Interpretations of the Closed Loop as an "Ensemble" The primary idea motivating the Closed Loop project is the presumption or postulate that all conceptual structure - every idea expressed in any language - emerges from the unmarked/undifferentiated continuum by a process of "motivated distinction", and takes the same general form. Ideas emerge because they are motivated - by something. They are given symbolic representation -- in sound or in writing -- because maintaining a library of distinctions is convenient in a social context. By the term "ensemble", we are suggesting a set of primary definitions, traditionally defined in a "stand alone" way, which are are proposing to define and understand as implications of a single basic common underlying general form, which we call the "closed loop". We are proposing to understand the closed loop as a general common foundation for all types of analytic thinking and models, suggesting that all types of thinking can be understood within a common framework, and thereby linked to one another through this common foundation. We are exploring an interpretation of this primary "strip", which we propose to interpret as a comprehensive integration of numerous fundamental definitions common to mathematics, ontology and semantics. We are generating a list of these fundamental definitions, which we are describing as an "ensemble". Each of these terms is found in common usage, but are generally defined independently, in ways which are not consistent and not based on a fundamental underlying commonality. We are proposing an underlying common form which we believe does adequately describe and define each of these concepts, but does so in a way which shares an underlying common definition which we are proposing generates or contains all these more specific definitions, which see as implications of the general form. Unit Strip This image shows the hierarchical decomposition of the unit interval in the general form of a "strip" defined in X and Y axes. List the primary elements of ontology, semantics and ontology, and fit these definitions into the dimensions of the Closed Loop How a comprehensive interpretative theory of ontology, epistemology and semantics can be fitted into a single algebraic semantics defined on the Loop.

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

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

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

Taxonomy Placeholder | Back essentially the same general form as abstraction Thu, May 6, 2021 Closed Loop Ensemble Synthetic dimensionality

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

Limit Placeholder | Back Boundary Sun, Mar 14, 2021

Unit interval Placeholder | Back the top level in the strip – the undifferentiated “one” at the top of the hierarchy of layers, defined in a recursive way like a “holon”, such that every nested interval is also a unit interval Sun, Apr 4, 2021 Reference In mathematics, the unit interval is the closed interval [0,1], that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted I (capital letter I). In addition to its role in real analysis, the unit interval is used to study homotopy theory in the field of topology. In the literature, the term "unit interval" is sometimes applied to the other shapes that an interval from 0 to 1 could take: (0,1], [0,1), and (0,1). However, the notation I is most commonly reserved for the closed interval [0,1]. URL https://en.wikipedia.org/wiki/Unit_interval Closed loop interval ontology Closed Loop Ensemble Core vocabulary

Unit Placeholder | Back universal undifferentiated one, can be defined in multiples, can be differentiated Sun, Mar 14, 2021 Reference universal undifferentiated one, can be defined in multiples, can be differentiated

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

Amount Placeholder | Back quantity (multiple) of some dimension or unit amount, value, quantity -- multiples of some unit Fri, Feb 12, 2021

Quantity Placeholder | Back amount as defined in some dimension Tue, Feb 16, 2021 Closed Loop Ensemble Core vocabulary

Quality Placeholder | Back a composite of abstract values where quantity is defined by intentional stipulation Sun, Mar 14, 2021

Roundoff error Placeholder | Back Bounded range of values Fri, Feb 12, 2021 Reference A roundoff error, also called rounding error, is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. Rounding errors are due to inexactness in the representation of real numbers and the arithmetic operations done with them. This is a form of quantization error. URL https://en.wikipedia.org/wiki/Round-off_error

Acceptable error tolerance Placeholder | Back An acceptable error tolerance is a bounded range -- from lower to higher in some range of values. bounded range of values The semantics of meaning involves error tolerance in stipulated dimensions. negotiation also involves acceptable error tolerance You say you are going to paint the house grey. What shade of grey? Sun, Apr 4, 2021 Closed Loop Ensemble Semantics

Hierarchy Placeholder | Back this form is a hierarchy of layers defined in the A-C / B-D axis Tue, Feb 16, 2021 Closed Loop Ensemble Core vocabulary