Guest Doron Shadmi Posted October 8, 2004 Share Posted October 8, 2004 A non technical explanation of Monadic Mathematics' '+' and '-' operations Let me explain my framework without technical formal definitions. Monadic Mathematics exists between two opposites which are its limitations. By limitations I mean that it cannot work beyond these opposites. The opposites are Emptiness (which is notated as {}) and Fullness (which is notated as {__}). {} or {__} cannot be used as inputs in Monadic Mathematics framework. Let us call Monadic Mathematics MM. The fundamental concepts that are used in MM are Length, Direction, Quantity. If we examine R from MM point of view, then we get this picture: Each number is a unique Length with a unique Direction. The most simple case is the length of a Point {.} which its length is 0 and it has no direction. The other useful case is a Segment {._.} where each Segment has its unique Length and Direction. Any Segment that starts at Point 0 can have at least two possible directions, for example: Number one can be 0_1 and this case is equivalent to +1 or number one can be 1_0 and this case is equivalent to -1 Addition and subtraction operations: There are 4 possible results by '+' or '-' operations: 1) Concatenation (which a part of it is equivalent to standard addition): For example: 0_1 + 0_1 = 0__2 0_1 + 1_0 = 1_0_1 0 + 0_1 = 0_1 0 + 0 = 0 In this case the second value is called Urelement Cardinal. 2) Sets Addition: Can be operated only between sets contents, for example: {0} + {0} = {0, 0} {1_0} + {0} + {0_1} = {1_0, 0, 0_1} 3) Elimination (which a part of it is equivalent to standard subtraction): For example: 0_1 - 0_1 = {} 0_1 - 1_0 = 0 0 - 0_1 = 0_1 0 - 0 = {} 0__2 - 0_1 = 0_1 4) Sets Subtraction: Can be operated only between sets contents, for example: {0} - {0} = {} {1_0} - {0} - {0_1} = {1_0} {1_0, 0, 0_1} - {0} = {1_0, 0_1} More details can be fount in http://www.geocities.com/complementarytheory/My-first-axioms.pdf I'll appreciate very much your remarks and insights, thank you. ---------------------------------------------------------------------------- EDIT (16/10/2004): One of the efficient ways to change some framework is to ask new fundamental questions that maybe lead us to new frontiers. For example, the fundamental question of the language of Mathematics framework is: ”How many?”. Let us try another question, for example: “What do we have?” A closer look of these questions shows that ”How many?” questions are mostly about the Quantity concept, where “What do we have?” questions are mostly about the Structure concept. Let us check if “What do we have?” questions can be a fruitful ground for mathematical development. First we have to define the minimal concepts that can be used under the structure concept. In other words, these concepts when used, determine our framework’s domain, for example: Let use say that the two main concepts that are related to the structure concept are Length and Direction. It means that by using these concepts, we can define the building-blocks of our mathematical framework. We also know that by using these building-blocks we suppose to get some input that can be used by us to develop our framework. So, the next question is “What are the limits that beyond them no input can be found?” The lowest “no input” state is Emptiness, where Length and Direction do not exist. The highest “no input” state is Fullness, where Length and Direction are beyond measurement. So the useful elements have measurable Length and/or Direction, that enable us to use them as some input. Let us represent these ideas in a table: Monadic Mathematics build-blocks -------------------------------- V = Available X = Not Available +-----------------------------+ | Measurement of | | | | Length | Direction | |--------------|--------------| | | | {} | | | Emptiness | | | | | | |--------------|--------------| | | | {.} | V | | Point | | | | | | |--------------|--------------| | | | {._.} | V | V | Segment | | | | | | |--------------|--------------| | | | | | V | Tendency | | | | | | |--------------|--------------| | | | {__} | X | X | Fullness | | | | | | +-----------------------------+ General: {' and '}' are the notations of a framework which we call a set. Between these notations we can put our examined concepts and then try to find out what we can do with each one of these concepts, and also what interactions can be found between concepts and themselves and/or concepts with other concepts. The concept of Nothingness or Emptiness is notated in this framework by {}. Any examined concept in my framework is examined by its structural properties and also by its Quantitative properties. The most basic Structural property in my framework is based on the Length concept. From this point of view, a Point {.} has exactly 0 Length. The Length concept has no meaning in my framework when it is related to the Emptiness concept (which is notated as {}), therefore 0 cannot be connected directly to the Emptiness concept. Another option to define 0 is to ask: How many things there are in {}? By 'how many?' question we actually define the Cardinal concept, which is notated by using '|' and '|' . In this case, the cardinal of {}, which is notated as |{}|, is equal to 0. In Standard Mathematics framework (when Ordinality is omitted) the one and only one option to connect between the Number concept and the Set concept, is by using only the Quantity concept, and in this case |{}| = 0. But as you see, in Monadic Mathematics framework, there are two kinds of cardinals, where one of them is the standard Quantitative Cardinal, but the second type of cardinal is what I call the Urelement Cardinal, which is based on the Length concept (which is the structural property of the Number concept). In Monadic Mathematics, the structural property of a number is more basic then its quantitative property. According to what I wrote above, when a Point eliminates itself, then the result is Emptiness. In other words Point - Point = Emptiness, or in other representation: 0 - 0 = {}. Standard Mathematics takes '0' notation as something which is first of all related to the Quantity concept. In this framework 0 - 0 = 0 Monadic Mathematics takes '0' notation as something which is first of all related to the Structure concept. In this framework 0 - 0 = {} A non technical explanation of Monadic Mathematics' '+' and '-' operations: The most basic question of Standard Mathematics is “How many?”. The most basic question of Monadic Mathematics is “What do we have?” The basis of a “How many?” question is the Quantity concept. The basis of a “What do we have?” question is the Structure concept. It means that the Number concept in Monadic Mathematics, is first of all based on the Structure concept. Question: What are the minimal, and distinguished, structural forms in Monadic Mathematics? Answer: Emptiness (notated as {}), Point (notated as {.}), Segment (notated as {._.}), Fullness (notated as {__}). {} and {__} are the weak ({}) and the strong ({__}) limits of Monadic Mathematics framework. It means that they cannot be used as inputs in Monadic Mathematics. So, Monadic Mathematics operations are based on {.} and {._.}. The two basic structural properties, which related to the Number concept, are Length and/or Direction. {.} has 0 length and no directions. {._.} has 0_x length and at least two opposite directions, which are 0_x or x_0. The most basic arithmetical operations between these elements are ‘+’ and ‘-‘. Important: If '-' sign is used not as a binary operation, for example: -x_0, then it is a negation symbol and not a binary Elimination operation. Therefore -x_0 = 0_x or -0_x = x_0. Also {} - x_0, which is a vacuous binary operation (because {} or {__} cannot be used as inputs) is equal to -x_0 = 0_x, etc. ‘+’ is understood as Concatenation, for example: 1_0 + 1_0 = 2__0 (._. + ._. = .__.) 0_1 + 1_0 = 1_0_1 (._. + ._. = ._._.) 0 + 0 = 0 ( . + . = . because Concatenation of 0 Length is 0 Length) 0__2 + 0_1 = 0___3 (.__. + ._. = .___.) 2__0 + 1_0 = 3___0 (.__. + ._. = .___.) 0_1 + 2__0 = 2__0_1 (._. + .__. = .__._.) etc … ‘-’ is understood as Elimination, for example: 1_0 - 1_0 = {} (._. - ._. = {} because both Length and Direction are identical) 0_1 - 1_0 = 0 (._. - ._. = . because only Length is identical) 0 - 0 = {} ( . - . = {} because Length is identical and there is no Direction) 0__2 - 0_1 = 0_1 (.__. - ._. = ._.) 2__0 - 1_0 = 1_0 (.__. - ._. = ._.) 0_1 - 2__0 = 1_0 (._. - .__. = ._.) etc … More detailed information can be found in: http://www.geocities.com/complementarytheory/My-first-axioms.pdf According to Monadic Mathematics the Number concept is the fundamental building-block of a non-destructive interaction between opposites, which is based on Included-middle reasoning (which is the logic that can be found between at least two opposites that define their middle domain). For more details please read: http://www.geocities.com/complementarytheory/GaloisDialog.pdf Link to comment Share on other sites More sharing options...
haggy Posted October 8, 2004 Share Posted October 8, 2004 Do your arms get tired from all the hand-waving? Link to comment Share on other sites More sharing options...
matt grime Posted October 8, 2004 Share Posted October 8, 2004 And he's not even demonstrated that there is a model of R in Monadic Maths either. Or in fact that there is anything that is a model of anything that exists in a model of MM, never mind anything as complicated as the real numbers, or addition. Link to comment Share on other sites More sharing options...
Guest Doron Shadmi Posted October 8, 2004 Share Posted October 8, 2004 Matt, I do more then that, please read pages 22 - 31 of http://www.geocities.com/complementarytheory/My-first-axioms.pdf where I clearly and simply demonstrate the priority of the Monadic Mathematics arithmetic over the Standard arithmetic. Link to comment Share on other sites More sharing options...
Guest Doron Shadmi Posted October 8, 2004 Share Posted October 8, 2004 Do your arms get tired from all the hand-waving? Dear Haggy, a special post for you. {' and '}' are the notations of a framework which we call a set. Between these notations we can put our examined concepts and then we try to find out what we can do with each one of these concepts, and also what interactions can be found between concepts and themselves and/or concepts with other concepts. The concept of Nothingness or Emptiness is notated in this framework by {}. Any examined concept in my framework is examined by its structural properties and also by its Quantitative properties. The most basic Structural property in my framework is based on the Length concept. From this point of view, a Point {.} has exactly 0 Length. The Length concept has no meaning in my framework when it is related to the Emptiness concept (which is notated as {}), therefore 0 cannot be connected directly to the Emptiness concept. Another option to define 0 is to ask: How many things there are in {}? By 'how many?' question we actually define the Cardinal concept, which is notated by using '|' and '|' . In this case, the cardinal of {}, which is notated as |{}|, is equal to 0. In Standard Mathematics framework (when Ordinality is omitted) the one and only one option to connect between the Number concept and the Set concept, is by using only the Quantity concept, and in this case |{}| = 0. But as you see, in Monadic Mathematics framework, there are two kinds of cardinals, where one of them is the standard Quantitative Cardinal, but the second type of cardinal is what I call the Urelement Cardinal, which is based on the Length concept (which is the structural property of the Number concept). In Monadic Mathematics, the structural property of a number is more basic then its quantitative property. According to what I wrote above, when a Point eliminates itself, then the result is Emptiness. In other words Point - Point = Emptiness, or in other representation: 0 - 0 = {}. Standard Mathematics takes '0' notation as something which is first of all related to the Quantity concept. In this framework 0 - 0 = 0 Monadic Mathematics takes '0' notation as something which is first of all related to the Structure concept. In this framework 0 - 0 = {} Can you understad what I am talking about? Link to comment Share on other sites More sharing options...
haggy Posted October 8, 2004 Share Posted October 8, 2004 Wow, Doron, your arms are so big. I envy you. Link to comment Share on other sites More sharing options...
Guest Doron Shadmi Posted October 8, 2004 Share Posted October 8, 2004 No Haggy I envy you, because all your troubles in this forum are limited to arms' observation. Link to comment Share on other sites More sharing options...
Guest Doron Shadmi Posted October 8, 2004 Share Posted October 8, 2004 A special post for Matt Grime, Let us go another step and ask: If the contradiction concept does not exist in included-middle reasoning, then how can we check the consistency of its axioms? After all, if there is no contradiction then there is no limit to anything and we cannot determine the consistency of anything in this framework. My answer is this: In an Included-Middle reasoning any product is the result of constructive interactions between at least two opposites, so if something exists because of this interaction, it cannot be but a consistent product of this interaction, or in other words, inconsistent products simply do not exist in this framework, and all we have is only consistent elements. An axiomatic system which is based on an included-middle reasoning, is based on the identity of a thing to itself, which is the new and simple meaning of the tautology concept in an included-middle reasoning framework ('if, then' propositions are not needed here). In short, all the "Energy" in an included-middle reasoning goes to research what we can do with our existing elements, and we do not spend our "Energy" to check the existence of each element in our framework, because if it survives the interaction between two opposites, it cannot be but an existing (and consistent) thing in our mathematical framework. If you cannot get it then please look at this included-middle reasoning axiomatic system: http://www.geocities.com/complementarytheory/My-first-axioms.pdf and see for yourself how we can do Math according to it. Link to comment Share on other sites More sharing options...
123rock Posted October 9, 2004 Share Posted October 9, 2004 ...0_1 and this case is equivalent to +1 ...1_0 and this case is equivalent to -1 0_1 + 1_0 = 1_0_1 So 1_0_1=0=0_0?!??!?!? Link to comment Share on other sites More sharing options...
Guest Doron Shadmi Posted October 9, 2004 Share Posted October 9, 2004 So 1_0_1=0=0_0?!??!?!? Not at all' date=' the meaning of '+' here is [b']structural[/b] concatenation. Therefore 0_1 + 1_0 = 1_0_1 . Please read posts #1, #5, #8 for better understanding, thank you. Link to comment Share on other sites More sharing options...
Guest Doron Shadmi Posted October 12, 2004 Share Posted October 12, 2004 In Standard Math the most basic question is “How many?”. In Monadic Math the most basic question is “What do we have?” The basis of “How many?” question is the Quantity concept. The basis of “What do we have?” question is the Structure concept. In short, in Monadic Mathematics, the Number concept is first of all based on the Structure concept. Question: What are the minimal and distinguished structural forms in Monadic Mathematics? Answer: Emptiness (notated as {}), Point (notated as {.}), Segment (notated as {._.}), Fullness (notated as {__}). {} and {__} are the weak ({}) and the strong ({__}) limits of Monadic Mathematics framework. In means that they cannot be used as inputs in Monadic Mathematics. So, we do math which is based on {.} and {._.}. The two basic structural properties, which are related to the Number concept, are: Length, Direction. {.} has 0 length and no directions. {._.} has 0_x length and at least two opposite directions, which are 0_x or x_0. The most basic arithmetical operations between these elements are ‘+’ and ‘-‘. Important: If '-' sign is used not as a binary operation, for example: -x_0, then we have to look at it as a negation symbol and not as a binary Elimination operation, therefore -x_0 = 0_x. The same holds for {} - x_0, which is a vacuous binary operation (because {} or {__} cannot be used as inputs) and has to be taken as if we wrote -x_0. ‘+’ is firstly understood as Concatenation, for example: 1_0 + 1_0 = 2__0 (._. + ._. = .__.) 0_1 + 1_0 = 1_0_1 (._. + ._. = ._._.) 0 + 0 = 0 ( . + . = . because Concatenation of 0 length is 0 length) 0__2 + 0_1 = 0___3 (.__. + ._. = .___.) 2__0 + 1_0 = 3___0 (.__. + ._. = .___.) 0_1 + 2__0 = 2__0_1 (._. + .__. = .__._.) and so on … ‘-’ is firstly understood as Elimination, for example: 1_0 - 1_0 = {} (._. - ._. = {} because both Length and Direction are identical) 0_1 - 1_0 = 0 (._. - ._. = . because only Length is identical) 0 - 0 = {} ( . - . = {} because Length is identical and there is no direction) 0__2 - 0_1 = 0_1 (.__. - ._. = ._.) 2__0 - 1_0 = 1_0 (.__. - ._. = ._.) 0_1 - 2__0 = 1_0 (._. - .__. = ._.) and so on … Link to comment Share on other sites More sharing options...
123rock Posted October 13, 2004 Share Posted October 13, 2004 By the laws that you have set forward, your own statements contradict you: If 1=1, in which is the only case that these monads can be stable, then 1-1=0, and thus 1_0+0_1=1_0_1=0 What is 1_0_1? Nobody knows from what you've explained. Please elaborate on how this can describe nature? Also, if 1_0=1_0, then 1_0+0_1=0_1+1_0, and thus 0_1_1_0=0_1_0=1_0_1 Link to comment Share on other sites More sharing options...
Guest Doron Shadmi Posted October 14, 2004 Share Posted October 14, 2004 By the laws that you have set forward' date=' your own statements contradict you: If 1=1, in which is the only case that these monads can be stable, then 1-1=0, and thus 1_0+0_1=1_0_1=0 [/quote'] Sorry dear 123rock but you still do not get it, because you use the basic question ”How many?” instead of “What do we have?” One of the efficient ways to change some framework is to ask new fundamental questions that maybe lead us to new frontiers. For example, the fundamental question of the language of Mathematics framework is: ”How many?”. Let us try another question, for example: “What do we have?” A closer look of these questions shows that ”How many?” questions are mostly about the Quantity concept, where “What do we have?” questions are mostly about the Structure concept. Let us check if “What do we have?” questions can be a fruitful ground for mathematical development. First we have to define the minimal concepts that can be used under the structure concept. In other words, these concepts when used, determine our framework’s domain, for example: Let use say that the two main concepts that are related to the structure concept are Length and Direction. It means that by using these concepts, we can define the building-blocks of our mathematical framework. We also know that by using these building-blocks we suppose to get some input that can be used by us to develop our framework. So, the next question is “What are the limits that beyond them no input can be found?” The lowest “no input” state is Emptiness, where Length and Direction do not exist. The highest “no input” state is Fullness, where Length and Direction are beyond measurement. So the useful elements have measurable Length and/or Direction, that enable us to use them as some input. Let us represent these ideas in a table: Monadic Mathematics build-blocks -------------------------------- V = Available X = Not Available +-----------------------------+ | Measurement of | | | | Length | Direction | |--------------|--------------| | | | {} | | | Emptiness | | | | | | |--------------|--------------| | | | {.} | V | | Point | | | | | | |--------------|--------------| | | | {._.} | V | V | Segment | | | | | | |--------------|--------------| | | | | | V | Tendency | | | | | | |--------------|--------------| | | | {__} | X | X | Fullness | | | | | | +-----------------------------+ General: {' and '}' are the notations of a framework which we call a set. Between these notations we can put our examined concepts and then try to find out what we can do with each one of these concepts, and also what interactions can be found between concepts and themselves and/or concepts with other concepts. The concept of Nothingness or Emptiness is notated in this framework by {}. Any examined concept in my framework is examined by its structural properties and also by its Quantitative properties. The most basic Structural property in my framework is based on the Length concept. From this point of view, a Point {.} has exactly 0 Length. The Length concept has no meaning in my framework when it is related to the Emptiness concept (which is notated as {}), therefore 0 cannot be connected directly to the Emptiness concept. Another option to define 0 is to ask: How many things there are in {}? By 'how many?' question we actually define the Cardinal concept, which is notated by using '|' and '|' . In this case, the cardinal of {}, which is notated as |{}|, is equal to 0. In Standard Mathematics framework (when Ordinality is omitted) the one and only one option to connect between the Number concept and the Set concept, is by using only the Quantity concept, and in this case |{}| = 0. But as you see, in Monadic Mathematics framework, there are two kinds of cardinals, where one of them is the standard Quantitative Cardinal, but the second type of cardinal is what I call the Urelement Cardinal, which is based on the Length concept (which is the structural property of the Number concept). In Monadic Mathematics, the structural property of a number is more basic then its quantitative property. According to what I wrote above, when a Point eliminates itself, then the result is Emptiness. In other words Point - Point = Emptiness, or in other representation: 0 - 0 = {}. Standard Mathematics takes '0' notation as something which is first of all related to the Quantity concept. In this framework 0 - 0 = 0 Monadic Mathematics takes '0' notation as something which is first of all related to the Structure concept. In this framework 0 - 0 = {} A non technical explanation of Monadic Mathematics' '+' and '-' operations: The most basic question of Standard Mathematics is “How many?”. The most basic question of Monadic Mathematics is “What do we have?” The basis of a “How many?” question is the Quantity concept. The basis of a “What do we have?” question is the Structure concept. It means that the Number concept in Monadic Mathematics, is first of all based on the Structure concept. Question: What are the minimal, and distinguished, structural forms in Monadic Mathematics? Answer: Emptiness (notated as {}), Point (notated as {.}), Segment (notated as {._.}), Fullness (notated as {__}). {} and {__} are the weak ({}) and the strong ({__}) limits of Monadic Mathematics framework. It means that they cannot be used as inputs in Monadic Mathematics. So, Monadic Mathematics operations are based on {.} and {._.}. The two basic structural properties, which related to the Number concept, are Length and/or Direction. {.} has 0 length and no directions. {._.} has 0_x length and at least two opposite directions, which are 0_x or x_0. The most basic arithmetical operations between these elements are ‘+’ and ‘-‘. Important: If '-' sign is used not as a binary operation, for example: -x_0, then it is a negation symbol and not a binary Elimination operation. Therefore -x_0 = 0_x or -0_x = x_0. Also {} - x_0, which is a vacuous binary operation (because {} or {__} cannot be used as inputs) is equal to -x_0 = 0_x, etc. ‘+’ is understood as Concatenation, for example: 1_0 + 1_0 = 2__0 (._. + ._. = .__.) 0_1 + 1_0 = 1_0_1 (._. + ._. = ._._.) 0 + 0 = 0 ( . + . = . because Concatenation of 0 Length is 0 Length) 0__2 + 0_1 = 0___3 (.__. + ._. = .___.) 2__0 + 1_0 = 3___0 (.__. + ._. = .___.) 0_1 + 2__0 = 2__0_1 (._. + .__. = .__._.) etc … ‘-’ is understood as Elimination, for example: 1_0 - 1_0 = {} (._. - ._. = {} because both Length and Direction are identical) 0_1 - 1_0 = 0 (._. - ._. = . because only Length is identical) 0 - 0 = {} ( . - . = {} because Length is identical and there is no Direction) 0__2 - 0_1 = 0_1 (.__. - ._. = ._.) 2__0 - 1_0 = 1_0 (.__. - ._. = ._.) 0_1 - 2__0 = 1_0 (._. - .__. = ._.) etc … More detailed information can be found in: http://www.geocities.com/complementarytheory/My-first-axioms.pdf According to Monadic Mathematics the Number concept is the fundamental building-block of a non-destructive interaction between opposites, which is based on Included-middle reasoning (which is the logic that can be found between at least two opposites that define their middle domain). For more details please read: http://www.geocities.com/complementarytheory/GaloisDialog.pdf Link to comment Share on other sites More sharing options...
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