Let one be the atom of itself.
Let cardinal be the number of atoms.
The cardinal of one is one.
Question: What is needed in order to get a cardinal beyond one?
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Let one be the atom of itself.
Let cardinal be the number of atoms.
The cardinal of one is one.
Question: What is needed in order to get a cardinal beyond one?
I think that for that to happen, one has to be the atom of itself no more. As long as one is the atom of itself, the cardinal can't be more then one. (The cardinal is the number of atoms of one & one is the atom of itself.) But if one wasn't the atom of itself but something bigger, the cardinal in one could be higher.
But if all remains as it is now, it's impossible to get a cardinal beyond one as long as one remains the atom of itself in my opinion.
In other words, in order to get cardinality beyond one, one has to be related to a thing that is beyond itself, which means that one is an open system.Quote:
Originally Posted by Jeroun
By using the word "system" we mean that the one still exists when it goes beyond itself, and we get an organic structure, which is both a one thing that has many leafs.
In that case the one (or the system) is not "a one of many " element, where a leaf is "a one of many " element.
If we research the logical foundation that stands in the basis of a thing that is not "a one of many " element, we can see that it is non-local by nature, and understood as NXOR or Not XOR connective.
In order to see it, let us research logically the concept of Membership, where its minimal condition is based on in,out relations:
Here is an example from set theory:
|{}| = 0 because only whet's in {} is counted (only the "one of many ..." objects, or their absence (as can be seen in this particular example)) where the invariant whole (the container) is not counted.
It is o.k. that it is not counted, but not because it is ignored, but because it is the whole (the NXOR product, which is not "a one of many " object)
As for the truth table of NXOR:
in out
0 0 → T (in , out are the same) = { }
0 1 → F
1 0 → F
1 1 → T (in , out are the same) = {}
As can be seen, we get T only if in , out are the same, and it does not matter if it is empty or full.
In this case we define a common logical basis for emptiness and fullness (they are two representations of the non-local element, or the whole, where the whole is not made of XOR products).
The logical basis of a part (a leaf) is based on XOR connective:
The truth table of locality is:
in out
0 0 → F
0 1 → T (in , out are not the same) = { }_
1 0 → T (in , out are not the same) = {_}
1 1 → F
The organic structure is based on NXOR\XOR logic.
Let system Z be the complementation between NXOR(non-locality) and XOR(locality).
The truth table of Z is:
in out
0 0 → T (in , out are the same) = { }
0 1 → T (in , out are not the same) = { }_
1 0 → T (in , out are not the same) = {_}
1 1 → T (in , out are the same) = {}
By system Z we may fulfill Hilbert's organic paradigm of the mathematical language. Quoting Hilberts famous Paris 1900 lecture:
The problems mentioned are merely samples of problems, yet they will suffice to show how rich, how manifold and how extensive the mathematical science of today is, and the question is urged upon us whether mathematics is doomed to the fate of those other sciences that have split up into separate branches, whose representatives scarcely understand one another and whose connection becomes ever more loose. I do not believe this nor wish it. Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts.
A clearer version of this stuff can be seen in http://www.geocities.com/complementa...digm-Shift.pdf .
Indeed this is a great philosophical question. It is something like I have read in eastern thought.Quote:
Originally Posted by DoronShadmi
Everything is in Brahma and indeed Brahma is in everything. And we too are in Brahma, and Brahma is also within us. The problem is of course with our understanding with this and nothing else.
Phylosophy have always look for the unity.
Human mind seems to dislike multiplicity.
jonics asked about the arkhé ,Aristotle explained the Substance (ousía),Neoplatonics talked about the ONE.
Leibniz thought about the Monade.
Even if we experience a multiple reality we look for the truly singular.
Most of philosopher consider multiplicity as appearence and that unity as real.
New ideas on Geometry and Maths changes this a bit.But not for to long.
Mathematicians have no trouble creating higher cardinalities, if you are using the same term that I am. Kantor was the first to get that ball rolling.
I was more referring to Lovachesky and Reeman and ther objection to the Euclidean Geometry and how theyr theories guide EISNTEIN to relativity matters.The result it is a multidimansional reallity.But it seems to be more undestandable as an abstraction or formula .it has profound concecuences on science and phylosophy.Quote:
Originally Posted by Walter
But not for too long.Cause we re always reason for singularity.
Kantor was the Concjunts one?:crash:
I think that mathematicians do not understand the universal principle, which is Symmetry\Asymmetry synthesis.Quote:
Originally Posted by Walter
Let us explore the traditional point of view about numbers (the values) and numerals (their representations).
According to the traditional point of view, any given R member (some real-number) is a unique value along the real-line (and so is some complex number (C member) in the complex plane).
This unique value can be represented by a unique and single symbol (for example let us use "@" for the ratio 1/4), but it is not useful to give a unique symbol for each given number, so during the years we developed several methods that produce numerals (representations of values) by systematic methods (actually this is similar to the informal way, which uses a finite amount of symbols in order to represent words and sentences according to certain rules).
The most useful one is the place value method that systematically uses a finite amount of basic symbols (their quantity is changed by the base value, for example: base-two has two basic symbols "0","1". Base-three has three basic symbols "0","1","2". Base-n has n basic symbols, ... etc.)
It is well known that the number of symbols that represent some value (some R member, in this case) can be changed if we use different bases to represent this value, and it does not matter if this value is a fraction or a whole number.
For example: in order to represent number five in base ten we need a single symbol, which is "5", but in order to represent number five in base three we need at least three symbols "101".
So, according to the traditional point of view "five", "5" or "101" are no more than different representations of the same value (where the value is the number itself).
Furthermore, it does not matter if we need infinitely many symbols or finitely many symbols in order to represent some value (some number) for example:
0.7[base 10] = 0.10110011001100110011001100110011001100110011... [base 2]
1 = 0.111...[base 2] = 0.222...[base 3] = ... = 0.999...[base 10] = ...
My mathematical framework is a complementation between VSL and ASL, where Mathematics of the past 2500 years is mostly based on ASL.
Here is a comparison between ASL and VSL, based on Dr. Linda Kreger Silverman's research (please look at the difference between what is called Visual Special Learning and Auditory Sequential Learning ( http://www.visualspatial.org/Articles/intro.pdf , http://www.visualspatial.org/Articles/idvsls.pdf ). ):
http://www.geocities.com/complementa...ry/VSL-ASL.jpg
Things are changed if Symmetry is used as the universal principle of logic.
Let a 2-valued framework be represented by A B.
A B relation is:
A B
A A
A B
B A
B B
These relations can be reduced (without a loss of generality) to
symmetric (AA or BB) \ asymmetric (AB or BA) relations, represented as:
AB
XX is AB symmetry --> A=B (AB is the same)
XY is AB asymmetry --> A≠B (AB is not the same)
A=B can be reduced (without a loss of generality) to a single value X.
A≠B cannot be reduced to a single value without a loss of detail X or Y.
Furthermore, in order to conclude that A≠B, they must share the same realm.
So 2-valued framework is at least X (symmetry or sameness) \ XY (asymmetry or difference) relation.
Let us use SA (Symmetry\Asymmetry relation) on 2-valued Logic:
T F NXOR
F F considered
F T not considered
T F not considered
T T considered
T F XOR
F F not considered
F T considered
T F considered
T T not considered
T F NXOR\XOR
F F considered
F T considered
T F considered
T T considered
T F generalization of NXOR\XOR
X X considered
X Y considered
Let us examine 3-valued logic:
http://www.geocities.com/complementa.../SymmLogic.jpg
Both cases are reduced to SA (Symmetry\Asymmetry relation) without a loss of generality:
3-valued logic can be extended to any x-valued logic where x is any standard or non-standard value of [0,1]. By using Symmetry\Asymmetry relation, we define the universal principle of any given logical system, standard or non-standard.Code:A B C
X X X --> X (symmetry)
X X Y --> X Y
X Y Z --> X Y Z (asymmetry)
By Symmetry, Symmetry is Asymmetry.
By Asymmetry, Asymmetry is_not Symmetry.
If there is no common basis, then anything is totally isolated, and nothing can be compared in order to define some difference. If there is only a common thing, then anything is totally connected, and nothing can be found beyond total unity. In both cases we do not get the researchable. So, the researchable is at least a unified isolation. Let us represent this notion by using an ASL\VSL representation method.
Let "=" be is
Let "≠" be is_not
Let "S" be Symmetry
Let "A" be Asymmetry
S=A from S point of view.
A≠S from A point of view, but in order to compare and conclude that A≠S, there must be a common basis that enables the comparison. So, the researchable is at least a unified isolation, represented as:
http://www.geocities.com/complementarytheory/OP.jpg
http://www.geocities.com/complementarytheory/SA.jpg
A particular value is defined only by NXOR(unification)\XOR(isolation) relation.
Since any particular value is at least NXOR\XOR relation, then any number is a particular path along the symmetric ur-element real-line.
For example:
http://www.geocities.com/complementa.../base-frac.jpg
In this case 0.01[base 2] is a particular path and 0.25[base 2] is another particular path.
By the traditional point of view a particular value (a number) is only the particular case of zenith-path along the symmetric ur-element real-line.
Because of this arbitrary limitation the non-zenith path 0.111...[base 2] is equal to the non-zenith path
0.222...[base 3] and both of them are defined as some representations of the zenith-path 1.000...
But when path is a general concept, then 0.111...[base 2] < 0.222...[base 3] < 1.000... as can be clearly seen in:
http://www.geocities.com/complementa...ry/base2_3.jpg
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This dialog is about the independency of mathematical\logical generalization of any particular representation method.
Here is some VSL example of the power of this independency:
I showed how 1.000 does not exist as distinct object if we deal with actual infinity, which is not less than total symmetry that does not allow the existence of any distinct object.
The school of Modern Mathematics simply does not understand this universal principle, and by mistake it thinks that it can save the identity of a distinct object when it deals with actual infinity.
If you disagree with me then please show how 1.000 exists as a distinct limit of a non-finite geometric series like 0.999 [base 10] that since it has the power of actual infinity (according to the traditional school of thought) there is no difference between 0.999 [base 10] and 1.000 , but somehow 1.000 still exists as distinct object.
Please look at this simple diagram:
http://www.geocities.com/complementarytheory/SA.jpg
By this diagram we clearly see that a non-finite series like 0.999 [base 10] (represented by magenta curve) is 1.000 (represented by a blue line) only in the case of total symmetry, which does not allow the existence of any distinct object.
The ignorance of the actual non-finite in order to get the requested result in the case of 0.999...[base 10] is no more than some trick that defines the conditions in order to solve some particular technical problem, which avoiding the full value of the researched mathematical system.
One can say that by using this trick we have developed the modern technology.
I say that this technology is nothing when compared to the technology of the consciousness, and this technology cannot be developed without understanding the non-finite, so the non-finite must not be ignored or misunderstood (in the case of the Cantorean system) if we are going to develop the technology of the consciousness in such a way that will not violate the Copernican principle ( http://en.wikipedia.org/wiki/Copernican_principle )
Only if False and True can be compared with each other, and it can be done only if there is a common basis (Symmetry) to different things (Asymmetry).Quote:
Logic is tautology
Only if you are limited to this particular representation method, but then the problem is not in the other method but it is simply your own limitation to get abstract notions which are independent of any particular representation method.Quote:
And that particular (ASL) representation method is more simple and straight-forward than another particular representation method.
As I said, the current school of thought is no more than a community of ASL-only persons that are unable to generalize mathematical\logical concepts that is not depend on ASL.
They cannot get off their ASL-only glasses and live under the illusion that Mathematics\Logic cannot be generalized unless ASL is used.
I regret if you went to all that effort only to respond to my post.
Much of what you say is elementary; much other uses verbal argumentation with too many different terms and meanings for me to assimilate. So I will readily grant that I may be one of the people who cannot understand your principle or what the problem is that you are trying to illuminate. However I sense, perhaps incorrectly given my limitations, that you are drawing a dichotomy between VSL and ASL as ways of thinking/learning. It is my impression that is a false dichotomy on the face of it, since mathematicians and analysts I have ever been exposed to use both. But, please, rest easy in your beliefs. My comments are much too casually thought out to to be worth engaging with the full armament of your system.
Ask any professional mathematician and you will find that he rejects any VSL representation as a legitimate formal method and accepts formalism only if it based on ASL representation.Quote:
Originally Posted by Walter
I am totally against this dichotomy.
:sick: !!!