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Progressive Thinkers as of 12/1/2022
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Many of us would accept the idea that Nature uses numbers or number patterns to express itself such as in the case of the Fibonacci number sequence. Yet, it can not be said too often that Mathematics itself gets in the way of understanding basic numbers. It is more properly understood as a means of understanding relationships of numbers placed into regimented systems of articulation which must conform to exercises called operations and axioms. Like the rules of a card game printed on a piece of paper or small cardboard accompanying a deck of cards instruction people how to play a particular game, axioms and operations of play are neither numbers nor number theory. They are confinements of how numbers are permitted to be arranged and what the numbers are supposed to mean in the context of the game. Numbers as patterns is much like art which at times is incorporated into some equations making them more of an artistic expression than what one might think of a stoic, scientific model of mathematics. Mathematics should not be viewed as a Mother hen sitting on a brood of numbers. Numbers are like eggs which came before the chicken. As you will see in the following excerpt, a discussion of numbers gets lost and becomes associated with mathematics:
Humanity has had a love-hate relationship with numbers from the earliest times. Bones dating from perhaps 30,000 years ago show scratch marks that possibly represent the phases of the Moon. The ancient Babylonians observed the movements of the planets, recorded them as numbers, and used them to predict eclipses and other astronomical phenomena. The priesthood of ancient Egypt used numbers to predict the flooding of the Nile. Pythagoreanism, a cult of ancient Greece, believed that numbers were the basis of the entire universe, which ran on numerical harmony. The Pythagoreans' ideas were a mixture of prescience (the numerical features of musical sounds) and mysticism (3 is male, 4 is female, and 10 is the most perfect number).
Numbers were associated with names for magical purposes: the biblical "number of the beast," 666, is probably an example of this practice. More recently, cranks have sought the secrets of the universe in the dimensions of the Great Pyramid of Giza, an aberration so common that it even has a name—pyramidology. Millions of otherwise rational people are terrified of the number 13, to the extent that hotels omit it from their floors, airplanes do not have a row 13, and the numbers for Formula 1 racing cars skip from 12 to 14 so that, for example, 22 cars would be numbered from 1 to 23. Learned tomes are written about the significance of such stalwarts as the golden number (1.618034), which does occur in flowering plants and modern architecture but does not occur in the shell of the nautilus and ancient Greek architecture, despite endless myths to the contrary. Many religions have their sacred numbers, as do organizations such as Freemasonry; Wolfgang Amadeus Mozart's music, notably the Magic Flute (1791), includes many intentional references to Masonic numerology.
Mathematics is the study of numbers, shapes, and related structures. Number mysticism belongs elsewhere and is generally categorized as numerology. Numerology sheds light on the innermost workings of the human mind but very little on the rest of the universe. Mathematics, meanwhile, sheds light on much of the universe but, as yet, very little on human psychology. Between the two lies fruitful scientific ground, yet to be cultivated extensively.
Let's rephrase the foregoing comment about Mathematics being, in particular (for the present discussion about language) a study of numbers. One type of study perhaps, but not "THE" (sole) study of numbers. Mathematics (speaking for some mathematicians) presents us with one type of interpretation about numbers. It goes without saying that numbers can have different interpretations and representations, whether you personally agree with them or not. There is no singular type of interpretation or usage. Whereas you may call your usage "THE" best way, the truth, the only way numbers should be used, other may disagree because numbers can be viewed artistically, which includes music as well as any dimension of conventional art or craft you are inclined to bring to mind. Likewise, language can be viewed artistically and mathematically. While some study language with a rigor akin to an electron-microscope or multi-million dollar Very Large Array telescoping system, others prefer the focal lens of imagination, accompanied with an artificial inebriation or not.
"Language" should not automatically be interpreted or defined as representing sounds called words. Nor should numbers be automatically interpreted or defined in terms of Mathematics. Since there are what we call "number-words" (one, two, three, four, five, six, seven, eight, nine, ten, etc...), we can also say the reverse that words can be used for numbers and those numbers identified not only as sounds but as quantities... even if you prefer to include other attributes.
And though I will mention it again, it needs to be clearly identified that researchers who may spend multiple decades... if not their entire adult lives (or more) on a given subject, routinely resort to some use of numbering or system of identification of materials with given numbers. These numbers typically entail the use of small numbers and these small numbers are categorized by a recognition of repetition. No less, when they observe other information, they typically arrange it in accordance with their prevailing dominant system of thinking so as to incorporate it or keep it out of their current thesis, because it is felt to be irrelevant or incapable of offering any added feature to one's hypothesis. No matter how open-minded a researcher may think themselves to be, you might well find some type of bias or even discrimination if your scrutinize too closely. It is a feature of research that many researchers are aware of and make allowance for so as to keep some measure of open communication with others whose views and interests are not exactly their own. Besides, a researcher must be open to the possibility that they are wrong... not as a character of crippling or mortally wounding themselves with self-doubt, but by keeping one foot in the door of the type of insanity everyone else shares and needs to find companionship with so as to maintain some semblance of normalcy, whether or not this period in history will be viewed as an Age of Irrationality or not.
When we come across some pattern identified with a repetition, be it a single number or sequential numbers or even different numbers that collectively repeat, it is common for that sequence to be assigned with the label that "it means something" or "it represents meaning". We may not know what it supposedly means, but by so labeling, we provide ourselves with the impetus to continue looking at and for the pattern, as well as collect instances of it; particularly when it appears to crop up in what might otherwise be described as different places and occasions, if not the observations of one or more others.
One such pattern is the already mentioned Fibonacci series:
Nature's numbers
Many aspects of the natural world display strong numerical patterns, and these may have been the source of some number mysticism. For example, crystals can have rotational symmetries that are twofold, threefold, fourfold, and sixfold but not fivefold—a curious exception that was recognized empirically by the ancient Greeks and proved mathematically in the 19th century.
An especially significant number is the golden ratio, usually symbolized by the Greek letter Φ. It goes back to early Greek mathematics under the name "extreme and mean ratio" and refers to a division of a line segment in such a manner that the ratio of the whole to the larger part is the same as that of the larger part to the smaller. This ratio is precisely (1 + Φ5)/2, or approximately 1.618034. The popular name golden ratio, or golden number, appears to have been introduced by the German mathematician Martin Ohm in Die reine Elementarmathematik (1835; "Pure Elementary Mathematics"). If not, the term is not much older and certainly does not go back to ancient Greece as is often claimed.
In art and architecture the golden number is often said to be associated with elegance of proportion; some claim that it was used by the Greeks in the design of the Parthenon. There is little evidence for these claims. Any building has so many different lengths that some ratios are bound to be close to the golden number or for that matter to any other ratio that is not too large or small. The golden number is also often cited in connection with the shell of the nautilus, but this too is a misunderstanding. The nautilus shell has a beautiful mathematical form, a so-called logarithmic (or equiangular) spiral. In such a spiral each successive turn is magnified in size by a fixed amount. There is a logarithmic spiral associated with the golden number, and in this case the fixed amount is precisely Φ. However, the spiral of the nautilus does not have the ratio Φ. Logarithmic spirals exist with any given number as their ratio, and the nautilus ratio has no special significance in mathematics.
The golden number is, however, legitimately associated with plants. This connection involves the Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,...), in which each number, starting with 2, is the sum of the previous two numbers. These numbers were first discussed in 1202 by the Italian mathematician Leonardo Pisano, who seems to have been given the nickname Fibonacci (son of Bonaccio) in the 19th century. The ratio of successive Fibonacci numbers, such as 34/21 or 55/34, gets closer and closer to Φ as the size of the numbers increases. As a result, Fibonacci numbers and Φ enjoy an intimate mathematical connection.
Fibonacci numbers are very common in the plant kingdom. Many flowers have 3, 5, 8, 13, 21, or 34 petals. Other numbers occur less commonly; typically they are twice a Fibonacci number, or they belong to the "anomalous series" 1, 3, 4, 7, 11, 18, 29,..., with the same rule of formation as the Fibonacci numbers but different initial values. Moreover, Fibonacci numbers occur in the seed heads of sunflowers and daisies. These are arranged as two families of interpenetrating spirals, and they typically contain, say, 55 clockwise spirals and 89 counterclockwise ones or some other pair of Fibonacci numbers.
This numerology is genuine, and it is related to the growth pattern of the plants. As the growing tip sprouts, new primordia–clumps of cells that will become special features such as seeds—arise along a generative spiral at successive multiples of a fixed angle. This angle is the one that produces the closest packing of primordia, and for sound mathematical reasons it is the golden angle: a fraction (1 - 1/?) of a full circle, or roughly 137.5 degrees. ("number symbolism." Encyclopædia Britannica.)
In the foregoing the author mentions the absence of "fivefold" rotational symmetries in crystals. We can also make note that not all chemicals follow the "octet rule". In fact there are three exceptions. We also find 1 start and 3 stop codons amongst a retinue of nonsense codons... nonsense can also be viewed as expressed exceptions that have not been removed. In other words, in evolution we find that a standard recurring form has been standardized by the elimination of past structures which have been found untenable. Yet, not all structures need to be removed automatically. They may reside in a life form as some residual component of a previous time like those structures we call vestigial organs. Vestigial organs are exceptions. Being residual may mean that the initial sources of development still remain because the overall environment requires for them to, since the environment may be poised to return to a former model of itself where such vestigial forms were of value. I make note of this to myself in order to follow up on the idea that we need to look closer at the exceptions. The exceptions may be Nature's way of expressing its "Method of Exhaustion". "Exhaustion" need not mean total absence... just the resemblance of non-functionality evincing what can be described as exceptions or alternatives. It is of some interest to reference the presence of the "3" or some fraction thereof, when speaking of Exhaustion in a Mathematical/Numerical sense, and the recurrence of the "3" being used in multiple references... as if humanity is exercising a cognitive model of exhaustion. If a repetition of threes is an expression of an exercised Exhaustion, what is being exhausted... yet lingers on? Present cognitive activity?
(Image: Eudoxus calculated the volume of a pyramid with successively smaller prisms that "exhausted" the volume.) The method of exhaustion, also due to Eudoxus, was a generalization of the theory of proportions. Eudoxus's idea was to measure arbitrary objects by defining them as combinations of multiple polygons or polyhedra. In this way, he could compute volumes and areas of many objects with the help of a few shapes, such as triangles and triangular prisms, of known dimensions. For example, by using stacks of prisms (see figure), Eudoxus was able to prove that the volume of a pyramid is one-third of the area of its base B multiplied by its height h, or in modern notation Bh/3. Loosely speaking, the volume of the pyramid is "exhausted" by stacks of prisms as the thickness of the prisms becomes progressively smaller. More precisely, what Eudoxus proved is that any volume less than Bh/3 may be exceeded by a stack of prisms inside the pyramid, and any volume greater than Bh/3 may be undercut by a stack of prisms containing the pyramid. Hence, the volume of the pyramid itself can be only Bh/3—all other possibilities have been "exhausted." Similarly, Eudoxus proved that the area of a circular disk is proportional to the square of its radius and that the volume of a cone (obtained by exhausting it by pyramids) is also Bh/3, where B is again the area of the base and h is the height of the cone. ("analysis." Encyclopædia Britannica.)
In several pages I am going to be speaking about language within the context of my "threes" research orientation. It is not an orientation typical of conventional discussions that may be trying to emphasize a singular objective within an academic-related setting. Nor do I intend to execute the rigor of attestations of a writing style one would use in producing a Doctoral thesis to coincide with a demeanor suggesting one is sitting with a suit or other business attire. I am in a dusty field where scorpions, side-winder snakes, ground squirrels, coyotes, and predatory insects abound... at least before too cold of weather sets in. typically I am in lounge-around apparel removed from folded stacks of multiple items that I actually never get around to wearing all of them, and some even have their purchase tags still attached. My life is filled with make-shift boxes arranged in an organizational system that rivals the desk of any multi-faceted thinking professor, industrial plant manager or arts and crafts time of any first grade class... though not as bad as a lunch room food brawl. One must have their moral limits.
And yet, much of the material used by those engaged in what are considered to be important, serious, no-nonsense tasks of writing about a subject they may or may not want to specialize in and make a career out of; is available to anyone for perusal and collection... unless of course one only expects to get the best information from behind pay-wall publications... many of which are little more than a "two-cents worth" of knowledge, because they say nothing more than what can be gathered from sources created by those who may have had only a passing interest in a topic and took a moment to create a relatively invisible internet blog about. More often than not I come across research by authors who are seriously misinformed, seriously deluded, or seriously trying to sound as dumb as all the others creating written about a topic no one is actually discussing because everyone is working in their own version of some solitary enclave of hermitage or nunnery. Many researchers don't come across pertinent information of others who may not be in their immediate purview of research, and the needed exchanges of ideas are not taking place except for by way of some email which acts as a truncated message attached to some pigeon leg where a decoder ring is needed but only available if you take a time machine back to the fifties and sixties where you could easily purchase one as a reward accompanying some product whose ingredients caused many a cavity in a now toothless or false-teeth generation.
For some of us, there are so many research interests that often require way-too narrow a focus that, in speaking for myself, limit the types of cognitive flavorings more conducive for my tastes, since I characteristically like to add what I believe to be attendant interesting points of departure (from other subjects) related to my "threes" research program. In playing the part of a Devil's Advocate, since I come up with the same objections and suggestions and digressions as others do when involved in an effort to find, collect, and deuce a given pattern occurring a) amongst, b) instead of or c) absent from other patterns; the amount and types of information needed run the gamut... (let us also say gauntlet) of subjects at our human disposal. Whereas you can side with me and call it creatively eclectic or take an opposite approach and call me a bona fide loony. (You might not be able to imagine the reaction of some to my discussing the topic of "threes". It's either as if I were an extra-terrestrial, an escaped lunatic, or someone with a disease that they must make an excuse to depart from. However, others are familiar with such a pattern recognition but not necessarily with the intensity to which I express an interest in the topic.) As one grows older they either become more conservative or start wearing ideas that reflect the hairstyle Einstein is famous for, though I would never stick my tongue out as one of his photographs show, for fear it might get stuck that way and my already jumbled level of; articulation would be that much worse. HA!
Seriously though, the study of language can be approached from multiple venues and invites different writers as well as speakers to choose multiple kinds of attendant materials if they want to make a given point about jokes, public speaking, wooing, sales presentations, persuasion, singing... and of course the infamy of rap and the inclination of many movie script writers to cover up a lack of imagination and intelligence by having characters cuss, stick an alcoholic beverage or cigarette in someone's mouth because they have nothing of value to say, or create bedroom scenes so that barnyard expressions are to be interpreted as some sort of artistic expression because they have no actual literary model to provide viewers.
Whereas some speak of the "dumbing down" of education, one must wonder why it has affected so very many script writers, comedians, journalists, song writers, social policy makers, as well as the butcher, baker and candlestick maker. Indeed, if one wanted to speak of the old Biblical notion of the Tower of Babel, one need only try to get a point out in any political rally. Then again, this has been a standard issue for many centuries, which only means humans are not collectively getting appreciably smarter by any notion of "leaps and bounds" one might otherwise expect from improved sanitation, improved nutrition, (supposedly) improved education methods, and improved means of communication which cover the planet. Thus, we need to rethink how we are thinking not only about language but human cognitive activity as well. We desperately need a new paradigm for our human existence as both individuals and a collective species. The current models of how we think about individual subjects as well as when we gather information from multiple subjects in an effort to create a better appreciation of what might otherwise be viewed as disconnected elements of perception into a workable composite so that multiple types of perception might well operate synergistically as viable contributors to a global society all of humanity belongs to.
Date of (series) Origination: Saturday, 14th March 2020... 6:11 AM
Date of Initial Posting (this page): 9th January 2023... 11:31 AM AST (Arizona Standard Time); Marana, AZ.