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The art of the impossible
Bookshelves are stuffed with volumes that expound the successes of the mind and the silicon chip. We expect science to tell us what can be done and what is to be done. Governments look to scientists to improve the quality of life and safeguard us from earlier 'improvements'. Futurologists see no limit to human inquiry, while social scientists see no end to the raft of problems it spawns. The contemplation by our media of science's future path is dominated by our expectations of great interventions: cracking the human genetic code, curing all our bodily ills, manipulating the very atoms of the material universe, and, ultimately, fabricating an intelligence that exceeds our own. Human progress looks more and more like a race to manipulate the world around us on all scales, great and small.
It would be easy to write such a scientific success story. But we have another tale to tell: one that tells not of the known but of the unknown; of things impossible; of limits and barriers which cannot be crossed. Perhaps this sounds a little perverse. Surely there is little enough to say about the unknown without dragging in theunknowable? But the impossible is a powerful and persistent notion. Unnoticed, its influence upon our history has been deep and wide; its place in our picture of what the Universe is like at its deepest levels is undeniable. But its positive role has escaped the critics' attention. Our goal is to uncover some of the limits of science: to see how our minds' awareness of the impossible gives us a new perspective on reality.
When we are young we think we know everything. But if we grow wiser as we grow older we will gradually discover that we know less than we thought. The poet W.H. Auden wrote of human development that
between the ages of twenty and forty we are engaged in the process of discovering who we are, which involves learning the difference between accidental limitations which it is our duty to outgrow and the necessary limitations of our nature beyond which we cannot trespass with impunity.
Our collective knowledge of the nuts and bolts of the Universe matures in a similar way. Some knowledge is simply the accumulation of more facts, broader theories, and better measurements by more powerful machines. Its rate of growth is always limited by costs and practicalities that we steadily overcome by attrition, little by little. But there is another form of knowledge. It is the awareness that there are limits to one's theories even when they are right. While the modest investigator might always suspect that there are things that will remain beyond our reach, this is not quite what we have in mind. There is a path of discovery that unveils limits that are an inevitable by-product of the knowing process. Discovering what they are is a vital part of understanding the Universe. This means that the investigation of the limits of our knowledge is more than a delineation of the boundaries of the territory that science can hope to discover. It becomes a crucial feature in our understanding of the nature of this collective activity of discovery that we call science: a paradoxical revelation that we can know what we cannot know. This is one of the most striking consequences of human consciousness.
There is an intriguing pattern to many areas of deep human inquiry. Observations of the world are made; patterns are discerned and described by mathematical formulae. The formulae predict more and more of what is seen, and our confidence in their explanatory and predictive power grows. Over a long period of time the formulae seem to be infallible: everything they predict is seen. Users of the magic formulae begin to argue that they will allow us to understand everything. The end of some branch of human inquiry seems to be in sight. Books start to be written, prizes begin to be awarded, and of the giving of popular expositions there is no end. But then something unexpected happens. It's not that the formulae are contradicted by Nature. It's not that something is seen which takes the formulae by surprise. Something much more unusual happens. The formulae fall victim of a form of civil war: they predict that there are things which they cannot predict, observations which cannot be made, statements whose truth they can neither affirm nor deny. The theory proves to be limited, not merely in its sphere of applicability, but to be self-limiting. Without ever revealing an internal inconsistency, or failing to account for something we have seen in the world, the theory produces a 'no-go' statement. We shall see that only unrealistically simple scientific theories avoid this fate. Logical descriptions of complex worlds contain within themselves the seeds of their own limitation. A world that was simple enough to be fully known would be too simple to contain conscious observers who might know it.
Of faces and games
I'm not young enough to know everything.
Complete knowledge is a tempting pie in the sky. Although it appears in some commentator's minds as the obvious goal of science, it is a concept largely unknown within the writings of contemporary science. It is the hallmark of many varieties of pseudo-science, just as it pervades countless ancient myths and legends about the origin and nature of the world. These stories leave nothing out: they have an answer for everything. They aim to banish the insecurity of ignorance and provide a complete interlinked picture of the world in which human beings play a meaningful role. They remove the worrying idea of the unknown. If you are at the mercy of the wind and the rain it helps to personify those unpredictable elements as the character traits of a storm god. Even today, many spurious attempts to explain the world around us still bear this hallmark. Horoscopes seek to create a spurious determinism that links our personalities to the orientations of the stars. Uncertainties about tomorrow can be hidden behind vague generalities about the future course of events. It is strange how many inhabitants of modern democracies feel no qualms about living under an astral dictatorship that would plan their every thought and action.
This desire for complete seamless explanation infests most examples of crank science. When somebody mails me their explanation of the architecture of the Universe derived from the geometry of the Great Pyramid, or the cipher of the Kabbalah, it will usually display a number of features: it will be entirely a work of explanation; there will be no predictions, no tests of its correctness; and nothing lies beyond its encompass. It is not the beginning of any research programme. Beyond refutation, it is always the last word.
This desire to link all things together is a deep human inclination. It is not a modern fashion that arrived with the word processor. Its most famous ancient manifestation is to be found in the work of the ancient Pythagorean sect who mingled mathematics with mysticism. They thought that number was a unifying principle in the Universe, so that anything that could be numbered was ultimately linked to other things with the same number. Numbers had meanings apart from their relationships with other numbers. Thus, musical harmony was linked to the motions of the heavenly bodies. The discovery that there were numbers that could not be represented by fractions precipitated a crisis so deep that these numbers had to be called 'irrational'. They appeared to lie beyond the complete arithmetic pattern of the Universe that the Pythagoreans had embroidered.
This unifying inclination of ours is a by-product of an important aspect of our intelligence. Indeed, it is one of the defining characteristics of our level of self-reflective intelligence. It allows us to organize knowledge into categories: to know vast numbers of thing by knowing rules and laws which apply in an infinite number of circumstances. We do not need to remember what the sum of every possible pair of numbers is: we need know only the principle of addition. The ability to seek and find common factors behind superficially dissimilar things is a prerequisite for memory and for learning from experience (rather than merely by experience). Some cultures have grown content with religious views of the world which are far less unified than others and have gods for every facet of life and Nature. In this sense, monotheistic faiths offer the most economical theological conception: by contrast, faiths with many disparate deities vying for influence seem less appealing.
All human experience is associated with some form of editing of the full account of reality ('we cannot bear too much reality'). Our senses prune the amount of information on offer. Our eyes are sensitive to a very narrow range of frequencies of light, our ears to a particular domain of sound levels and frequencies. If we gathered every last quantum of information about the world that impinged upon our senses they would be overwhelmed. Scarce genetic resources would be lopsidedly concentrated in information-gatherers at the expense of organs which could exploit a smaller quantity of information in order to escape from predators or to prey on sources of food. Complete environmental information would be like having a one-to-one scale map. For a map to be useful it must encapsulate and summarize the most important aspects of the terrain: it must compress information into abbreviated forms. Brains must be able to perform these abbreviations. This also requires an environment that is simple enough and displays enough order, to make this encapsulation possible over some dimensions of time and space.
Our minds do not merely gather information; they edit it and seek particular types of correlation. They have become efficient at extracting patterns in collections of information. When a pattern is recognized it enables the whole picture to be replaced by a briefer summary form which can be retrieved when required. These inclinations are helpful to us and expand our mental powers. We can retrieve the partial picture at other times and in different circumstances, imagine variations to it, extrapolate it, or just forget it. Often, great scientific achievements will be examples of one extraordinary individual's ability to reduce a complex mass of information to a single pattern. Nor does this inclination to abbreviate stop at the door of the laboratory. Beyond the scientific realm we might understand our penchant for religious and mystical explanations of experience as another application of this faculty for editing reality down to a few simple principles which make it seem under our control. All this gives rise to dichotomies. Our greatest scientific achievements spring from the most insightful and elegant reductions of the superficial complexities of Nature to reveal their underlying simplicities, while our greatest blunders often arise from the oversimplification of aspects of reality that subsequently prove to be far more complex than we realized.
Our penchant for completeness is closely associated with our liking for symmetry. We have a natural sensitivity for pattern and an appreciation of symmetry that quickly picks up subtle deviations from perfect symmetry. Our desire for a full and perfect description of the world owes much to this curious sensitivity. Where does it originate?
A powerful means of understanding why we possess many odd abilities is to recognize that our mental faculties evolved several million years ago in environments that were very different from those in which we now live. In that primitive environment certain sensitivities would tend to enhance the survival prospects of those that possessed them with respect to those who did not. Those attributes which made survival more probable would be the expression of some complex genetic cocktail with no predetermined purpose. Although one feature of an attribute might aid survival, there might be by-products of this attribute which showed up subsequently in all sorts of unexpected ways. Many of our aesthetic sensitivities have arisen in this indirect manner. Accordingly, we can identify good evolutionary reasons why we might be expected to have developed an acute appreciation for symmetry. If we look at the natural environment we see that lateral (left-right) symmetry is a very effective discriminator between living and non-living things in a crowded scene. You can tell when a living creature is looking at you. This sensitivity has a clear survival value. It enables you to recognize potential predators, mates, and meals. This biological source of our appreciation of symmetry is supported by the fact that our most acute sensitivity for symmetry is manifested in our appreciation of the human form, especially the face (Fig. 1). Symmetry of bodily form--especially that of the face--is our most common initial indicator of human beauty, and we go to enormous lengths to enhance it and protect it. In lower animals it is an important indicator of mates. In humans it has had all manner of by-products which influence our aesthetic appreciation and underlie our acute sensitivity to patterns, symmetry, and form. Remarkably, no computer has yet managed to reproduce our many levels of visual sensitivity to patterns.
This sensitivity means that deviations from symmetry are quickly identified and have a sophisticated interpretation all their own. Because they capture our attention so dramatically they are much used in (English) humour. Try the effect of the following classic deviation from the traditional anapaestic symmetry of the limerick form:
There was a young man of Milan
Whom rhymes they never would scan;
When asked why it was,
He said, 'It's because
I always try to cram as many words into the last line as ever I possibly can.'
A microcosm of our attitudes towards completeness can be found in the world of games. Simple games, like noughts and crosses, are entirely predictable. With a little thought you can devise a strategy that prevents you from ever losing, no matter who goes first and what moves your opponent makes. Draughts and chess (or Chinese chess) are games that are more satisfying because they lack this completely predictable completeness. The simplest game which could continue for ever is claimed to be Edward De Bono's L-Game. Each player has an L-shaped token which can be placed anywhere on the small board. After placing the L-piece, either one, two, or neither of the black spots may be placed on the empty squares. The aim of the game is to prevent your opponent from moving his L-shape on the next move. The starting positions and a typical winning configuration are shown in Fig. 1.2.
Some games with deceptively simple rules, like John Horton Conway's Game of Life, possess so many developments of great complexity that it is impossible to determine all the possible configurations that could arise. In fact, this game has been shown to share the same level of complexity as the whole of arithmetic. We might wonder whether our investigations of the natural world will eventually be completed in any sense. Perhaps all the laws of Nature might be found, even if all their outworkings might not be listable? Like the perennial noughts-and-crosses addict, would we then cease to be surprised by anything we found in the natural world? In later chapters we shall return many times to look at this question from a variety of different angles.
Those for whom all things are possible
With men this is impossible; but with God all things are possible.
The notion of the impossible has a history bound up with our religious desires. Most human cultures have displayed a desire to worship or acknowledge beings or spirits greater than themselves. These 'gods' are usually credited with superhuman powers: that is what distinguishes them from mortal men and women. Their powers may be exaggerated human ones, or powers that humans do not possess in any measure at all. In the most extreme case the gods may possess limitless powers which enable them to do anything at all and to know everything.
This deceptively simple idea is not without its problems. We can see that it is attractive for the adherents of a particular deity to believe in their god's limitless powers, if only to avoid subservience to the god next door. But looking a little deeper, we see that if their god's actions were limited in some way, then whatever, or whoever, was doing the limiting would have a greater claim to be in control of events than the god. If your god has no jurisdiction over the wind, then the wind has a justifiable claim to be a superior deity. Eventually, someone will appeal to the superior power of the wind.
Although a deity of limited powers has a credibility problem, one of limitless power seems to have far deeper problems of principle. How can there exist a Being for whom nothing is impossible? For whom 2 + 2 = 5; whose existence can be terminated; who is not bound by the laws of logic? Surely some things must be impossible or chaos and contradiction beckons? If a deity has defining characteristics then there must exist opposites of those attributes which define impossible actions for him or her. Few traditional religions now grapple with these hard questions, yet they are questions that clearly trouble many scientists. The late Heinz Pagels tells how this question was decisive in destroying his early belief in God:
When I was in high school I remember reflecting on what kind of being God could possibly be--I was curious...I also remember asking that if God was all-powerful, could he do things like change the laws of logic? If he could change the laws of logic, then he was a kind of lawless Being incomprehensible to the human mind. On the other hand if he couldn't change the laws of logic, he wasn't all-powerful. These alternatives left me dissatisfied...this 'teenage theology' left me with the feeling that either God was not subject to the laws of logic, in which case there was no point thinking rationally about God, or he was subject to the laws of logic, in which case he was not a very impressive God.
Some are content with the notion of a 'miracle', an event which defies the rules by which Nature operates (or, at least, of our experience of them), but none elevate violations of the laws of logic or mathematics to the same evidential status.
Ancient authorities tried to distinguish more finely between actions which were in character and those which were out of character, regarding the latter as logically impossible for a being with the attributes of deity. But these distinctions seem rather slippery to modern ears. Some apologists for the miraculous stress the incompleteness of our knowledge of what is possible in the Universe, and have sought to accommodate God's action in exceptions to the laws of Nature, while others have tried to explain it by our inability to determine the future course of chaotically sensitive situations.
If we look at a religious tradition like the Judaeo-Christian one, we find that God's ability to do the humanly 'impossible' is a defining characteristic. 'To believe only possibilities is not faith, but mere philosophy', as Thomas Browne argued back in the seventeenth century. This feature also serves to establish one of the defining differences between God and mankind: human limits are what fix the great gulf between God and humanity. Thus, when magicians and shamans arise they seek confirmation of their status by demonstrating apparently miraculous powers and by their ability to perform acts which are impossible for the rest of us. They endorse a view of the Universe in which there is a hierarchy of beings whose status rises as the limitations on their actions grow fewer and weaker.
Our religious traditions reveal that restrictions on human thoughts and actions are often imposed by the gods. These are not limits which our mortal nature prevents us surpassing: they are like the motorway speed limit rather than the law of gravity. They are presented as taboos that we ignore at our peril. A huge range of human cultures have taboos, whether it be on naming gods, visiting certain places, or counting their populations. Just as earthly rulers distinguish themselves from their subjects by the imposition of constraints upon their behaviour which are not of any obvious benefit to the rulers, except to impress their subjects, so it is imagined that the deity must follow similar practices. The habit of obedience is thought to be a valuable lesson for everyone to learn--a notion that any army sergeant-major will heartily endorse. Thus we see that the notion of impossibility has lodged itself effortlessly at the heart of our religious thinking in many different ways.
The forbidden fruit of the 'Tree of Knowledge of Good and Evil' in the book of Genesis is an interesting example because it entwines two notions that are often separated: forbidden actions and forbidden knowledge. Eating from the Tree of Knowledge was forbidden in order to prevent awareness of some new form of knowledge. The term 'forbidden fruit' has since become a byword for any sort of taboo on human actions.
It is quite common to encounter forbidden actions: our legal systems abound with them. Forbidden knowledge is a more controversial idea. All modern states have secrets and we keep some information concealed from certain people for various reasons--security, confidentiality, financial advantage, malice, surprise, and so on--but there are many who believe that there should be complete freedom of information whatever form it takes--as a fundamental human right, like the right to justice and education. This issue has run into controversy with the imposition of restrictions on the Internet and on the attitudes of some governments to the availability of simple encryption programs like PGP ('Pretty Good Privacy') which are beyond the means of any government's computer system to break. Alternatively, one can adopt the (British) compromise position that knowledge is not special. Like any human activity or possession (guns, cars, etc.) it may need to be subject to some democratically imposed restrictions for the common good (just as you wouldn't like your credit card PIN number published each day in the papers).
Religious taboos are usually framed in order to maintain the exclusivity of the gods. Some things must be impossible for everyone else if omnipotence is to have any advantage for its possessor. In some Islamic cultures there was a reluctance to produce perfect patterned mosaics because this would trespass into the realm of perfection that is the sole preserve of Allah. Thus, whereas in some religions there are things which humans cannot know because of their finiteness and mortality, in others there are things which they know how to do but must not do, for fear of offending the exclusivity of the gods.
Alan Cromer has argued that the great monotheistic faiths like Islam and Judaism created environments in which science found it hard to develop primarily because they were focused upon deities for whom there was no sense of impossibility:
Belief in impossibility is the starting point for logic, deductive mathematics, and natural science. It can originate only in a mind that has freed itself from belief in its own omnipotence.
By contrast, the presence of an omnipotent, interventionist being who is unrestricted by laws of Nature undermines faith in the consistency of Nature. A concept of impossibility seems to be a necessary prerequisite for a scientific understanding of the world. This is an interesting argument because it has also been claimed that monotheism provided an environment in which science could flourish because it gave credence to the idea of universal laws of Nature. The decrees of an omniscient deity gave rise to belief in laws imposed on things from outside which govern the workings of the world, in opposition to the idea that the things in the world behaved as they did because of their immanent properties. The distinction is significant. If every stone behaves in a manner dictated by its inward nature, or so as to produce harmony with other stones, then every stone should behave differently and there is little motivation to search for habitual behaviours shared by all moving stones. A feature of this position is that while it is consistent with the growth of abstract science and the concept of externally imposed laws of Nature, it does not ensure it. Although there is strong evidence from ancient China that the absence of a monotheistic view hindered the development of the mathematical sciences and led to a waning of faith in the underlying unity and rationality of Nature, it is not possible to demonstrate that Western science was an inevitable consequence of the Judaeo-Christian and Islamic cultures in the sense that it would not have developed in the absence of their monotheistic beliefs. It may well have been an unexpected by-product of a theistic world-view, but the aims and approaches to the world of these two cultures can be very different. Perhaps, as Oscar Wilde once remarked in a rare moment of seriousness, 'Religions die when they are proved true. Science is the record of dead religions.'
We began this section by introducing the familiar idea of a god who is omniscient: someone who knows everything. This possibility does not immediately ring alarm bells in our brains; it is plausible that such a being could exist. Yet, when it is probed more closely one can show that omniscience of this sort creates a logical paradox and must, by the standards of human reason, therefore be judged impossible or be qualified in some way. To see this consider this test statement:
THIS STATEMENT IS NOT KNOWN TO BE TRUE BY ANYONE.
Now consider the plight of our hypothetical Omniscient Being ('Big O'). Suppose first that this statement is true and Big O does not know it. Then Big O would not be omniscient. So, instead, suppose our statement is false. This means that someone must know the statement to be true; hence it must be true. So regardless of whether we assume at the outset that this statement is true or false, we are forced to conclude that it must be true! And therefore, since the statement is true, nobody (including Big O) can know that it is true. This shows that there must always be true statements that no being can know to be true. Hence there cannot be an Omniscient Being who knows all truths. Nor, by the same argument, could we or our future successors, ever attain such a state of omniscience. All that can be known is all that can be known, not all that is true.
As an aside, we note that the American political scientist, Stephen Brams, has carried out a fascinating analysis of many traditional theological questions relating to God's action in the world, for example the problem of suffering. Brams uses the methods of 'game theory', a branch of mathematics designed to ascertain whether there are optimal strategies for individuals who have different courses of action open to them. The word 'game' is used to describe any situation where two or more participants have a choice of strategies with associated costs and benefits. Brams sought to discover whether we could glean any evidence that the moral nature of the Universe reflects the optimal strategy of an omniscient being. The results were illuminating. Evil and suffering can be inevitable aspects of an optimal strategy to do good. It can turn out the deduction of an omniscient being's existence is logically undecidable if certain strategies are being adopted.
The limitations that this lack of omniscience ensures should not be seen solely in a negative light. Errors and inconsistencies play an important role in our learning process. We learn by our mistakes. If we encounter inconsistencies we re-evaluate the situation as a whole and re-examine the assumptions we have made. It is far from clear to what extent machine intelligence will emulate us in this respect. At some stage in the evolutionary process we began to develop the faculty of imagination. This enabled us to learn about the impossible as well as the possible. Our ability to understand the world thereby increased significantly in scope and speed. Remarkably, we are able to conceive of things that are impossible. Indeed, most of us live our daily lives confident that all manner of impossible things are not merely possible, but actual. Most of us have more interest in the possible than the impossible (this attitude is sometimes called 'pragmatism'); but some people take a greater interest in the impossible. Nor are the latter simply idealists or fantasists. Whole genres of fantastic literature and art have sprung from the challenges posed by linguistic and visual impossibilities.
A paradox is truth standing on its head to attract attention.
The word 'paradox' is a synthesis two Greek words, para, beyond, and doxos, belief. It has come to have a variety of meanings: something which appears contradictory but which is, in fact, true; something which appears true but which is, in fact, contradictory; or a harmless chain of deductions from a self-evident starting point which leads to a contradiction. Philosophers love paradox. Indeed, Bertrand Russell once remarked that the mark of good philosophy is to begin with a statement that is regarded as too obvious to be of interest and from it deduce a conclusion that no one will believe.
While some paradoxes may be trivial, others reflect profound problems about our ways of thinking and challenge us to re-evaluate them or so seek out unsuspected inconsistencies in the beliefs that we held to be self-evidently true. Anatol Rapoport, an international authority on strategic analysis--an arena where paradoxical results often result from innocuous beginnings--draws attention to the stimulating role that the recognition of paradox has played in many areas of human thinking:
Paradoxes have played a dramatic role in intellectual history, often foreshadowing revolutionary developments in science, mathematics, and logic. Whenever, in any discipline, we discover a problem that cannot be solved within the conceptual framework that supposedly should apply, we experience shock. The shock may compel us to discard the old framework and adopt a new one. It is to this process of intellectual molting that we owe the birth of many of the major ideas in mathematics and science. Zeno's paradox of Achilles and the tortoise gave birth to the idea of convergent infinite series. Antinomies (internal contradictions in mathematical logic) eventually blossomed into Godel's theorem. The paradoxical result of the Michelson-Morley experiment on the speed of light set the stage for the theory of relativity. The discovery of wave-particle duality of light forced a reexamination of deterministic causality, the very foundation of scientific philosophy, and led to quantum mechanics. The paradox of Maxwell's demon, which Leo Szilard first found a way to resolve in 1929, gave impetus more recently to the profound insight that the seemingly disparate concepts of information and entropy are intimately linked to each other.
Excerpted from Impossibility by John D. Barrow Copyright ©1999 by John D. Barrow. Excerpted by permission.
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1. The Art of the impossible