I think what we want to do is -- fold this information and theme group into the Core Vocabulary group, while starting to consider an assembling sequence *** The basic terms and primary vocabulary of the Closed Loop model of epistemology and ontology can be mapped to the unit strip, which is the foundation for all definitions. The Closed Loop model is a systematic interpretation of this form, which opens to its full power and meaning under the twist, shown in the animated graphic in the header. Every definition is created by boundaries. Every category is created by boundaries. Boundaries are created by stipulation (human intention). Boundaries create intervals. Each term needs its own diagram. Real number line Boundary Interval Decimal system Numbers Abstraction

Real number line Note | Back We interpret the closed loop strip as a representation of the Real Number Line, because its elements are consistent with the common definition as a linear continuum. There is much to say on this subject, and much to follow. Mon, May 10, 2021 Reference In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set R of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one. It can be thought of as a vector space (or affine space), a metric space, a topological space, a measure space, or a linear continuum. The real line is a linear continuum under the standard < ordering. Specifically, the real line is linearly ordered by <, and this ordering is dense and has the least-upper-bound property. In addition to the above properties, the real line has no maximum or minimum element. It also has a countable dense subset, namely the set of rational numbers. It is a theorem that any linear continuum with a countable dense subset and no maximum or minimum element is order-isomorphic to the real line. URL https://en.wikipedia.org/wiki/Real_line Closed Loop Ensemble Core terms on the strip

Boundary Placeholder | Back Any of the edges in this form, each of which functions as a limit Mon, Mar 15, 2021 Reference Any of the edges in this form, each of which functions as a limit In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S not belonging to the interior of S. An element of the boundary of S is called a boundary point of S. The term boundary operation refers to finding or taking the boundary of a set. Notations used for boundary of a set S include bd(S), fr(S), and {\displaystyle \partial S}\partial S. Some authors (for example Willard, in General Topology) use the term frontier instead of boundary in an attempt to avoid confusion with a different definition used in algebraic topology and the theory of manifolds. Despite widespread acceptance of the meaning of the terms boundary and frontier, they have sometimes been used to refer to other sets. For example, Metric Spaces by E. T. Copson uses the term boundary to refer to Hausdorff's border, which is defined as the intersection of a set with its boundary.[1] Hausdorff also introduced the term residue, which is defined as the intersection of a set with the closure of the border of its complement.[2] URL https://en.wikipedia.org/wiki/Boundary_(topology) Closed Loop Ensemble Core vocabulary Core terms on the strip

Interval Placeholder | Back Every value is an interval -- a bounded range in X or Y Qualitative value is related to stipulation from the top down That relates value to hierarchy A range of values between boundaries -- can be vertical (Y axis) or horizontal (X axis) interval - value - hierarchy - stipulation - boundary starting point in definition chain interval Fri, Apr 2, 2021 Reference In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers x satisfying 0 < x is an interval which contains 0, 1, and all numbers in between. Other examples of intervals are the set of numbers such that 0 < x < 1, the set of all real numbers, the set of nonnegative real numbers, the set of positive real numbers, the empty set, and any singleton (set of one element). URL https://en.wikipedia.org/wiki/Interval_(mathematics) Closed Loop Ensemble Core vocabulary Core terms on the strip

Decimal system Placeholder | Back differentiation of a unit into multiple subunits, which can be binary (as above) or decimal, with no significant topological difference Tue, Feb 16, 2021 Closed Loop Ensemble Core terms on the strip

Numbers Placeholder | Back Wikipedia defines number as "a mathematical object used to count, measure, and label" We want to define "number" in terms mapped to the Closed Loop framework -- and connected through it to all other major mathematical and semantic definitions. Numbers are going to be defined as "intervals" -- as bounded ranges in a linear/sequential "totally ordered" structure like a row or a vector, and they are given symbolic names/labels, which we call "numerals" (0,1,2,3,4,5,6...) We are going to construct those definitions out of synthetic dimensions *** Newer idea as this project grows -- Start with the "natural numbers" Map natural numbers to objects -- like "apples" -- 1 apple, 2 apples, three apples -- first apple, second apple, third apple Integer numbers Rational numbers Sun, May 2, 2021 Reference A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can be represented by symbols, called numerals; for example, "5" is a numeral that represents the number five. As only a relatively small number of symbols can be memorized, basic numerals are commonly organized in a numeral system, which is an organized way to represent any number. The most common numeral system is the Hindu–Arabic numeral system, which allows for the representation of any number using a combination of ten fundamental numeric symbols, called digits. In addition to their use in counting and measuring, numerals are often used for labels (as with telephone numbers), for ordering (as with serial numbers), and for codes (as with ISBNs). In common usage, a numeral is not clearly distinguished from the number that it represents. URL https://en.wikipedia.org/wiki/Number Closed Loop Ensemble Core vocabulary Core terms on the strip Digital geometry

Abstraction Placeholder | Back Abstraction is a process of creating general categories that combine attributes of specific categories. It is defined as a hierarchical process in ascending layers (from specific class to general class in ascending layers in the vertical axis). Sat, Mar 20, 2021 Reference Abstraction in its main sense is a conceptual process where general rules and concepts are derived from the usage and classification of specific examples, literal ("real" or "concrete") signifiers, first principles, or other methods. "An abstraction" is the outcome of this process—a concept that acts as a common noun for all subordinate concepts, and connects any related concepts as a group, field, or category. Conceptual abstractions may be formed by filtering the information content of a concept or an observable phenomenon, selecting only the aspects which are relevant for a particular subjectively valued purpose. For example, abstracting a leather soccer ball to the more general idea of a ball selects only the information on general ball attributes and behavior, excluding, but not eliminating, the other phenomenal and cognitive characteristics of that particular ball. In a type–token distinction, a type (e.g., a 'ball') is more abstract than its tokens (e.g., 'that leather soccer ball'). Thinking in abstractions is considered by anthropologists, archaeologists, and sociologists to be one of the key traits in modern human behaviour, which is believed to have developed between 50,000 and 100,000 years ago. Its development is likely to have been closely connected with the development of human language, which (whether spoken or written) appears to both involve and facilitate abstract thinking. Abstraction involves induction of ideas or the synthesis of particular facts into one general theory about something. It is the opposite of specification, which is the analysis or breaking-down of a general idea or abstraction into concrete facts. Abstraction can be illustrated with Francis Bacon's Novum Organum (1620), a book of modern scientific philosophy written in the late Jacobean era[3] of England to encourage modern thinkers to collect specific facts before making any generalizations. Bacon used and promoted induction as an abstraction tool, and it countered the ancient deductive-thinking approach that had dominated the intellectual world since the times of Greek philosophers like Thales, Anaximander, and Aristotle.[4] Thales (c. 624–546 BCE) believed that everything in the universe comes from one main substance, water. He deduced or specified from a general idea, "everything is water", to the specific forms of water such as ice, snow, fog, and rivers. Modern scientists can also use the opposite approach of abstraction, or going from particular facts collected into one general idea, such as the motion of the planets (Newton (1642–1727)). When determining that the sun is the center of our solar system (Copernicus (1473–1543)), scientists had to utilize thousands of measurements to finally conclude that Mars moves in an elliptical orbit about the sun (Kepler (1571–1630)), or to assemble multiple specific facts into the law of falling bodies (Galileo (1564–1642)). In mathematics Main article: Abstraction (mathematics) Abstraction in mathematics is the process of extracting the underlying structures, patterns or properties of a mathematical concept or object,[19] removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena. The advantages of abstraction in mathematics are: It reveals deep connections between different areas of mathematics. Known results in one area can suggest conjectures in another related area. Techniques and methods from one area can be applied to prove results in other related area. Patterns from one mathematical object can be generalized to other similar objects in the same class. The main disadvantage of abstraction is that highly abstract concepts are more difficult to learn, and might require a degree of mathematical maturity and experience before they can be assimilated. URL https://en.wikipedia.org/wiki/Abstraction Basic concepts of ontology Closed Loop Ensemble Core vocabulary Core terms on the strip