Chapter 11: Scale, Proportion, and Time
• “The quality without a name.” • Relationships must be at a scale we can comprehend. • Proportion and “the sense of fitness.” • The whole is experienced in relation to our physical being. • Space and time and the arts. The four-dimensional arts. • Communal works of art built through time. • The Golden Section, the “Divine Proportion.” Other “incommensurate” relationships. • The Fibonacci sequence • Science now addresses realities far out of scale with our own. • Cancelling infinities by introducing others. • Science is one form taken by human desire to understand the universe.
A thought which reveals itself without word or sound, but solely by means of shapes which stand in a certain relationship to one another….[Y]ou have established certain relationships which have aroused my emotions. This is architecture.…The purpose of architecture is to move us. Architectural emotion exists when the work rings within us in tune with a universe whose laws we obey, recognize and respect.
[Le Corbusier, Towards a New Architecture, p 156]
Scale is really proportion. Who can teach proportion? Without a sense of proportion, no one should attempt to build. This gift of sense must be the diploma Nature gave to the architect.
[Frank Lloyd Wright, In the Cause of Architecture, 1925]
The deepest meaning of art, that which both awakens and restores us, is available to the degree that the work manifests a complex dynamic equilibrium, an enlivened harmony, specific to that instance, among the parts and between each of them and all of them together—that quality represented here by the symbol < >´. Twentieth century French architect (and sometime abstract painter) Le Corbusier writes that the presence or absence of “architecture” (implicitly contrasted with mere building) is decided by the presence or absence of “certain relationships,” which arouse emotion. I am saying that a somehow blessed pattern of relationships is the defining quality of art of any kind. British-American architect Christopher Alexander has called it “the quality without a name,” which in architecture is effected by a multitude of relationships within and among buildings and their surroundings, and between buildings and human beings, resulting in “beautiful, timeless buildings, whose sense of wholeness unites thought and feeling and which are tiny but complete pieces of a larger, universal order.”[1] He writes that
[A]t the core of all successful acts of building and at the core of all successful processes of growth, even though there are a million different versions of these acts and processes, there is one invariant feature, which is responsible for their success. Although this has taken a thousand different forms at different times, in different places, still, there is an unavoidable, invariant core to all of them.
[Christopher Alexander, The Timeless Way of Building, p 8]
To produce this quality, this “invariant feature,” in what we make, it is not necessary to learn new processes, but rather to set free in ourselves a process which is part of us already. The fact that it is natural, however, does not mean that it is always, or even often, easy. “[T]his capacity in us is not accessible until we first go through the discipline which teaches us to let go of our fears,” Alexander says.[2]
Is it likely that all the people who through the centuries have spoken and written of marvelous and mysterious conditions of relationship, un-nameable and difficult of attainment, have been talking about different phenomena? I think not.
For us to experience such relationships, and the joyous sense of being rightfully at home in the universe which they allow us, the object or work must be at a scale we can comprehend. Great music must be embodied in sound waves we are equipped to hear. The Hubble telescope shows us the Horsehead galaxy reduced by distance to a scale we can imagine; images of the exquisite traceries made by the movements of subatomic particles must be vastly enlarged before we can see and appreciate them.
Architecture is the art form occupying comparatively large amounts of physical space, and the importance of scale is felt almost physically there. Scale in architecture or any of the other arts consists in the relationship between the size of the work and its parts and the size of human beings, that human size being inevitably our experiential reference for everything else.
I once knew a two-year-old child who was infuriated when she couldn't climb into her wicker doll-carriage, raging at an invisible barrier as real as if it were made of plate glass. She had to learn about scale—her own as different from that of her dolls. In contemporary architecture, when buildings must be designed to hold great numbers of machines and the infrastructure supporting them and the people attendant on them, the question of scale is particularly difficult. Buildings which are “out of scale”—meaning that we are unable to feel ourselves in comfortable relationship to them—are not too small for us, like a doll carriage as vehicle; they are too big. Inside them, people work in huge, floor-sized rooms divided into identical cramped (too small) employee cubicles decorated with bits of personal trivia. The monstrous interiors of “Big Box” stores assail shoppers with similarly inhuman spaces.
We feel in some way alienated from reality within or beside such buildings. We say that they are “impersonal,” “dehumanizing”; they don't relate to us. Of course they don't—they relate to the machines or merchandise they house and to the motives of the owner, the architect, the developer. They relate to profit, to efficiency, to marketability in dollars-per-square-foot, and in some cases to the building as a sign, a “logo” for a corporation. They place themselves massively, immovably, between us and the natural world, within which we feel ourselves to be in perfect scale even when dwarfed by its magnificence. (In the affluent parts of cities, in an effort to disguise this “lack of scale,” retail spaces are often inserted at ground level; they serve to distract us from the looming and alien monoliths above.)[3] Cities are clogged with buildings grossly out of scale both with earlier buildings and with us, and the countryside is more and more afflicted with McMansions designed primarily to advertise the financial success of the owner. By contrast, the gardens of Versailles are vast, and the palace is large indeed, but we are continually brought into a relationship with the amazing whole by the scale and artistry of the parts (paths, statues, fountains, clipped trees); we experience grandeur without alienation.[4]
The word “scale” originally meant a means of measurement and comparison, a way of measuring relationship between things, as in an architect’s “scale.” Some of its meanings contain the idea of a graduated or ordered series (as in the musical scale, or the “gray scale” in art), and there is also within it the idea of “proper or intended size,
proportion, and relationship with reference to other elements or to the whole.”[5]
According to Webster’s dictionary, proportion involves “the relation of one part to another or to the whole with respect to magnitude, quantity, or degree.” But it is also, less obviously perhaps, the relation of one part to another with respect to color, tone, shape, area, complexity, texture, density, loudness, kinetic intensity, melodiousness, and on and on—of relationships that can’t be measured.[6] In a painting, this red blob is in a harmonious relationship with this small black spiky element and with the white space that surrounds them.
When parts are “in proportion” to one another, says the dictionary, they are related to each other and to the whole “as a sense of fitness demands.”[7] Here we are again, with an immeasurable and undefinable human attribute: the “sense of fitness.” “Fitness” is unquantifiable, because it is a web of comparison and relationship, not of fixed or mathematically definable quantities, and every small relationship affects every other and the whole. It is, in fact, concinnity. However, that whole is experienced in relation to a specific entity—the human being—as ancient Greeks and Renaissance humanists understood. To say that “man is the measure of all things” is only to claim that human beings are, for ourselves only, the ultimate criterion for measurement in the universe, since we experience the world from inside our own skins, in relation to our own physical size and shape and capabilities, and to our preconceptions, sensory capacity, and emotional makeup. We can experience only as we are, not as God may or a beetle does. We exist both as individuals and as members of a species, and it is obvious that there are many ways in which our individual experience is similar or identical to that of other human beings, both living and dead. We live in essentially the same bodies, are equipped with essentially the same sensory capacities. Humans are “one of the most genetically homogenous species.”[8] We live in the same world, sky above, earth below; we drink water, eat food, breathe air, fear death.
Since we are part of the natural world, we naturally feel ourselves to be in scale with it. People often say that they feel small and insignificant when confronted by very large objects like mountains, but because in nature these things are harmonious, their parts relating coherently with the whole, they are not alienating and destabilizing, no matter how awe-inspiring or frightening. When the proportional relationships among the parts are such that the miracle of < >´ is achieved—but only if we are receptive—we experience the transcendent, whether in nature or in the arts. What about the starry heavens, the trail of a comet? The star-pattern or the comet appears to be in scale with us until we imagine the scale at which it actually exists. To imagine things at another scale in space-time is a mental adventure, but we cannot live there.
In everyday life, time and physical space are experienced as different entities, and the “space-time” of physics can be difficult to grasp. But in listening to music—or in asking, “How far is it to Boston?” and answering, as we habitually do, something like “about two hours”—we experience space-time. It is clear that a play or a piece of music has duration, and exists within that frame of time just as a two-dimensional painting exists within its frame, or as we live within the boundaries of a certain place, with another place “about two hours” away. The relationships among the parts and the whole of an oratorio or a novel occur within a space which is time. Physicist Stephen Hawking has written that “It is often helpful to think of the four coordinates of an event as specifying its position in a four-dimensional space called space-time. It is impossible to imagine four-dimensional space. I personally find it hard enough to visualize three-dimensional space!”[9] But in music and theatre, dance and film, we experience the fourth dimension as the space in which art occurs. Such four-dimensional art objects cannot be owned or hoarded or traded on a market; they require the skilled and inspired work of performing artists in order to exist fully. The composer, playwright, or choreographer constructs the form of the work, the fundamental proportional relationships among the parts in time, the details and embellishments. The essential subtleties and nuances of the work as performed are dependent on the artistry—the craft, technique, intelligence, aesthetic judgement, and heart—of the performers (and the director or conductor), as individuals or as part of a group. Time is the space in which the work occurs and is experienced. Literature and poetry, painting and sculpture, architecture and landscape architecture also exist in space-time; it becomes clear that everything does.
Early in the twentieth century the Cubists tried to incorporate the idea of time, the fourth dimension, in painting and sculpture by representing aspects of the material subject remembered or imagined, or brought into view simply by moving the head or the body. This allowed them and those who came after them visual adventures in the vast reaches outside the mathematical linear perspective of the single fixed eye. Time is required, in varying amounts—from moments to years—for the making of any work of art, and time must be allowed for the experience of it, for a painting or poem as surely as for a sonata or a pas-de-deux or a novel. The relationships among the parts, their scale in relation to us, their proportional harmonies and dissonances, can saturate our consciousness only within the dimension of time. John Ruskin advised that rather than going and looking at a painting, one should go and “watch” a painting for a goodly period of time.
Craftsman David Pye writes about another way in which art occurs in the dimension of time:
There are some classes of work done, we must believe, without any conscious intention of producing what is beautiful yet which do so in a high degree. Moreover some of the best of these, having been produced co-operatively by the work of successive generations must, in their appearance, differ widely from what their earlier contributors intended, for all that they provided the indispensable foundation and ground plan of what we see now.
[David Pye, The Nature & Aesthetics of Design, p 113]
He goes on to say that works of beauty produced by members of a cultural group over generations, through time periods longer than a single human life, must be the result of all the work having been made within the same matrix, that matrix being “the history and environment dealt out to us by the tradition of the society we are bred in.” [10]
Political, social, and religious structures—whole cultures, in fact—can be “works of beauty” built up in the same kind of multi-generational space/time. We show that we know this by being willing to fight and die for them; they are works of human craftsmanship and art, and they are ours. It is folly to throw away such patrimony without anxious and careful thought, and without a conscious humility.
Certain town planners attempt to collapse the time necessary for the natural process of architectural accretion while nevertheless garnering some of its benefits—the spontaneity, variety and even beauty produced by organic growth through time. They hope to achieve something of the organic harmony, the inherently lively “fitness” of earlier buildings and communities—built over generations by coherent cultures—in the design of new housing complexes, “manufactured communities,” constructed within a short period in the United States today.
I'll tell you how Kentlands was designed. We played a sort of game. There were five existing farm buildings, and each designer sequentially added buildings and spaces. In a sense, we were trying to compress history: to achieve in a short time what would happen over a longer period. Today this method is part of our process. The way we design is that [my partner] or I do the parti, [starting point, concept] very crudely; we then hand it to the first designer, who has it for two hours before handing it over to the next designer, and so forth…. Kentlands has authentic variety, built-in disjunctions and inconsistencies….Our town plans have a history, they're not finite designed places—mistakes are inscribed and, of course, they continue to accrue as the towns are built out.
[Andres Duany, quoted in Harvard Design Magazine, Winter/Spring 1997, p 50]
A common culture can provide the matrix in which harmonious organic change can occur in the built environment, but such a culture is more and more rare. We must substitute another “matrix,” and in much of the world today commerce and a desire for grandeur, however ersatz, appear to be what we have in common. As Robert Campbell has said, “[B]y looking at a building you know what the society values”—a fact that is rarely mentioned in the overflowing abundance of contemporary architectural theory. [11]
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The benefits to the human psyche of certain specific proportional relationships are generally ignored today, though they’ve been well known for thousands of years and we still choose them instinctively. One such set of relationships is of course the Golden Mean, Golden Proportion, or Golden Section. The ancient Greeks considered it to be the most glorious proportion, a cornerstone of the cosmos and of art. In the Middle Ages it was known as the Divine Proportion, and was associated with the Trinity.
In the Golden Section, the whole is divided into two parts so that the smaller of the two exists in precisely the same relationship to the larger that the larger bears to the whole. As we have seen, that relationship can only be expressed as an irrational number—that is, a number that never ends, and cannot be accurately measured with a ruler (a measured “scale”). It is represented by the Greek symbol Ø (phi), or numerically as 1.61803398…. As far as we know, the decimal positions in this number continue forever, with no repeating patterns of numbers in the sequence. When we try to represent it in numbers we can write only a closer and closer approximation of it. In this it is like the relationship between the circumference of a circle and its diameter (π, or pi), or between a square and its diagonal (√2). The parts in such relationships are said to be incommensurate with one another, which means that they cannot be measured accurately on the same ruler or scale. Many fundamental relationships within the natural world can be expressed in whole numbers, as 1:2, 2:3, and so on, but it has been known at least since the time of Pythagoras that the Golden Proportion cannot. It can of course be produced geometrically, in the real world, using straight edge and compass, just as the diagonal of a square, or the diameter of a circle, can be found easily by drawing it.[12]
Late in the 20th century Gregor and David Churnodsky, brother mathematicians from Latvia, spent more than ten years attempting to calculate the exact numerical value of the irrational number pi —in other words, to see whether it is in fact a rational number, though one with a vast number of decimal places. To arrive at the necessary computational power, they assembled and combined various elements, both improvised and purchased, until their New York apartment itself became a kind of giant computer. By 1992, they had calculated the value of pi, or π, in a string of numbers long enough to reach from Manhattan to Chicago, without finding a repeated numerical pattern of any kind or length. They came to think that it might take a computer as big as the universe itself to calculate pi, which of course represents one of the most fundamental physical and mathematical relationships in nature.[13] A story like this may bring us a ghost of some of the wonder and unease which Pythagoras and his followers must have felt when they encountered the qualities of such numbers. In 2011 the value of pi had been calculated to 10 trillion digits to the right of the decimal point.[14] There is no evidence that the discovery of finer and finer values of pi will ever arrive at an end to the string of numbers or detect any regular pattern within it.[15]
We are told that although there is an infinite number of rational numbers, somehow there is an infinitely larger number of irrational numbers.[16] (Since “infinity” means a mathematical quantity larger—or smaller—than any fixed assignable quantity, we find ourselves here again attempting to imagine the unimaginable.) And we cannot, so far, generate irrational numbers; we can only find them and investigate them, one by one. The great choreographer George Balanchine observed, “God creates. We can only discover.” Today, lesser artists and designers habitually claim to have “created” all sorts of wonders, and even our computers are said to “create” files and folders and musical scores.
Life forms in which the parts are related to one another and to the whole in the golden proportion are easily found in the natural world; in flowers and insects, or trees and their leaves.
Consider the shape of an egg, of one wing of a butterfly, of the chambered nautilus, of the calico surfperch. These asymmetrical [that is, not bilaterally symmetrical] forms also possess a beautiful balance in their forms which has come to be known as dynamic symmetry….The golden rectangle continually generates other golden rectangles and thus outlines the equiangular spiral [as found in the structure of DNA]….
…Nature has many forms of packaging—squares, circles, hexagons, triangles. The golden rectangle and equiangular spiral are two of the most aesthetically pleasing forms…found in starfish, shells, ammonites, the chambered nautilus, seedhead arrangements, pinecones, pineapples, and even the shape of an egg. …The shape of the golden rectangle or the proportion of the golden mean can be found in all shapes with dynamic symmetry.
[Theoni Pappas, The Joy of Mathematics, pp 154, 105-6, 155]
Such relationships are common among the works of human beings as well, whether arrived at instinctively or with conscious intent, for instance in Greek temples, the sand and rock garden at Ryoan-ji Temple, near Kyoto, Druidic stone circles, and the US dollar bill.
The use of the golden mean in art has come to be labelled as the technique of dynamic symmetry. Albrecht Dürer, George Seurat, Pietter Mondrian, Leonardo da Vinci, Salvador Dali, George Bellows all used the golden rectangle in some of their works to create dynamic symmetry….Mondrian is said to have approached every canvas in terms of the golden rectangle.
[Theoni Pappas, The Joy of Mathematics, p 154]
American artist Mark Rothko’s large paintings of stacked soft-edged rectangles of color are often described as expressing a monumental spiritual power, as “creating a total environment, a unified atmosphere of all-encompassing, awe-inspiring spirituality.”[17] By means of the simplest geometric analysis, many of these paintings can be seen to be made up of golden rectangles, often one within the other, as in two such rectangles established within a bounded ground which is also a golden rectangle. However, myriad other relationships within these works—of color, light and dark, dispersal and density, shape and contour and so on and on—though crucial to the whole—are beyond analysis. Such complex multiplicity of essential internal relationships is of course common to every thing that exists, whether it is manifest in actions, words, sound waves, or in the realm of the visual.
Attempts to find identifiable proportional relationships within masterpieces—of painting, literature, music and so on—are fairly common and can be interesting. Photographs of paintings are reproduced with superimposed diagrams showing the golden relationships, or squares, triangles and circles, within the work. The interweaving of themes and imagery in poems and stories is identified and analyzed. There are now computer programs which, it is said, can produce architectural plans in the style of Palladio, or, as noted earlier, music in the style of Bach. It is not clear whether these amount to anything more than curiosities; certainly the webs of relationship within great masterpieces are hugely complex, and computer-generated music is unconvincing so far—and often notably nasty. Perhaps that which I am calling < >´ is the product of a vast number of golden relationships in two and three and four or more dimensions, in tone and pitch and volume, in activity and stillness, in color, intensity and lightness and darkness as well as in shape, occurring at every level of complexity and nuance in a work.
As we have seen, the great pre-Socratic philosopher Heraclitus built his philosophical system on his understanding of the Greek word Logos, which in its most profound meaning signified the infinite and probably unknowable totality of relationships in and underlying the Universe as a whole, with its generative foundation in the very thought of God. In our own time, scientists like Albert Einstein have attempted to find a simple (and thus “elegant” and “beautiful”) formula for it, which would constitute “The Final Theory.” In the previous chapter I quoted Roger Lipsey, who wrote that in the craft traditions of the world one finds that “[t]here is an intuition of the Logos informing both Nature and human nature, accompanied by recognition that the Logos hides and must be perseveringly sought.”[18]
Professor Lipsey’s word “inform” is an interesting one. Webster’s dictionary gives five meanings for the word before the sixth, which is “communicate.” We appear to have lost our understanding of the word as meaning “to give shape to,” or “to form the mind of”; every one of the synonyms provided refers to a verbal informing. The word comes from the Latin informare, which is “to shape, fashion, or describe.” The first two meanings given indicate actions firmly rooted in the real, physical world, in making rather than in verbal symbols: they carry an insight into the truth, which is that our minds are “informed”—literally physically shaped—by experience in the real world. Today it is presumed that words are our principal means of informing and communicating, yet with words we only point toward the reality experienced by our faculties. There is in fact a constant play, in our whole-mind, between experienced reality and symbol. To choose the second in preference to the first—or worse, to substitute it for the first—is to undermine, or sabotage, our capacity for wisdom. As Heraclitus taught, “The Lord whose oracle is at Delphi neither speaks nor conceals, but gives signs.” (fragment 93)[19]
The fundamental importance of the relationships we can identify in both the natural world and in the works of human beings (for example those of phi-related systems such as the Fibonnaci numerical sequence) is well recognized. However, though we are capable of achieving works of great subtlety and complexity in which such relationships can be found, each new totality, in the end, must be arrived at by insight, using our five senses and our still-mysterious capacity for non-verbal thought. Further, our appreciation of great works requires those same sub-conscious resources. If we are talking, within our own minds or to others—or if someone is talking to us—access to these abilities is curtailed; we will see more of this in later chapters.
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Philosophies, religions, and mythologies are sets of stories by which we attempt to teach ourselves how best to live our lives, so that we may develop fully both as individuals and in fruitful relationship with one another and the natural world. In the modern era, the success of reductionist science in making the lives of most of us healthier and more comfortable has led to a general reverence for science as the ultimate teller of truth. We have seen that by the middle of the twentieth century many believed that science and the technologies it made possible would in time solve all of our problems by providing rational and empirically testable explanations for all the vast variety of forces at work, from the weather to our own psyches, thereby allowing us to control those forces. Such faith becomes less and less tenable as time passes, but its most passionate adherents are not daunted. They believe that the devastation caused by the byproducts of technology today will be set right by the science of tomorrow (or Gee Whiz, we’ll move to Mars!), and that religions are useful only as temporary explanations for those aspects of reality which science so far cannot measure and explain. (Theoretical science seeks to perform that same function, of course.)
Some scientists, like Northwestern University professor Greg Olson, are less certain of the future accomplishments of science as it now defines itself:
[When doing graduate work at MIT] Olson recognized the importance of science’s quest for value-free facts, but he now thinks that objective knowledge must be tempered with emotional and intuitive ways of approaching the world. “The goal of science is to be like Vulcan and suppress your humanity,” he says. “The way we do scientific observation is reptilian. A reptile can observe you and eat you without experiencing any emotion. We are training scientists to behave that way. And we’re damaging them. When it’s taken too far, I think science does brain damage.”
Olson is serious about the neurology of the matter. “The mammalian brain, compared to the reptilian brain, is structured for emotion. Our higher-level skills of synthesis and evaluation are controlled by the limbic system, where our emotions are seated. It’s where the power is. Call it spirituality, call it what you want, but we have this awesome power that exists only in mammals.”
[Wired magazine. Feb. 2001, p 139]
It is only by means of our “higher levels of synthesis and evaluation” that we recognize < >´; it is indeed an “awesome power” and could be called spirituality. In the second epigraph to this chapter Frank Lloyd Wright calls it “this gift of sense,” the accreditation Nature gives to the architect, and it is the kind of thought capable of arousing
le Corbusier’s emotions: “a thought which [in architecture] reveals itself without word or sound.”[20]
Many of the most powerful scientific theories now address realities that are entirely out of scale with our own. The life experience of a butterfly has much more relevance to ours in space and time than does physical Relativity, since the latter makes a measureable difference only at speeds approaching that of light—186,000 miles per second—and over distances longer than three miles. So far, and for the foreseeable future, what happens at the speed of light has little relation to our lives and to our thinking about how best to live, and the choices we must make. And yet Einstein's General Theory of Relativity is quoted as support for claims that all truth is relative, or that there is no such thing as Truth or the Good as our ancestors aspired to them. Quantum mechanical theory, another great intellectual achievement of the first half of the twentieth century, deals with phenomena occurring at unimaginably small scales, such as a millionth of a millionth of an inch. These two great theories don't by any means give us the whole picture—for one thing, they are still mutually incompatible—but they have changed our lives in the technologies that they make possible, such as nuclear power and the microchip. As we do all our scientific discoveries and theories, we’ll use them as we can, for better and for worse.
Stephen Hawking writes:
Any physical theory is always provisional, in the sense that it is only a hypothesis: you can never prove it.… In practice, what often happens is that a new theory is devised that is really an extension of the previous theory…. Einstein's general theory predicted a slightly different motion from Newton's theory…. However we still use Newton's theory for all practical purposes because the difference between its predictions and those of general relativity is very small in the situations that we normally deal with.
[Stephen Hawking, A Brief History of Time, p 10]
Later in the same book he provides an example of scientific “fudging.”
[S]eemingly absurd infinities occur…but in all these cases the infinities can be cancelled by a process called renormalization. This involves cancelling the infinities by introducing other infinities. Although this technique is rather dubious mathematically, it does seem to work in practice.…
[Ibid. p 157]
This is honest science at work—its flexibility and pragmatism, its constant questioning, its uncertainties, its improvisations and temporary assumptions and leaps of faith. We (both scientists and laypersons) avidly desire the certainties it habitually promises. But as British philosopher Mary Midgley has written, “Thinking out how to live is a more basic and urgent use of the human intellect than the discovery of any fact whatsoever.”[21]
Most of us know very little about science. We used to think that God understood the mysteries; now the general belief seems to be that scientists will one day explain the few little mysteries still remaining. [22] Many scientists are even more sure in this belief than are lay people, but how can we feel confident of scientific certainty when, for instance:
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i) the widely accepted Big Bang theory for the origin of our universe depends on the assumption that more than 95% of its substance is made up of entities known as “cold dark matter” and “cold dark energy,” about which nothing whatever is known, except that they appear to exert gravitational force. We can only begin to think about, analyze, measure, 5% of the universe? Now there’s a “fudge factor”! The ancient Buddhist concept of Emptiness, a generative Nothingness, would do just as well.[23]
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ii) Science takes its fundamental authority in part from the symbolic language in which it thinks, from mathematics—and mathematics is clearly susceptible to manipulation. As Professor Hawkings pointed out, it can be used in “mathematically dubious” ways, in artful ways in fact, toward pragmatic ends.[24]