.
>
> The conjunction of chemical elements creates molecules containing all the parts of each element.
>
> Such conjunctions beget spatial objects.
>
> An element may serve as a branching point for the graph of the molecule, it may signify 1,2,3,4,... branches into 3D space.
>
> The additions of further conjuncts is not exhaustible; no multitude of individual atoms exhausts the individuals.
>
> Yet, the branched structure, as a consequence of valence, is a set of nodes and lines representing the parts of the atoms. It is not necessarily a circle but circles are not excluded.
>
> Does this concept of conjunction conform to CSP's definition of a continuum based on the concept of exhausting individuals?
>
>
> Cheers
>
> Jerry
> No virus found in this message.
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