The emergence of new infinite system (new 'infinite' idea, new number system and new limit theory) determines the production of "new mathematical analysis". The mathematical analysis based on the classical infinite system is called "classical mathematical analysis", and the mathematical analysis based on the new infinite system is called "new mathematical analysis (the fourth generation of mathematical analysis)".
Three major differences between new and classical mathematical analysis in treating "X ----> 0 infinite mathematical things"
1 Different "infinite ideas" and different "treating objects"
In the foundation of classical mathematical analysis, "infinite" is composed by two concepts of "potential infinite and actual infinite" which can not be well defined and are inevitably confused. So, it is impossible to know clearly what on earth those "X----> 0 infinite mathematical things" are for people to treat (numbers or non-numbers? potential infinite mathematical things or actual infinite mathematical things? potential infinite numbers or actual infinite numbers? ...?)------they are non-number "variables" that are forever in change.
In new mathematical analysis, "infinite" is composed by two concepts of "abstract infinite law" and "the carriers of 'abstract infinite law'", which can be well defined and cannot be confused. According to the new number system, all of those "X----> 0 infinite mathematical things" for people to treat are clearly known as "the mathematical carriers of 'abstract infinite law'" ------they are numbers.
2 Different "infinite number theories and infinite number forms"
In classical mathematical analysis, people can not get rid of the confusion and bondage of "potential infinite number and actual infinite number" and have had to define all the "infinite numbers" participating in any quantitative calculations as "non-number variables of both potential infinite number and actual infinite number that are forever in change" mixed by the "potential infinite number and actual infinite number" derived by the confused "potential infinite and actual infinite" concepts. But, there is never such "variables" in existing classical number system at all; so, whenever the "infinite numbers" are calculated mistakenly as "potential infinite numbers in forever change", paradoxes have been inevitably produced (as a members of zeno's paradox family, harmonic series paradox is the most typical case).
In new mathematical analysis, all the "infinite numbers" participating in any quantitative calculations are "mathematical carriers of abstract infinite law" with characteristics of number sense representing the existence of abstract infinite law, which are clearly defined as numbers with quantitative nature and within the new number system. And, "abstract infinite law in forever change" will never be taken as number in any quantitative calculations. In new number system, those "X----> 0 infinite mathematical things" belong to “inter--number” and is called “intersimal”
3 Different processing theory and operations to "infinite mathematical things"
In classical mathematical analysis, the defects of existing classic number system unable people to understand the nature of "variables", making its quantitative cognitive theory of "infinite mathematical things" can be nothing to do with existing classical number system and nothing to do with the specific quantitative properties of "infinite mathematical things". So, theoraticaly or operationaly, it is impossible to established essential rules of "mathematical analysis" to know exactly what "X----> 0 infinite mathematical things" being processed are and what quantitative properties they have (because there is no such number form in the existing classic number system), some kinds of formal languages and pipeline operations are ok (such as the completely equivalent languages and pipeline operations in three generations of classical mathematical analysis: "let infinitesimal be 0" in pre--standard analysis, "take the limit" in standard analysis and "take the standard number" in non-standard analysis) [1]. In many situations, some unified pipelining operation theories can be used for the processing of all the "infinite mathematical things" (the most typical situation is: whether in calculus or in harmonic series, all the "X----> 0 infinite mathematical things" are with exactly the same quantitative nature without any differences). And such pipeline operation theories, processes and results in the "potential infinite--actual infinite" based three generations of classical mathematical analysis (pre--standard analysis, standard analysis, non-standard analysis) are completely equivalent, inevitably producing exactly the same "infinitesimal--infinity quantitative cognition paradox families".
In new mathematical analysis, the quantitative cognitive theory of "infinite mathematical things" is a targeted operation theory closely related to the new number system as well as the specific quantitative properties of "infinite mathematical cariers". Before calculating, all the "X----> 0 infinite mathematical things" participating in any quantitative calculations must be truly "mathematical analyzed" according to the "theory of infinite law carrier" to know what quantitative meanings and properties they have so as to perform the targeted operations to them. And, such targeted operation theories, processes and results are nonequivalent to any of the three generations of classical mathematical analysis in many situations--------it is impossible to have those suspended "infinitesimal--infinity related paradox families"
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https://www.journals.elsevier.com/journal-of-mathematical-analysis-and-applications/
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Systems analysis of the disciplines of engineering and computer sciences, which deals with the analysis of groups of interacting entities for their control.
Is of great importance in several areas, including operational research.
It is also "a clear formal investigation to assist someone, referred to as the decision maker, determine the best course of action and make a better decision than anything earlier
The analysis of synthesis and synthesis terms comes from the classical Greek, where they mean respectively (to be involved) and that (cooperates), and these terms are used in scientific disciplines of logic, mathematics, economics and psychology to denote similar investigation procedures. In general, analysis is defined as the procedure for which we have to analyze the whole intellectual or large to parts or components. The synthesis is defined as a countermeasure: to combine separate elements or components to form a coherent
The systems discussed that systems analysis could be in any area such as: industrial processes, management, decision making, environmental protection processes, etc. The brothers Howard Odem and Eugene Odim began applying the systems view of ecology in 1953 based on the work of Raymond (1942) and Arthur Transley (1935).
Why there sure be infinite related “quantitative cognizing paradox families in present classical mathematical analysis?
The emergence of new infinite system (new 'infinite' idea, new number system and new limit theory) determines the production of "new mathematical analysis". The mathematical analysis based on the classical infinite system is called "classical mathematical analysis", and the mathematical analysis based on the new infinite system is called "new mathematical analysis (the fourth generation of mathematical analysis)".
There is no infinite related mathematical analysis without the “infinite related number forms”. So, the “quantitative cognizing work on infinite related number forms” is an important and unavoidable task for mathematical analysis. But our studies have proved that in present mathematical analysis, people have been admitting the being of “potential infinite, actual infinite” concepts, unable to deny their qualitative differences and the important roles they play in the foundation of present classical infinite theory system and, unable to deny that the present classical mathematical analysis is basing on present classical infinite theory system. The fact is: on the one hand, present classical mathematical analysis can not avoid the constraining of “potential infinite--actual infinite” concepts and their relating “potential infinite number forms--actual infinite number forms”; on the other hand, no clear definitions for these two concepts of “potential infinite--actual infinite” and their relating “potential infinite number forms--actual infinite number forms” have been constructed since antiquity, thus naturally lead to following two unavoidable fatal defects in present classical mathematical analysis:
（1）it is impossible (unable) to understand theoretically what the important basic concepts of “potential infinite, actual infinite” and their relating “potential infinite number forms--actual infinite number forms” are. So, in many “qualitative cognizing activities on infinite relating mathematical things (infinite relating number forms)” in present classical mathematical analysis, many people actually don’t know or even deny the being of “potential infinite, actual infinite” concepts and their relating “potential infinite number forms--actual infinite number forms”--------the “qualitative cognizing defects on infinite relating mathematical things (infinite relating number forms)”.
（2）it is impossible (unable) to understand operationally what kind of relationship among the important basic concepts of “potential infinite, actual infinite”, their relating “potential infinite number forms--actual infinite number forms” and all the“infinite number forms as well as their quantitative cognizing operations” are. So, in many “quantitative cognizing activities on infinite things (infinite number forms)” in present classical mathematical analysis, many people have been unable to know whether the infinite relating number forms being treated are “potential infinite number forms” or “actual infinite number forms”, no one has been able to avoid the confusing of “potential infinite number forms” and “actual infinite number forms”, no one has been able to know whether or not treating “potential infinite number forms” or “actual infinite number forms” with the same way or different ways. What is more, many people actually don’t know or even deny the being of “potential infinite number forms” and “actual infinite number forms”--------the “quantitative cognizing defects on infinite relating mathematical things (infinite relating number forms)”.
The above two fatal defects have decided since antiquity the absence of “infinite carrier theory” and the confusion of “abstract infinite concept and its concrete infinite mathematical carrier” which have been making us impossible to construct the scientific and systematic theory of “infinite related concrete number form (the mathematical carriers of abstract infinite concept)”, impossible to construct the “potential infinite--actual infinite” relating number forms as well as the “potential infinite--actual infinite” relating number system and its relating treating theory, thus unavoidable forming an insurmountable obstacle in the “quantitative cognizing process on infinite things (infinite number forms)” in present classical mathematical analysis. And, it is impossible to know what those “’both potential infinite number forms and actual infinite number forms’, ‘neither potential infinite number forms nor actual infinite number forms’ mathematical things” are, and can only treat all of them with the “flow line (pipelining)” unified approach （such as the three formal languages in three generations of mathematical analysis: before standard analisis, standard analisis, non-standard analisis）. So, many members of different infinite related paradox families in present “potential infinite--actual infinite” based classical mathematical analysis have been produced one by one naturally (the newly discovered “strict mathematical proven” Harmonic Series Paradox is a typical example), forming a thousands-year old suspended huge black cloud of paradoxes over present classical infinite related science and mathematical analysis.
Five of my published papers have been up loaded onto RG to answer such questions:
1，On the Quantitative Cognitions to “Infinite Things” (I)
2，On the Quantitative Cognitions to “Infinite Things” (II)
3，On the Quantitative Cognitions to “Infinite Things” (III)
4 On the Quantitative Cognitions to “Infinite Things” (IV)
5 On the Quantitative Cognitions to “Infinite Things” (V)
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