- #36
Doron Shadmi
Hi Carla,
My overview describes a non-conventional perception of the continuum concept, and the minimal structure of a NUMBER.
As I wrote to you in the previous post, through my point of view, Math Langauge must include our cognition’s abilities to develop it, as a legal part of its research.
By doing this, we may avoid some possible hidden assumptions that can be in the basis of our axiomatic systems.
Through this attitude, I have found that the minimal conditions that gives us the ability to identify and count elements, is strongly based on our ability to associate between some counted elements and our memory.
If ,by analogy, "some elements" means beads, and our memory is a string, then any number which is not 0, can't be less than some necklace.
Most of the Modern Math axiomatic systems are based on the SET concept.
A SET is a finite or infinite collection of objects in which order has no significance, and multiplicity is generally also ignored.
We notate a SET by {}.
The simplest set is the empty set = {} , which means a SET with on objects.
By Math language we use the word "members" instead of "objects".
So, any non-empty set includes some finite or infinite collection of members, and its name depends on some common property of these members.
When mathematicians (the first one was Cantor) researched the properties of some infinite collections of numbers, they discovered that there is more than one level of infinite.
Actually there are infinitely many levels of infinities.
The first infinite level is called the countable infinity and the second type that have been discovered is called the uncountable infinity.
The continuum is the uncountable infinity and above it there are infinitely many levels of infinities, that Modern Math research tries to find and define.
Another question in Modern Math is the Continuum Hypothesis (CH), which tries to find if there is or there is not some infinite set between the countable and the uncountable sets.
Modern Math describes the Continuum as: "Infinitely many points with no gaps between them"
Through my research I have found that it is impossible to define the Continuum by a tool, which has exactly it opposite property.
A continuous line is a non-localized element, and a point is a localized element, so any exploration of a continuous line by infinitely many points, is as if we say:
"If you want to see the darkness, please turn on the lights".
Instead of forcing the continuum to be expressed by its opposite, I associate them without forcing one opposite property on the other, and got a new points of view on these concepts:
Continuum, Discreteness, Number, Infinity, Information, Symmetry and Cognition's abilities to create Math, as a part of Math research.
Professional mathematicians will not accept it, because I change a lot of fundamental paradigms of Modern Math, by this point of view.
Yours,
Doron
My overview describes a non-conventional perception of the continuum concept, and the minimal structure of a NUMBER.
As I wrote to you in the previous post, through my point of view, Math Langauge must include our cognition’s abilities to develop it, as a legal part of its research.
By doing this, we may avoid some possible hidden assumptions that can be in the basis of our axiomatic systems.
Through this attitude, I have found that the minimal conditions that gives us the ability to identify and count elements, is strongly based on our ability to associate between some counted elements and our memory.
If ,by analogy, "some elements" means beads, and our memory is a string, then any number which is not 0, can't be less than some necklace.
Most of the Modern Math axiomatic systems are based on the SET concept.
A SET is a finite or infinite collection of objects in which order has no significance, and multiplicity is generally also ignored.
We notate a SET by {}.
The simplest set is the empty set = {} , which means a SET with on objects.
By Math language we use the word "members" instead of "objects".
So, any non-empty set includes some finite or infinite collection of members, and its name depends on some common property of these members.
When mathematicians (the first one was Cantor) researched the properties of some infinite collections of numbers, they discovered that there is more than one level of infinite.
Actually there are infinitely many levels of infinities.
The first infinite level is called the countable infinity and the second type that have been discovered is called the uncountable infinity.
The continuum is the uncountable infinity and above it there are infinitely many levels of infinities, that Modern Math research tries to find and define.
Another question in Modern Math is the Continuum Hypothesis (CH), which tries to find if there is or there is not some infinite set between the countable and the uncountable sets.
Modern Math describes the Continuum as: "Infinitely many points with no gaps between them"
Through my research I have found that it is impossible to define the Continuum by a tool, which has exactly it opposite property.
A continuous line is a non-localized element, and a point is a localized element, so any exploration of a continuous line by infinitely many points, is as if we say:
"If you want to see the darkness, please turn on the lights".
Instead of forcing the continuum to be expressed by its opposite, I associate them without forcing one opposite property on the other, and got a new points of view on these concepts:
Continuum, Discreteness, Number, Infinity, Information, Symmetry and Cognition's abilities to create Math, as a part of Math research.
Professional mathematicians will not accept it, because I change a lot of fundamental paradigms of Modern Math, by this point of view.
Yours,
Doron
Last edited by a moderator: