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Frame of Reference - Index

Septimus Stele


1) Looking back on my school experience, I wonder if learning mathematics would have been easier had I know that math is just another language, like French or Kiswahili. Somehow I was given the impression that math was more like 'scientific truth' or religious studies, that the base 10 number system we take for granted, the decimal system, is something sacred and better than counting by sevens or twelves. Algebra and geometry are so counter-intuitive for most people, at least initially, that learning these techniques of thinking, this new language, actually changes our world view in some fundamental way.

Example: "A function of a variable x is a rule f that assigns to each value of x a unique number f(x), called the value of the function at x." Now excuse me, but this sentence is completely unintelligible to someone who has not been trained in the language of mathematics or does not have a tutor standing right there to explain it and show how the rule of a function is to be used. Once the language is learned and you know how to follow the rule your whole life is changed!

2) It's understood that a continuous region of mathematical space has an infinite number of points, i.e. a line, no matter what length, is composed of infinite possible points. When we think of physical space in Universe, however we should not be so hasty in assuming that every property of the abstract mathematical space we use to organize our experiences is an actual property of the concrete physical space we live in. What is space then? If not mathematical, what? Is it the space of material objects? Is it the space of physics or perceptions? In terms of material objects or of perceptions, the hypothetical points mentioned do not exist; any material or perceptual phenomenon is spread over a certain finite region of space-time. So when we look for infinity in the smallest point in matter we do not ask whether matter consists of an infinity of unoberservable points, but rather, whether matter is infinitely divisible. So what kind of answer do we want? What linguistic context are we trying to satisfy?

Infinite precision, the precise location of an electron at any moment in time for example, is basically a nonphysical notion, but any desired finite degree of precision [determined by the given context] is, in principle, obtainable. The precision with which something can be measured is thus a good example of something that is potentially infinite, but never actually infinite. It is sometimes thought that quantum mechanics proves that there is a smallest size of particle that could exist. This is not true. Quantum mechanics insists only that in order to 'see' very small particles, we must use very energetic processes to look for them. It is the same with ethical decisions, the more certain we want to be that our decisions are correct, the more effort required to secure our answers. (see Vicesimus Alter Stele: Ethical Decisions, verse 19)

The most recent conclusions of science seem to suggest that there is a limit to smallness where particles become instead waves such as the duality exhibited by photons (light). Therefore, if the smallest point becomes a segment of gravitational force, for example, all matter is fundamentally composed of motion (energy) as its simplest state.

3) The man who wishes to attain human perfection should study Logic first, next Mathematics, then Physics, and lastly Metaphysics. (Maimonides, Guide to the Perplexed)

"The notion of set was consciously introduced only at the turn of the [20th.] century. All the objects that mathematicians discuss -- functions, graphs, integrals, groups, spaces, relations, sequences - all can be represented as sets. One can say that mathematics is the study of certain features of the universe of set theory... A set is obtained when we take a thought and abstract from it all the emotive content, keeping only the abstract relational structure. A set is the form of a possible thought... According to set theorists, there certainly are infinite sets. Indeed, there is to be an endless hierarchy of infinities..." (Rudy Rucker, Infinity and the Mind, 1982)

4) "In this form of meditation [on the formless and infinite aspect of God] the aspirant is not supposed to arrive at complete blankness of mind, but focus on a significant symbol... the infinity one imagines is not mentally externalized as if it were an unlimited stretch of something outside... but as within the aspirant. After picturing infinity within, the aspirant should give himself the strong suggestion of his identity with infinity by mentally repeating, 'I am as infinite as the sky within' or 'I am as infinite as the ocean within:' or it may be even more helpful to use the bare formula 'I am the Infinite within' and, while mentally repeating this formula, to grasp and realize the significance of infinity through the image that has been chosen. It is not necessary to repeat the formula in so many words; it is enough to cling to the thought expressed in the formula." (Meher Baba, Discourses, 1967)

If one can do this while walking backwards on a balance beam with closed eyes, this exercise will improve your math skills. (I suggest you use a very low beam, close to the ground. I did this on the curb of the sidewalk while walking home from school, not that it did me any good. Maybe that's how I became an Atheist, seeing things backwards.) Infinity is beyond our comprehension and talk of beginnings or creation runs into the limits of language and is trapped in paradox. Or so it would seem. The failure to understand infinity, or better yet, to acquiesce to the impossibility of understanding infinity, leads people to accept religion that offers a convenient abstraction of infinity, god.

5) 'Innumeracy' is the inability to deal comfortably with the fundamental notions of numbers and probabilities. The consequences of this shortcoming are not usually as obvious as functional illiteracy, halitosis or bad grammar. Much of Innumeracy is related to anxiety, fear of looking stupid, or the back-lash against looking too smart. For many people, the consequence of Innumeracy is the belief in pseudoscience (and of course religious dogma). In our society there are so many technical advancements, some that conflict directly with historical teachings of religion and popular superstitions (throwing salt over your shoulder just makes a mess on the floor), it's surprising to see how many people still believe in Tarot cards, channeling mediums and crystal power. Well-educated people who condemn one form of superstition are quite comfortable accepting another, such as Christianity.

Further, Innumeracy is, in part, responsible for the huge growth in organic gardening, acceptance of herbal medicines and naturopathy. Scientists' assessments of various risks in food additives are often dismissed by the absolutist who fears any risk that is quantified, but ignores risks every day that are not so well publicized, disease organisms, salmonella or E. coli in organic food sources. Innumerate people tend to accept anecdotal and personal evidence rather than summaries of information that explain statistical significance, sample size and probabilities. (Some people claim that brown eggs taste better than white eggs.) Fallacies of logic can catch the most wary in deceptive snares, sometimes these traps are set deliberately by politicians or by business people for commercial gain. Knowing that these traps are possible, and the limitations of first impressions and intuition, is the first step toward critical thinking. You don't have to be a mathematician to think clearly and act forthrightly.

6) Famous women in mathematics are few, but the real significance of this fact is that women have been disadvantaged when it comes to participation in science and in academia in general. Hypatia of Alexandria is another hero for Atheists because she was stoned to death in 415 AD for her controversial views. She was praised for her work in mathematics by her contemporaries, but stoned by intolerant, fearful Christian monks for her pagan beliefs.

The Marquise de Chatelet translated Sir Isaac Newton's Principia Mathematica into French in the early 1700's. In 1750 the Italian scholar Maria Gaetana Agnesi became known for her achievements in differential calculus and she became the first woman professor of mathematics.

More recently Sophie Germain had to fight hard against the prejudices of her family, friends and co-workers to become an accomplished mathematician. She taught herself mathematics and physics and produced original work in number theory and the theory of elasticity. She believed that her contributions to science would stand the test of time but had to defy social prejudice to accomplish much of her work on her own. Etiquette demanded that she obtain a letter of invitation (in 1809 France) each time she wished to visit an institution. Her host was required to provide transportation and escorts. These formalities restricted her freedom to discuss topics with scientists. Only later in the 1820's, did Germain receive the collaboration she needed with the help of Jean Baptiste Joseph Fourier and she was able to publish her important contributions to mathematics. These restrictions against women had their beginnings in religion.

7) Eternity is infinity with soul. "Eternity defies man's imagination and comprehension. [Then why bother trying to understand it?] It is not an object, nor a place, it is not a period of time, it has a beginning for mankind, but it has no end. All men from Adam to the last one born will be in it... Man is a creature of time... Time, however, as important as it is, has no relation to eternity. Eternity is unique and incomparable. It has no measurable length, breadth, depth nor height. It has no options and is unalterable. It is everlastingly the same... The restrictions of distance and time found in earth life are absent in eternity... in life, while there is yet time, our eternal destiny can be decided... are you ready for eternity? [Do we have a choice?] ... One prepares for this great meeting by accepting Jesus Christ as the Savior of man, who died on Calvary's cross [a human sacrifice] for man's sins [to appease god] and was raised for his justification. (Romans 4:25)" (Published by the Church of God in Christ, Mennonite)

The confusion expressed here is understandable considering it is that same befuddlement that pervades any discussion of infinity. The failure to pay attention to hundreds or thousands of years of scientific information, however, is more than just curious, it is absurd.

8) One of the earliest rational thinkers was John Philoponus (circa. 580-640 AD), born a Christian in Caesarea. He held the chair of philosophy in Alexandria during that erstwhile golden age of enlightenment. He was the first to make a thorough attack on the tenets of Aristotle's physics, mathematics and cosmology. He reasoned that Sun and stars must be composed of fire, and their different colors indicate differences in material composition (as with different colors of fire on Earth). More revolutionary, the different individuality of each of the celestial bodies proves their perishable and changing nature, (thus an argument against biblical eternity) and these all possess a tridimensional extension in common with terrestrial bodies (the first suggestion of three dimensional graphic shapes in the cosmos). Philoponus conceived Universe as a vast mechanism subject to physical laws with which matter was imbued by a god at the moment of creation.

Philoponus conjectured an 'immaterial kinetic power' that kept projectiles in motion once this power was imparted by the thrower. This impetus keeps the projectile moving until it is consumed, whereupon it resumes its natural movement downward. He anticipated conceptually, the arithmetic of artillery shell trajectories by about 1,000 years. In a similar way, he suggested that light was an impetus emitted from a luminous body and propagated to the eye according to the laws of geometrical optics. The ideas of Philoponus were not elaborated or bettered until Galileo, another heretic. He was simply ahead of his time.

9) What is the probability of living without a certain disease or succumbing to a certain hazard? Not being killed in a car accident may be 99% certain, 98% will avoid household accidents, escaping lung disease is 95% certain, avoiding dementia only 90%, cancer just 80% safe, heart disease less at 75% and the list goes on. While the chance of avoiding any particular disease or accident may be encouraging, the probability of avoiding all the above misfortunes is less than 50%. Once you consider your family history your probability of succumbing to one of the above diseases may be even greater. Life expectancies are increasing because people often survive their first disease attack, but fewer survive the second or third episode. How long do you expect to live? Its nice to have a plan just in case you beat the odds and live to be one hundred or more.

The use of mathematics in statistics is no less important than the application of human rationalizations. Consider the changes in the prices on the stock market. The daily ups and downs aren't completely random, but it's safe to say that there is a very large element of chance involved. However, commentators always seem to have a way to explain the rallies or declines, even those of a minor nature. There's the familiar profit-taking, or lower employment, or higher employment (both reports can cause the same changes) the federal deficit report, reports of improved corporate earnings and mass institutional buying. Almost never does a commentator say that the market's activity for the day or even the week was largely a result of random fluctuations, or a non-significant statistical change.

CHANCE occurs when an event must happen, it's just a matter of time of when will it happen, like tossing a coin which must be either heads or tails.

RISK, is involved in the determination of probability for events that may never happen, and only might happen; that lightening might strike a certain person.

10) Mathematicians contribute fundamental and essential technology to solving the mysteries of Universe. One recent development relates to the "superstring theory" which invisions the origin of Universe in ten or more dimensions during the moment of the big bang. This math is capable of handling concepts relating to phenomena otherwise handled separately by relativity and quantum physics. This "hyperdimensional geometry" is the pathway to unification of all physical laws, and even more importantly, superstring theory suggests that all matter and energy exists and operates due to the fundamental hyperdimensional geometry of Universe. (This might just as well have been written in Greek.)



Life is unique.
Experience is priceless.
Reality is nothing.
-Life is so because there is only one.
It is our only spark of consciousness and thought. We have no other.

Life is unique.
-Experience is so because it is unchangeable.
It is our basis and foundation.
Experience is priceless.
-Reality is so because it is.
It is only what we experience through life.
Reality is nothing.
Since life is so, we must savor it and get the most
satisfaction from it as possible.

For life is unique.
Since experience is so, we must build as much of it
as possible.
For Experience is priceless.
Since reality is so, we must not even consider it.
For reality is nothing.
Finally, life is so because it happens and then
disappears to never happen again.

Only life is unique.
Finally, experience is so because it is built
and is there forever.
Only experience is priceless.
Finally reality is so because it is only
made from life and experience.
Only reality is nothing.

(Kenneth Baba Jacob, 1992)

12) Mathematics is a mental construct that we agree to use in a certain way, like a language, and it makes sense because we can be taught to operate within this special language. Formal systems were devised (the prime example being Russell and Whitehead, Principia Mathematica) in which theorems, following strict rules of inference are developed from axioms -- which were the primordial thoughts, the Ur-theorems, from which all others sprang -- and these theorems were thought sacrosanct.

Kurt Godel (1906-1978) by thinking in a very original way suggested that theorems were patterns of symbols, and that a statement in a formal system could not only talk about itself, but also deny its own theoremhood. The consequence of his queer thinking is that the goal of formalization of mathematics is revealed as illusory. All formal systems -- at least ones that are powerful enough to be of interest -- turn out to be incomplete because they are able to express statements that say of themselves that they are unprovable. In 1931 Godel demonstrated the "incompleteness of mathematics." It's not really math that is incomplete (it is certainly useful), but any formal system that attempts to capture all the truths of mathematics in its finite set of axioms and rules is likewise incomplete.

Math has never been the same since. (This is the moral equivalent of proving that a certain god does not exist.) Godel also invented the theory of recursive functions, which today is the basis of a powerful theory of computing. If one were to develop a system of ethics based on the methods of mathematics, a science of ethics, let's say, as envisioned by John Locke (in the seventeenth century), someone would eventually point out that this formal system was incomplete, and that would be very disheartening. (see Septimus Decimus Stele, Morals, verse 12) Thus we are all still left to our own devices.

13) "Innumeracy and pseudoscience are often associated, in part because of the ease with which mathematical certainty can be invoked to bludgeon the innumerate into a dumb acquiescence. Pure mathematics does indeed deal with certainties, but its applications are only as good as the underlying empirical assumptions, simplifications, and estimations that go into them... Any bit of nonsense can be computerized--astrology, biorhythms, the I Ching--but that doesn't make the nonsense any more valid." (John Allen Paulos, Innumeracy, 1988) The old saying: 'figures don't lie, but economists sure can figure' is too true in many walks of life, not just economics. In politics it's called 'spin.' In religion it's called 'belief.' In psychiatry its called 'analysis.'

14) There was no greater mathematician at the beginning of the sixteenth century than Galileo Galilei (1564-1642, he died the same year Isaac Newton was born). He calculated the parabolic motion of projectiles, described the motion of objects rolling down an inclined plane, invented a military compass and devised the pendulum clock. He is best know for his work in astronomy, because he introduced competent mathematics into the study. He built a telescope to monitor the solar system and discovered the four large moons of Jupiter, that Venus had phases and that Sun had spots; all of this was against the Catholic Church doctrine. Universe "...is written in the language of mathematics ... without which it is humanly impossible to understand a single word of it; without these, one is wandering about in a dark labyrinth." The surprising thing is that there are still so many people wandering, as he described. He was condemned by the inquisition, forced to recant and live under house arrest. (Another hero for Atheists.)

15) "All four Yana dialects used a single counting system with almost identical numeral names:" One, baiyu; two, uhmitsi; three, bulmitsi; four, daumi; five, djiman. Beginning again at six, baimami; seven, uhmami; eight, bulmami; nine, daumima; ten, hadjad. Later Ishi identified: twenty, uhsiwai; forty, daumistsa; sixty, baimamikab; eighty, bulmamikab. "The system was quinary [something like the Maya system] that is to say there were basic numeral names up to five, and from five to ten there was additions to these [the roots] with further additions from ten to twenty. When twenty was reached, it became a new unit as one hundred is with us, the twenties or scores being given names not built on the smaller numeral names." (Theodora Kroeber, Ishi In Two Worlds, 1961) Fluency with numbers was something of a sacred trust among the native Americans, since numbers often had mystical connections as in Numerology today.

16) There is a lot of acceptance for predictive dreams as a form of extra-sensory perception with which nearly everyone has had experience. When one has such a dream and the predicted event happens, or even close, it's hard not to believe in precognition. The story of Moses has encouraged many religious people to take dream interpretation seriously. What is the probability that any certain dream matches some portion of real life in the future? Since dreams are most often about events in life anyway, it is only a small leap of faith to assume that dreams might be associated with the future as much as with the past.

If we assign a low probability to a dream matching the future, say one in 10,000, the probability of having a non-matching dream is very high. For a year the probability of having non-matching dreams is .964, very high, but this leaves 3.6% probability that a matching dream will occur sometime during a year. When you add all the people in the USA that means that 3.6 percent of the people will likely have a dream that seems realistic every day. There's no need to invoke any special parapsychological abilities; the ordinariness of apparently predictive dreams does not need any explanation. What would need explaining would be the absence of apparently predictive dreams. If you reduce the probability to one in a million (rather than one in 10,000) there will still be many such dreams occurring in a population the size of the USA. (Not to mention the fact that the memory of the dream is altered after the real occurance so that when the story is related the storyteller makes it seem more predictive than it probably actually was.)

The same numbers game applies to the occurrence of other coincidences.

17) "The first material which is used for numbers is the series of ten rods used for lengths, a ready part of the education of the senses; the rods are graduated in length from one to ten. The shortest rod is 10 mm., the second 20 mm. and so on, up to the tenth which is 10 cm., or one metre. When they are to be used in teaching numbers, the rods are no longer all of the same color, as when they are used as sense-material intended to make the eye estimate graduated lengths. Here, the various segments of 10 mm. are colored alternately in red and blue... Many small children can count, reciting from memory, the natural series of numbers, but they are confused when dealing with the quantities corresponding to them.

"Counting the fingers, the hands and the feet certainly forms something more concrete for the child... the difficulty is to know why, if the hand has five fingers, he should have to say about the same object -- "One, two, three, four, five." This confusion, which the rather more mature mind corrects, interferes with numeration in the earlier years. The extreme exactness and concreteness of the child's mind needs help which is precise and clear. When numerical rods are in use we find out that the very smallest children take the keenest interest in numbers... These numbers, [rods] which can be handled and compared, lend themselves at once to combinations and comparisons... the moving and combination of quantities... is the beginning of arithmetical operations... the child repeats the exercise(s) many times of his own accord because it interests him." (Dr. Maria Montessori, The Discovery of the Child, 1948)

Roman numerals were an equivalent way that Western Society learned numeracy.

18) The idea that mathematics is a physical property of nature, and simply has to be unveiled, like knowledge of geology, has been a common assumption. The concepts of the theory of natural numbers and the axioms and definitions of algebra are to be regarded as "...platonist, in the sense that both numbers and sets or sequences of numbers were treated as existing in themselves [ontological?] ...the present sharp distinction between pure and applied mathematics is a rather recent development... An axiomatic theory would then consist of just those statements which are deducible by purely logical means from a certain limited set of statements and of the statements which can be obtained from these by definitions expressible purely logically in terms of the primitives." (Charles Parsons, Encyclopedia of Philosophy, 1967) So mathematics, instead of pre-existing, is more like a Michaelangelo statue partially carved from a single rock, emerging as if it pre-existed in the rock, but in reality it is a construction of the rational mind, albeit a very talented mind, of Man.

19) A more obscure yet fundamental part of mathematics is the concept of 'following rules.' It remained to a 20th century philosopher, Ludwig Wittgenstein, to identify and explain this essential phenomenon. As suggested in verse 17 above, young children are not born with the ability to count their fingers, they have to learn the rules of counting: that each new number does not stand for one finger but for an accumulation, these are to be numbered individually, in sequence, and that five is not only the last, but identifies the total quantity. As an adult it is easy to perform this task, but it is so easy that the fact that there are important rules to follow is nearly lost. Wittgenstein argued: How do you know if someone understands the rules? If they can follow repeatedly, by their showing and doing the same thing as taught, not by their ability to repeat the rule verbally.

It is more obvious in other intellectual pursuits such as logic, telling time, mechanics, less obvious in morality, that what a person does is the proof of their understanding the rule. But knowing the significance of a rule, is another matter. If someone can apply a rule in many different contexts and make allowances and explain their action, we can say they understand the rule. Converting temperature from Centigrade to Fahrenheit is an example that is easy to understand. Knowing what heat is and how it transfers, and knowing how to convert from C to F shows we understand heat and how to use the rule.

The opposite is also true: "It would also be possible to imagine such a training in a sort of arithmetic. Children could calculate, each in his own way -- as long as they listened to their inner voice and obeyed it. Calculating in this way would be like a sort of composing." (Ludwig Wittgenstein, Philosophical Investigations I, 1953) How often do we hear in ethics or religion, "Listen to your inner voice for guidance?" How random might this be? (see Vicesimus Alter Stele: Ethical Decisions, verse 10)

20) Is all matter made of space, in Strings of curved space? The Big Bang was to have incorporated 10 dimensions, as incomprehensible as that is, at the onset of the Universe inflation. Six of these dimensions were somehow reduced to four by "phase transition" as the moments elapsed and the expanding matter cooled. In August, 1984, John Schwarz and Michael Green collaborated "...producing a self-consistent string theory that incorporated supersymmetry: anomalies disappeared when one calculated one-loop amplitudes with either of two internal gauge symmetry groups."

"...Are subatomic particles -- electrons for example -- particles or waves? One can write equations that accurately predict the behavior of electrons by using either wave or particle equations. The two approaches are mathematically equivalent and both will yield reliable results. If you run electrons through an apparatus designed to detect waves, you see waves, and if you run them through an apparatus designed to detect particles, you see particles. This is the wave-particle duality and it threatens to make a hash of the belief that there is an objective reality out there.

"... We live in a Universe that presents two complementary aspects. One obeys locality and is large, old, expanding, and in some sense mechanical. The other is nonlocal, is built on forms of space [i.e. string theory] and time unfamiliar to us, and is everywhere interconnected. We peer through the keyhole of quantum weirdness and see a little of this ancient, original side of the cosmos." (Timothy Ferris, The Whole Shebang, 1997)

21) In Mesopotamia (modern Iraq) an advanced mathematics had existed since at least the time of Hammurabi (c. 1700 BC). His writing includes explanation of many problems in arithmetic and algebra and facts of elementary geometry. Measuring in increments was well established. It is reasonable to assume that much of this knowledge was distributed to Greece, Egypt and elsewhere.

22) "Probability enters our lives in a number of different ways... first... dice, cards and roulette wheels. Later we become aware that births, deaths, accidents, economic and even intimate transactions all admit of statistical descriptions [not statistical causation]. Next we come to realize that any sufficiently complex phenomenon, [weather] will often be amenable only to probabilistic simulation. Finally, we learn from quantum mechanics that the most fundamental microphysical processes are probabilistic in nature.

"Not surprisingly, then, an appreciation for probability... is a mark of maturity and balance... Statistical tests and confidence intervals, the difference between cause and correlation, conditional probability, independence, and the multiplication principle, the art of estimating and the design of experiments, the notion of expected value and of a probability distribution, as well as the most common examples and counter-examples of all of the above, should be much more widely known. Probability, like logic, is not just for mathematicians anymore, it permeates our lives." (John Allen Paulos, Innumeracy, 1988)

23) "What is infinity? It isn't something that corresponds to the interrogative pronoun 'what.' It's a fallacy of logic and language to attempt to give an explanation. Like dividing by zero, it leads to nonsense, he realized. This conclusion was not nonsense, it was the only concept that made sense.

"...It may sound inconsequential, even simple, but it [giving up on infinity] leads to other more important conclusions. For example, if we have this nagging question about the nature of infinity [or an infinite god] yet we still manage to live, then we can live without knowing other answers, such as: 'What does God look like?' or 'How was God created?' If we can live without the fundamental answer about infinity then we can live just fine without a whole sequence of answers. Just because we can't explain something, some mystery, it doesn't mean we should make up an answer or accept an answer that just happens to satisfy our psychic needs.

"Now Jack knew: 'It doesn't have to make sense!'

"'Why?' Religion didn't need to answer all the mysterious questions, because there would always be this huge failure to understand infinity. There could always be another prior or subsequent question, so why contrive answers and explanations for all the rest?

"Jack had been a doubter, not a cynic. He had wanted to believe, only his doubts were based on more than just a personality type... Jack's doubts had been legitimate questions about the logic and soundness of all the rationalizations and myths of the Mormon religion... There were many aspects of religion that just didn't make sense. People throughout history had repeatedly constructed elaborate hierarchies and complicated morals and dressed these in sanctitiy. It kept order in society, most of the time. [As mathematics does for science.] ...but now Jack saw clearly that these fairy stories were little more than elaborate rouses." (IJ, Jack and Lucky, 1968) Math is a logical construction of man that makes sense, religion fails this same test.

24) "The discovery of fractals is one of the Twentieth Century's major advances in the understanding of Nature. Since the 1970's many natural patterns have been shown to be fractal. Examples include coastlines, clouds, lightning, trees and mountain profiles. Fractals are referred to as a new geometry because they look nothing like the more traditional shapes such as triangles, circles and squares... fractals consist of patterns which recur on finer magnifications, building up shapes of immense complexity... given people's continuous visual exposure to Nature's fractals, do we possess a fundamental appreciation of these patterns -- an affinity independent of conscious deliberation?" (Richard Taylor and Ben Newell, "A Resonance Between Art and Nature," 2001) For some, finding this kind of elegance in nature is akin to seeing the hand of god. But of course such knowledge was never revealed to Man, nor was any other aspect of mathematics.

On to Octavus Stele
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