Here is a list of my axioms, which are related to R: A definition for a point: A singleton set p that can be defined only by tautology ('='), where p has no internal parts. A definition for an interval (segment): A singleton set s that can be defined by tautology ('=') and ('<' or '>'), where s has no internal parts. The axiom of independency: p and s cannot be defined by each other. The axiom of complementarity: p and s are simultaneously preventing/defining their middle domain (please look at http://www.geocities.com/complementarytheory/CompLogic.pdf to understand the Included-Middle reasoning). The axiom of minimal structure: Any number which is not based on |{}|, is at least p_AND_s, where p_AND_s is at least Multiset_AND_Set. The axiom of duality(*): Any number is both some unique element of the collection of minimal structures, and a scale factor (which is determined by |{}| or s) of the entire collection. The axiom of completeness: A collection is complete if an only if both lowest and highest bounds are included in it and it has a finite quantity of scale levels. The Axiom of the unreachable weak limit: No input can be found by {} which stands for Emptiness. The Axiom of the unreachable strong limit: No input can be found by {__} which stands for Fullness. The Axiom of potentiality: p {.} is a potential Emptiness {}, where s {._.} is a potential Fullness {__}. The Axiom of phase transition: a) There is no Urelement between {} and {.}. b) There is no Urelement between {.} and {._.}. c) There is no Urelement between {._.} and {__}. Urelement (http://mathworld.wolfram.com/Urelement.html). The axiom of abstract/representation relations: There must be a deep and precise connection between our abstract ideas and the ways that we choose to represent them. (*) The Axiom of Duality is the deep basis of +,-,*,/ arithmetical operations. Tautology means x is itself or x=x. Singleton set is http://mathworld.wolfram.com/SingletonSet.html . Multiset is http://mathworld.wolfram.com/Multiset.html . Set is http://mathworld.wolfram.com/Set.html . (By the way the diagrams in my papers are also proofs without words http://mathworld.wolfram.com/ProofwithoutWords.html ) More detailes about my work can be found in: http://www.geocities.com/complementarytheory/No-Naive-Math.pdf The Axiom of the paradigm-shift: Within any consistent system, there is at least one well-defined set, which its content cannot be well-defined within the framework of the current system. Let us stop here to get your remarks.
Dear James R, I thought about something and I'll be glad to know your opinion. It goes like this: When two violins in the same room are tuned with each other, if we play on one of them we find that the strings of the other violin are also vibrate. Now let as say that intuition is our tuned instrument, and if a person expresses its intuitions by developing a way of thinking, the people that embrace this way of thinking probably share the same intuitions. On the basis of these common intuitions a community can be established. Let us say that this community is the first organization that deals with some part of the human knowledge, so in these early stages this community has no comparators on this part of the human knowledge. Quickly this community becomes the most developed organization, which holds this part of the human knowledge, and other parts of human civilization look at this organization as the one and only one possible intuition which standing in the basis of a one and only one way (school) of thought (and I am not talking about variations, which are actually different brunches of the same way of thought, or the same school of thought if you like). 2500 years are passing and this school of thought survives because of two main reasons: 1) This way of thought was fitting to the needs of the human civilization along these 'slow' (linear) years. 2) Any other alternative intuitions (if they where at all) where put aside because: a) They where not useful in their time. b) And if they where useful and also a real alternative to the current school, then the current school used its power and money to block this alternative intuition by forcing its educational methods on the public. We have to understand that intuitions cannot be learned, but a lot of external power can distort them until they lost their ability to be the source of a new school of thought. The 120 century is the time where our civilization moved from linear time to a non-linear time. In this time the power of few holds the destiny of our civilization, and most of their power is based on the technical abilities that where developed by this school of thought, that was established 2500 years ago. But our technical achievements, which are not balanced by another ways of thought, are like a government with no opposite. We have learned that evolution needs diversity; otherwise we quickly get a dead planet. The field of evolution in our non-linear time splits to "hardwhere" and "softwhere" parallel pathes, where the hardwhere side is our technology and the softwhere side is our morality. We can clearly see that there is no balance between the levels of these two paths, and this lack of balance in a non-linear time can quickly lead us to a dead-end street. Therefore I think that we have to do the best we can to find the balance between our morality level and our technical abilities. The first place that binds both paths is the language of mathematics. In my opinion people which learn this powerful language, must first of all to develop their moral abilities by opening themselves to another intuitions which are not their intuitions and let them flourish in their communities. By this way we develop our tolerance and learn how to live side by side, and if other intuitions are better then our intuition in this period of time, we do our best to help them flourish instead of trying our best to shut them down. And we have the motivation to do that because we understand that we are all in the same boat. My intuitions and ideas about the language of mathematics are different then the standard school of mathematics. But in my opinion the most important difference, which I think fits to our non-linear time (more then the standard school) is that I include the mathematician cognition's ability to develop Math as a part of the mathematical research. By this self-reference attitude I hope to develop the gateway that can connect between our moral abilities to our technical abilities. And for that I need your help. What do you think?
Doron, I am sympathetic to your aims, but I don't agree with you that mathematics has any obvious connection to morality. In fact, I can't think of how such a connection could arise in any straightfoward way. You seem to have put a lot of thought, and presumably a lot of time, into developing your ideas. Axioms in mathematics are assumed truths which are thought not to require proof because they are, in some sense, obvious. A system of axioms demonstrates its usefulness in terms of the results which can be proved within the system. For example, from Euclid's 5 axioms, the whole of plane geometry can be proved. Interestingly, if we replace Euclid's axiom of parallelism, we open up whole new realms of non-Euclidean geometries. Of course, both Euclidean and non-Euclidean geometries are immensely useful ideas, since a wealth of theory and applications flow from these systems. As a mathematician, I am happy to accept your axioms (though I really should check to see if they are self-consistent) as a possible basis for a mathematical system. Presumably, you have some aim in mind in using this set of axioms rather than some other set. So, I ask you again: what can you prove using your axioms? What useful results follow from them?
Dear James R, First, thank you for your reply. As I wrote, my aim goes beyond technical proofs, so to understand me better, let us look at both moral and technical abilities, which may be exist in this powerful and beautiful language. I started my private research about 25 years ago, by asking myself, what are the minimal conditions that give us our abilities to count. You can find this research in: http://www.geocities.com/complementarytheory/ONN1.pdf http://www.geocities.com/complementarytheory/ONN2.pdf http://www.geocities.com/complementarytheory/ONN3.pdf An overview of my basic perceptions of the Language of Mathematics and its reasoning, can be found in: http://www.geocities.com/complementarytheory/No-Naive-Math.pdf On the other hand, here is an example of how we maybe find a gateway to our morality by using methods, which are based on mathematical thinking: http://www.geocities.com/complementarytheory/Moral.pdf As for your question about the usefulness of my system: Today's number system is a quantity-only system, which ignores the internal complexity of the natural numbers (and I do not mean to the differences between primes, non-primes, odds , evens, partitions, permutations, etc..., which are all based on 0_redunduncy_AND_0_uncertainty building-blocks), which are the building-blocks of the entire standard system. In short, my number system is based on the information concept, where each building-block in it has an internal structure that cannot fully described only by quantitative-only and 0_redunduncy_AND_0_uncertainty approach of the standard system. The main concept of my new number system is based on the complementary relations that exist between symmetry level and information's clarity-level, and these relations are based on what I call complementary-logic, which is based on included-middle reasoning, and both excluded-middle reasoning and fuzzy logic are limited proper sub-systems of it. By my system we get these benefits: 1) Each building-block has a unique internal complexity, that can be the basis for infinitely many unique building-blocks, which can be found upon infinitely many different scales. 2) There are infinitely many unique internal structures that can be found in some particular scale level. 3) There can be infinitely many complex structures, that are based on (1) and (2) building- blocks. 4) These complex structures are much more accurate models then any model which is based on the quantitative-only standard number system, and some of the reasons are: a) The structure that is based on the complementary relations between symmetry and information concepts (where redundncy_AND_uncertainty are useful properties of them) is inherent property of my new system, and gives it the ability to understand the deepest principles of any dynamic/structural abstract or non-abstract complex object, without first reducing it to quantitative model (which is inevitable when we use the standard quantitative-only number system). b) The new natural numbers (which are now taken as topological information's building-blocks) are ordered as Mendeliev-like table, which gives us the ability to define their deep topological connections, even before we use them in some particular model. These deep topological connections can be used as gateways between so-called different models, and expending our understanding about these explored models. c) My number system is the first number system, which is based on our cognition’s ability to count, as an inherent property of the abstract concept of a number. By this research I have found and described how the number concept is based on the interactions between our memory and some abstract or non-abstract elements. Through this approach our own cognition is included in the development of the Language of Mathematics, and we are no longer observers, but full participators where our own congenital abilities are legitimate parts of the mathematical research itself. For example: What is called a function is first of all a reflection of our memory on the explored elements. A function is the property that gives us the ability to compare things and get conclusions that are based on this comparison. If something is compared by us to itself, we get the self identity of an element to itself by tautology (x=x). If more then one element is compared, then we get several information clarity degrees that describe several possible interactions between our memory and the explored objects, and these several possible interactions can be ordered by their internal symmetrical degrees. In this case multiplication and addition operations are complementary operations, where multiplication can be operated only between identical elements (redundancy_AND_uncertainty > 0) and addition is operated between non-identical elements (redundancy_AND_uncertainty = 0). Because any function (which is not based on self reference of an element to itself) is a connection between at least two elements, its minimal abstract model cannot be less then a pointless line-segment, which is used as a connector between the examined elements. In this case no interval (memory) can be described in terms of points (objects) and vise versa, and we get these four independent building-blocks of the language of Mathematics (which now includes the mathematician’s cognition-abilities as a legitimate part of it): {}, {.}, {._.}, {__} By this new approach we can build, for example, a totally new Turing-like machine, that can change forever our abilities to deal with complexity which is based in simplicity. Please look at my website http://www.geocities.com/complementarytheory/CATpage.html if you want to understand more. Let us examine some different interpretations between my approach and the conventional approach: Doron: The main idea behind the integers (unless we choose to change it) is to look on the number line as if it has a one and only one scale factor, which its value is 1 and only 1. In this case any arbitrary interval cannot be but 1 (or -1 if we take zero's left side). For example: ...___-2___-1___0___1___2___3___4___5___6___... ...___-2___-1___0___1___2___3___4___5___6___... 3___4 < 2___3___4___5 --> [3,4]<[2,5] by the new approach. By this approach no proper subset of N can be put in 1-1 correspondence with the entire N, for example N and its odds: Code: ...___1___2___3___4___5___6___7___8___9___... (Entire N) | | | | | ...___1_______3_______5_______7_______9___... ( Entire Odds) In the standard way the interval {.__.} is omitted and we get: Code: ... 1 2 3 4 5 6 7 8 9 ... (Entire N) | | | | | | | | | ... 1 3 5 7 9 11 13 15 17 ... ( Entire Odds) As we can clearly see, standard math does not find 1-1 map between numbers, but between their represented notations, and we can clearly see that the standard point of view does not distinguish between a number and its represented notation. Also: 2 <-> 3 5 <-> 4 and in this case (where {._.} is omitted) [3,4] = [2,5] by standard math. ------------------------------------------------------------------------- When [3,4] and [2,5] are taken as R members then the inifinitely many elements that exist between 3 to 5 and 2 to 5 in infinitely many different scales, can be put in 1-1 and onto, and in this case [3,4]=[2,5] because of the duality of each R member, which is clearly explained here: http://www.geocities.com/complementarytheory/No-Naive-Math.pdf ------------------------------------------------------------------------- Infinitely many elements in infinitely many scales have bigger cardinality then infinitely many elements that can be found in a one and only one particular scale (scale 0 is excluded in both cases). Therefore |N|<|Q|<|R| where each number is at least {.}_AND_{._.} (as can be seen in http://www.geocities.com/complementarytheory/No-Naive-Math.pdf).
A chain of shadows Please look at the attached pdf http://www.geocities.com/complementarytheory/Roots-Chain.pdf . By this model we can see that √1 is the "shadow" of √2 and √2 is the "shadow" of √3. I think that we can conclude that √3 is the "shadow" of √4 ... and so on. In short, I am talking about roots which each one of them is the diagonal of its dimension level, where each n_dim diagonal is the "shadow" of n+1_dim diagonal. We have a chain of "shadows" between infinitely many diagonals in |N| dimension levels. Do you think that this "Chain of Shadows" has any mathematical/physical meaning?
Doron, I don't understand your claim that, according to standard mathematics, [3,4]=[2,5]. I would say that this expression is undefined in standard mathematics. If, by the notation [3,4] you mean the interval which includes all numbers from [3,4], then to compare this to any other interval you need to define a measure of the interval. For example, one possible measure would be: f([x,y]) = |x-y| By this measure, we would have: f([3,4]) = |3-4| = 1, and f([2,5]) = |2-5| = 3. We can then validly compare the measures and say f([2,5]) > f([3,4]) Alternatively, we might require that [x,y] = [p,q] iff x=p and y=q, in which case it is only possible to test for equality or inequality of two intervals. It is not clear to me by what process you say that [3,4] < [2,5]. Perhaps you can explain.
It is very simple: [3,4] and [2,5] are closed intervals. By standard Math if these two intervals are related only to N members, then there is nothing between the numbers of each interval and we get 1-1 and onto map of: 3 <--> 2 4 <--> 5 In short, we get the same cardinality for both closed intervals. And the reason for this is: In standard Math an interval {._.} is defined by points {.}, where in my new system I have this axiom: The axiom of independency: p and s cannot be defined by each other. Since in my system {.} and {._.} are independed, then [3,4] < [2,5]. Please remember that in both cases we are talking only on N members.
Doron, You seem to be using the word "interval" in a strange way. When you refer to the "interval" [2,5], do you mean the set of integers {2,3,4,5}, or do you mean all real numbers between 2 and 5, inclusive? If you mean the set of integers, then the set has cardinality 4, which is obviously greater than the cardinality of the set of two integers {3,4}. If, on the other hand, you mean all real numbers between 2 and 5, then the cardinality of [2,5] is the same as the cardinality of [3,4], which would be c, or Aleph<sub>1</sub>, if you believe the continuum hypothesis. Of course, that's standard mathematics, and perhaps what you are doing is suggesting a different way to determine the cardinality of a set of real numbers. That's what I'm not sure about.
[2,5] is {2,5}. [3,4] is {3,4}. It means that all we have is the lowest and uppest bounds where nothing exists between them. This is the way of how by standard Math we can find 1-1 and onto between the entire N members and some proper subset of N members Please read this again: The main idea behind the integers (unless we choose to change it) is to look on the number line as if it has a one and only one scale factor, which its value is 1 and only 1. In this case any arbitrary interval cannot be but 1 (or -1 if we take zero's left side). For example: ...___-2___-1___0___1___2___3___4___5___6___... ...___-2___-1___0___1___2___3___4___5___6___... 3___4 < 2___3___4___5 --> [3,4]<[2,5] by the new approach. By this approach no proper subset of N can be put in 1-1 correspondence with the entire N, for example N and its odds: Code: ...___1___2___3___4___5___6___7___8___9___... (Entire N) | | | | | ...___1_______3_______5_______7_______9___... ( Entire Odds) In the standard way the interval {.__.} is omitted and we get: Code: ... 1 2 3 4 5 6 7 8 9 ... (Entire N) | | | | | | | | | ... 1 3 5 7 9 11 13 15 17 ... ( Entire Odds) As we can clearly see, standard math does not find 1-1 map between numbers, but between their represented notations, and we can clearly see that the standard point of view does not distinguish between a number and its represented notation.
james, i think what he is trying to show that standard math does not find a 1-1 map between numbers (as envisioned by him using a number line concept - that's how i think of it). i think i can illustrate his definition of one-to-one. imagine a number line like a ruler, and if you take two rulers together, one-to-one means that the numbers line up directly to 1,2,3,4,5. since his definition of enumeration states that a ruler like number line must line up with 1,2,3,4,5 so he says that there can be no one-to-one relation between all integers, and all odd integers, since 3 lines up to 3 and not 2 and thus leaves 2 without a member to enumerate. therefore, if you believe, like he does, that numbers are more than just symbols but that they have some inner structure, say this structure is represented by the length of a part of the ruler around the number, then standard math cannot find one-to-one map because it compares the symbols 1,2,3,4, and not the actual numbers themselves (represented here by the length of the ruler around the symbol 1,2,3,4). then it is clear... i think Please Register or Log in to view the hidden image!
i still don't understand what is the benefit of this concept? it seems to reduce the abstractness of mathematics. after all, if you cannot enumerate numbers, what can we do?
Dear fallen angel, this is a perfect explanation of my idea about numbers. Let us think about Mandelbrot set http://aleph0.clarku.edu/~djoyce/julia/julia.html . The set itself is the black areas, where no information can be found. This black area is the invariant or the constant side of Mandelbrot set, but the other side of it is its border area, where the interesting information is created when Mandelbrot set gradually disappearing at infinity. No one of these sides can be ignored if we want to understand what is a Mandelbrot set. The same approach has to be used if we want to understand what is R collection. Any R member is a unique (invariant and constant) element in the collection, but on the same time each constant is a scale factor of the entire R collection. It means that the entire R collection exists between two opposite states (minus is the mirror -not the opposite- of plus side). In one state, when 0 is the scale factor, no R member except 0 can be found. On the other state No R member can be found when we reach oo (as clearly can be shown here: http://www.geocities.com/complementarytheory/RiemannsLimits.pdf ). Furthermore, because of this duality of any R member, we get a system which is both absolute (when a single scale is examined) and relative (where the same place of the real line is examined simultaneously on several different scales). Another example: Pi = the relations between the perimeter and the diameter of a circle. Pi is invariant in any arbitrary given scale, but when several scale levels a simultaneously compared, we can clearly see that each circle has a different curvature. If our system is a circle, then if we want to understand what is a circle, then both its invariant and variant properties cannot be ignored. (We also have to be aware to the fact the no circle can be found when Diameter or Perimeter = 0, or Diameter or Perimeter = oo. In short, our basic approach is to find the gateways between opposite properties, and the best way to do it, is by an including-middle logical reasoning (http://www.geocities.com/complementarytheory/CompLogic.pdf). By my number system, you do not have to reduce any abstract or non-abstract element to a quantitative model, in order to analyze and conclude some meaningful and useful things about it. And the reason is, my number system is both structural/quantitative information system, where each element of it is examined by the Symmetry concept, which is the most powerful tool of the language of Mathematics. In my system Symmetry degree and information clarity degree have complementary relations, where each concept simultaneously preventing/defining the other concept. The result of these complementary interactions is infinitely many gateways to infinitely many topologies, where each one of them can be a building-block for another logical reasoning. If we ignore the structural property of a given number and look only on its quantity, then this point of view is not more abstract then the structural/quantitative point of view, but more trivial then the structural/quantitative point of view. We don't have to be happy if there is a way to take two complex systems like, for example, two persons, and then to say that we have two objects. This is a trivial point of view of these persons, and by my number system we can choose if we want to ignore or not their internal complexity. Therefore the result of this kind of view is richer then the standard quantitative-only point of view, for example: By this new approach we can build, for example, a totally new Turing-like machine, that can change forever our abilities to deal with complexity which is based in simplicity. Please read post #4 of this thread, and also the list of my axioms (post #1) thank you.
No dear James R you are in a very god company. I did not meet yet a professional mathematician, which allowed himself to understand this stuff, because at the moment that he understand my theory his point of view will immediately be changed by a paradigm-shift.
Fist let us write again our last post: Tautology: x implies x (An example: suppose Paul is not lying. Whoever is not lying, is telling the truth Therefore, Paul is telling the truth) http://en.wikipedia.org/wiki/Tautology. (tautology is also known as the opposite of a contradiction). Set: A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is also ignored. Multiset: A set-like object in which order is ignored, but multiplicity is explicitly significant. Singleton set: A set having exactly one element a. A singleton set is denoted by {a} and is the simplest example of a nonempty set. Urelement:(no internal parts) An urelement contains no elements, belongs to some set, and is not identical with the empty set http://mathworld.wolfram.com/Urelement.html. {.} is both a Singleton set and a Urelement. A definition for a point: A singleton set p that can be defined only(*) by tautology ('='), where p has no internal parts. (*) only by tautology means: the minimal possible existence of a non-empty set. ----------------------------------------------------------------------------------------------- Now let us move to the next step in order to define what is a number in my system. First let us examine a well-known relation between mathematical objects and their relations. =>> is ‘represented by’ |{}|=>>0 ; |{{}}|=>>|{0}|=>>1 ; |{{},{{}}}|=>>|{0,{0}}|=>>|{0,1}|=>>2 ; |{{},{{},{{}}}}|=>>|{0,{0,{0}}}|=>>|{0,1,2}|=>>3 ; … A definition for an interval (segment): A singleton set s that can be defined by tautology ('=') and ('<' or '>'), where s has no internal parts. (Sign '<' means that we look at the segment from left to the right. Sign '>' means that we look at the segment from right to the left. When both '<' , '>' are used then we have a directionless segment.) By the definition of a segment we get {._.}, which is the indivisible singleton set that exists between any two {.}. Now we have the minimal building-blocks that allows us to define the standard R members. The axiom of independency: p and s cannot be defined by each other. By the above axiom {.} and {._.} are independed building blocks. The axiom of complementarity: p and s are simultaneously preventing/defining their middle domain (please look at http://www.geocities.com/complementarytheory/CompLogic.pdf to understand the Included-Middle reasoning). By the above axiom we define the basic property of the middle domain between {.} and {._.} The axiom of minimal structure: Any number which is not based on |{}|, is at least p_AND_s, where p_AND_s is at least Multiset_AND_Set. The above axiom allows us to: 1) To define the internal structure of standard R members. 2) To define the internal structures of my new number system. The axiom of duality(*): Any number is both some unique element of the collection of minimal structures, and a scale factor (which is determined by |{}| or s) of the entire collection. The above axiom allows us to construct a collection of R members and also a collection of my new number system. First, let us see how we use my method to construct a collection of R members. R members are constructed like this: 1) First let us examine how we represent a number by my system: =>> is ‘represented by’ a) |{}|=>>0 b) There is 1-1 and onto between ‘0’ and the left point of {._.} and we get {‘0’_.} c) |{{}}|=>>|{0}|=>>1 e) There is 1-1 and onto between ‘1’ and the right point of {._.} and we get {‘0’_’1’} In short, {.} is the initial place of R collection, which is represented by ‘0’, where {‘0’_.} is the initial place of the second place of R collection, which is represented by ‘1’, and we get our first two must-have building-blocks of R collection. 2) When we get {‘0’_’1’} we have our two must-have numbers, which are ‘0’ and _’1’. Be aware that ‘0’ is the representation of {.} where ‘1’ is the representation of {._.}. 3) If we get {.}_AND_{._.}, then and only then we have the minimal must-have information to construct the entire R collection because: a) We have ‘0’ AND _’1’ that give us the to basic scale factors 0 and _1. b) We also have our initial domain _1, which standing in the basis of any arbitrary scale factor that is determined by the ratio between the initial domain _1 and another segment that is smaller or bigger than the initial domain _1 , for example: Code: 0 = . 1 = 0[COLOR=Blue]______1[/COLOR] 2 = 0[COLOR=DarkRed]____________2[/COLOR] 3 = 0[COLOR=Green]___________________3[/COLOR] .5 = 0[COLOR=Red]__.5[/COLOR] pi = 0[COLOR=Magenta]______________________pi[/COLOR] The negative numbers are the left mirror image of the above numbers. There is no division in my number system because both {.} and {._.} are indivisible by definition. In short, any segment is an independent element, that clearly can be shown in the above 2-D representation. If we use a 1-D representation, we get the standard Real-line representation, but then we can understand that division is only an illusion of an overlap of independent elements when they are put on top of each other in a 1-D representation, for example: Code: 0[COLOR=Red]__.5[/COLOR] [COLOR=Blue]__1[/COLOR][COLOR=DarkRed]_____2[/COLOR][COLOR=Green]_____3[/COLOR][COLOR=Magenta]__pi[/COLOR] (*) The Axiom of Duality is the deep basis of +,-,*,/ arithmetical operations. Since in my system nothing is divisible, then '/' stands for a ratio between at least any given two (indivisible) numbers. ----------------------------------------------------------------------------------- Let us stop here (before we continue to my new number system) to get your remarks.
