~ The Study of Threes ~
http://threesology.org
Visitors as of August 8th, 2022
| Page 1 | Page 2 | Page 3 | Page 4 | Page 5 |
| Page 6 | Page 7 | Page 8 | Page 9 | Page 10 |
| Page 11 | Page 12 | Page 13 | Page 14 | Page 15 |
| Page 16 | Page 17 | Page 18 | Page 19 | Page 20 |
One might use multiple other terms to describe memory. For example, one might playfully describe "hammer" and "nail" memory or motorcycle and car memory, or Math and Literature memory, or god and human memory, or fruit and vegetable memory. While some terms may appear to be silly and others taken from some off-the-wall graffiti encounter; the labeling of memory can take on color words, music symbols, terms used in boxing, from a fishing tackle, horse racing, etc... It all depends on one's imagination to describe what one believes to be basic mental operations (that may or may not be contrasted with feats of exceptional memory skill). For example, we might claim someone to have a "blanket' memory because it appears to cover most or many subjects. Then again, as a contrast, one might use the term pillow memory, or mattress memory, etc... Just because so-called experts are left to the task of naming observations doesn't mean their labels are particularly any good or promise to assist in a further understanding and development out of which different labels are constructed.
And while we look at memory (often described in dichotomies) as viewed in psychology and then compare them to the ideas found in Mathematics, one might also correlate activities occurring in (for example) genetic mutations, to see what (to me) looks like properties to be found in mathematics, where transitional stages might well be overlooked due to the labels employed for describing what one thinks they see taking place:
If we look at the labeling the properties of basic math operations term as associative, commutative, distributive, zero, closure, inverse, transitive, reflexive, symmetric, substitution, without too much imagination it would be able to view them as types of memory, if they were presented to a group of people who were not familiar with them being used in mathematics and that they were not familiar with existing labels for memory. Each of the labels could be used as references to properties of memory. Likewise if they were presented with the labels addition, subtraction, multiplication, division, reflexive, symmetric, transitive, substitution, and square root and told they represented properties of memorization. Some might well want to compare all of them to find some common denominator such as I have described as a dichotomy, with an associated developmental theme represented by the generalized idea of number (counting) development expressed by some as 1, 2, 3, or one- two- many, or as 1- 2- 3 with stops, transitions, starts, stumblings, back stepping, skipping, etc., in order to describe a non-smooth transition may take place.
With respect to Mathematics, one might well encounter the notion of static and dynamic, though it is not certain to what degree one should actually use the words since different perceptions can use different definitions for specific applications. Whereas I think that Mathematics can and has expressed some level of dynamism, this does not mean it is an ongoing developmental movement, though... due to application of mathematics by millions of people, one might want to think such a manipulative usage describes an active "dynamic-ness", which is true, but not necessarily in the sense of developing towards an advanced species of Mathematics where patterns-of-two do not affect a dominant character trait such as is described by the term "Diploblastic" organism contrasted to more complex "triploblastic" life forms.
And if we look at the labeling used in describing different Mathematical models, what we find is the use of dichotomies (for example):
- Linear vs. nonlinear
- Static vs. dynamic
- Explicit vs. implicit
- Discrete vs. continuous
- Deterministic vs. probabilistic (stochastic)
- Deductive, inductive, or floating (an embellished dichotomy or an expressed transitional formula?)
- Strategic vs non-strategic
- Mathematical (equation) vs. Physical (practical, workable, applicable)
- Philosophical, Metaphysical, Spiritual vs. Everyday use
There are of course other ways of looking at the idea of Mathematical models (for example):
- An Introduction to Mathematical Modelling
- Mathematical Modeling
- Mathematical Models, Computers, and Platonism
Naming observations to suit a give language can often lead to misinterpretation amongst other researchers in different language cultures as well as misrepresentation since even so-called experts can be wrong... yet education institutions demand that a given idea is repeated for consistent teaching purposes... requiring researchers to develop a different type of dialogue outside academia in order for progress to be made. Such a dialogue is needed for developing an assumed form of mathematics which supersedes the present models. Such searching is standard practice in all human endeavors, or we wouldn't have different ideas and products. But in describing a Dynamic or advanced Mathematics, I am not speaking in terms of some marginal change brought on by mere change in terms or arrangement of terms. I see the development of a species of mathematics which is comparable to the development of three Germ layers, though some adventurous readers might want to magnify this idea by inserting some other numerical value without realizing I am using the analogy in a general sense of comparison, whereby to speak in terms of 4,5,6, etc. germ layers is superfluous and has more to do with the person's ego than providing additional supportive evidence-through-analogy.
While the dichotomies of teaching pre-math concepts can be easily understood (with examples described on the previous page as: big/small, tall/short, heavy/light, full/empty, less/more, in/out, long/short, thick/thin); the use of opposites applied to mathematics may become less visible as we attempt to describe basic concepts of a process involving higher mathematics, particularly if the person involved in discussing such basic concepts has not come to the realization there is an underlying usage of dichotomization, be it called pairing, coupling, duality, opposite, parallelism, etc., or by using language which obscures the presence of such a simple (basic) structure. Take for example this discussion on The five basic concepts in Pure Mathematics, where the five ideas are labeled Natural number, unknown, postulate, function and group. While each of these give the impression of a singular orientation of specificity, one necessarily resorts to some level of recognition by way of pairing them with or against an opposing idea. Yet, the oppositional pairing for recognition may not come to the forefront of discussion and more often takes place "automatically" by way of practiced training which no longer expends a deliberated conscious effort to associate a given idea with some oppositional or contrast with which to measure its identity for use.
Someone well skilled by way of multiple practices in the use of mathematics as a way of expressing one or more given ideas, no longer expends an analytical effort to restate basic concepts unless engaged in trying some step-wise progressive analysis for a given point of conjecture concerning rules-of-thumb regarding procedure or attending to axiom references as guides. Going over one's tracks while doing an attempted progressive math excursion, may lead one to momentarily encounter some earlier, more basic notion which does explicitly outline a pattern-of-two composite like tracing over a path which one may or may not have laid out breadcrumbs not eaten by the shadow of time nipping at one's heels in their flight to discover an unexplored trail of intellectualization being blue-printed with the tracings of visualization which may or may not find their way into an equation where condensation and abbreviation may obscure a larger message of realization requiring a different type of mathematical expression altogether, because one's current vocabulary of expression is like using grunts or cryptic runes of descriptiveness without an accompanying dictionary being written since the creator may think that stopping to create such will either interrupt the flow of insight that might be lost due to an assumed fleeting nature of wistful approximation being presented to someone who is primed to be sensitively aware but could be easily distracted from the vista if they don't pursue it... much like a child chasing a butterfly or firefly or pointing out some image in an overhead cloud.
While some argue that an exploration of higher mathematics requires a particular set of visual and language skills coupled to the ability and knowledge of placing given math symbols into a given arrangement; the assumed problem in thinking that only a few select individuals have the capacity of appreciating such views lay not in this being fact, but that the fact the common person has not learned how to describe the same perspective with a language and representation which mathematicians can grasp as being the same view as seen from a different vantage point, and not that they are particularly gifted intellectually. One of the problems in distinguishing whether or not a person has the capacity to grasp a perception and related means of expressing their grasp of some presumed advanced view, lays in the possibility that a person may well mix and match different levels of interpretation and articulation in a medium and manner they are most comfortable with. Whereas one part of a single view may seem to represent a person who is "well beyond their years" (and perhaps labeled as an "old or ancient soul"), and yet another part of the same view appears to represent an adolescent or childish comprehension; all of which may also be illustrated with varying levels of expression along a spectrum of both master and novice... otherwise denoted as a professional or beginner. A person may well have the capacity to grasp complex ideas but do not have the exposure to the language (vocabulary) and tools by which such a distinction can be appropriately ascertained by those in a position to pass judgment, for the ill or good of a given individual. Some individuals may truly be not only beyond their age group in learning, but also beyond the perspicacity of the so-called masters or professionals of a given subject who use their positions of authority/dominance to keep a person from being able to express the alternative view exceeding themselves. On the other hands, this view might well be used as a rationale by those wanting to assert a view that is indeed an alternative orientation, but is not an advance in human understanding, comprehension or expression for both general and specific uses.
When doing even a cursory search for terms being used by others to describe basic math concepts, we routinely find ideas that are or can be reduced to the activity of pairing. Take for example these from this article entitled Basic Concepts in Mathematics:
- Number sense (order and value)
- Addition and Subtraction (1st and 2nd arithmetical operations taught and learned)
- Multiplication and Division (3rd and 4th operations frequently taught and learned)
- Decimals and Fractions (5th and 6th operations frequently taught and learned)
While this is not to say such operations are always taught in such a sequence in all classrooms, it is a general rule of thumb for teachers to be taught their craft that children are to be taught "higher" or more difficult/complex mathematical operations in a stepwise fashion, though multiple types of analogy with day-to-day instruments may be used such as a ruler, a clock, a calendar; which themselves may be associated with cultural orientations such as observing some holiday, celebration or other social activity such as a birthday, wedding, funeral, hours of operation for a given business, attendance at some social event, meal time, television program, etc...
Because humans are being subjected to more and more instances of artificiality, basic concepts... particularly the (presumptive) origin thereof, can be overlooked and replaced by some imagined artificiality such as easily seen when advertisers use concepts suggesting a pure, or real, or wholesome or natural reality to obscure the actual instance of something that was manufactured for a profit... but many come to believe in the falsehoods generated multiple times in multiple ways by those who are lying... in many instances because they either don't know the truth or think that their efforts are an acceptable and standard practice. Unfortunately, this same substitution of reality takes place in all subjects... including Mathematics.
The development of the use of Triads in the I-Ching which followed the Yin-Yang patterns-of-two idea, is similar to the development of a higher order of thinking about sets of items which have been paired. Whereas the I-Ching boasts the notion of triads, what we see is actually an embellished collection of 8 biads. The I-Ching triads takes off from the pairing of the male-female aspects noted in the Yin/Yang idea expressed with lines... sort of like a tally sheet. The so-called collection of triads is a tally sheet for trying to arrange the male-female duality in different patterns. And like the development of the DNA formula in which it was noted that used two amino acids would not meet the requirement for generating the discovered quantity of amino acids, a series of three tally marks were used, but only a pattern-of-two limit of symbolism was incorporated. Whereas a single tally mark could represent the male... or more specifically the generative aspect of the male penis, and a two-line configuration could be used to illustrate the female aspect of generation of the female vagina; the underlying focus on preserving the pattern-of-two schema of the Yin/Yang forced the developers to resort to using the single or double-line configurations by way of repetition. Hence, since the philosophy of the Yin/Yang did not permit the conceptualization of a third gender or non-gender or Hermaphroditic composite, we are presented with a conceptualization which is an embellished dichotomy and not a true trichotomy with a three-line configuration.
I bring this up because we are confronted by the same situation not only in an effort to create what we think will be a ternary computing system, but also in the present model of mathematics because both of them have an underlying structural basis involving patterns-of-two. Whereas we see a lot of embellishments taking place in mathematics to give the impression we are moving beyond into a higher mathematics just as the creators of the supposed triads in the I-Ching no doubt considered themselves to be creating a higher level of understanding and enlightened consciousness... as no doubt do those attempting to create a ternary computer system; but that which we design may be little more than an embellished dichotomy.
Yes, we imagine ourselves enabled to mathematically create a trinary computer language distinct from the current three-patterned embellishments applied to the binary language... (by way of using the 3 basic Boolean operators Using AND, OR, NOT); and qualify our efforts by creating a system in which the idea is best suited for and then claimed to be superior because most people come to use it in their daily lives; but have we actually created a better mouse trap (so-to-speak) if in making our design more profitable (in any sense... commercially, academically, conversationally, etc...), because we have also replaced the conventional mouse with a biologically mutated creature better suited to our design? Are not products, religions, governments, languages, sports, etc., simply mutations that can help one idea over another prosper more?
As humans, we are constrained by our biology subjected to the environmental constraints of a very dominant pattern-of-two called the night/day sequence from which may have arisen the recurring biological system of a mouth and anus, despite later biological developments such as the three germ layers and multiple three-patterned organizations in our physiology: (List of threes in Anatomy by Dr. McNulty and Associates). Because of this consideration, we must ask whether we can truly conceptualize beyond such a basic pattern or only fool ourselves into thinking we can or do by way of renaming and rearranging our perceptions.
For example, when we look at both a Checker and Chessboard, one might be swayed to think they are different due to complexity, color and shape of playing pieces and the social definition as well as recognition applied to players, as well as the made-up rules governing play. However, the basic configuration is the same. A board with 64 squares involving two opponents. All the rest is embellishment, just as are the boards themselves an embellishment of cognitive activity. And cognitive activity may be an embellishment of an underlying physiological process which is an embellishment of genetic activity and it an embellishment of atomic activity and atomic activity an embellishment of ???.
Very often we artificialize a basic structure, a basic pattern into a configuration which gives us the impression we are engaged in greater complexity and are therefore a superior thinking person... particularly if we create a name that is accepted by others as representing something they too can intellectually attach their own strivings for believing themselves having some unique quality of superiority, giftedness, genius, etc., though the underlying basis for their belief is based on a simple pattern that can be uncovered and divulged to others who may rightly recognize that they too share in an ability to conceptualize the same pattern(s), albeit they don't use the same terminology nor is it viewed as being important by a given sect of supposed professionals, professors, experts, or presumed leaders who want circumstances to remain so that their practiced configurations of thought remain viable and they can retain a position that can not be easily attacked or questioned due to socially or institutionally enforced barriers.
One of the biggest unsolved problems in Mathematics is Mathematics itself. We do not yet know how to use Mathematics to analyze Mathematics as the metaphor of a type of cognitive expression whose partitioners routinely resort to using practiced answers which conceal a jumbled array of excuses, detours, dead ends, deflections, sleight-of-hand responses, gaming, institutionalized stratagems, etc... involving symbols, number arrangements, words, applications, venues, personalities, institutions, etc., in an effort to defend themselves against any would-be detractors, such as those who view mathematics (in some respects) as a variant of artistic, musical, and poetic expression; where art, music and poetry are not seen as symbolic representations of a given level and type of cognitive processing whose basic pattern(s) are obscured by the "languages" of expression seen in these subjects, though other subjects may well be added to this list of expression forms. For example, the language of an insect used in the world of a given insect, if that world could be equally perceived by an animal... would be illustrated by the language tools most practiced by the animal and not that of an insect, though each may claim that any purposeful interpretation would result in a meaning that is "lost in translation", even if both actually experienced the other's world in similar terms of reflective and reflexive sensation. Indeed, what then is the superiority of one language model over another? Is one language superior to another because it is forcibly taught to be learned in all public education systems, while other forms of language may have to rely upon less enforced instruction governed by little or no public funding... thus surrendering to the impositions used by businesses, religions and customs?
If I say the words "one, two, three" (where spaces and commas are not mentioned), am I saying the number or the word, with or without a visualized representation? Hence, let us ask if mathematics exists outside human cognition as a product of existence, yet mathematics is the present form of metaphor, of analogy that humans are using to describe perceptions that may, for the most part, be obscured from the human form of physiology-based perception and culturally learned communication?
I'm inclined to consider that Mathematics has an ability to evolve, but that we need to recognize we might well be subjected to the presence of a different type of life form whose earlier models exhibit less than three Germ layers. Hence, we have lots of life forms just as we have lots of different maths, but unlike distinguishing life forms which have only two germ layers, the same is not so for those researching mathematics. They have not yet created a comparable ideology about a developmental sequence taking place with Mathematics, whereby different maths can be organized according to the dominant underlying growth pattern. If this dominant pattern is two-structure as I am pointing out, is it possible to recognize a fledgling or fully developed expression of a higher mathematics with its own simple expressions such as when we recognize animals from earth worms to humans all have three germ layers, differentiated from two germ layer life forms called Diploblasts? Do we have Monoploblastic, Diploblastic and triploblastic types of mathematics? Or are present researchers of Mathematics similar to early biologists with a mindset yet too primitive to even think in such a manner?
Does a triploblastic level of Mathematics exist, or is Mathematics in a Diploblastic developmental stage which is trying to move into a triploblastic realm by way of embellishments? While one may want to distinguish a gradient of biological development of Diploblastic life forms whereby we use labels such as primitive and advanced Diploblasts, and we can easily describe less to greater complexity in mathematics; are we nonetheless subjected to a Mathematics in a relative Diploblastic level... but striving to move beyond? In getting some idea about developmental transitions, there are multiple biology-focused articles on the subject, but not on the subject of Mathematics, though some refer to it with the biological phrase "evolutionary development". It is of need to look to biology to find the developed idea of evolutionary termed progress, particularly when the idea can be related in some way to artificiality. The words "artificiality" and "synthetic" can be viewed interchangeably, though not all authors may intend this parity. Let me provide the abstract of one view.
Using three examples drawn from animal systems, I advance the hypothesis that major transitions in multicellular evolution often involved the constitution of new cell-based materials with unprecedented morphogenetic capabilities. I term the materials and formative processes that arise when highly evolved cells are incorporated into mesoscale matter 'biogeneric', to reflect their commonality with, and distinctiveness from, the organizational properties of non-living materials. The first transition arose by the innovation of classical cell-adhesive cadherins with transmembrane linkage to the cytoskeleton and the appearance of the morphogen Wnt, transforming some ancestral unicellular holozoans into 'liquid tissues', and thereby originating the metazoans. The second transition involved the new capabilities, within a basal metazoan population, of producing a mechanically stable basal lamina, and of planar cell polarization. This gave rise to the eumetazoans, initially Diploblastic (two-layered) forms, and then with the addition of extracellular matrices promoting epithelial–mesenchymal transformation, three-layered Triploblasts. The last example is the fin-to-limb transition. Here, the components of a molecular network that promoted the development of species-idiosyncratic endoskeletal elements in gnathostome ancestors are proposed to have evolved to a dynamical regime in which they constituted a Turing-type reaction–diffusion system capable of organizing the stereotypical arrays of elements of lobe-finned fish and tetrapods. The contrasting implications of the biogeneric materials-based and neo-Darwinian perspectives for understanding major evolutionary transitions are discussed. (This article is part of the themed issue 'The major synthetic evolutionary transitions'.) 'Biogeneric' developmental processes: drivers of major transitions in animal evolution by Stuart A. Newman (Philos Trans R Soc Lond B Biol Sci. 2016 Aug 19; 371(1701): 20150443. doi: 10.1098/rstb.2015.0443)
Reading the foregoing as a philosophy piece on Mathematics, even though the author's intent is not related to mathematics, the idea of a developmental scenario for Mathematics is not difficult to understand. One problem though is that Mathematics is seen as a single life-form (entity... "living" species of human thinking processes). While most biology texts do not speak in terms of a precursor "mono"-ploblastic life form, as mentioned in on earlier page, I an inclined to view sponges in this way, though the idea is controversial. Nonetheless, the idea of a "1" germ cell development is not difficult to think about nor imagine having occurred, despite there being no clear-cut examples where biologists come to agree upon. Hence, the absence of a 1-germ layer expression does not mean it didn't occur, unless we opt for the idea a two-germ layer form originated first and a 3-layer model is second. Whereas in Mathematics we do see transitional stages of development such as from simple number counting to basic arithmetic, further transitions of complexity may be difficult for some to discern just like offshoots of plants and animals are sometimes argued over as to when they occurred... just as in the case of human fossils. Does such and such a species belong here in the overall development, or does it belong elsewhere, or is a mutation belonging to its own platform? Such can be the case when we look at Mathematics, though historically speaking, we can distinguish before and after developments even if several different people in different places develop a similar idea such as calculus.
It is not typical for many people to think of Mathematics in terms of biological ideas, though we should avail ourselves of such an opportunity since such an exploration may lead to useful comparisons... if not a tool to influence the development of a different model of mathematics. While many have applied mathematics to different subjects and ingeniously developed a useful mathematical tool for a given context, different subjects are not always used as a means to analyze mathematics in an effort to break its underlying constraints by way of analyzing its fundamental nature; which a standard psychology or philosophy course has in its purview of typical contemplations. Mathematics is all too often viewed as an unassailable form of thinking whose basic pattern(s) is/are numbers, but not that those numbers express a more basic pattern of cognitive activity, nor that counting is different from an exercise in arithmetic., and neither that subtizing is different than counting... though some may want to claim it is a form of counting, because they don't or can't think of it in an alternative way.
For example, if a bird lays 3 eggs and we take one away, if its behavior suggests to us that it knows it is missing one egg, is this the result of the bird's ability to count, or simply recognize a vacancy such as when a person recognizes they are missing a finger even if it is one they do not customarily use? A recognized absence does not mean an underlying measurement of counting is taking place. Similarly, the loss of a loved one does not mean the person is thinking of the loss in terms of quantity. And again, if the Moon or Sun were to disappear by some long term or short term event, it is not likely the loss would be commonly referred to in a numerical sense such as "the 1 Sun, the 1 Moon" is/are gone. In many cases no specific quantity may be mentioned. Instead, we opt for some generic reference like "a" for the quantity 1, or some word with an "s" to indicate plurality; thus signaling the primitive use of words for numbers and not some advanced equation involving calculus, trigonometry or geometry. Similarly, what comes to mind is the reference that while many people can do math well, or write poetry well, or play an instrument well, they can not be individually called a Mathematician, a Poet or a Musician. While it may be difficult for me to qualify what I mean when I am describing those who are someone exceeding those exhibiting a talent or knack for something; many a reader may come to agree that there is something each of us in our own way interprets about a person to claim they are something which emphasizes a rarity, though they engage in a behavior multiple others do, and that multiple others may even be better at doing in a given context. With this said, the way I view Mathematics as expressing a primitive state of development may be that which you describe as an exceptionally profound representation of supreme thinking. To me, Mathematics (exercises and expressions) is like viewing a society that practices a guild system of training artists. And like art, mathematics can be used for commercial applications as well as for decorative designs, but it is still art, no matter how it is named or used from one era to the next.
While it is a convention to think of subtizing in terms of knowing a quantity, it would perhaps be better to claim this view as a human form of subtizing, and not a generic, or more basic model when we include the observations of animals involved in behavior which suggests they are counting and can distinguish what we humans define as different quantities, without a reference to quality, though this may be greatly useful to some animals. Subtizing on a very basic level may have nothing to do with an enumerated quantity as described by humans. The ability to distinguish between what we humans describe as quantity, may not be what is occurring amongst animals. While it is convenient for humans to think in terms of counting, there may be some underlying biological context which creates parameters to delineate quantity even if a quantity is not the intended definition of an underlying biological process. For example, if we count the number (quantity) of organelles in an animal and plant cell, we can say the animal cells have less than twenty and plant cells have none. Yet, it depends on how we define an organelle and in which part of a cell we are looking. It there is no established commitment of looking at a cell in the same way, then interpretations can differ, though the same words are being used to portray a researcher's assumed orientation of analysis. Your own search may see varying quantities and discussions as well as something you yourself may devise as a means to make sense of the different quantities you are presented with and how they are interpreted. The same goes for Mathematics. If we can't agree upon what is to be meant by a basic form or formula, a large sort of different interpretations and explanations can ensue, leaving us in a landscape where some defer to reigning leaders in a given field, while others creating their own vistas of exploration that may or may not find further widespread agreement.
In speaking towards the need for development a dynamic calculus as opposed to a static one that is presently being used, it is necessary for the reader to come to some level of agreement that the present model of calculus... of Mathematics as a whole, is static... like the statue of a pretended life form that others project some imagined dynamic property to that some may roll down a hill in order to claim it is alive because it is moving under its own power.
Date of Origination: 24th August 2022... 4:07 AM
Date of Initial Posting: 2nd September 2022... 6:32 AM
Updated Posting: 2nd January 2023... 11:07 AM