~ The Study of Threes ~
http://threesology.org
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Permit me to restate the dominant view for the present course of pages in the Mathematical series, and along with it that so-called advanced mathematics or even mathematical behavior beyond a primary school level requires an increased practice of artificial thinking, otherwise if it were natural, mathematics might well be a commercialized sport or pursued with as much excitement and vigor as do other past-times:
We are confronted by a present day mathematics which relies heavily on patterns-of-two, just as we see being used in the binary code of computers utilizing zeroes and ones (due to the on/off constraints of a typical electric circuit). In the case of mathematics, there are words and symbols more often being used than numbers; whereby we do not see an applied repetition of a single two-patterned numerical set as one might see in a Janus-faced sculpture. Whereas we can find instances of monomial and polynomial (with specificities named trinomial, quadrinomial, quintinomial, hexinomial, etc., even if uncommon), let us note that the dominant orientation in mathematics appears to be patterns-of-two variously described (for example) as:
- add/subtract
- multiply/divide
- rational/irrational
- greater than/less than
- subscript/superscript
- real/imaginary
- fraction/whole
- positive/negative
- constant/variable
- word/empty word (i.e. 1 + 1 "1" is a word, a sign such as "+" is an empty word... numbers are words, symbols are empty, or non-words)
- open set/closed set
- parallel/series (such as seen in a battery pole arrangement regarding amps and volts)
- mononomial/polynomial (Strictly speaking, "mono-mial" refers to one item and "poly-nomial" refers to more than two, without counting variables.)
- Prime numbers/Composite numbers
- angle/straight
- curved/flat (such as curved or flat space)
- incongruent/in- or non-congruent
- symmetrical/asymmetrical
- (a,b) open interval/ [a.b] closed interval
- ordinal/cardinal
- etc...
(Note: Mathematicians claim the word "monomial" is a "polynomial" and then qualify this by saying a mononomial can have multiple variables. I think this is a cover-up concerning the use of the word "monomial" when the usage of the word is to be a reference for one or more variables, it should have its own terminology to distinguish it instead of relying solely upon where it is being applied. It is an error in logic that Mathematicians try to portray as a truth defined by extenuating circumstances. Mathematics fumbled the ball on using the word "monomial" as a polynomial, yet changed the linguistic rules used in philosophy to assuage self-recriminations for using a term incorrectly.)
While a mathematician working in their field may not verbally express (as a list) each and every one of the foregoing examples (or others), nor do other professionals in their respective fields making a point of highlighting the multiple dualities which exist, the point is that such groupings occur but do not necessarily use the same terminology. The underlying pattern-of-two represents a cognitive 'schema' (recurring structural component of brain activity) to be found in multiple subjects, but not necessarily to the degree we find in Mathematics and other ideas. Thus, let us ask if Mathematics little more than a sophisticated, elaborated, or convoluted means of pairing? Whereas we can list groups-of-two such as the foregoing list, making use of them in alternative ways involving numbers and symbols creates a system of activity, just like the pairing of amino acids help to produce proteins and from proteins multiple biological activities.
In several instances of those whose research has been on the history of Mathematics, we observe early counting systems using a doubling or pairing system, to the extent even today the use of the word "teen" references a repetition of the word "ten", as well as when we say 20 (two tens), 30 (three tens), 40 (four tens), etc... However, not only have words sometimes been repeated to express a higher quantity, but symbols themselves such as in the case of using a group of single lines to designate a group of four, and then a slanted one to represent a group of five (sometimes called tally marks).
Is the currency of dualities found in present day Mathematics a Westernized embellishment of the Easternized Yin/Yang ideology which apparently got its cue from observations of natural (or thought-to-be natural) events? What then does this not only describe about human conceptualization, but the effects of an environment exhibiting repetitious patterns (such as the night/day sequence) on a course of planetary incremental decay to which biology is impressed upon to accommodate by way of adaptation as a survival mechanism tied to a structure of eventual diminishment that biology must follow or remove itself from the influences thereof? Will we be able to recognize the deterioration being exhibited in the forms and function of Mathematics if we have not recognized it basic structuring in order to determine changes wrought over expanses of time?
It is of need to compare this organizational framework with other pattern-of-two sets such as the yin/yang philosophy, double-speak of politicians, Persistent Dichotomies used in a college course on psychology, and the sometimes noted Legal Doublets, though one might include other philosophical terms such as dichotomy, duplicity, dyad, duality, etc...
In the case of Legal doublets, some readers may claim they are not dichotomies in terms of opposites, but they nonetheless represent a similar cognitive pattern of "twoness", so I include them.
Other examples of patterns-of-two, whether viewed in terms of being opposite or not, is seen in the Binomial naming system of different life forms. And as seen on the previous page, we see the usage of two names as a typical exercise in the names of actors and actresses when the list of credits on a movie are seen. However, with regards to names, one needs to review the history of naming (such as for humans), where a single name may have been the first naming standard centuries ago (just like many of us use only one name to address a person), that was followed by two names and then like many people today, have three names, a first... middle... last. In a Chinese culture one might see the custom of using a person's last name first because this is foremost important for them in knowing some past history that may be connected with their family name, and then following the last name with the person's first name. In addition, whereas the use of two names may be dominant in some European places, there is an increased usage of three names when we get to the U.S. As if the U.S. is a "later born" offspring who evolved into using more "threes" references than European forbearers. The usage of "threes" or three-patterned ideas has been documented by anthropologists such as Alan Dundez in his 1967-68 book entitled "Every Man His Way". Here are links to a review, the book and the chapter about "threes" being used in the American culture:
- Every Man His Way: Readings in Cultural Anthropology Paperback – January 1, 1968 by Alan Dundes
- Every Man His Way: Review and Parentheses
- Number 3 in American Culture
- Number Three in the American Culture
With respect to Mathematics being viewed as a "culture", an opening excerpt from the Number 3 chapter is highly useful:
Students undertaking professional training in anthropology are rarely, if ever, required to formally study their own cultures. They must demonstrate competence in various topics and areas, but these do not normally include materials from their own cultures. They may be told that the identification and careful delineation of native categories may be crucial to a fuller understanding of that culture which they investigate, but their own native categories, the identification of which is equally important for an understanding of another culture, may not be considered at all. With our present knowledge of the cultural relativity of perception and cognition, it seems clear that students of anthropology should be encouraged to analyze their own native categories with the same care and methodological rigor that is demanded of them in their fieldwork in other cultures. If the reduction of ethnocentric bias is truly an ideal of anthropological scholarship, then anthropologists should go into the field with as comprehensive an understanding of the nature of their own culture as possible.
Mathematics is as much a culture as are different sports, music, acting, politics, baking, law making, praying, etc. It has its own language and particularized venues, despite the claim that it can be performed just about anywhere... just like singing, dancing and criminal behavior. Yet, it is the opposite view which is most often tendered when mathematics is discussed, to the extent we speak of Mathematics in one or another culture as if it is some sort of natural appendage instead of added-to activity as one might view the exercise of observing a holiday. How we interact with mathematics creates the present ostracism where many a people distance themselves from and may even say they hate mathematics. And what do mathematicians think about the current relationship has in different cultures of the world? Here is one example:
Why do so many people Hate math?
Duality, in mathematics, (is a) 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 (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. Britannica: Duality (Mathematics)
The idea that the dualities occurring in mathematics (as well as various forms of philosophy), may well stem from recurring (repetitious) patterns to be viewed (and sometimes felt) in nature... such as hot/cold, light/darkness, etc... is easy to understand. But this is not the case for citing early (primitive) notions of duality (such as male/female) with more abstract ideas (such as time and space.) Whereas the connection (if any) may be lost in time, not the fundamental characterization of "twoness". The primitive side of human conceptualization can recognize "twoness" as more than "oneness" and less than "threeness", but how does the brain of humanity accomplish this without resorting the label "subtizing"?
Apparently, the simplest origination of thinking in terms of number and then perhaps the development of an equation, is an instance of pairing one item with another item, even if a given individual does not label them with a number, nor intentionally (consciously) engage in any simple "plus, minus, subtract, multiply" activity with them. Whether we take a single item and then subject it alone to the "plus, minus, subtract, multiply" constraints of further activity whereby one might think of in terms of fractions to create some measure of a paired to to-be-compared result; the point is that we are faced with the presence of a repetition. Our present mathematics rests on repetition, though many a reader no doubt takes this observation for granted.
The same goes for developing a language to be used in computers based on the two-patterned on/off switching functionality of an electrical circuit. Necessarily so, because of the constraint, some observers might be inclined to think of the situation as being logical, normal and natural... particularly if they invest much of their livelihoods and self-identity with the form and functionality of such an existence. The idea of thinking alternatively beyond the boarders of the underlying constraints of a give "two" pattern may not come to mind... even to the extent that substitute symbols are adopted. In other words, instead of using zeros and ones for the basic binary code of computers, one would use some other number combination... even if moving away from simplicity would seem to be counter-intuitive and produce cumbersome situations.
Whereas we think of numbers and equations as being non-biological and thus have no inherent capability of self-movement, some will nonetheless come to say that a given equation is dynamic... that it represents or reflects a situation of movement. On the other hand, if we could make an equation motile, as if developing their own means of self-propulsion, would we also be speaking in terms of sentience, or merely the presence of some underlying functionality which becomes alive due to some emergent property that in biological circumstances might be defined in terms of chemistry? Hence, is it possible to alter the underlying form and function of Mathematics to make it develop into some primitive life-form?
Since it is thought by some that life emerged by way of some as yet unknown process by churning different chemical elements into a soup, broth or stew under particular environmental conditions, can it be that the elements of a new type of life form is possible if we could only place the correct math elements together? Is this a fanciful tale of some imaginative exercise or the contemplations of some otherwise comic book overture breeching the boundaries of convention for the sake of playing some one-upmanship game similar to a rock, paper, scissors configuration, because a two-element construction of such a game (rock/paper, scissors/paper, rock/scissors) is not as intellectually entertaining as when three items are used in games of chance? Whereby we additionally think of mathematics as box filled with different games of chance, whose results are dictated by the axiomatic rules being applied like the rules printed on a folded sheet of paper supplied with a board game by a manufacturer who may or may not insert their own rule variations for one or more players to achieve a certain outcome?
Rules of a game just like the axioms used in mathematics are constraints on how one is to think about playing pieces— where in mathematics are numbers and symbols instead of checker or chess pieces, unless the game pieces are like those found in a game of Monopoly. And like education (as well as established systems of commerce) which enforce the use of mathematics, so too do manufacturing plants which produce games with the same rules. The rules are standardized, traditionalized, and functionalized to exert a repetition so as to facilitate a means by which repetition can be used to bring costs down as well as get everyone to think alike... where variability of thought is only permitted in how well or poorly one plays a game in terms of strategy... despite the claims of luck... and of course the standardized idea that "the system is rigged" to favor the house, the government, the institution, the business, a given religious ideology, etc...
However, in thinking about the development of a truly self-motile (dynamic) form of mathematics, (let us call it a dynamic Calculus); we are confronted by the same impasse that chemical evolutionists are when trying to decide which elements— when placed together under which conditions —will produce a product one might describe as a life form, or at least some construction which sits at the threshold of becoming a dynamic creature... Whereas one might describe the precursor of life to be a cell, the elements (organelles) of a cell might well be argued to be a proto-life form, just as we might think in terms of a proto-mathematics or even proto numbers. In this sense, numbers could be viewed as separate organelles that... when combined, create an organization that one might describe as a number line that is encapsulated in cells or cellular structures called ones, tens, hundreds, thousands, millions, etc... Organelles within a cell, cells within an organ, organs within a body, bodies within a community, etc...
What then is Mathematics if the present analogy is describing a situation where without mathematics, many of the functional activities of a present day society are not viable? Is present Mathematics a type of overlooked disease which functions symbiotically in order to sustain its own viability, yet that viability is one akin to a weed out of which are described (intellectual) fruits sold in different market places, like a market originating in due course to take advantage of given commodities, though those commodities are less valuable than what could be found elsewhere if current social standards did not dictate to the populace that they must engage in the practices of slave trading, narcotics smuggling, and other less desirable activities so as to act as a detour, a dead end for the establishment of a new type of Mathematics?
Why is Mathematics so contrary to everyday thinking? Is the general rule-of-thumb for human cognition to be cited as being rather dysfunctional, or is it that mathematics has been contoured over time to present itself as a dominant means of thought whereby any underlying developmental fledgling of conceptualization is prevented from being exercised and practiced in the same developmental manner as mathematics was permitted to develop... like social laws being established to protect the invalid, the handicapped, the marginalized, the ostracized, and other weak, poor and the disenfranchised which make up most of society? No less, like an ancient religion kept alive by hidden practices and once viewed as an underdog of conceptualization; it has been treated with an abundance of "tax incentives" (social dispensations) which assisted in its fledgling development to produce what one might refer to as a rags-to-riches story.
Because so many ideas have had to go through a process of development where bullying, condemnation, and various disparagements arose, it is not far-fetched to consider that the growth of Mathematics did not share in this type of upbringing. While one might fully embrace comments found in an entitled The Perceptual Origins of Mathematics; as in most cases there is no description of Mathematics going through a trial and error process of acceptance, just as we encounter today in classrooms and those who attempt to use more math in their day-to-day activities, but can't seem to wrestle out a workable conceptual framework that is as easy to use as making out a grocery list or memorizing phone numbers... which is itself become a lost art since cellphones are used to memorize more than just numbers. In short, historians of Mathematics do not customarily speak in terms of how Mathematics was not only rejected, but is largely ignored even today. Whereas most people engage in simply models of calculation, more complex mathematical activities are not routinely turned to as one might when looking to a clock, calendar or other recording device of a dynamic situation. Indeed, humans... for the most part... have a terrible relationship with mathematics.
On the one hand we see many people creating collection boxes of tools, sewing implements, fishing tackle, memorabilia, coins, cards, string, models (aircraft, ships, spacecraft, battlefields), etc., it is not customary to find collections of different maths. While one may have a collection of books on different sciences, religions, crafts, sports, vehicle repair, etc., not so is the case for mathematics models. Mathematics is an esoteric interest, and not one that we might otherwise describe as being of commonplace interest and an everyday conversational topic. Understanding the psychological disposition of people towards Mathematics may be an avenue to explore in which to develop a "Higher" form, though such an idea may not sit well with many Mathematicians who favor being able to dominate a type of thinking that only a few seem to be capable of conceptualizing... as if it were a foreign language which helps a select few to hold and maintain control over a given club membership. Simplifying mathematics by way of unraveling its basic foundation in an attempt to see whether or not it rests on a firm foundation is like questioning a building inspector's survey where they may begin to second-guess as to whether or not they have been as thorough in their examination as they believe themselves to be, or whether they rest on the reviews of others who are themselves not routinely questioned... particularly not by the common people who learn how to unravel the jargonized expressions and idioms customarily seen amongst adolescents experimenting with secret codes which amount to embellished ignorance.
Before leaving this page I want to mention that there exist other recognizably repeating patterns associated with number such as for example the value "64". It is of some interest to note that it has been used to create the number of squares on checker and chess boards as well as syllogism constructions, all of which predate the finding of this same value in genetics... and one might want to think that the value 64 occurring with amino acids exposed itself when other ideas involving repetitious activity were developed, though not all such activities.
Date of Origination: 21st August 2022... 4:23 AM
Date of Initial Posting: 2nd September 2022... 6:27 AM
Updated Posting: 2nd January 2023... 11:02 AM