Threesology Research Journal: Math Perspective page 6
Mathematics Perspective: page 6

~ The Study of Threes ~

Flag Counter
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

If we view mathematics in the same manner as we do DNA, yet not as a life-building (or life manipulating) formula... but as a virus seeking a host... which has attached itself to multiple life forms as a means of attempting to ensure its own existence... either as a parasite or symbiotic organism where attachment can be beneficial (but does not necessarily mean it will); then we must not get carried away by thinking a grand achievement has been reached if we exclude looking at the functionality of the environment to which both are being exposed. In other words, is DNA to be praised or environmental circumstances which has enabled DNA to develop and flourish? Similarly, in light of the present context, the presence of mathematics... or even merely numbers, does not necessarily automatically equate with the overall situation as being beneficial to either the development of mathematics or the development of the organism it is attached to, by its various adaptations we may label as kinds or types of Mathematics such as geometry, trigonometry, statistics, etc...

An attachment to either humans or a constructed type of mathematics does not mean either will flourish or even be provided with a sustained representation of itself before mathematics and the life form ever met. For example: just because humans are using a tool called mathematics and its usage has provided multiple insights and vantage points that might not have been achieved or thought of, this does not mean the types of mathematics being used are not a series of sign posts leading humans along a trail towards a sink hole covered with a meadow whose bucolic scenery fills human with fanciful imaginations and romanticism. Nor does it mean that to which mathematics is being applied is thus improved. A case in point is particle physics. Are particles better off now that they have been attached with labels, numbers and equations? Is human life better now that statistics has been applied to it? While some may want to make an argument for individual cases where health and enumeration are used by some to evade being part of a usual medical statistic, let us speak in terms of the whole of humanity. If we say humanity is better off with than without mathematics, let us note that this "betterment" is due to an evolving mathematics and not the mathematics it started out with centuries ago. It is doubtful that if humanity were practicing the ancient forms of mathematics, the idea of "betterment" would not be that appealing. This of course begs the question of whether continued improvement in the human condition requires continued improvement in mathematics or a so-so type of a presumed human betterment will reach a point of (unsustainable) equilibrium dictated by an incrementally deteriorating environment that leaves us with no further growth. Hence, does mathematics have to be replaced by some different model of thinking, or is such even possible given the type of organic brain humanity has? This is of course not meant to imply a computer-like brain would be better.

Having a symbiotic relationship does not automatically mean the relationship is beneficial in the sense that both or one of the participants reaches their greater fulfillment, though the fulfillment reached may be more when together than either one would have been able to reach individually. Many people live impoverished lives and many who have gained a large financial nest egg do nothing with their wealth in terms of helping others. The rich are parasites on the poor. It is more the case where an older battery is attached to/with a new battery. While there may be a prolongation of a certain level of energy output, the energy value will necessarily be reduced to the lowest common denominator of the two combined energy sources as a sustained model of equilibrium. In other words, the older battery is a drag on the energy of the newer battery whose potential is lessened by the older battery due to its present state of character ability having deteriorated over time. Similarly, a man and woman may be attached to one another by way of a connection called love or respect, but no matter the strength of the connection, the state of physical, emotional and intellectual energy(s) may help one while the other is sacrificed and defines the sacrifices as an expression of love or respect; where the value of the two together may be interpreted as being more than the sum of the two as individuals, yet the equation being used is a "fudging the numbers" type of modeling. However, imperfect as my analogies may be at times, the underlying reference is nonetheless understandable.

Like love, Mathematics is often viewed in romanticized terms and relegated to some exalted pedestal such as a Monarchical role and viewed as a Queen of all sciences (with an unsurpassable type of logic) based on a value system designed by human ego which is a system of valuation that has not appreciably taken a look at itself in a non-fableistic mirror not being distorted by present applications and definitions based on practices and era-specific defined practicalities such as military- and commercial-oriented uses. Mathematics is not typically used to analyze itself because there are too many assumptions about it being wholly inviolable and those assumptions are relegated to the position of a sacred axiom written not necessarily on stone, but on archival paper, or some electronic storage and translated into all languages, used in all systems of education, and modes of commerce... thus creating a global dependency on it.

Viewing both DNA and Mathematics as adaptive organisms, necessitates (by way of a further analogy) looking at the Earth and its involvement with other celestial bodies, since the Earth itself can be viewed as an adaptive organism upon which humanity is claiming a supremacy of adaptation to, though not often in the form of a useful symbiotic organism. While it is not customary to look at planetary bodies as organisms which have "settled into place" or adapted to a particular role as a segmented functionality (such as when viewing the entire solar system as a type of organism with multiple segments), the idea is instructive to the extent of not excluding such a perspective from consideration, so as to note that mathematics itself can be viewed as a life form as well. The movement of the planetary segments has its own type of peristalsis with electro-magnetism that can be viewed as flagella. Mathematics on the other hand, is like a stationary life form deriving its sustenance from that which comes into its reach like organic matter floating by. If Mathematics is such a creature, than it may have reached or will shortly reach its ultimate form of development since its binary structure creates the circumstance of a maximizable limitation. It can not grow beyond its underlying pattern, though its underlying pattern can be seen as a pattern in life forms which are considered to be more complex. In other words, a complex life form such as humans also exhibits less complex organism structures in its biological design.

While some readers may think that using a biological model for describing Mathematics has its limitations, the point of the exercise has merit in an effort to not being obsessed in using mathematics to describe basic patterns of mathematics as a means of illustrating limitations due to its design.

The terrain of Earth has promoted growth involving a 1- 2- 3 developmental sequence, though each step may not be evident or even exist similarly in all life forms. (In the past I was found of saying that we are environmentally influenced to be genetically predisposed to (1) conceptualize many of ideas, (2) perform many of our activities, and (3) grow physiologically along a 1- 2- 3 maturational development sequence. By rearranging the contingencies of our environmental abilities, we can redirect the associative processes of the encephalized human organism. It deals simply with connjectural transparencies.) Since I have no routine habit of speaking in conversation identical with or similarly to the way I write, those with whom I have had the occasion to converse with are somewhat taken aback when I do so, and can't make heads nor tails of me. They think of me as being (let us say Autistically smart) while at the same time ordinarily common-place, if not a little slow and confused... in mundane mind-numbing jobs (because I neither do as they do nor have their same interests or preoccupations such as being primarily focused on relationships or financial gain.)

Of late, I frequently use the three germ layers as an example in different writing contexts because of their readily available transmissiblility to easily understood enumeration, since there are named life forms with three and two germ layers... and for a one germ layer example, I have proposed the sponges; since it is one of the earliest extant animal life forms that might be included in such a measurement; though not everyone agrees with the assessment concerning the sponges. If not the sponges and there is no living or fossilized evidence for a single germ layer-developing life form, we are left with considering that we need a different way of thinking about how the germ layers originated... if it occurred without a "1st-step germ layer" signification. In other words, did a single-germ layer development occur as a compartmentalization event and then "poof" or "shazam" a distinct two-germ layer modeling took place like a kid stepping over the first step as they ascend a flight of stairs because there is something on it they want to avoid? Was there no distinct one- germ layer world, or was the scenario a collage of occurrences we can not at present appreciate as such?

Nonetheless, much like the cognitive use of primary- secondary- tertiary, or mono- di- tri, or Major Premise (MP)- minor premise (mp)- Conclusion (C), and various other triadic structures conveying sequentiality, the idea of a 1-2-3 sequencing can be understood, even if we can not as yet agree up that to which we point our finger at in unison as a definitive example of a single germ layer-occurring life form. Surely life does not work off a set of axioms which excludes referencing a single Germ layer... or does the axiomatic program which Nature works off of a different animal of logic than that in use by humans because humans at present are particularly naive or even stupid? While in some cases only two references are labeled or mentioned (or even only a single item)... in such instances a closer look may reveal the existence of a prior existence of one or two previous forms (formulas). In other instances, take for example the single strandedness of RNA and double strandedness of DNA, we do not yet see a fully established triple stranded model. RNA with its single strandedness, coupled with the idea of an RNA world hypothesis as that preceding the advent of DNA with its double strandedness, provides some measure of confirmation of an existing 1-2-3 developmental sequencing.

An example of differentiated strandedness can be seen in a list of animal viruses categorized by (1) DNA, (2) RNA and (3) Reverse Transcriptase (Exhibiting single and double strandedness abbreviated as SS and DS). And let me also note that along with single/ double strandedness, we find the reference of "segmentation," as well as the geometries of 'linear and circular' gets attention. (While the idea of segmentation needs its own series of articles like that of hybrids, I mention the third item since I have organized some ideas into the three-patterned formula of "linear- circular- triangular": (Line, Circle, Triangle page 1)

Genome-Based Classification:

The nature of the nucleic acids in the genome is the criteria of the genome-based classification. Viruses that have RNA as a genome are called "RNA viruses," whereas viruses that have DNA as a genome are called "DNA viruses". An exception to this rule comprises viruses that replicate via reverse transcription, which are grouped separately as "reverse transcribing (RT) viruses," regardless of whether the genome is RNA or DNA. As a result, animal viruses can be largely classified into three groups based on the nucleic acids of the virus genome. (Discovery and Classification by Wang-Shick Ryu, in Molecular Virology of Human Pathogenic Viruses, 2017 (Genome-Based Classification)

DNA viruses (14K)

Reverse Transcriptase viruses(11K)

Let me repeat myself: As of late, I have been relying on the usage of the three germ layers development as a routine descriptive of a developmental 1-2-3 sequence. Previously, I was fond of saying: "We are environmentally influenced to be genetically predisposed to (1) conceptualize many of our ideas, (2) perform many of our activities, and (3) grow physiologically along a 1-2-3 maturational development sequence. It deals simply with conjectural transparencies." My use of number in a format of serialization would seem to have inevitably directed me along a course of taking a closer look at mathematics, so long as one is not dissuaded from doing so when confronted by the sentinels at the gate asking for the proper credentials by speaking in the vernacular of mathematics while exhibiting the required signs and symbols of the ordained cult members. However, being a mathematician does not automatically make one suited for the task of providing information which may enable the development of a new model of mathematics and is otherwise to be noted as a different way of thinking, defined as logic or otherwise.

While the idea of a one- two- (three) can easily be seen in the count of strands displayed by RNA and DNA, some perspectives may not make the allowance for placing the counts together (based on the notion that Nature itself engages in an identifiable form of counting if not geometry and other mathematics), just as when comparing the amino acid basis of both, the idea of a three-to-one ratio is not recognized in that both share the same (three) amino acids (adenosine- cytosine- guanine), and then are individually identified with either (one) thymine (DNA) or (ibe) uracil (RNA). Placing the two together and identifying the pattern is not a conventional priority because it has no meaning for most researchers not involved with cataloguing enumerations of different subject materials. Geneticists have their own logic prioritizations. This is similarly found in Mathematics. A good example is to look at Lambda calculus from a basic descriptional format (Notice the phrase "foundational" mathematics that relies primarily on mathematics as if it were accurate and honest enough as a tool to police itself... much in the manner of expecting the legal system to validate itself or permitting religions and businesses to police their own self-righteous assumptions... like the Catholic church in its hypocritical and criminal oversight of pedophiles in its ranks, or letting the Oil industry determine what is the best price people should pay.)


Lambda calculus (also written as ƛ-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation that can be used to simulate any Turing machine. It was introduced by the mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics.

Lambda calculus consists of constructing lambda terms and performing reduction operations on them. In the simplest form of lambda calculus, terms are built using only the following (three) rules:

Basic three rules of Lamda calculus (9K)

Producing expressions such as: (ƛx.ƛy. (ƛz.(ƛx.z x) (ƛy.z y)) (x y)). Parentheses can be dropped if the expression is unambiguous. For some applications, terms for logical and mathematical constants and operations may be included.

The (two) reduction operations include:

Two Lamda reduction operations (8K)

If De Bruijn indexing is used, then a-conversion is no longer required as there will be no name collisions. If repeated application of the reduction steps eventually terminates, then by the Church-Rosser theorem it will produce a ß-normal form.

Variable names are not needed if using a universal lambda function, such as Iota and Jot, which can create any function behavior by calling it on itself in various combinations.

Let us also note in the article the description of another (rather simplistic) pattern-of-two described as typed and untyped forms of the Lamda Calculus. It is a linguistically primitive two-patterned illustration consistent with many other yin/yang profiles being embellished (if not later governed) by a three-pattern system of "proper" or preferred usage, like the (assumed) I-ching triads applied to the yin/yang biads, though different terms are used. No less, it is of some interest to review what may have been the "logic" behind the adoption of the Lambda (ƛ) symbol. Will any reader of this notice the three-to-one ratio in the phrase of "eeny, meeny, miny, moe"?: (Three to one ratios page A)

Origin of (using) the lambda symbol

There is some uncertainty over the reason for Church's use of the Greek letter lambda (ƛ) as the notation for function-abstraction in the lambda calculus, perhaps in part due to conflicting explanations by Church himself. According to Cardone and Hindley (2006):

Origin of Lamda symbol usage (13K)

This origin was also reported in [Rosser, 1984, p.338]. On the other hand, in his later years Church told two enquirers that the choice was more accidental: a symbol was needed and ƛ just happened to be chosen.

Dana Scott has also addressed this question in various public lectures. Scott recounts that he once posed a question about the origin of the lambda symbol to Church's son-in-law John Addison, who then wrote his father-in-law a postcard:

Dear Professor Church,

Russell had the iota operator, Hilbert had the epsilon operator. Why did you choose lambda for your operator?

According to Scott, Church's entire response consisted of returning the postcard with the following annotation: "eeny, meeny, miny, moe".

(As an aside "three" note: an SEP article on Alfred North Whitehead says: "Whitehead conceives of the student's educational process of self-development as an organic and cyclic process in which each cycle consists of three stages: First the stage of romance, then the stage of precision, and finally, the stage of generalization.") The article should have included not only can we see a recurring process which appears to be organic... that is, developmental in terms of what may be described as an aging process, but also the presence of a serial enumeration that expresses a limitation or conservation in not only length but complexity and application.

From where I'm standing, Lambda calculus could be viewed as a precursor to the idea for needing a new type of Calculus, except that (or so I should think), the originator was not privy to the idea of Mathematic's foundation as exhibiting an over-riding (let us say over-bearing) reliance on a two-patterned yin/yang model of descriptiveness which took the later developed I-Ching model of calculus and elaborated as well as expanded upon it. Indeed, if you developed an observational means of enumerating natural events but had no formal or cultural interest in using numbers, how would you designate the process of seriation which confronted your consciousness with a pattern-of-two repetitions? Would you use words to list them, and then later on another person with a slightly developed mind/brain who thought of in an attempted three-patterned modeling; would you not use culturally designated expressions to illustrate an attempted variation of patterns-of-three, though by using only two symbols such as a long and two short lines configured in a three-patterned way?

It is rather humorous to note that Linguistics has adopted and expanded a usage of Lambda Calculus as a model of inquiry and understanding to a perceived foundational approach to elucidating linguistic patterning considered as a derived superior system of logistical apprehension (complete cognitive understanding) and apperception (basic cognitive process), and yet itself overlooks its own originating influences involving the recurring patterns-of-three in the human ear, such as the idea the three inner- middle- and outer ear parts develop from all three germ layers and neural crest cells. (The development of the mammalian outer and middle ear by Neal Anthwal, Hannah Thompson)

When it comes to an article by authors who reference a comment such as "a survey of the literature", meaning the respective literature concerning an item brought up as a topic used as an example, I can attest to the fact that my own personal survey of "the" literature (at least the literature I came across during a period in which I had to cull information by using a library's card catalog since there was no computer system in use at the time several decades ago), results in having a different set of results, though the information may have been by those who did not have the techniques of analysis possessed today. For example, when an author claims that the number of turns of the cochlea is 21/2 turns and the upper limit is three, my own research, having been taken up several decades ago, resulted in finding multiple other values being recited by different authors. Unfortunately, I took such information for granted and was not particularly diligent in citing reference locations or authors. However, the fact that I didn't simply put "3 as an upper limit" as the only value means I was inclined towards some measure of listing the truth of what I came across. Although upon re-reading the list, I think I engaged in a bit of assumptive playfulness but putting in values which I think might well occur as standard references if I were to engage in reading more references than I had already surveyed. I say this because I can not be quite certain since the initial recording was more than thirty years ago and I do not want to be drawn and quartered for what might appear to some readers as an intentional act of lying or deception. When I attempted to count the number of turns from looking at an illustration of the Cochlea, I could count "3", but since I thought this might be a stretch of the imagination due to a desire to see the "3", I mentioned alternative measurements as I then found them. I ended the count by saying "etc." because I thought additional reading would provide many other values than those I had come upon from sitting in a library for an unrecalled amount of time. The values I came across many decades ago were:

  • 2 3/8 turns
  • 2 1/4turns
  • 2 1/2 turns
  • 2 5/8 turns
  • 2 3/4 turns
  • 2 7/8 turns
  • etc., (One must wonder if these amounts reflect the idea representing the notion of an "almost three" developmental transition having been achieved but no actual three form or model. In other words, like many ideas being found which are either described as a "three" or are suggested as a possible three construction, we find that upon closer examination they are between a two and a three model. As if a three stage of development has not been fully achieved, but that which has and can be viewed as a fractional proportion exceeding a "tw0", is "rounded-up" like some accountants prefer to do.)
  • Apparently, three turns appears to be the limit: On the number of turns in human cochleae by Slavomir Biedron, Martin Westhofen, Justus Ilgner

In addition to the foregoing Ear image, let me list some additional ear-related patterns-of-three (from one of my earlier pages: ()) for those who think the image does not provide enough for them to be classified as "many" or serving to persuade, if not convince them that there is a dominant re-occurring instance of patterns-of-three... yet again, we do not see this in what we claim to be a "language"-dependent symbolic logic called Mathematics. In other words, why is there no dominant underlying pattern-of-three being used instead of a yin/yang (two-patterned) profiling, though logic recurringly testifies for the need to use a three-patterned format?:

Language development may be directly related to how we hear:

3-Patterned Ear Structure

3 overall divisions: Outer ear~ Inner ear~ Middle ear
3 middle ear divisions: Tympanum~ Epitympanum~ Mastoid antrum
3 eardrum membranes: Cutaneum~ Collagen fibers~ Mucosm
3 semi-circular canals: Used for balance (equilibrium)
3 bones: (ossicular chain) Incus~ Stapes~ Malleus
3 main malleus ligaments: Anterior~ Lateral~ Superior
3 incus anchorage points: Malleus~ Stapes~ Bony fossa wall
3 cochlea sections: (Scala) Vestibuli~ Tympani~ Cochlear duct
3 extrinsic muscles (Auricularis): Anterior~ Superior~ Posterior
3 sound conduction paths: Electrical~ Mechanical~ Fluid
or: Bone (solid)~ Air (gas)~ Fluid (liquid)
3 nerve stimulation paths: Mechanical~ Chemical~ Electrical
3 outer hair cell rows
typical in mammals
but some sources give 3, 4, or 5
Neurotrophin-3 (NT-3) is synthesized by inner and outer hair cells of the developing organ of Corti. Brain-derived neurotrophic factor (BDNF) is also synthesized. (Prestin is the motor protein of the outer hair cells.)
3 sound qualities: Pitch~ Volume (intensity)~ Tone
3 sound wave propagation processes: Diffraction~ Transmission~ Reflection
3 main forms of ossicular chain fixation: Fluid~ Mechanical~ Otosclerosis
3 classes of ossicular lever action: Force arm~ Resistance arm~ Fulcrum
3 acoustic distortion forms: Frequency~ Phase~ Amplitude
3 basic properties of vibrating bodies: Inertia~ Elasticity~ Dissipation
3 principal types of deafness: Conduction~ Nerve~Stimulation
3 types of hearing loss: Conductive~ Sensorineural~ Mixed
3 (inner ear) organs of balance: Semicircular canals~ Utricle~ Saccule
(collectively called the vestibular organ, which can be referred to as a {3-in-1} ratio.)

Arranged on the surface of the basilar membrane are orderly rows of the sensory hair cells, which generate nerve impulses in response to sound vibrations. Together with their supporting cells they form a complex neuroepithelium called the basilar papilla, or organ of Corti. The organ of Corti is named after the Italian anatomist Alfonso Corti, who first described it in 1851. Viewed in cross section the most striking feature of the organ of Corti is the arch, or tunnel, of Corti, formed by two rows of pillar cells, or rods. The pillar cells furnish the major support of this structure. They separate a single row of larger, pear-shaped, inner hair cells from three or more rows of smaller, cylindrical, outer hair cells. The inner hair cells are supported and enclosed by the inner phalangeal cells, which rest on the thin outer portion, called the tympanic lip, of the spiral limbus. On the inner side of the inner hair cells and the cells that support them is a curved furrow called the inner sulcus. This is lined with more or less undifferentiated cuboidal cells. ("ear, human." Encyclopædia Britannica. Article by Joseph E. Hawkins, Emeritus Professor of Otolaryngology (Physiological Acoustics), Medical School, University of Michigan, Ann Arbor. Editor of Otophysiology.)

Describing hair cells in the Organ of Corti (109K) (54K)

Because there is a recurring "three" theme to hearing processes, one must wonder what would be the result of language if the ear had some other dominant structural pattern such as two, four, five, etc... In addition, we must wonder how and why a three-patterned sequence of ear components influences the production of a symbolic (math) logic with a dominant two-patterned structure. Is there some interference pattern taking place and the double-slit experiment used in particle physics is an unrecognized externalized representation of a personal recognition being recreated, like a subconscious existence trying to make its way to consciousness? The structure of the ear reminds me of the three-germ layers model of "segmented" layering. (I emphasize the word "segmented" with quotes because the counting system: 1,2,3,4,5,6,7,8,9...) is a serial formula of segmentation we might alternatively use the word "metamerism" or even "tagamatism" with. Each word describes a particularity of organization with increasing emphasis of differentiation. Yet, they also express the application of a mechanistic... that is, mechanical or solidified model and not necessarily fluidic, gaseous or a plasmatic (plasma) state of existence, at least metaphorically speaking. In other words, numbers are predominantly viewed as card slots, like those positions used for depositing punch cards on the old computer versions back in the 1970s, though the use of punch cards apparently began as an operational device back in the 1700s to control textile looms.

While the words serialization, segmentation, slotting, and metamerism can in some measure advance the perception of a number line... one number following another; the word "tagmatism" or "tagmatization" or "tagmentalism" (though one might coin alternatives including compilations;, has the additional feature for providing an enumerated identification of segmentation (initiated in a biological context by the idea of fused segmentation); such as describing the (arthropod) structure (for example) of a spider's two-part body labeled as the Prosoma and Opisthosoma while an insect such as an ant has a three-part body labeled head- thorax- abdomen, each with their own respective parts and overall viewed as an expression of Bilateral symmetry... all of which are ideas seen in Mathematics– albeit expressed in the respective vernacular of formulaic signs, symbols, compartmenalizations of thought, axiomatic directories, definitives, exclusions, hyper-positioning, fulcrumatic extensions, etc... From a Penguin Biology dictionary (ISBN 978-0-141-01396-1) on page 652, we find the following definition that some will read as a specificity for Arthropodic-centered biology, yet I read as a Mathematics primer. Indeed, one can well pick up any text on any subject and see what is and what is not current in the contours of Mathematical thinking. (One can read a text as a specificity, as a generality or alternatively.)

Notice that the initial description barks loudly of a (rather humorous) Marxian-like sociological orientation with a Communistic (communalism) flavor:

Tagma (pl, tagmata)

Functional and anatomical regionalizations of the arthropod body associated with division of labor; thus appendages with similar functions tend to be on adjacent segments.

Tagmata include head, thorax, abdomen, but each class within the phylum has a characteristic pattern of tagmosis.

  • Insects have (a) distinct head, thorax, and abdomen;
  • Crustaceans generally lack a clear head-thorax distinction
    • (However, a three-part configuration may be present, but if not, the head and thorax fuse to produce a segment called the cephalothorax.)
  • Arachnids have Prosoma and Opisthosoma, etc.
  • Tagmosis in post-head region is closely related to mode of locomotion.

(continued on next page)

Date of Origination: 19th March 2022... 5:23 AM
Initial Posting: 8th August 2022, 9:10 AM
Updated Posting: 2nd January 2023... 10:48 AM