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So, in fact, all of these trajectories are simply parts of ellipses. We humans have gradually discovered many additional recurring shapes and patterns in nature, involving not only motion and gravity, but also electricity, magnetism, light, heat, chemistry, radioactivity and subatomic particles. These patterns are summarized by what we call our laws of physics.

Just like the shape of an ellipse, all these laws can be described using mathematical equations. The answer is 3, by placing them along the three edges emanating from a corner of your room. Where did that number 3 come sailing in from? We call this number the dimensionality of our space, but why are there three dimensions rather than four or two or 42? So what do we make of all these hints of mathematics in our physical world?

Most of my physics colleagues take it to mean that nature is for some reason described by mathematics, at least approximately, and leave it at that. I was quite fascinated by all these mathematical clues back in grad school. My starting assumption, the external reality hypothesis, states that there exists an external physical reality completely independent of us. But if we assume that reality exists independently of humans, then for a description to be complete, it must also be well-defined according to nonhuman entities — aliens or supercomputers, say — that lack any understanding of human concepts.

That brings us to the Mathematical Universe Hypothesis, which states that our external physical reality is a mathematical structure. Since the ball is made of elementary particles quarks and electrons , you could in principle describe its motion without making any reference to tennis balls:. That would be slightly inconvenient, however, because it would take you longer than the age of our universe to say it. It would also be redundant, since all the particles are stuck together and move as a single unit.

Although useful, such words are all optional baggage. All of this begs the question: Is it actually possible to find such a description of the external reality that involves no baggage? To answer this question, we need to take a closer look at mathematics. To a modern logician, a mathematical structure is precisely this: a set of abstract entities with relations between them.

This is in stark contrast to the way most of us first perceive mathematics — either as a sadistic form of punishment or as a bag of tricks for manipulating numbers. Modern mathematics is the formal study of structures that can be defined in a purely abstract way, without any human baggage. Think of mathematical symbols as mere labels without intrinsic meaning. The bottom line is that if you believe in an external reality independent of humans, then you must also believe that our physical reality is a mathematical structure.

Everything in our world is purely mathematical — including you. Above I described how we humans add baggage to our descriptions. Consider the sequence of chess moves that has become known as the Immortal Game, where white spectacularly sacrifices both rooks, a bishop and the queen to checkmate with the three remaining minor pieces. A game of chess can be represented a number of ways from left : a physical board, a drawn illustration, a computer simulation, and algebraic notation.

Each is a form of baggage representing the mathematically pure game. Chess involves abstract entities different chess pieces, different squares on the board and relations among them. For example, one relation that a piece may have to a square is that the former is standing on the latter. Similarly, a description of a chess position given purely verbally in English is equivalent to a description given purely verbally in Spanish.

The Immortal Game itself, percent pure, with no additives. The Mathematical Universe Hypothesis implies that we live in a relational reality, in the sense that the properties of the world around us stem not from properties of its ultimate building blocks, but from the relations among these building blocks. This crazy-sounding belief of mine that our physical world not only is described by mathematics, but that it is mathematics, makes us self-aware parts of a giant mathematical object.

Ultimately, this demotes familiar notions such as randomness, complexity and even change to the status of illusions; it also implies a new and ultimate collection of parallel universes so vast and exotic that all the above-mentioned bizarreness pales in comparison, forcing us to relinquish many of our most deeply ingrained notions of reality. When we drive language as far as we can toward the pole of precision and definitive, yes-or-no certainty, we arrive at formalisms such as mathematics, grammar, and logic.

But this does not prove that it corresponds to nature. Experience alone can decide on truth. Pure logical thinking cannot yield us any knowledge of the empirical world; all knowledge of reality starts from experience and ends in it. Propositions arrived at by purely logical means are completely empty as regards reality. But we still have to do the thinking, and we cannot bring the world into this thinking while remaining solely within the self-contained and reassuring purity of the templates.

The world breaks every fixed template into which we try to pour it. Referring back to sentence diagrams: if we can substitute one adjective for another without affecting the correctness of our structure, then we have to acknowledge that the diagram fails us badly as an adequate explication of our speech; it cannot distinguish between any of our meanings. Words hopelessly overflow the expressive power of the diagram. Despite the fact that purely mathematical thought and thought retaining observational content have very different character, they are intimately woven together by the scientist, even if the weaving does not often receive critical attention.

This is both necessary and proper. At least in the case of sentence diagrams we still have the meaningful text alongside the abstract grammatical structure indicated by the lines of the diagram. We can refer back to this text and relate our abstract construction to it at any time. We can re-enflesh the formal skeleton. Or, alternatively, we feed on feelings awakened by nature, yet feelings that are uninstructed by any disciplined grasp of what nature is really saying.

It would be healthier if we could begin questioning our scientific inheritance without losing ourselves in romanticism. It is hardly impertinent to point out that if, in the interest of precision, we narrow our technical language down to an empty formalism, then we are not discovering the world to be meaningless; we are insisting that it be meaningless.

There can be discomfort and threat in any confrontation with a speaking presence, and perhaps we should open ourselves to the possibility that much of our satisfaction in the unqualified rigor and precision of our science is really the satisfaction of curling up within the secure refuge of speechless quantity and logic, without having to venture too far out into the complex, soul-gripping presentations of the phenomenal world.

T he conflict between the official banishment of qualitative language from hard science on the one hand, and the inevitable reliance upon it on the other, has led to a strange sort of schizophrenia. I will offer illustrations in a moment. But to see what is going on here, first consider a familiar case. At a time when scientists were learning to observe and measure very high velocities, Einstein was led to the startling and unexpected theory of relativity. Relativity did not so much negate Newton as confirm his results and extend them to cover more extreme conditions.

And this is true! The revolution becomes apparent as soon as we try to hear the meanings that alone enable our grammatical refinements to speak of the world. Physicist David Bohm reminds us that. Likewise, referring to relativistic effects upon mass, the late physicist Richard Feynman writes:. Our entire picture of the world has to be altered even though the mass changes only by a little bit. Emphasis in original. But it leaves no room for anything else, either. The reductionist worldview is chilling and impersonal.

It has to be accepted as it is, not because we like it, but because that is the way the world works. O nce you have sacrificed meaning in order to arrive at your well-behaved grammatical abstractions, there is no way to recover the lost meaning from the abstractions alone.

Such speculation leads the Scientific American copy-writer in the familiar direction taken by so many scientists when they try to explain themselves to a popular audience — namely, toward language that is almost mystical. Their divergent speculations would make the most levitated medieval metaphysician blush. These speculations go far beyond parallel universes and tend to arise whenever researchers try to explain what sort of world their equations are about. Can time flow backward?

Is mathematics an effective way to describe the world?

Are there wormholes that take a shortcut through spacetime, linking two different places and times? Is there a shadow universe sharing gravity, but no other forces, with our own universe? Does observation create reality? Does consciousness create reality? The ground under our feet becomes even less stable when we consider how even the most basic terms of routine scientific explanation are more or less blank.

This is just one example of the general fact that there is not enough reality in a formal grammar, or in a formalism of any sort, to constrain our understanding of the content expressing itself through the formalism. And this science, dominated in its meaningful aspects by untethered human fancy, is the same science, so we are continually told, that has displaced the human being from his cherished place at the center of the world.

T here is, however, one reality principle in the hard sciences, and it rules with a vengeance. It is found in the uncompromising and perfectly healthy, in its place demand for devices that actually work. What the researcher proposes does not become a part of science until it leads to an experimental apparatus that suffers predictable change under a specified set of circumstances. This technological imperative, with its useful and striking consequences for our daily life, accounts for much of the popular conviction that science must have succeeded in connecting us to reality.

And so it must in one way or another. Our science brings us very real manipulative skills. But the skills enabling us to manipulate a thing are not necessarily the skills yielding deep insight into its nature. In fact, in a world of speech and expression consider your relations to family and friends , manipulation tends to work directly against understanding.

In concerning ourselves with the mechanistic logic we can lay bare in an object, we are throwing a veil over its distinctive expressive character. The following reflection may help to clarify the point. If you wanted to create a manageable, bounded, relatively self-contained realm embodying your conviction that the world is driven and controlled by a kind of formal necessity — by a pure structure of logic — you could hardly do better than to invent the computer.

The properties of the materials that implement the computation are hidden by careful engineering. This physical nature begins to seem irrelevant. This machine, with its externally imposed formal purity unsullied by the peculiarities of its material embodiment, a machine whose admired logic gives us virtually no understanding of the physical device itself, has become our reigning model for understanding the physical world. Whatever the wonders we have produced within the closed system of technology, they do not testify to the disciplining of our understanding by physical reality except in a highly impoverished way.

The magic of the digital machine is that by squinting at it in just the right way, we can drop the material device from view altogether and see only the clean, universal, eternal pattern of lawfully articulated logical bits that we ourselves have impressed upon the machine. This logic certainly does not picture for us the inherent lawfulness of copper, silicon, glass, and all the rest.

Despite this, we are ever more inclined to view the natural world through the mental grid or chain-link fence constituted by our logic-machine ideal, and we thereby reinforce our impossible desire for a universal grammar of nature that somehow explains and determines everything that happens. T he displacement of meaning by our grammatical fixation helps us to understand the curious ambiguity in our modern sense of alienation from the world.

On the one hand, we imagine a kind of ironclad necessity imposed by universal physical laws. Almost the opposite: the typical human complaint in the scientific era has been one of meaninglessness , which is a kind of hopeless non-determinism. The scientific account of the world lacks enough significant order, enough pattern and coherence of the speaking sort, enough sense and intention — in sum, enough textual meaning — to provide a context for our own meaningful existence. The problem is not so much that we are cogs in some inexorable machine suborning us to its own purposes, as that our science would allow this machine no purpose at all.

And so we become lost atoms moving senselessly in the void. We are not predetermined but undetermined, not fixed but aimless.

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The more the universe seems comprehensible, the more it also seems pointless. Yes, the more we reduce our comprehension of the universe to a mere grammar, the more it seems pointless. When we try, we end up supplying our own content, however crude and unrecognized. As for those who do look at the world, they may see elements of farce or hostility in limited contexts, but they certainly see a great deal more.

T he emptiness of scientific language, just so far as it fulfills the reigning quantitative and logical ideal, is scarcely open to dispute. It has been recognized, as we have seen, by prominent scientists and philosophers. If you still want to declare the world cold and impersonal, indifferent to human hopes and feelings, relentless and implacable in its mindless obedience to physical necessity — well, that is certainly your privilege. But the mathematical precision, certainty, determinacy, and universality of scientific laws is simply not an adequate or convincing ground for this contention.

At least, it remains unavailable until one elucidates a path from empty formalism to a revelatory description of the world, and then demonstrates what the precision, certainty, determinacy, and universality of modern science mean for the enfleshed world. Amazingly, this paradox has been little considered by working scientists, especially in the hard sciences.

One physicist who at least refers to the problem is Richard Feynman. Alluding to the same theme we have traced here, he reminds the mathematically inclined physicist of the necessity for a step beyond formalism toward real-world meaning:. Mathematicians are only dealing with the structure of reasoning, and they do not really care what they are talking about.

They do not even need to know what they are talking about, or, as they themselves say, whether what they say is true But the physicist has meaning to all his phrases. That is a very important thing that a lot of people who come to physics by way of mathematics do not appreciate It is necessary at the end to translate what you have figured out into English, into the world, into the blocks of copper and glass that you are going to do the experiments with. But this should not be taken simplistically.

It cannot be merely a matter of translating from the language of pure mathematics to the meaning of the physicist because, as Feynman has just acknowledged, there is nothing we can say mathematics is about — no content available for translation. This content finds its way into our thinking by processes distinct from the abstract ruminations of the pure mathematician.

The mathematics derives from the content, not the other way around. A formalism itself cannot direct us to any specific content capable of embodying the formalism. How then do we find the content of our science? The ease with which this question has been ignored stands as one of the most stunning features of our science-committed culture.

Science historian E. Yet the fact is that we cannot even see a thing, let alone determine its relations, without taking it to be a certain kind of thing possessed of its own characteristic qualities. The question is only whether we will accept uncritically our half-conscious assumptions about the substantial nature of things — as when, for example, we imagine subatomic particles to be very tiny bits of the qualitatively familiar stuff we deal with every day an imagination that has caused no end of grief to physicists — or whether we will instead raise these notions to full consciousness, where we can subject them to proper criticism.

T ry sitting outdoors in a natural landscape for half an hour. After quieting yourself and becoming as receptive as possible to the surrounding world, consider this: Is there any content here beside the purely qualitative? How many of us, during years or decades of creative work, will put such a problem to ourselves in this direct, observational, scientifically sanctioned way, as opposed to thinking about the problem in our studies or laboratories, with our thought mediated by a vast network of mental abstractions?

Now try subtracting from the content of your observation everything qualitative. In the case of the tree over there, remove the green of the foliage, the gray of the bark, the smell of sap, the rustling of leaves in the breeze, the felt hardness of the trunk Nothing at all. You do not even have geometric form, since without light and color there is no visible form, and without the different qualities of touch there is no felt form.

Form is not something independent that we proceed to flesh out with qualities; it subsists in nothing but the qualities themselves. You may want to say that the quantities we abstract from our qualitative experience of the world point us toward a more substantial reality hidden behind the world of our perceptions.

Is mathematics an effective way to describe the world?

But unless you can say something about this hidden reality — unless you can characterize it, giving your quantitative constructions some sort of content — where is your science? And how will you characterize this content without appealing to qualities? A true scientist would investigate the qualitative world in its own terms.

These terms are not particularly obscure; they simply refuse to conform to our preferred scientific stance. An elementary quality such as red proves maddeningly elusive when our aim is to pin it down.

My red shirt turns out to be a different color depending on the lighting and on the other colors around it, as well as on the state of my own eyes. Similarly with the qualitative nature of an entire complex organism: we recognize a single species-nature in a lowland spruce tree and an alpine one, but this common nature comes to dramatically different expression in the two cases. They give us a more or less satisfying, a more or less penetrating, insight into and feel for what a phenomenon is like.

When we are reckoning with qualities, questions of similarity are more central than questions of identity. Often, however, we can read very little of this character in a frozen snapshot. What they are is their inner movement, their manner of exchange and mutual interaction, so that we can catch them only in flight — by moving in a like manner along with them.

They leave behind every effort to grasp them and pin them down. A sculptor of stone succeeds only by suggesting movement. Even in depicting a massive rock as a rock we must somehow capture a movement of profound rest, an ageless silence that is itself speech. We can certainly learn to know qualities. However, our inner activity in taking into ourselves a particular quality involves much more than the play of abstractions over the surface convolutions of our brains.

We experience qualities with our whole being, discovering, for example, that this color has something cool about it, that one something aggressive, and the other one something calming — characteristics of the sort that great artists have always had an ability to work with. What melody of its own is this phenomenon picking out upon the mathematically tuned world-lyre? The two usages are closely intertwined.

The way we reduce the world to atomic things without qualities is by reducing our descriptive language to the atomic terms of logic without meaning. That is, we can obscure the qualitative character of the world only by obscuring the meaningful character of our words. But we never fully succeed in this. The world remains word-like because it is full of the meanings of language, just as our words remain world-like because they are full of the qualities of the world. In the world, our consciousness meets something like its own activity, something akin to its own nature.

With our wide-ranging potential for conscious experience, we are ourselves expressions of the cosmos. Is it surprising, then, that we should be able to give conscious expression to what speaks in the world? In some scientific quarters, such a thought is seen as outrageous, while at the same time some of those most envied of scientists, the physicists, speak casually of consciousness as in one way or another fundamental to the cosmos.

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As Sir Arthur Eddington wrote in It is a knowledge of structural form, and not knowledge of content. All through the physical world runs that unknown content, which must surely be the stuff of our consciousness. Modern science is generally unwilling to admit of consciousness in its understanding of nature — an odd fact when you consider how many authorities in different fields loudly disavow the Cartesian diremption of matter from mind. The situation becomes more understandable only when we realize how thoroughly Cartesian these authorities remain: they take their stand firmly astride the fractured Cartesian bedrock, and then hope only to make one side of the divide disappear by reducing it to the terms of the other.

A real solution will be found only when we go back and refuse the split altogether, finding another way forward. And this way will include the recognition that the world has a word-like character. Only in language do we find the marriage of inner and outer in a way that overcomes all the conundrums of the mind-body dichotomy. But appreciating this solution can require agonizingly hard work when you have been raised, as nearly everyone in our culture has, upon Cartesian habits of thought.

My own path away from these habits was blazed by the philologist and historian of meaning, Owen Barfield. The prevailing refusal to set aside our Cartesian blinders can hardly be disputed. When we analyze sound — whether of a volcano or a musical performance — solely in terms of air waves, our terms are, strangely enough, as fully available to a deaf person as to someone with good hearing a point once made by the German physicist and educator, Martin Wagenschein. In fact, the ideal of rigor within the hard sciences generally aims, rather impossibly, for the use of terms understandable by someone who has no conscious perception of the world whatsoever.

Such a person, if he actually existed, would have no world in need of understanding. If, however, we do have a world to understand, it is a world whose nature is to present itself within our consciousness. W e gain a kind of unqualified crystalline clarity by filtering our perceptions of the world through a web of logically precise abstractions.

Even space and time become, through analysis, a collection of discrete points or discrete instants of time. But the quantifiable crystalline clarity we thereby achieve belongs to our perceptual filter and not to the world. Dazzled by this clarity and fixity, we become blind to context. Go out again into a natural setting, sit down, and spend a while taking in everything you can see, hear, feel, and smell. Then ask yourself: does this world, in any meaningful sense, consist of discrete points or instants of time?

You will be hard put to find any justification in observation for these abstract notions. The world and its events present themselves — stunningly, when you compare your actual experience to the various theoretical ways of thinking about the world — as one seamless whole. Points and instants flow into each other, participate in each other, and cannot be clearly separated from each other. Pick any visible object — a tree, say — and try to isolate it cleanly and without ambiguity from everything around it. It cannot be done. Again, this is hardly controversial. The entire discipline of ecology was founded upon the awareness that organisms are an expression of their environment, and the environment is an expression of its organisms.

But it is not only organisms that require a contextual understanding. Every entity, process, and law described by physics gains its real content only by means of its context. We may be able to discern in a process a certain quantitative lawfulness that is invariant from one context to another — because the quantities have had all context and phenomenal content stripped from them. But while this absolute sort of lawfulness may be abstractable from the physical process, the observable content itself is never invariant or subject to necessity in the way we take our universal laws to be.

But in the open atmosphere the air usually gets colder with height. You can understand the difference only by considering the two different contexts, one of which limits the upward movement of air, while the other does not. The different effects extend even to the question of whether your bones will be subject to a dangerous loss of mass — something of concern to long-term inhabitants of orbiting space stations.

These examples will seem either trivial or profound, depending on our ability to discern the subtle distinction they require. All you need to do is to change the context, and a different behavior results. This is true of any law presuming to specify, in unqualified terms, what real things will actually do. Such laws will be thoroughly contextual, so that in different contexts the phenomena will bring their lawfulness to a different expression.

Every particle of matter in the universe attracts every other particle with a force directly proportional to the product of the masses of the particles and inversely proportional to the square of the distance between them. Real bodies moving according to this grammar may approach each other, spiral around each other, or move directly away from each other. The actual behavior of things in the world is always an expression of context. The coherence to expect is more like the coherence of a picture or image than of isolated entities. It is more like the coherence of a sentence in a story which plays into and colors all the words around it than the coherence of a logical proposition.

Qualities already imply context. Discrete, qualitatively featureless particles can exist only in nameless, side-by-side aggregation; they can never give us the kind of contextual unity that plays into, modifies, and binds together the various elements of the context. In order for there to be a true context, something must reach across and penetrate all the elements, shaping each of them to the character of the whole. But the many conundrums into which this inquiry leads us — conundrums widely recognized whenever scientists temporarily extricate themselves from their dense mesh of theoretical abstractions, face nature herself, and try to understand what they have been talking about — will remain insuperable obstacles to progress until we can begin to explore what this new, contextual, and more imaginal form of understanding might require of us.

If you add in all these other laws, then, at least in principle, you will understand everything that happens. What can you point to that escapes this all-encompassing lawfulness? The short answer is a simple reminder: I have not been suggesting that anything needs to violate the universal laws of physics — no more than a meaningful sentence needs to violate the rules of grammar.

Where we have such content, it speaks forth its own coherent meaning, and while this meaning may always respect an underlying formal grammar, it can never be reduced to such a grammar. But this short answer requires expansion. Think of the movements of the heavens, which perhaps are what most naturally come to mind when we imagine the determinism of physical law.

Perfectly timed eclipses, precisely targeted space probes, the regular rhythms of day, month, and year — certainly these are real phenomena, and we commonly manage to predict them with extraordinary accuracy. Could any phenomena be more fully determined by mathematical law than these? Well, again, the point is not that mathematical laws must be violated. Nor is it that there must be some element of randomness or wild, lawless disorder in the cosmos. Contextual coherence, after all, is not randomness and disorder.

But neither can its significance be expressed in purely quantitative or formal terms. It is a different — a meaningful — kind of order. These bodies have become in our minds little more than reifications of their governing equations.

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The distinctive nature of the bodies doing the movement has completely dropped from view. You need only reflect upon all the scientific disciplines arising from our experience on earth to realize that, when we think of the moon or sun as mere points in motion, we have blocked from our view virtually all the reality of these bodies — all the reality we would have to account for if somehow we were adapted and sensitive to their alien conditions.

If you ignore everything except points in motion — everything constituting the expressive reality of a phenomenon — then it is not the phenomenon you are describing. You are simply using an image say, of the moon as a token to stand in your imagination for the lawful grammar that, as we have already recognized, can be abstracted from the physical world. Earlier generations spoke of various influences streaming in from the heavens, and of the humanly relevant dispositions of celestial bodies or beings , and of the lunatic or mercurial nature of people or events, and of the heavens declaring the glory of God.

E ven if we start with the words we commonly use in stating our most rigorously quantitative physical laws, and if we take these words as really meaning something, we are immediately carried toward a richly qualitative world. Can we gain an adequate scientific understanding of gravity except by referring to the willful use of our muscles or our experience of pressure?

True, many scientists will react initially to the question by citing the purely objective relationships of moving masses — relationships given in strictly mathematical terms. But the word relationship here turns out to be rather pregnant. It conceals — so long as we are willing to avert our gaze — what sort of connection between things we really have in mind. Objects changing their positions in space may give us certain mathematically describable relationships, but so, too, can points on a piece of graph paper.

No one takes these points to be exerting a physical force upon each other. Neither could we think of planets as exerting a force upon each other unless we had an independent concept of force. As the graph paper illustrates, the mathematical relationships alone do not give us such a concept. Think about it all you wish, but a force is something real in the world, and you will never find a concept for it except through your own experience of the world. This experience, like all our experience, occurs within consciousness.

Such possibilities may be crazy or not, but confirming or refuting them would require a kind of devotion to our experience of the world that we long ago lost interest in sustaining. In any case, without some sort of experience of force within the inner domain of our own consciousness, we have no meaning for the scientific concept of force.

But this shape is not taken to be a mere distribution of mathematical points within a spatial grid; rather, it is the overall expressive gesture of the thing. This older conception of cause points us toward the qualitative form or meaningful patterns, the governing unity, according to which phenomena unfold rather as a spoken sentence progressively unfolds to express an antecedent governing idea — an idea that informs and transforms the individual words, shaping them to itself.

The usual notion of cause and effect in science can be understood as a more or less distant approximation of formal causation.

We shift from imaginal thinking to abstraction, from recognition of qualitative expression and the mutual interpenetration of elements to the search for isolated, well-defined parts. We redeem the approximations by realizing that they are approximations and by allowing them to clarify details which we then enliven by bringing them back into qualitative connection with the meaningful whole.

Goldstein looked at the various ways we analyze organisms into rule-based, mechanical parts and then try to reconstruct the whole from these parts. It never works. Goldstein showed that the reflex is an artifact of our own stance as researchers, whereby we conceptually and experimentally isolate one part of an organism, cutting the part off from its whole. Moreover, he finds that higher organisms, including human beings, are much more likely to show approximations of reflexes, because it is we who can allow parts of ourselves to become isolated and de-centered.