Primary conceptual building blocks of the Closed Loop dimensional system. We are just beginning with this. How can "everything" be constructed from these fundamental structures? How are these structures themselves represented? What are they "made out of"? Generally, the thesis is that fundamental symbolic representation is derived (emerges) under some basic and "primal" motivation, creating symbolic abstractions that can be replicated and communicated and serve as building blocks for communication among people who share a common reality, and have a need to describe its facets. This vocabulary is intended to include Foundations of mathematics Theory of conceptual structure Semantics Cognitive science Epistemology Synthetic dimensionality We might augment this vocabulary by going through major books on these subjects, looking for terms that should be included. Interval Value Boundary Hierarchy Stipulation Quantity Numbers Unit interval Synthetic dimensionality topic Making a point Set - and paradox Point Line Primitives Intersection (Euclidean geometry) Taxon Function Abstraction Abstract object Assembling sequence Matrix Abstract object, concrete object

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

Value Placeholder | Back All values are defined within intervals, as a bounded range in that interval, can be defined in A and Y Quantitative values are defined by stipulation in the hierarchy of abstraction -- in a hierarchy -- in a descending cascade Tue, Feb 16, 2021

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

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

Stipulation Placeholder | Back Qualitative values are stipulated within intervals across a descending hierarchy/cascade of levels Stipulation is intentional, and involves selection and specification When ask "What do you mean by beauty?" -- a person can respond by defining their meaning of beauty. This might be considered rather advanced, bu this is how people actually do it. They define beaty according to their values, which they stipulate (or assert or affirm, as a free-will personal choice) To affirm, to state, to specify, to select The word follows the famous Humpty-Dumpty theory of meaning, as where he states that "words mean what I want them to mean, nothing more, nothing less" The "nothing more, nothing less" show the value coefficient in the meaning Tue, Feb 16, 2021 Reference This word does not have good semantic-based definitions in a quick search

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

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

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

Synthetic dimensionality topic Placeholder | Back Recursive definition of dimension, where values of the dimension are built from synthetic dimensions A dimension where the values are defined in terms of dimensions A recursive dimension, intended to define complex semantic cascades in common algebraic concept that expresses every part of the structure A "composite" dimension A "holistic" dimension A "synthetic" (Put together from pieces) dimension Thu, May 6, 2021 Core vocabulary Synthetic dimensionality

Making a point Placeholder | Back We define a set in terms of boundary values in dimensions. A set is all the points or objects that are constrained by the intersection of the defining dimensions. So a set is created by the intersection of multiple dimensions... [?] Ordinarily, a set is any collection of elements -- put together for any reason When somebody is "making a point" in a sentence, we want to look at the question of how that sentence or paragraph or composite of semantic elements combine to establish a "point" in abstract space that is significant. This is what I want to talk about... Let's say that normal common conversation is generally conducted in "synthetic dimensions" -- some of which might be "quantitative" -- but most of which is usually not. So we want to make the argument that The object of speech /language is to make a point -- to make points So we are talking in a complex array of usually-qualitative dimensions We have generally agreed on the innate/inherent dimensionality of the words we use -- whether they are verbs or nouns So we compile these nested distinctions into (linear - sequential) strings -- and then intersect these dimensions The "point we are making" is a point constrained within the intersection of the simultaneous dimensionality we just invoked What makes that point "significant"? Why does it "matter"? We know these things instinctively, and we use these ideas every day. But how does it actually work? We say this "point" is an intersect like a Venn diagram in synthetic dimensional values These values are complex objects defined in boundary value ranges These dimensions are a range of values - and we specify those dimensions, and the values we want to affirm for those dimensions And we include several of them in "making a point" If we specify five qualitative dimensions, we define the point in those five dimensions -- each one of which is qualitatively defined in a range, and which combine in an intersect which established this "point" at the intersect of these dimensions and the values we have assigned/stipulated ***** The argument for semantics is Human conversation and writing is intended to "make points" -- as in "what is your point?" If you are vague and confused -- your comments are "pointless" "What is the point of doing this?" Fri, Apr 16, 2021 Core vocabulary Semantics

Set - and paradox Placeholder | Back A set is a collection of points or objects contained by the intersection of a number of dimensions. Objects in the set meet the constraints (boundary values) of all the dimensions. Sun, Mar 14, 2021 Reference In mathematics, a set is a collection of distinct elements.[1][2][3] The elements that make up a set can be any kind of things: people, letters of the alphabet, numbers, points in space, lines, other geometrical shapes, variables, or even other sets.[4] Two sets are equal if and only if they have precisely the same elements.[5] Sets are ubiquitous in modern mathematics. The subject called set theory is part of the foundations of mathematics.[4] *** Naïve set theory Main article: Naive set theory The foremost basic property of a set is that it can have elements, which are also called its members. Another essential property of sets is that two sets are equal (one set equals the other, so the two sets are in fact one and the same) if and only if every element of each set is an element of the other. This property is called the extensionality of sets.[11] [this gets into the question of identity and equality -- is item A exactly the same as item B -- or only "similar"?] The simple concept of a set has proved enormously useful in mathematics, but it suffers from inconsistencies at the most fundamental level. A loose notion that allows any property without restriction to define a collection, leads to several paradoxes, most notably: Russell's paradox – It shows that the "set of all sets that do not contain themselves," i.e. the "set" {x|x is a set and x ? x} does not exist. Cantor's paradox – It shows that "the set of all sets" cannot exist. Naïve set theory defines a set as any well-defined collection of distinct elements. Problems arise from the vague meaning of the term well-defined. URL https://en.wikipedia.org/wiki/Set_(mathematics)

Point Placeholder | Back What is a point? It is the intersection of (at least) two dimensions -- usually X and Y. Because we are defining a "line" as having "width", a point is actually a matrix cell. ORIGIN is probably a point defined in this way. Sun, Mar 14, 2021

Line Placeholder | Back We are taking up a new approach to the foundations of Euclidean geometry. In Euclid -- a line is Euclid described a line as "breadthless length" which "lies equally with respect to the points on itself" Sun, Mar 14, 2021 Reference In geometry, the notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth. Lines are an idealization of such objects, which are often described in terms of two points (e.g., {\displaystyle {\overleftrightarrow {AB}}}{\displaystyle {\overleftrightarrow {AB}}}) or referred to using a single letter (e.g., {\displaystyle \ell }\ell ).[1][2] Until the 17th century, lines were defined as the "[...] first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which [...] will leave from its imaginary moving some vestige in length, exempt of any width. [...] The straight line is that which is equally extended between its points."[3] Euclid described a line as "breadthless length" which "lies equally with respect to the points on itself"; he introduced several postulates as basic unprovable properties from which he constructed all of geometry, which is now called Euclidean geometry to avoid confusion with other geometries which have been introduced since the end of the 19th century (such as non-Euclidean, projective and affine geometry). In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic geometry, a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation, but in a more abstract setting, such as incidence geometry, a line may be an independent object, distinct from the set of points which lie on it. When a geometry is described by a set of axioms, the notion of a line is usually left undefined (a so-called primitive object). The properties of lines are then determined by the axioms which refer to them. One advantage to this approach is the flexibility it gives to users of the geometry. Thus in differential geometry, a line may be interpreted as a geodesic (shortest path between points), while in some projective geometries, a line is a 2-dimensional vector space (all linear combinations of two independent vectors). This flexibility also extends beyond mathematics and, for example, permits physicists to think of the path of a light ray as being a line. URL https://en.wikipedia.org/wiki/Line_(geometry)

Primitives Placeholder | Back Eucline defined a line as "breadthless extension" Sun, Mar 14, 2021 Reference All definitions are ultimately circular in nature, since they depend on concepts which must themselves have definitions, a dependence which cannot be continued indefinitely without returning to the starting point. To avoid this vicious circle, certain concepts must be taken as primitive concepts; terms which are given no definition.[4] In geometry, it is frequently the case that the concept of line is taken as a primitive.[5] In those situations where a line is a defined concept, as in coordinate geometry, some other fundamental ideas are taken as primitives. When the line concept is a primitive, the behaviour and properties of lines are dictated by the axioms which they must satisfy. In a non-axiomatic or simplified axiomatic treatment of geometry, the concept of a primitive notion may be too abstract to be dealt with. In this circumstance, it is possible to provide a description or mental image of a primitive notion, to give a foundation to build the notion on which would formally be based on the (unstated) axioms. Descriptions of this type may be referred to, by some authors, as definitions in this informal style of presentation. These are not true definitions, and could not be used in formal proofs of statements. The "definition" of line in Euclid's Elements falls into this category.[6] Even in the case where a specific geometry is being considered (for example, Euclidean geometry), there is no generally accepted agreement among authors as to what an informal description of a line should be when the subject is not being treated formally. URL https://en.wikipedia.org/wiki/Line_(geometry)

Intersection (Euclidean geometry) Placeholder | Back Sun, Mar 14, 2021 Reference In geometry, an intersection is a point, line, or curve common to two or more objects (such as lines, curves, planes, and surfaces). The simplest case in Euclidean geometry is the intersection of two distinct lines, which either is one point or does not exist if the lines are parallel. The red dot represents the point at which the two lines intersect. Determination of the intersection of flats – linear geometric objects embedded in a higher-dimensional space – is a simple task of linear algebra, namely the solution of a system of linear equations. In general the determination of an intersection leads to non-linear equations, which can be solved numerically, for example using Newton iteration. Intersection problems between a line and a conic section (circle, ellipse, parabola, etc.) or a quadric (sphere, cylinder, hyperboloid, etc.) lead to quadratic equations that can be easily solved. Intersections between quadrics lead to quartic equations that can be solved algebraically. URL https://en.wikipedia.org/wiki/Intersection_(Euclidean_geometry)

Taxon Placeholder | Back Taxon -- plura taxa -- for genus and/or species Mon, Mar 15, 2021 Reference In biology, a taxon (back-formation from taxonomy; plural taxa) is a group of one or more populations of an organism or organisms seen by taxonomists to form a unit. Although neither is required, a taxon is usually known by a particular name and given a particular ranking, especially if and when it is accepted or becomes established. It is very common, however, for taxonomists to remain at odds over what belongs to a taxon and the criteria used for inclusion. If a taxon is given a formal scientific name, its use is then governed by one of the nomenclature codes specifying which scientific name is correct for a particular grouping. Initial attempts at classifying and ordering organisms (plants and animals) were set forth in Linnaeus's system in Systema Naturae, 10th edition, (1758) as well as an unpublished work by Bernard and Antoine Laurent de Jussieu. The idea of a unit-based system of biological classification was first made widely available in 1805 in the introduction of Jean-Baptiste Lamarck's Flore françoise, of Augustin Pyramus de Candolle's Principes élémentaires de botanique. Lamarck set out a system for the "natural classification" of plants. Since then, systematists continue to construct accurate classifications encompassing the diversity of life; today, a "good" or "useful" taxon is commonly taken to be one that reflects evolutionary relationships. URL https://en.wikipedia.org/wiki/Taxon

Function Placeholder | Back My old definition of function was "an input/output relationship" Put X into the function, out comes Y Mon, Mar 15, 2021 Reference In mathematics, a function is a binary relation between two sets that associates to each element of the first set exactly one element of the second set. Typical examples are functions from integers to integers, or from the real numbers to real numbers. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of set theory, and this greatly enlarged the domains of application of the concept. A function is a process or a relation that associates each element x of a set X, the domain of the function, to a single element y of another set Y (possibly the same set), the codomain of the function. It is customarily denoted by letters such as f, g and h.[1] If the function is called f, this relation is denoted by y = f?(x) (which reads "f of x"), where the element x is the argument or input of the function, and y is the value of the function, the output, or the image of x by f.[2] The symbol that is used for representing the input is the variable of the function (e.g., f is a function of the variable x).[3] A function is uniquely represented by the set of all pairs (x, f?(x)), called the graph of the function.[note 2][4] When the domain and the codomain are sets of real numbers, each such pair may be thought of as the Cartesian coordinates of a point in the plane. The set of these points is called the graph of the function; it is a popular means of illustrating the function. Functions are widely used in science, and in most fields of mathematics. It has been said that functions are "the central objects of investigation" in most fields of mathematics. URL https://en.wikipedia.org/wiki/Function_(mathematics)

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

Abstract object Placeholder | Back Generally we mean something represented by a symbol in a medium -- a "map" A "concrete object" is the territory that the map (symbol) is pointing to (representing) The dimensional construction of abstract objects -- a subject we want to explore as foundational to this method Thu, Apr 8, 2021 URL https://en.wikipedia.org/wiki/Abstract_and_concrete Core vocabulary Formal systematic definitions

Assembling sequence Placeholder | Back Sequence of dimensional assembly point line (bounded, two edges, convergence, limits either side) strip matrix row cell cell contents boundary dimension set set object / set member Thu, Mar 25, 2021

Matrix Placeholder | Back rows and columns Thu, Mar 25, 2021 Reference https://en.wikipedia.org/wiki/Matrix_(mathematics) In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns.[1][2] For example, the dimension of the matrix below is 2 × 3 (read "two by three"), because there are two rows and three columns: {\displaystyle {\begin{bmatrix}1&9&-13\\20&5&-6\end{bmatrix}}.}{\displaystyle {\begin{bmatrix}1&9&-13\\20&5&-6\end{bmatrix}}.} Provided that they have the same dimensions (each matrix has the same number of rows and the same number of columns as the other), two matrices can be added or subtracted element by element (see conformable matrix). The rule for matrix multiplication, however, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second (that is, the inner dimensions are the same, n for an (m×n)-matrix times an (n×p)-matrix, resulting in an (m×p)-matrix). Even when two matrices have dimensions allowing them to be multiplied in either order, the results need not be the same. That is, matrix multiplication is not, in general, commutative. Any matrix can be multiplied element-wise by a scalar from its associated field. Matrices are often denoted by capital roman letters such as {\displaystyle A}A, {\displaystyle B}B and {\displaystyle C}C.[3] ****** Important A matrix is a rectangular array of numbers (or other mathematical objects) for which operations such as addition and multiplication are defined. [8] Most commonly, a matrix over a field F is a rectangular array of scalars, each of which is a member of F.[9][10] Most of this article focuses on real and complex matrices, that is, matrices whose elements are respectively real numbers or complex numbers. More general types of entries are discussed below. For instance, this is a real matrix: {\displaystyle \mathbf {A} ={\begin{bmatrix}-1.3&0.6\\20.4&5.5\\9.7&-6.2\end{bmatrix}}.}{\displaystyle \mathbf {A} ={\begin{bmatrix}-1.3&0.6\\20.4&5.5\\9.7&-6.2\end{bmatrix}}.} The numbers, symbols, or expressions in the matrix are called its entries or its elements. The horizontal and vertical lines of entries in a matrix are called rows and columns, respectively. Size The size of a matrix is defined by the number of rows and columns that it contains. A matrix with m rows and n columns is called an m × n matrix, or m-by-n matrix, while m and n are called its dimensions. For example, the matrix A above is a 3 × 2 matrix. Matrices with a single row are called row vectors, and those with a single column are called column vectors. A matrix with the same number of rows and columns is called a square matrix.[11] A matrix with an infinite number of rows or columns (or both) is called an infinite matrix. In some contexts, such as computer algebra programs, it is useful to consider a matrix with no rows or no columns, called an empty matrix. URL https://en.wikipedia.org/wiki/Matrix_(mathematics)

Abstract object, concrete object Placeholder | Back Most of the mathematics we are exploring has to do with abstract objects -- by which we mean some symbolic representation -- usually in an alphabet -- and actually existing in some medium -- such as words written on printed on a piece of paper. Is a generalization or "universal" an abstract object? I would say yes, in that its only concrete actuality/reality is that it points to a class of specific objects -- which in some cases might also be abstract or universal Fri, Mar 26, 2021 Reference In metaphysics, the distinction between abstract and concrete refers to a divide between two types of entities. Many philosophers hold that this difference has fundamental metaphysical significance. Examples of concrete objects include plants, human beings and planets while things like numbers, sets and propositions are abstract objects.[1] There is no general consensus as to what the characteristic marks of concreteness and abstractness are. Popular suggestions include defining the distinction in terms of the difference between (1) existence inside or outside space-time, (2) having causes and effects or not, (3) having contingent or necessary existence, (4) being particular or universal and (5) belonging to either the physical or the mental realm or to neither.[2][3][4] Despite this diversity of views, there is broad agreement concerning most objects as to whether they are abstract or concrete.[1] So under most interpretations, all these views would agree that, for example, plants are concrete objects while numbers are abstract objects. Abstract objects are most commonly used in philosophy and semantics. They are sometimes called abstracta in contrast to concreta. The term abstract object is said to have been coined by Willard Van Orman Quine.[5] Abstract object theory is a discipline that studies the nature and role of abstract objects. It holds that properties can be related to objects in two ways: through exemplification and through encoding. Concrete objects exemplify their properties while abstract objects merely encode them. This approach is also known as the dual copula strategy.[6] The type–token distinction identifies physical objects that are tokens of a particular type of thing.[7] The "type" of which it is a part is in itself an abstract object. The abstract–concrete distinction is often introduced and initially understood in terms of paradigmatic examples of objects of each kind: URL https://en.wikipedia.org/wiki/Abstract_and_concrete