~ The Study of Threes ~

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The current model of Human Mathematics is that of an embellished Dichotomy attempting to let us say, transgress... or to evolve into a cognitive trichotomy as is consistent with the seriality of numbers (from 1 to 2 to 3... and so forth), which can be describe in instances of biological development such as from the single-stranded RNA to the double-stranded DNA, as well as the 2-germ layer life form model to the 3-germ life form model. Whether one thinks in terms of a specific subject matters such as biology, philosophy, mathematics, etc., or in terms of a basic cognitive profile expressing itself within the scope (vocabulary, symbols, extent of application, etc.,) of a given subject, it is valuable to assess the overall dimensions of ideas expressing themselves in a manner conducive to the use of an attempted singular means of identifying diverse illustrations of such a pattern(s). Here are some examples from other subjects:

- Yin/Yang dichotomy has "transgressed" its primary "two" state of expression by way of being presented with a presumed cognitive trichotomy through the philosophy of the I-Ching and the so-called triads (though they are actually embellished biads).
- Double-helix and paired amino acids of DNA "transgressed" from this "two" or dual pattern state of expression by the adoption of a triplet code.
- Checkers and Chess games with their two-sided oppositions overlayed on a 64 square parameter of executable play, was transformed into using a three-option diagonal- vertical- horizontal activity.
- One might include the good/evil (god/devil) dichotomy paired with the existence of three "great" Monotheistic religions (Christianity, Judaism, Islam).

This is not to say that all dichotomies achieve some lasting viability in the development of a third phase, such as we see in the development of Germ layers, where a two-germ layers existence can be seen in different (simple) life forms and where a distinct three-germ layers is identifiable in multiple (complex) life forms. This is assuming of course if the human brain's expression of cognitive activity has an inherent orientation towards developing along a scenario we can identify with a serial model of enumeration such as 1... 2... 3..., whether or not we define this as progress, as a consequence of adaptation to environmental pressures, or simply some activity of mental activity one might describe as imposed pattern creation... where even the lack of pattern can be labeled a pattern, so as to comply with a human brand of cognitive activity not fully understood.

If we use any of the basic interactive properties better known as add, subtract, multiply, divide; some observers may not see this as a list of two dichotomies, and instead prefer to reference them as a group-of-four. Assigning them a "group-of-four" (or as two sets of dichotomy) does not in themselves advantage us in recognizing additional similar patterns unless one actively searches for them, or for some reason a person's mind necessarily becomes practiced towards identifying the presence of one or more patterns, be it geometric, numerical, sound, weight, height, absence, etc...

Let us take for example the simple 1 + 1 = 2. We can call this a pattern of 3 numbers and 2 symbols or an overall pattern of 5 as a set. If we say there is a sequence of three numbers grouped in a linear fashion, we might well want to look for other expressions which suggest to us this same identity... and identity that we made up and assigned so as to give direction to observations from which a dialogue with our own mind can be established, and perhaps shared with others who may simply agree, or we might wait to reveal after we think we have developed what we believe to be is a defensible position of our efforts that some may want to argue about or take time to whittle away because of some unrecognized ax they wish to grind, but is nonetheless valuable for reaching a greater (enlarged) knowledge base of that which we pursue... and may well be a trail from which we diverge and come to redefine as more information becomes available and alternative perspectives are advanced from different vantage points.

Since I frequently resort to making reference to "Cognitive Activity", it is of need to address how I am using the phrase in the present context of reviewing what I claim to be basic patterns thereof, related to a Mathematical context. This is not to say that all subjects have their own basic patterning, though some viewers might insist this is the case, because they are looking at patterns derived primarily from the language- labels being used. However, someone might argue that using the words "one, two, three..." etc., are word labels that can have multiple interpretations, though the intent is to describe a non-reducible reference. This is particularly true if one subscribes to the idea that math is itself a metaphor of actual brain activity transformed to comply with an era-specific, culture-specific, gender-specific allowance of expression. In such a view, all vocal utterances, regardless of the language being used in a given context, are generalities of actual brain activity, or that brain activity is simply being aligned to correspond to the relative vocabulary of a given time and place, whereby the most pristine basic cognitive activity is simply variable chemical events occurring among neurons in whatever state of nutrition and development a given brain is in.

Irrespective of the different stances one might take on explaining basic cognitive activity, I am using the phrase connected with numbers used to identify and catalog both repetitions and lack thereof associated with ideas whose lengths in sentence and paragraph representations may be long and rely upon a non-commonplace vocabulary, but nonetheless can be interpreted as displaying simple enumeration. For example, if the content of a long sentence speaks of a contrast, this necessarily represents a pattern-of-two. Such might also be true if a question is asked which hints at a known or unknown alternative consideration. Some readers might not see a contrast or recognize that a contrast is being presented as a consideration, though in many instances the act of using a contrast is applied without any consciously prolonged observation thereof.

If we say that all languages rely heavily (that is "predominantly") on patterns-of-two and is therefore a representation of basic cognitive behavior throughout the world, what then do we say when confronted by multiple expressions which apparently defy this pattern? Whereas we could be dismissive or turn to some religious or other colloquially practiced digression to prevent ourselves from further exploration; or we could take some time to catalogue the different underlying patterns based on a simple process of association. For example, some might use the punctuation mark called a period most often, and not make any reference to the question mark or exclamation point, if they do not use them or use them very infrequently. Another researcher however would prefer to catalogue all three and admit that some punctuation marks are used more often than others, but that if examined closely over time, each of the three could be associated with a numerical (quantitative) value describing usage... which may vary from person to person and not necessarily be expressed solely by the spoken word as opposed to the written or typed versions.

Since Mathematics relies heavily on language and the words of the language being used frequently represent a variety of patterns-of-two that others may describe as dualities, dichotomies, contrasts, pairings, squared, combinations, etc., we need to take stock of this behavior and describe it as a type of cognitive activity. When we compare the quantity of patterns-of-two being used in Mathematics with other subjects and note that some, but not all also rely prominently on such a pattern, it is of need to use the tool of further comparison with respect to (for example) biological development, since the history of using numbers (later called Mathematics) has followed in its development with that of human thinking related to biology and sociology aligned with representative philosophical perspectives.

In multiple cases (with respect to philosophy), we find a recurrent usage of patterns-of-two another person might label as dualities or dichotomies, as well as attributing a particular example to a given subject such as music, art, religion, politics, etc., without even mentioning the word "philosophy" or "philosophical", whereby we can find different contrasting ideas becoming isolated and "owned" by a given subject instead of combining all of them and labeling them different flavors or colors or instances of a similar cognitive activity being show-cased in different contexts. Claiming some level of ownership to an idea sporting some contrast typically aligned with a given subject's vocabulary, often leads some observers to overlook the same type of general pattern of thinking taking place in all subjects, but the presence of a recurring use of dichotomies is not being contrasted with other recurring patterns as a means of identifying an overall compilation to denote whether there is a change in frequency over time and place... if not subject context.

- Persistent Dichotomies (Occurring in Psychology courses)
- Rationalism vs. Empiricism
- Take the Dichotomy test
- Dichotomies as a Prerequisite for Existence by Mallika Vasak, May 2nd, 2020
- The Dichotomy of Nature by W. H. Sheldon, July 6th, 1922
- Duality in Mathematics and Physics by Sir Micharel F. Atiyah
- Encyclopedia of Mathematics: Duality
- Zeno’s Paradoxes (for example, singularity versus plurality)
- Subjective-objective dichotomy
- Nature versus Nurture (A philosophical debate in Psychology.)
- Heaven Versus Hell (A philosophical debate in Religion.)
- List of Aesop's Fables (Dealing most frequently with a contrast.)
- Dualism (for example, monism versus dualism)
- Mind/ Body dualism
- Dichotomies
- Category:Dichotomies
- Category talk: Dichotomies
- Dichotomy vs. Juxtaposition: What Is the Difference?
- Dichotomies in biology and the origin of life problem
- Beware Dichotomies by Charles J. Kowalski, Adam J. Mrdjenovich; Perspectives in Biology and Medicine,
Johns Hopkins University Press, Volume 59, Number 4, Autumn 2016, pp. 517-535, 10.1353/pbm.2016.0045

__Abstract__This essay combines our thoughts concerning the generally destructive practice of dichotomization with a selective review of the literature supporting our critique. The apparent simplicity of dichotomous thinking encourages its use even when a dyadic representation is totally inadequate to understanding complex situations, and this "simpler is better" mantra continues to stymie our understanding of many of the world's complexities. The identification and naming of two distinct, opposing categories often results in their being seen as in opposition to one another, and that it is somehow incumbent upon us to choose one or the other. This either/or orientation reinforces the original split, confusing explanans and explanandum. We begin by considering dichotomization in general terms, and then turn to brief descriptions of several particular dichotomies. Some of these persist despite what might well be considered sufficient evidence to deny their usefulness, and this often deflects attention away from the more fertile, interesting, and important questions that may be directed to the points at which they intermingle.

- The Prokaryote-Eukaryote Dichotomy: Meanings and Mythology by Jan Sapp, DOI: https://doi.org/10.1128/MMBR.69.2.292-305.2005
- Dichotomy and Dualism – Human Geography UPSC by Lotus Arise, Jan. 15, 2021
- Why does human mind tend to understand everything in terms of duality or binary states rather than continuum of states?
- Dualism and Mind by Scott Calef
- Dualism in Cosmology (Various orientations described as Religion, though some may prefer to adopt a label that does not suggest the idea of Religion.)
- The False Dichotomy in Mathematics Education Between Conceptual Understanding and Procedural Skills: An Example from Algebra by Carolyn Kieran, 1st Jan. 2013 [Knowing that vs. Knowing how]
- Discrete and continuous: a fundamental dichotomy in mathematics by James Franklin, Journal of Humanistic
Mathematics, 7 (2):355-378 (2017)

The distinction between the discrete and the continuous lies at the heart of mathematics. Discrete mathematics (arithmetic, algebra, combinatorics, graph theory, cryptography, logic) has a set of concepts, techniques, and application areas largely distinct from continuous mathematics (traditional geometry, calculus, most of functional analysis, differential equations, topology). The interaction between the two – for example in computer models of continuous systems such as fluid flow – is a central issue in the applicable mathematics of the last hundred years. This article explains the distinction and why it has proved to be one of the great organizing themes of mathematics.

- Purifying applied mathematics and applying pure mathematics: how a late Wittgensteinian perspective sheds light onto the (pure vs. applied mathematics) dichotomy, by José Antonio Pérez-Escobar & Deniz Sarikaya
- Are There Absolutely Unsolvable Problems? Gödel's Dichotomy by
Solomon Feferman, 9th Jan. 2006

This is a critical analysis of the first part of Gödel's 1951 Gibbs lecture on certain philosophical consequences of the incompleteness theorems. Gödel's discussion is framed in terms of a distinction between objective mathematics and subjective mathematics, according to which the former consists of the truths of mathematics in an absolute sense, and the latter consists of all humanly demonstrable truths. The question is whether these coincide; if they do, no formal axiomatic system (or Turing machine) can comprehend the mathematizing potentialities of human thought, and, if not, there are absolutely unsolvable mathematical problems of diophantine form.

*Either – the human mind – infinitely surpasses the powers of any finite machine, or else there exist absolutely unsolvable diophantine problems.* - Duality in Mathematics

Duality, in mathematics, principle whereby one true statement can be obtained from another by merely interchanging two words. It is a property belonging to the branch of algebra known as lattice theory, which is involved with the concepts of order and structure common to different mathematical systems. A mathematical structure is called a lattice if it can be ordered in a specified way (see order). Projective geometry, set theory, and symbolic logic are examples of systems with underlying lattice structures, and therefore also have principles of duality.

Projective geometry has a lattice structure that can be seen by ordering the points, lines, and planes by the inclusion relation. In the projective geometry of the plane, the words "point" and "line" can be interchanged, giving for example the dual statements: "Two points determine a line" and "Two lines determine a point." This last statement, sometimes false in Euclidean geometry, is always true in projective geometry because the axioms do not allow for parallel lines. Sometimes the language of a statement must be modified in order that the corresponding dual statement be clear; the dual of the statement "Two lines intersect in a point" is vague, while the dual of "Two lines determine a point" is clear. Even the statement "Two points intersect in a line," however, can be understood if a point is considered as a set (or "pencil") containing all the lines on which it lies, a concept itself dual to the idea of a line being considered as the set of all points that lie on it.

There is a corresponding duality in three-dimensional projective geometry between points and planes. Here, the line is its own dual, because it is determined by either two points or two planes.

In set theory, the relations "contained in" and "contains" can be interchanged, with the union becoming the intersection and vice-versa. In this case, the original structure remains unchanged, so it is called self-dual.

In symbolic logic there is a similar self-duality if "implied" and "is implied by" are interchanged, along with the logical connectives "and" and "or."

Duality, a pervasive property of algebraic structures, holds that two operations or concepts are interchangeable, all results holding in one formulation also holding in the other, the dual formulation.

__Examples of two-patterned topics in research on Mathematics:__

- teaching math vs. learning math
- those who hate math vs. those who love math
- pure mathematics vs. applied mathematics
- qualitative math vs. quantitative math
- rote learning vs. meaningful/active/smart learning

In this youtube lecture: The essential dichotomy underlying mathematics, the speaker provides the dichotomy of "nothing/something". However, if we then suggest it is a fundamental representation of basic human cognition, we run into the problem of overlooking other models of dichotomy, if we rely on a strict observance of these words with the value of zero and one (0/1); because the closest representation with a similar characterization is the binary code of computers. Other dichotomies do not easily yield themselves to being assigned the values of zero and one. For example, take any of the dual examples from a yin/yang list, and there is no clear means of assigning such values to them. The male/female dichotomy nor the night/day, nor the hot/cold groups (sets), etc..., become routinely or logically assessed in terms of the zero and one values being used as numerical references for nothing and something. Use of such words can create an obstacle towards analyzing other subjects. However, even the words duality, dichotomy, paired, parallel, opposite, twin, opposite, etc., can cause misunderstanding if flexibility is not permitted, because one' ego is too tightly wound around a particular concept and label which is meant more for effecting an exclusive specificity than an inclusive generality.

Similar to the previous "nothing/something" idea (yet with an attempt to explicate the notion of a step beyond duality) can be found here: Laws of Form and the Logic of Non-Duality by Louis H. Kauffman, UIC

If you use a word such as "dual" to define the parameters of one's search for basic patterns in Mathematic's, the following list might be found on the Wikipedia (List of Dualities):

- Alexander duality
- Alvis–Curtis duality
- Artin–Verdier duality
- Beta-dual space
- Coherent duality
- De Groot dual
- Dual abelian variety
- Dual basis in a field extension
- Dual bundle
- Dual curve
- Dual (category theory)
- Dual graph
- Dual group
- Dual object
- Dual pair
- Dual polygon
- Dual polyhedron
- Dual problem
- Dual representation
- Dual q-Hahn polynomials
- Dual q-Krawtchouk polynomials
- Dual space
- Dual topology
- Dual wavelet
- Duality (optimization)
- Duality (order theory)
- Duality of stereotype spaces
- Duality (projective geometry)
- Duality theory for distributive lattices
- Dualizing complex
- Dualizing sheaf
- Eckmann–Hilton duality
- Esakia duality
- Fenchel's duality theorem
- Hodge dual
- Jó nsson–Tarski duality
- Lagrange duality
- Langlands dual
- Lefschetz duality
- Local Tate duality
- Opposite category
- Poincaré duality
- Twisted Poincaré duality
- Poitou–Tate duality
- Pontryagin duality
- S-duality (homotopy theory)
- Schur–Weyl duality
- Series-parallel duality[1][2]
- Serre duality
- Spanier–Whitehead duality
- Stone's duality
- Tannaka–Krein duality
- Verdier duality
- Grothendieck local duality

It is unfortunate for some researchers who may come upon such a list to be combined with their own list of patterns-of-two as an attempted argument against other patterns (such as 3 or 7 or 4, etc...) as being more prevalent, without taking a moment to step back and not only recognize that it is only a handful of number-related patterns that humans are using repetitively, but that the absence of any pattern and the presence of another number(s) pattern is not being considered as an overall cognitive set in itself. Some people are so consumed with trying to establish a given number pattern as being more dominant or more special, etc., they don't consider that the variable presence and absence between different subjects illustrates a cognitive map humanity (for the most part) neither recognizes nor is able to define the terrain(s) being exposed and points of interest that humanity could benefit from exploring. If you think you have found an interesting trail, then let me know and I will grab a backpack, compass, canteen, rifle, first aid kit, rope, etc., for the journey into some uncharted territory. Unless of course you are someone who doesn't want to share or think that it is a path to be taken only by some assumed worthiness, skill and other preparation to insure control of the findings to fit with some pre-conception being proposed as a definitive truth suggested in the terms denoted as "the way", "the right view", etc...., originating from impoverished conditions seeking sense and personal measure in an otherwise difficult time and place.

But the foregoing list does not necessarily create for "some-one" (a term meaning plurality and singularity combined), an added ability to perceive other patterns-of-two unless the parameters of search are expanded to make allowance for a wider breadth of representation as to how such a pattern can be illustrated by different people living under different conditions, times, and places. However, if we were to combine the word dual or duality to such examples as the following short list, one might then more easily appreciate the task being undertaken to show that Mathematics is saturated more so with patterns-of-two than other patterns, and this well relate to an underlying reliance of art-aligned visualizations and less on what we might otherwise call linguistic-based logic illustrated with symbols... since symbols are often an abbreviated model of artistic renditions of expression.

- add/subtract duality
- multiply/divide duality
- rational/irrational duality
- every number has an opposite (In fact, every number has two opposites: the additive inverse and the reciprocal—or multiplicative inverse.)
- positive (absolute)/negative (opposite) duality
- opposite angels: When any two straight lines intersect each other, there are different pairs of angles that are formed. The angles that are directly opposite to each other are known as opposite angles.
- Algorithm vs logarithm difference:
*"An algorithm is a process for calculating complex numbers. In other words, it is a logical procedure that produces a particular result. A logarithm has a base number. A base-ten logarithm has three bases. Hence, the base of a logarithm is a power. The exponent of a number is the base of the logarithm. An algorithm is an order-of- magnitude that is a power of ten. An algorithm, on the other hand, is a method for calculating the common logarithm of a number. The latter is often called a heuristic. It is a mathematical algorithm. Its inverse is an exponential. If you want to compare the two, you must first find out which one is more efficient.* - finite/ infinite duality
- amplitude/ frequency duality
- hypotenuse/ opposite side duality
- equality/ inequality duality
- odd/ even duality
- Opposite of pi? The word pi typically refers to the mathematical constant classically defined as the ratio of a circle's circumference to its diameter, equal to approximately 3.14159, and represented with the Greek character π. There are no categorical antonyms for this word.
- Perimeter versus Area: Perimeter is the distance around the outside of a shape. Area measures the space inside a shape.
- less than/greater than duality
- single bar graph/ multi-bar graph (two or more bars is related to the concept of "many", hence... we have a "one-many" concept seen also in the "E Pluribus Unum" phrase as part of the U.S. presidential symbol)
- monogon/ polygon
__Reference of "two" for three types of circles:__(Identify Circle Components Components, circumference, and area of circles.)- Tangent Circles: Two or more circles that intersect at one point.
- Concentric Circles: Two or more circles that have the same center, but different radii.
- Congruent Circles: Two or more circles with the same radius, but different centers.

- mononomial/polynomial duality
- superscript/subscript duality
- polybinuak/ non-polynomial (Division by a variable (this can lead to negative exponents); A variable with a negative exponent, ...with a fractional exponent, ...inside a radical: What Cannot Be A Polynomial? )

Let me also make reference to the word "polynomial" which is like the name of a family were individuals may be called Mononomial, Binomial, Trinomial, and
that even though all are considered to be a "polynomial", the word can be referenced as *"A monomial, or two or more monomials, combined by addition or
subtraction".* (Identifying
Characteristics of Polynomials.) The composite reminds me of an individual with a first, second and last name distinction which may be a cognitively
expressed image projection occurring in other subjects as well, but not commonly referred to as having analogical similarities in other subjects. In other
words, the same pattern arises in other subjects in different guises (symbolism, words, etc...), but is overlooked as a recurring cognitive theme which has
developed over time and may well be altered with further time advancing humans in an incrementally deteriorating environment.

In a sense, mathematics is a person in the sense that it discriminates, isolates, and is very emotional, though some see it as having a stoic demeanor. Equations can be long (tall), short, simple, complicated, rambling, part of a group, be explorative, be insecure, project a dominance and stalwartness, preface a believed in truth, playfulness, etc....

With respect to a pattern-of-three such as the three germ layers, let us call one germ layer as a singularity, two germ layers, a duality and three germ layers a triplicity. While the mouth/anus association might well be called a duality, it persists into the development of a triplistic germ layer modeling platform. Hence, we might want to say this is a developmental overlap, and think that perhaps in a one-germ layer state of development, such an organizational pairing does not exist or is just beginning to show itself as a differentiation. In other words, let us not overlook the presence of overlappings taking place in Mathematics nor that development necessarily means moving completely away from some earlier functionality, though in later stages of development an earlier function that was predominant, shows itself as if it were like a vestigial organ whose used has been lost to time and there for any clarity of definitive explanation for use or application that was initially developed.

__Date of Origination:__30th August 2022... 5:01 AM

__Date of Initial Posting:__10th September 2022... 5:58 AM

__Updated Posting:__2nd January 2023... 11:09 AM