Jabberwocky. Or: Grand Unified Theory of Uncertainty???

Jabberwocky, Lewis Carroll's whimsical nonsense poem, uses made-up words to create an atmosphere and to tell a story. "Billig", "frumious", "vorpal" and "uffish" have no lexical meaning, but they could have. The poem demonstrates that the realm of imagination exceeds the bounds of reality just as the set of possible words and meanings exceeds its real lexical counterpart.

Uncertainty thrives in the realm of imagination, incongruity, and contradiction. Uncertainty falls in the realm of science fiction as much as in the realm of science. People have struggled with uncertainty for ages and many theories of uncertainty have appeared over time. How many uncertainty theories do we need? Lots, and forever. Would we say that of physics? No, at least not forever.

Can you think inconsistent, incoherent, or erroneous thoughts? I can. (I do it quite often, usually without noticing.) For those unaccustomed to thinking incongruous thoughts, and who need a bit of help to get started, I can recommend thinking of "two meanings packed into one word like a portmanteau," like 'fuming' and 'furious' to get 'frumious' or 'snake' and 'shark' to get 'snark'.

Portmanteau words are a start. Our task now is portmanteau thoughts. Take for instance the idea of a 'thingk':

When I think a thing I've thought,
I have often felt I ought
To call this thing I think a "Thingk",
Which ought to save a lot of ink.

The participle is written "thingking",
(Which is where we save on inking,)
Because "thingking" says in just one word:
"Thinking of a thought thing." Absurd!

All this shows high-power abstraction.
(That highly touted human contraption.)
Using symbols with subtle feint,
To stand for something which they ain't.

Now that wasn't difficult: two thoughts at once. Now let those thoughts be contradictory. To use a prosaic example: thinking the unthinkable, which I suppose is 'unthingkable'. There! You did it. You are on your way to a rich and full life of thinking incongruities, fallacies and contradictions. We can hold in our minds thoughts of 4-sided triangles, parallel lines that intersect, and endless other seeming impossibilities from super-girls like Pippi Longstockings to life on Mars (some of which may actually be true, or at least possible).

Scientists, logicians, and saints are in the business of dispelling all such incongruities, errors and contradictions. Banishing inconsistency is possible in science because (or if) there is only one coherent world. Belief in one coherent world and one grand unified theory is the modern secular version of the ancient monotheistic intuition of one universal God (in which saints tend to believe). Uncertainty thrives in the realm in which scientists and saints have not yet completed their tasks (perhaps because they are incompletable). For instance, we must entertain a wide range of conflicting conceptions when we do not yet know how (or whether) quantum mechanics can be reconciled with general relativity, or Pippi's strength reconciled with the limitations of physiology. As Henry Adams wrote:

"Images are not arguments, rarely even lead to proof, but the mind craves them, and, of late more than ever, the keenest experimenters find twenty images better than one, especially if contradictory; since the human mind has already learned to deal in contradictions."

The very idea of a rigorously logical theory of uncertainty is startling and implausible because the realm of the uncertain is inherently incoherent and contradictory. Indeed, the first uncertainty theory - probability - emerged many centuries after the invention of the axiomatic method in mathematics. Today we have many theories of uncertainty: probability, imprecise probability, information theory, generalized information theory, fuzzy logic, Dempster-Shafer theory, info-gap theory, and more (the list is a bit uncertain). Why such a long and diverse list? It seems that in constructing a logically consistent theory of the logically inconsistent domain of uncertainty, one cannot capture the whole beast all at once (though I'm uncertain about this).

A theory, in order to be scientific, must exclude something. A scientific theory makes statements such as "This happens; that doesn't happen." Karl Popper explained that a scientific theory must contain statements that are at risk of being wrong, statements that could be falsified. Deborah Mayo demonstrated how science grows by discovering and recovering from error.

The realm of uncertainty contains contradictions (ostensible or real) such as the pair of statements: "Nine year old girls can lift horses" and "Muscle fiber generates tension through the action of actin and myosin cross-bridge cycling". A logically consistent theory of uncertainty can handle improbabilities, as can scientific theories like quantum mechanics. But a logical theory cannot encompass outright contradictions. Science investigates a domain: the natural and physical worlds. Those worlds, by virtue of their existence, are perhaps coherent in a way that can be reflected in a unified logical theory. Theories of uncertainty are directed at a larger domain: the natural and physical worlds and all imaginable (and unimaginable) other worlds. That larger domain is definitely not coherent, and a unified logical theory would seem to be unattainable. Hence many theories of uncertainty are needed.

Scientific theories are good to have, and we do well to encourage the scientists. But it is a mistake to think that the scientific paradigm is suitable to all domains, in particular, to the study of uncertainty. Logic is a powerful tool and the axiomatic method assures the logical consistency of a theory. For instance, Leonard Savage argued that personal probability is a "code of consistency" for choosing one's behavior. Jim March compares the rigorous logic of mathematical theories of decision to strict religious morality. Consistency between values and actions is commendable says March, but he notes that one sometimes needs to deviate from perfect morality. While "[s]tandard notions of intelligent choice are theories of strict morality ... saints are a luxury to be encouraged only in small numbers." Logical consistency is a merit of any single theory, including a theory of uncertainty. However, insisting that the same logical consistency apply over the entire domain of uncertainty is like asking reality and saintliness to make peace.

Picking a Theory is Like Building a Boat at Sea

"We are like sailors who on the open sea must reconstruct their ship

 but are never able to start afresh from the bottom." 

Otto Neurath's analogy in the words of Willard V. Quine

Engineers, economists, social planners, security strategists, and others base their plans and decisions on theories. They often argue long and hard over which theory to use. Is it ever right to use a theory that we know is empirically wrong, especially if a true (or truer) theory is available? Why is it so difficult to pick a theory?

Let's consider two introductory examples.

You are an engineer designing a robot. You must calculate the forces needed to achieve specified motions of the robotic arms. You can base these calculations on either of two theories. One theory assumes that an object comes to rest unless a force acts upon it. Let's call this axiom A. The other theory assumes that an object moves at constant speed unless a force acts upon it. Let's call this axiom G. Axiom A agrees with observation: Nothing moves continuously without the exertion of force; an object will come to rest unless you keep pushing it. Axiom G contradicts all observation; no experiment illustrates the perpetual motion postulated by the axiom. If all else is the same, which theory should you choose?

Axiom A is Aristotle's law of inertia, which contributed little to the development of mechanical dynamics. Axiom G is Galileo's law of inertia: one of the most fruitful scientific ideas of all time. Why is an undemonstrable assertion - axiom G - a good starting point for a theory?

Consider another example.

You are an economist designing a market-based policy to induce firms to reduce pollution. You will use an economic theory to choose between policies. One theory assumes that firms face pure competition, meaning that no single firm can influence market prices. Another theory provides agent-based game-theoretic characterization of how firms interact (without colluding) by observing and responding to price behavior of other firms and of consumers.

Pure competition is a stylized idealization (like axiom G). Game theory is much more realistic (like axiom A), but may obscure essential patterns in its massive detail. Which theory should you use?

We will not address the question of how to choose a theory upon which to base a decision. We will focus on the question: why is theory selection so difficult? We will discuss four trade offs.

"Thanks to the negation sign, there are as many truths as falsehoods;

we just can't always be sure which are which." Willard V. Quine

The tension between right and right. The number of possible theories is infinite, and sometimes it's hard to separate the wheat from the chaff, as suggested by the quote from Quine. As an example, I have a book called A Modern Guide to Macroeconomics: An Introduction to Competing Schools of Thought by Snowdon, Vane and Wynarczyk. It's a wonderful overview of about a dozen theories developed by leading economic scholars, many of them Nobel Prize Laureates. The theories are all fundamentally different. They use different axioms and concepts and they compete for adoption by economists. These theories have been studied and tested upside down and backwards. However, economic processes are very complex and variable, and the various theories succeed in different ways or in different situations, so the jury is still out. The choice of a theory is no simple matter because many different theories can all seem right in one way or another.

"The fox knows many things, but the hedgehog knows one big thing." Archilochus

The fox-hedgehog tension. This aphorism by Archilochus metaphorically describes two types of theories (and two types of people). Fox-like theories are comprehensive and include all relevant aspects of the problem. Hedgehog-like theories, in contrast, skip the details and focus on essentials. Axiom A is fox-like because the complications of friction are acknowledged from the start. Axiom G is hedgehog-like because inertial resistance to change is acknowledged but the complications of friction are left for later. It is difficult to choose between these types of theories because it is difficult to balance comprehensiveness against essentialism. On the one hand, all relevant aspects of the problem should be considered. On the other hand, don't get bogged down in endless details. This fox-hedgehog tension can be managed by weighing the context, goals and implications of the decision. We won't expand on this idea since we're not considering how to choose a theory; we're only examining why it's a difficult choice. However, the idea of resolving this tension by goal-directed choice motivates the third tension.

"Beyond this island of meanings which in their own nature are true or false

lies the ocean of meanings to which truth and falsity are irrelevant." John Dewey

The truth-meaning tension. Theories are collections of statements like axioms A and G in our first example. Statements carry meaning, and statements can be either true or false. Truth and meaning are different. For instance, "Archilochus was a Japanese belly dancer" has meaning, but is not true. The quote from Dewey expresses the idea that "meaning" is a broader description of statements than "truth". All true statements mean something, but not all meaningful statements are true. That does not imply, however, that all untrue meaningful statements are false, as we will see.

We know the meanings of words and sentences from experience with language and life. A child learns the meanings of words - chair, mom, love, good, bad - by experience. Meanings are learned by pointing - this is a chair - and also by experiencing what it means to love or to be good or bad.

Truth is a different concept. John Dewey wrote that

"truths are but one class of meanings, namely, those in which a claim to verifiability by their consequences is an intrinsic part of their meaning. Beyond this island of meanings which in their own nature are true or false lies the ocean of meanings to which truth and falsity are irrelevant. We do not inquire whether Greek civilization was true or false, but we are immensely concerned to penetrate its meaning."

A true statement, in Dewey's sense, is one that can be confirmed by experience. Many statements are meaningful, even important and useful, but neither true nor false in this experimental sense. Axiom G is an example.

Our quest is to understand why the selection of a theory is difficult. Part of the challenge derives from the tension between meaning and truth. We select a theory for use in formulating and evaluating a plan or decision. The decision has implications: what would it mean to do this rather than that? Hence it is important that the meaning of the theory fit the context of the decision. Indeed, hedgehogs would say that getting the meaning and implication right is the essence of good decision making.

But what if a relevantly meaningful theory is unprovable or even false? Should we use a theory that is meaningful but not verifiable by experience? Should we use a meaningful theory that is even wrong? This quandary is related to the fox-hedgehog tension because the fox's theory is so full of true statements that its meaning may be obscured, while the hedgehog's bare-bones theory has clear relevance to the decision to be made, but may be either false or too idealized to be tested.

Galileo's axiom of inertia is an idealization that is unsupported by experience because friction can never be avoided. Axiom G assumes conditions that cannot be realized so the axiom can never be tested. Likewise, pure competition is an idealization that is rarely if ever encountered in practice. But these theories capture the essence of many situations. In practical terms, what it means to get the robotic arm from here to there is to apply net forces that overcome Galilean inertia. But actually designing a robot requires considering details of dissipative forces like friction. What it means to be a small business is that the market price of your product is beyond your control. But actually running a business requires following and reacting to prices in the store next door.

It is difficult to choose between a relevantly meaningful but unverifiable theory, and a true theory that is perhaps not quite what we mean.

The knowledge-ignorance tension. Recall that we are discussing theories in the service of decision-making by engineers, social scientists and others. A theory should facilitate the use of our knowledge and understanding. However, in some situations our ignorance is vast and our knowledge will grow. Hence a theory should also account for ignorance and be able to accommodate new knowledge.

Let's take an example from theories of decision. The independence axiom is fundamental in various decision theories, for instance in von Neumann-Morgenstern expected utility theory. It says that one's choices should be independent of irrelevant alternatives. Suppose you are offered the dinner choice between chicken and fish, and you choose chicken. The server returns a few minutes later saying that beef is also available. If you switch your choice from chicken to fish you are violating the independence axiom. You prefer beef less than both chicken and fish, so the beef option shouldn't alter the fish-chicken preference.

But let's suppose that when the server returned and mentioned beef, your physician advised you to reduce your cholesterol intake (so your preference for beef is lowest) which prompted your wife to say that you should eat fish at least twice a week because of vitamins in the oil. So you switch from chicken to fish. Beef is not chosen, but new information that resulted from introducing the irrelevant alternative has altered the chicken-fish preference.

One could argue for the independence axiom by saying that it applies only when all relevant information (like considerations of cholesterol and fish oil) are taken into account. On the other hand, one can argue against the independence axiom by saying that new relevant information quite often surfaces unexpectedly. The difficulty is to judge the extent to which ignorance and the emergence of new knowledge should be central in a decision theory.

Wrapping up. Theories express our knowledge and understanding about the unknown and confusing world. Knowledge begets knowledge. We use knowledge and understanding - that is, theory - in choosing a theory. The process is difficult because it's like building a boat on the open sea as Otto Neurath once said. 

Fog of War

"War is the realm of uncertainty;

three quarters of the factors on which action in war is based 

are wrapped in a fog of greater or lesser uncertainty."

Carl von Clausewitz, On War

What makes a great general?

Hannibal changed Carthaginian strategy from naval to land warfare, and beat the Romans in nearly every encounter. Julius Caesar commanded the undying loyalty of his officers and soldiers. Napoleon Bonaparte invented the modern concept of total war with a citizen army. Was their genius in strategy, or tactics, or logistics, or charisma? Or was it crude luck? Or was it the exploitation of uncertainty?

War is profoundly influenced by technology, social organization, human psychology and political goals. Success in war requires understanding and control of these factors. War consumes vast human and material resources and demands "genius, improvisation, and energy of mind" as Winston Churchill said. And yet, Clausewitz writes: "No other human activity is so continuously or universally bound up with chance."

Why? What does this imply about the successful military commander? What does it mean for human endeavor and history in general?

Clausewitz uses the terms "chance" and "uncertainty", sometimes interchangeably, to refer to two different concepts. An event occurs by chance if it is unexpected, or its origin is unknown, or its impact is surprising. Adverse chance events provoke "uncertainty, the psychological state of discomfort from confusion or lack of information" (Katherine Herbig, reference below).

Chance and uncertainty are dangerous because they subvert plans and diminish capabilities. Soldiers have been aware of both the dangers and the advantages of surprise since they first battered each other with sticks. Conventional military theorists aimed to avoid or ameliorate chance events by careful planning, military intelligence, training and discipline, communication, command and control. Clausewitz also recognized that steadfast faithfulness to mission and determination against adversity are essential in overcoming chance events and the debilitating effect of uncertainty. But "Clausewitz dismisses as worse than useless efforts to systematize warfare with rules and formulas. Such systems are falsely comforting, he says, because they reduce the imponderables of war to a few meagre certainties about minor matters" (Herbig). Clausewitz' most original contribution was in building a systematic theory of war in which the unavoidability of chance, and its opportunities, are central.

Why is uncertainty (in the sense of lack of knowledge) unavoidable and fundamental in war? Clausewitz' answer is expressed in his metaphor of friction. As Herbig explains:

"Friction is the decremental loss of effort and intention caused by human fallibility, compounded by danger and exhaustion. Like the mechanical phenomenon of friction that reduces the efficiency of machinery with moving parts, Clausewitz' friction reduces the efficiency of the war machine. It sums up all the little things that always go wrong to keep things from being done as easily and quickly as intended. ...

"What makes friction more than a minor annoyance in war is its confounding with chance, which multiplies friction in random, unpredictable ways."

War, like history, runs on the cumulative effect of myriad micro-events. Small failures are compounded because war is a coordinated effort of countless local but inter-dependent occurrences. Generals, like symphony conductors, choose the score and set the pace, but the orchestra plays the notes. A mis-tuned violin, or a drummer who mis-counts his entry, can ruin the show. Moses led the children of Israel out of Egypt, but he'd have looked pretty funny if they had scattered to the four winds. Moses' genius as a leader wasn't plied against Pharaoh (Moses had help there), but rather against endless bickering and revolt once they reached the desert.

Uncertainty originates at the tactical rather than the strategic level. The general can't know countless local occurrences: a lost supply plane, failed equipment here, over-reaction there, or complacency someplace else. As an example, the New York Times reported on 27 November 2011:

"The NATO air attack that killed at least two dozen Pakistani soldiers over the weekend reflected a fundamental truth about American-Pakistani relations when it comes to securing the unruly border with Afghanistan: the tactics of war can easily undercut the broader strategy that leaders of both countries say they share.

"The murky details complicated matters even more, with Pakistani officials saying the attack on two Pakistani border posts was unprovoked and Afghan officials asserting that Afghan and American commandos called in airstrikes after coming under fire from Pakistani territory."

Central control is critical, but also profoundly limited by the micro-event texture of history.

Conversely, uncertainty can be exploited at the tactical level by flexible and creative response to random opportunities. The field commander has local knowledge that enables decisive initiative: the sleeping sentinel, the bridge not destroyed, the deserted town. The general's brilliance is in forging a war machine whose components both exploit uncertainty and are resilient to surprise.

Uncertainty is central in history at large, like in war, because they both emerge from the churning of individual events. In democratic societies, legislatures pass laws and executive branches formulate and implement policies. But only active participation of the citizenry brings life and reality to laws and policies. Conversely, citizen resistance or even apathy dooms the best policies to failure. This explains the failure of democratic institutions that are imported precipitously to countries with incompatible social and political traditions. Governments formulate policy, but implementation occurs in the context of social attitudes and historical memory. You can elect legislatures and presidents but you can't elect the public. Non-centralized beliefs and actions also dominate the behavior of industrial economies. The actions of countless households, firms and investors can vitiate the best laid plans of monetary and fiscal authorities. All this adds up to Clausewitz' concept of friction: global uncertainty accumulating from countless local deviations.

In peace, like in war, the successful response to uncertainty is to face it, grapple with it, exploit it, restrain it, but never hope to abolish it. Uncertainty is inevitable, and sometimes even propitious. The propensity for war is the ugliest attribute of our species. Nonetheless, what we learn about uncertainty from the study of war applies to all our endeavors: in business, in politics and beyond. Waging peace demands the same staunchness, determination and inventive flexibility in the face of the unknown, as the successful pursuit of war.

Main source:

Katherine L. Herbig, 1989, Chance and Uncertainty in On War, in Michael Handel, ed., Clausewitz and Modern Strategy, Frank Cass, London, pp.95-116.

See also:

Peter Paret, 1976, Clausewitz and the State: The Man, His Theories, and His Times, re-issued 2007, Princeton University Press. 

Can We Replay History?

After the kids' party games and the birthday cake came the action-packed Steve McQueen movie. My friend's parents had rented a movie projector. They hooked up the reel and let it roll. But the high point came later when they ran the movie backwards. Bullets streamed back into guns, blows were retracted and fallen protagonists recoiled into action. The mechanism that pulls the celluloid film forward for normal showing, can pull the film in the reverse direction, rolling it back onto the feeder reel and showing the movie in reverse.

If you chuck a round pebble off a cliff it will fall in a graceful parabolic arch, gradually increasing its speed until it hits the ground. The same pebble, if shot from the point of impact, at the terminating angle and speed, will gracefully and obligingly retrace its path. (I'm ignoring wind and air friction that make things a bit more complicated.)

Deterministic mechanisms, like the movie reel mechanism or the law of gravity, are reversible.

History is different. Peoples' behavior is influenced by what they know. You pack an umbrella on a trip to the UK. Google develops search algorithms not search parties because their knowledge base is information technology not mountain trekking. Knowledge is powerful because it enables rational behavior: matching actions to goals. Knowledge transforms futile fumbling into intelligent behavior.

Knowledge underlies intelligent behavior, but knowledge is continually expanding. We discover new facts and relationships. We discover that things have changed. Therefore tomorrow's knowledge-based behavior will, to some extent, be unpredictable today because tomorrow's discoveries cannot be known today. Human behavior has an inherent element of indeterminism. Intelligent learning behavior cannot be completely predicted.

Personal and collective history does not unfold like a pre-woven rug. Human history is fundamentally different from the trajectory of a pebble tossed from a cliff. History is the process of uncovering the unknown and responding to this new knowledge. The existence of the unknown creates the possibility of free will. The discovery of new knowledge introduces indeterminism and irreversibility into history, as explained by the philosophers G.L.S. Shackle and Karl Popper.

Nonetheless history is not erratic because each increment of new knowledge adds to the store of what was learned before. Memory is not perfect, either of individuals or groups, but it is powerful. History happens in historical context. For instance, one cannot understand the recent revolutions and upheavals in the Arab world from the perspective of 18th century European revolutions; the historical backgrounds are too different, and the outcomes in the Middle East will be different as well. Innovation, even revolution, is spurred by new knowledge laid over the old. A female municipal official slapped a Tunisian street vendor, Mohamed Bouazizi. That slap crystalized Mr Bouazizi's knowledge of his helpless social impotence and lit the match with which he immolated himself and initiated conflagrations around the Mideast. New knowledge acts like thruster engines on the inertial body of memory. What is emerging in the Mideast is Middle Eastern, not European. What is emerging is the result of new knowledge: of the power of networking, of the mortality of dictators, of the limits of coercion, of the power of new knowledge itself and the possibilities embedded in tomorrow's unknowns.

Mistakes are made, even with the best intentions and the best possible knowledge. Even if analysts knew and understood all the actions of all actors on the stage of history, they still cannot know what those people will learn tomorrow and how that new knowledge will alter their behavior. Mistakes are made because history does not unwind like a celluloid reel.

That's not to say that analysts are never ignorant, negligent, stupid or malicious. It's to say that all actions are, in a sense, mistakes. Or, the biggest mistake of all is to think that we can know the full import of our actions. We cannot, because actions are tossed, like pebbles, into the dark pit of unknown possible futures. One cannot know all possible echoes, or whether some echo might be glass-shatteringly cataclysmic.

Mistakes can sometimes be corrected, but never undone. History cannot be run backwards, and you never get a second chance. Conversely, every instant is a new opportunity because the future is always uncertain. Uncertainty is the freedom to err, and the opportunity to create and discover. 

The Language of Science and the Tower of Babel

And God said: Behold one people with one language for them all ... and now nothing that they venture will be kept from them. ... [And] there God mixed up the language of all the land. (Genesis, 11:6-9)

"Philosophy is written in this grand book the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and to read the alphabet in which it is composed. It is written in the language of mathematics." Galileo Galilei

Language is power over the unknown. 

Mathematics is the language of science, and computation is the modern voice in which this language is spoken. Scientists and engineers explore the book of nature with computer simulations of swirling galaxies and colliding atoms, crashing cars and wind-swept buildings. The wonders of nature and the powers of technological innovation are displayed on computer screens, "continually open to our gaze." The language of science empowers us to dispel confusion and uncertainty, but only with great effort do we change the babble of sounds and symbols into useful, meaningful and reliable communication. How we do that depends on the type of uncertainty against which the language struggles.

Mathematical equations encode our understanding of nature, and Galileo exhorts us to learn this code. One challenge here is that a single equation represents an infinity of situations. For instance, the equation describing a flowing liquid captures water gushing from a pipe, blood coursing in our veins, and a droplet splashing from a puddle. Gazing at the equation is not at all like gazing at the droplet. Understanding grows by exposure to pictures and examples. Computations provide numerical examples of equations that can be realized as pictures. Computations can simulate nature, allowing us to explore at our leisure.

Two questions face the user of computations: Are we calculating the correct equations? Are we calculating the equations correctly? The first question expresses the scientist's ignorance - or at least uncertainty - about how the world works. The second question reflects the programmer's ignorance or uncertainty about the faithfulness of the computer program to the equations. Both questions deal with the fidelity between two entities. However, the entities involved are very different and the uncertainties are very different as well.

The scientist's uncertainty is reduced by the ingenuity of the experimenter. Equations make predictions that can be tested by experiment. For instance, Galileo predicted that small and large balls will fall at the same rate, as he is reported to have tested from the tower of Pisa. Equations are rejected or modified when their predictions don't match the experimenter's observation. The scientist's uncertainty and ignorance are whittled away by testing equations against observation of the real world. Experiments may be extraordinarily subtle or difficult or costly because nature's unknown is so endlessly rich in possibilities. Nonetheless, observation of nature remorselessly cuts false equations from the body of scientific doctrine. God speaks through nature, as it were, and "the Eternal of Israel does not deceive or console." (1 Samuel, 15:29). When this observational cutting and chopping is (temporarily) halted, the remaining equations are said to be "validated" (but they remain on the chopping block for further testing).

The programmer's life is, in one sense, more difficult than the experimenter's. Imagine a huge computer program containing millions of lines of code, the accumulated fruit of thousands of hours of effort by many people. How do we verify that this computation faithfully reflects the equations that have ostensibly been programmed? Of course they've been checked again and again for typos or logical faults or syntactic errors. Very clever methods are available for code verification. Nonetheless, programmers are only human, and some infidelity may slip through. What remorseless knife does the programmer have with which to verify that the equations are correctly calculated? Testing computation against observation does not allow us to distinguish between errors in the equations, errors in the program, and compensatory errors in both.

The experimenter compares an equation's prediction against an observation of nature. Like the experimenter, the programmer compares the computation against something. However, for the programmer, the sharp knife of nature is not available. In special cases the programmer can compare against a known answer. More frequently the programmer must compare against other computations which have already been verified (by some earlier comparison). The verification of a computation - as distinct from the validation of an equation - can only use other high-level human-made results. The programmer's comparisons can only be traced back to other comparisons. It is true that the experimenter's tests are intermediated by human artifacts like calipers or cyclotrons. Nonetheless, bedrock for the experimenter is the "reality out there". The experimenter's tests can be traced back to observations of elementary real events. The programmer does not have that recourse. One might say that God speaks to the experimenter through nature, but the programmer has no such Voice upon which to rely.

The tower built of old would have reached the heavens because of the power of language. That tower was never completed because God turned talk into babble and dispersed the people across the land. Scholars have argued whether the story prescribes a moral norm, or simply describes the way things are, but the power of language has never been disputed.

The tower was never completed, just as science, it seems, has a long way to go. Genius, said Edison, is 1 percent inspiration and 99 percent perspiration. A good part of the sweat comes from getting the language right, whether mathematical equations or computer programs.

Part of the challenge is finding order in nature's bubbling variety. Each equation captures a glimpse of that order, adding one block to the structure of science. Furthermore, equations must be validated, which is only a stop-gap. All blocks crumble eventually, and all equations are fallible and likely to be falsified.

Another challenge in science and engineering is grasping the myriad implications that are distilled into an equation. An equation compresses and summarizes, while computer simulations go the other way, restoring detail and specificity. The fidelity of a simulation to the equation is usually verified by comparing against other simulations. This is like the dictionary paradox: using words to define words.

It is by inventing and exploiting symbols that humans have constructed an orderly world out of the confusing tumult of experience. With symbols, like with blocks in the tower, the sky is the limit.

The End of Science?

Science is the search for and study of patterns and laws in the natural and physical worlds. Could that search become exhausted, like an over-worked coal vein, leaving nothing more to be found? Could science end? After briefly touching on several fairly obvious possible end-games for science, we explore how the vast Unknown could undermine - rather than underlie - the scientific enterprize. The possibility that science could end is linked to the reason that science is possible at all. The path we must climb in this essay is steep, but the (in)sight is worth it.

Science is the process of discovering unknowns, one of which is the extent of Nature's secrets. It is possible that the inventory of Nature's unknowns is finite or conceivably even nearly empty. However, a look at open problems in science, from astronomy to zoology, suggests that Nature's storehouse of surprises is still chock full. So, from this perspective, the answer to the question 'Could science end?' is conceivably 'Yes', but most probably 'No'.

Another possible 'Yes' answer is that science will end by reaching the limit of human cognitive capability. Nature's storehouse of surprises may never empty out, but the rate of our discoveries may gradually fall, reaching zero when scientists have figured out everything that humans are able to understand. Possible, but judging from the last 400 years, it seems that we've only begun to tap our mind's expansive capability.

Or perhaps science - a product of human civilization - will end due to historical or social forces. The simplest such scenario is that we blow ourselves to smithereens. Smithereens can't do science. Another more complicated scenario is Oswald Spengler's theory of cyclical history, whereby an advanced society - such as Western civilization - decays and disappears, science disappearing with it. So again a tentative 'Yes'. But this might only be an interruption of science if later civilizations resume the search.

We now explore the main mechanism by which science could become impossible. This will lead to deeper understanding of the delicate relation between knowledge and the Unknown and to why science is possible at all.

One axiom of science is that there exist stable and discoverable laws of nature. As the philosopher A.N. Whitehead wrote in 1925: "Apart from recurrence, knowledge would be impossible; for nothing could be referred to our past experience. Also, apart from some regularity of recurrence, measurement would be impossible." (Science and the Modern World, p.36). The stability of phenomena is what allows a scientist to repeat, study and build upon the work of other scientists. Without regular recurrence there would be no such thing as a discoverable law of nature.

However, as David Hume explained long ago in An Enquiry Concerning Human Understanding, one can never empirically prove that regular recurrence will hold in the future. By the time one tests the regularity of the future, that future has become the past. The future can never be tested, just as one can never step on the rolled up part of an endless rug unfurling always in front of you.

Suppose the axiom of Natural Law turns out to be wrong, or suppose Nature comes unstuck and its laws start "sliding around", changing. Science would end. If regularity, patterns, and laws no longer exist, then scientific pursuit of them becomes fruitless.

Or maybe not. Couldn't scientists search for the laws by which Nature "slides around"? Quantum mechanics seems to do just that. For instance, when a polarized photon impinges on a polarizing crystal, the photon will either be entirely absorbed or entirely transmitted, as Dirac explained. The photon's fate is not determined by any law of Nature (if you believe quantum mechanics). Nature is indeterminate in this situation. Nonetheless, quantum theory very accurately predicts the probability that the photon will be transmitted, and the probability that it will be absorbed. In other words, quantum mechanics establishes a deterministic law describing Nature's indeterminism.

Suppose Nature's indeterminism itself becomes lawless. Is that conceivable? Could Nature become so disorderly, so confused and uncertain, so "out of joint: O, cursed spite", that no law can "set it right"? The answer is conceivably 'Yes', and if this happens then scientists are all out of a job. To understand how this is conceivable, one must appreciate the Unknown at its most rambunctious.

Let's take stock. We can identify attributes of Nature that are necessary for science to be possible. The axiom of Natural Law is one necessary attribute. The successful history of science suggests that the axiom of Natural Law has held firmly in the past. But that does not determine what Nature will be in the future.

In order to understand how Natural Law could come unstuck, we need to understand how Natural Law works (today). When a projectile, say a baseball, is thrown from here to there, its progress at each point along its trajectory is described, scientifically, in terms of its current position, direction of motion, and attributes such as its shape, mass and surrounding medium. The Laws of Nature enable the calculation of the ball's progress by solving a mathematical equation whose starting point is the current state of the ball.

We can roughly describe most Laws of Nature as formulations of problems - e.g. mathematical equations - whose input is the current and past states of the system in question, and whose solution predicts an outcome: the next state of the system. What is law-like about this is that these problems - whose solution describes a progression, like the flight of a baseball - are constant over time. The scientist calculates the baseball's trajectory by solving the same problem over and over again (or all at once with a differential equation). Sometimes the problem is hard to solve, so scientists are good mathematicians, or they have big computers, (or both). But solvable they are.

Let's remember that Nature is not a scientist, and Nature does not solve a problem when things happen (like baseballs speeding to home plate). Nature just does it. The scientist's Law is a description of Nature, not Nature itself.

There are other Laws of Nature for which we must modify the previous description. In these cases, the Law of Nature is, as before, the formulation of a problem. Now, however, the solution of the problem not only predicts the next state of the system, but it also re-formulates the problem that must be solved at the next step. There is sort of a feedback: the next state of the system alters the rule by which subsequent progress is made. For instance, when an object falls towards earth from outer space, the law of nature that determines the motion of the object depends on the gravitational attraction. The gravitational attraction, in turn, increases as the object gets closer. Thus the problem to be solved changes as the object moves. Problems like these tend to be more difficult to solve, but that's the scientist's problem (or pleasure).

Now we can appreciate how Nature might become lawlessly unstuck. Let's consider the second type of Natural Law, where the problem - the Law itself - gets modified by the evolving event. Let's furthermore suppose that the problem is not simply difficult to solve, but that no solution can be obtained in a finite amount of time (mathematicians have lots of examples of problems like this). As before, Nature itself does not solve a problem; Nature just does it. But the scientist is now in the position that no prediction can be made, no trajectory can be calculated, no model or description of the phenomenon can be obtained. No explicit problem statement embodying a Natural Law exists. This is because the problem to be solved evolves continuously from previous solutions, and none of the sequence of problems can be solved. The scientist's profession will become frustrating, futile and fruitless.

Nature becomes lawlessly unstuck, and science ends, if all Laws of Nature become of the modified second type. The world itself will continue because Nature solves no problems, it just does its thing. But the way it does this is now so raw and unruly that no study of nature can get to first base.

Sound like science fiction (or nightmare)? Maybe. But as far as we know, the only thing between us and this new state of affairs is the axiom of Natural Law. Scientists assume that Laws exist and are stable because past experience, together with our psychological makeup (which itself is evolutionary past experience), very strongly suggests that regular recurrence can be relied upon. But if you think that the scientists can empirically prove that the future will continue to be lawful, like the past, recall that all experience is past experience. Recall the unfurling-rug metaphor (by the time we test the future it becomes the past), and make an appointment to see Mr Hume.

Is science likely to become fruitless or boring? No. Science thrives on an Unknown that is full of surprises. Science - the search for Natural Laws - thrives even though the existence of Natural Law can never be proven. Science thrives precisely because we can never know for sure that science will not someday end. 

Squirrels and Stock Brokers, Or: Innovation Dilemmas, Robustness and Probability

Decisions are made in order to achieve desirable outcomes. An innovation dilemma arises when a seemingly more attractive option is also more uncertain than other options. In this essay we explore the relation between the innovation dilemma and the robustness of a decision, and the relation between robustness and probability. A decision is robust to uncertainty if it achieves required outcomes despite adverse surprises. A robust decision may differ from the seemingly best option. Furthermore, robust decisions are not based on knowledge of probabilities, but can still be the most likely to succeed.

Squirrels, Stock-Brokers and Their Dilemmas

Decision problems.

Imagine a squirrel nibbling acorns under an oak tree. They're pretty good acorns, though a bit dry. The good ones have already been taken. Over in the distance is a large stand of fine oaks. The acorns there are probably better. But then, other squirrels can also see those trees, and predators can too. The squirrel doesn't need to get fat, but a critical caloric intake is necessary before moving on to other activities. How long should the squirrel forage at this patch before moving to the more promising patch, if at all?

Imagine a hedge fund manager investing in South African diamonds, Australian Uranium, Norwegian Kroners and Singapore semi-conductors. The returns have been steady and good, but not very exciting. A new hi-tech start-up venture has just turned up. It looks promising, has solid backing, and could be very interesting. The manager doesn't need to earn boundless returns, but it is necessary to earn at least a tad more than the competition (who are also prowling around). How long should the manager hold the current portfolio before changing at least some of its components?

These are decision problems, and like many other examples, they share three traits: critical needs must be met; the current situation may or may not be adequate; other alternatives look much better but are much more uncertain. To change, or not to change? What strategy to use in making a decision? What choice is the best bet? Betting is a surprising concept, as we have seen before; can we bet without knowing probabilities?

Solution strategies.

The decision is easy in either of two extreme situations, and their analysis will reveal general conclusions.

One extreme is that the status quo is clearly insufficient. For the squirrel this means that these crinkled rotten acorns won't fill anybody's belly even if one nibbled here all day long. Survival requires trying the other patch regardless of the fact that there may be many other squirrels already there and predators just waiting to swoop down. Similarly, for the hedge fund manager, if other funds are making fantastic profits, then something has to change or the competition will attract all the business.

The other extreme is that the status quo is just fine, thank you. For the squirrel, just a little more nibbling and these acorns will get us through the night, so why run over to unfamiliar oak trees? For the hedge fund manager, profits are better than those of any credible competitor, so uncertain change is not called for.

From these two extremes we draw an important general conclusion: the right answer depends on what you need. To change, or not to change, depends on what is critical for survival. There is no universal answer, like, "Always try to improve" or "If it's working, don't fix it". This is a very general property of decisions under uncertainty, and we will call it preference reversal. The agent's preference between alternatives depends on what the agent needs in order to "survive".

The decision strategy that we have described is attuned to the needs of the agent. The strategy attempts to satisfy the agent's critical requirements. If the status quo would reliably do that, then stay put; if not, then move. Following the work of Nobel Laureate Herbert Simon, we will call this a satisficing decision strategy: one which satisfies a critical requirement.

"Prediction is always difficult, especially of the future." - Robert Storm Petersen

Now let's consider a different decision strategy that squirrels and hedge fund managers might be tempted to use. The agent has obtained information about the two alternatives by signals from the environment. (The squirrel sees grand verdant oaks in the distance, the fund manager hears of a new start up.) Given this information, a prediction can be made (though the squirrel may make this prediction based on instincts and without being aware of making it). Given the best available information, the agent predicts which alternative would yield the better outcome. Using this prediction, the decision strategy is to choose the alternative whose predicted outcome is best. We will call this decision strategy best-model optimization. Note that this decision strategy yields a single universal answer to the question facing the agent. This strategy uses the best information to find the choice that - if that information is correct - will yield the best outcome. Best-model optimization (usually) gives a single "best" decision, unlike the satisficing strategy that returns different answers depending on the agent's needs.

There is an attractive logic - and even perhaps a moral imperative - to use the best information to make the best choice. One should always try to do one's best. But the catch in the argument for best-model optimization is that the best information may actually be grievously wrong. Those fine oak trees might be swarming with insects who've devoured the acorns. Best-model optimization ignores the agent's central dilemma: stay with the relatively well known but modest alternative, or go for the more promising but more uncertain alternative.

"Tsk, tsk, tsk" says our hedge fund manager. "My information already accounts for the uncertainty. I have used a probabilistic asset pricing model to predict the likelihood that my profits will beat the competition for each of the two alternatives."

Probabilistic asset pricing models are good to have. And the squirrel similarly has evolved instincts that reflect likelihoods. But a best-probabilistic-model optimization is simply one type of best-model optimization, and is subject to the same vulnerability to error. The world is full of surprises. The probability functions that are used are quite likely wrong, especially in predicting the rare events that the manager is most concerned to avoid.

Robustness and Probability

Now we come to the truly amazing part of the story. The satisficing strategy does not use any probabilistic information. Nonetheless, in many situations, the satisficing strategy is actually a better bet (or at least not a worse bet), probabilistically speaking, than any other strategy, including best-probabilistic-model optimization. We have no probabilistic information in these situations, but we can still maximize the probability of success (though we won't know the value of this maximum).

When the satisficing decision strategy is the best bet, this is, in part, because it is more robust to uncertainty than another other strategy. A decision is robust to uncertainty if it achieves required outcomes even if adverse surprises occur. In many important situations (though not invariably), more robustness to uncertainty is equivalent to being more likely to succeed or survive. When this is true we say that robustness is a proxy for probability.

A thorough analysis of the proxy property is rather technical. However, we can understand the gist of the idea by considering a simple special case.

Let's continue with the squirrel and hedge fund examples. Suppose we are completely confident about the future value (in calories or dollars) of not making any change (staying put). In contrast, the future value of moving is apparently better though uncertain. If staying put would satisfy our critical requirement, then we are absolutely certain of survival if we do not change. Staying put is completely robust to surprises so the probability of success equals 1 if we stay put, regardless of what happens with the other option. Likewise, if staying put would not satisfy our critical requirement, then we are absolutely certain of failure if we do not change; the probability of success equals 0 if we stay, and moving cannot be worse. Regardless of what probability distribution describes future outcomes if we move, we can always choose the option whose likelihood of success is greater (or at least not worse). This is because staying put is either sure to succeed or sure to fail, and we know which.

This argument can be extended to the more realistic case where the outcome of staying put is uncertain and the outcome of moving, while seemingly better than staying, is much more uncertain. The agent can know which option is more robust to uncertainty, without having to know probability distributions. This implies, in many situations, that the agent can choose the option that is a better bet for survival.

Wrapping Up

The skillful decision maker not only knows a lot, but is also able to deal with conflicting information. We have discussed the innovation dilemma: When choosing between two alternatives, the seemingly better one is also more uncertain.

Animals, people, organizations and societies have developed mechanisms for dealing with the innovation dilemma. The response hinges on tuning the decision to the agent's needs, and robustifying the choice against uncertainty. This choice may or may not coincide with the putative best choice. But what seems best depends on the available - though uncertain - information.

The commendable tendency to do one's best - and to demand the same of others - can lead to putatively optimal decisions that may be more vulnerable to surprise than other decisions that would have been satisfactory. In contrast, the strategy of robustly satisfying critical needs can be a better bet for survival. Consider the design of critical infrastructure: flood protection, nuclear power, communication networks, and so on. The design of such systems is based on vast knowledge and understanding, but also confronts bewildering uncertainties and endless surprises. We must continue to improve our knowledge and understanding, while also improving our ability to manage the uncertainties resulting from the expanding horizon of our efforts. We must identify the critical goals and seek responses that are immune to surprise. 

Beware the Rareness Illusion When Exploring the Unknown

Here's a great vacation idea. Spend the summer roaming the world in search of the 10 lost tribes of Israel, exiled from Samaria by the Assyrians 2700 years ago (2 Kings 17:6). Or perhaps you'd like to search for Prester John, the virtuous ruler of a kingdom lost in the Orient? Or would you rather trace the gold-laden kingdom of Ophir (1 Kings 9:28)? Or do you prefer the excitement of tracking the Amazons, that nation of female warriors? Or perhaps the naval power mentioned by Plato, operating from the island of Atlantis? Or how about unicorns, or the fountain of eternal youth? The Unknown is so vast that the possibilities are endless.

Maybe you don't believe in unicorns. But Plato evidently "knew" about the island of Atlantis. The conquest of Israel is known from Assyrian archeology and from the Bible. That you've never seen a Reubenite or a Naphtalite (or a unicorn) means that they don't exist?

It is true that when something really does not exist, one might spend a long time futilely looking for it. Many people have spent enormous energy searching for lost tribes, lost gold, and lost kingdoms. Why is it so difficult to decide that what you're looking for really isn't there? The answer, ironically, is that the world has endless possibilities for discovery and surprise.

Let's skip vacation plans and consider some real-life searches. How long should you (or the Libyans) look for Muammar Qaddafi? If he's not in the town of Surt, maybe he's Bani Walid, or Algeria, or Timbuktu? How do you decide he cannot be found? Maybe he was pulverized by a NATO bomb. It's urgent to find the suicide bomber in the crowded bus station before it's too late - if he's really there. You'd like to discover a cure for AIDS, or a method to halt the rising global temperature, or a golden investment opportunity in an emerging market, or a proof of the parallel postulate of Euclidean geometry.

Let's focus our question. Suppose you are looking for something, and so far you have only "negative" evidence: it's not here, it's not there, it's not anywhere you've looked. Why is it so difficult to decide, conclusively and confidently, that it simply does not exist?

This question is linked to a different question: how to make the decision that "it" (whatever it is) does not exist. We will focus on the "why" question, and leave the "how" question to students of decision theories such as statistics, fuzzy logic, possibility theory, Dempster-Shafer theory and info-gap theory. (If you're interested in an info-gap application to statistics, here is an example.)

Answers to the "why" question can be found in several domains.

Psychology provides some answers. People can be very goal oriented, stubborn, and persistent. Marco Polo didn't get to China on a 10-hour plane flight. The round trip took him 24 years, and he didn't travel business class.

Ideology is a very strong motivator. When people believe something strongly, it is easy for them to ignore evidence to the contrary. Furthermore, for some people, the search itself is valued more than the putative goal.

The answer to the "why" question that I will focus on is found by contemplating The Endless Unknown. It is so vast, so unstructured, so, well ..., unknown, that we cannot calibrate our negative evidence to decide that whatever we're looking for just ain't there.

I'll tell a true story.

I was born in the US and my wife was born in Israel, but our life-paths crossed, so to speak, before we were born. She had a friend whose father was from Europe and lived for a while - before the friend was born - with a cousin of his in my home town. This cousin was - years later - my 3rd grade teacher. My school teacher was my future wife's friend's father's cousin.

Amazing coincidence. This convoluted sequence of events is certainly rare. How many of you can tell the very same story? But wait a minute. This convoluted string of events could have evolved in many many different ways, each of which would have been an equally amazing coincidence. The number of similar possible paths is namelessly enormous, uncountably humongous. In other words, potential "rare" events are very numerous. Now that sounds like a contradiction (we're getting close to some of Zeno's paradoxes, and Aristotle thought Zeno was crazy). It is not a contradiction; it is only a "rareness illusion" (something like an optical illusion). The specific event sequence in my story is unique, which is the ultimate rarity. We view this sequence as an amazing coincidence because we cannot assess the number of similar sequences. Surprising strings of events occur not infrequently because the number of possible surprising strings is so unimaginably vast. The rareness illusion is the impression of rareness arising from our necessary ignorance of the vast unknown. "Necessary" because, by definition, we cannot know what is unknown. "Vast" because the world is so rich in possibilities.

The rareness illusion is a false impression, a mistake. For instance, it leads people to wrongly goggle at strings of events - rare in themselves - even though "rare events" are numerous and "amazing coincidences" occur all the time. An appreciation of the richness and boundlessness of the Unknown is an antidote for the rareness illusion.

Recognition of the rareness illusion is the key to understanding why it is so difficult to confidently decide, based on negative evidence, that what you're looking for simply does not exist.

One might be inclined to reason as follows. If you're looking for something, then look very thoroughly, and if you don't find it, then it's not there. That is usually sound and sensible advice, and often "looking thoroughly" will lead to discovery.

However, the number of ways that we could overlook something that really is there is enormous. It is thus very difficult to confidently conclude that the search was thorough and that the object cannot be found. Take the case of your missing house keys. They dropped from your pocket in the car, or on the sidewalk and somebody picked them up, or you left them in the lock when you left the house, or or or .... Familiarity with the rareness illusion makes it very difficult to decide that you have searched thoroughly. If you think that the only contingencies not yet explored are too exotic to be relevant (a raven snatched them while you were daydreaming about that enchanting new employee), then think again, because you've been blinded by a rareness illusion. The number of such possibilities is so vastly unfathomable that you cannot confidently say that all of them are collectively negligible. Recognition of the rareness illusion prevents you from confidently concluding that what you are seeking simply does not exist.

Many quantitative tools grapple with the rareness illusion. We mentioned some decision theories earlier. But because the rareness illusion derives from our necessary ignorance of the vast unknown, one must always beware.

Looking for an exciting vacation? The Endless Unknown is the place to go. 

The Pains of Progress

To measure time by how little we change is to find how little we've lived, 

but to measure time by how much we've lost is to wish we hadn't changed at all. Andre Aciman

The last frontier is not the Antarctic, or the oceans, or outer space. The last frontier is The Unknown. We mentioned in an earlier essay that uncertainty - which makes baseball and life interesting - is inevitable in the human world. Life will continue to be interesting as long as the world is rich in unknowns, waiting to be discovered. Progress is possible if propitious discoveries can be made. Progress, however, comes with costs.

The emblem of my university entwines a billowing smokestack and a cogwheel in the first letter of the institution's name. When this emblem was adopted (probably in 1951) these were optimistic symbols of progress. Cogwheels are no longer 'hi-tech' (though we still need them), and smoke has been banished from polite company. But our emblem is characteristic of industrial society which has seared Progress on our hearts and minds.

Progress is accompanied by painful tensions. On the one hand, progress is nurtured by stability, cooperation, and leisure. On the other hand, progress grows out of change, conflict, and stress. A society's progressiveness reflects its balance of each of these three pairs of attributes. In the most general terms, progressiveness reflects social and individual attitudes to uncertainty.

Let's consider the three pairs of attributes one at a time.

Change and stability. Not all change is progress, but all progress is change. Change is necessary for progress, by definition, and progress can be very disruptive. The disruptiveness sometimes arises from unexpected consequences. J.B.S. Haldane wrote in 1923 that "the late war is only an example of the disruptive result that we may constantly expect from the progress of science." On the other hand, progressives employ and build on existing capabilities. The entrepreneur depends on stable property rights before risking venture capital. The existing legal system is used to remove social injustice. Watt's steam engine extended Newcomen's more primitive model. The new building going up on campus next to my office is very disruptive, but the construction project depends on the continuity of the university despite the drilling and dust. Even revolutionaries exploit and react against the status quo, which must exist for a revolutionary to be able to revolt. (One can't revolt if nothing is revolting.) Progress grows from a patch of opportunity in a broad bed of certainty, and spreads out in unanticipated directions.

Conflict and cooperation. Conflict between vested interests and innovators is common. Watt protected his inventions with extensive patents which may have actually retarded the further development and commercialization of steam power. Conflict is also a mechanism for selecting successful ideas. Darwinian evolution and its social analogies proceed by more successful adaptations replacing less successful ones. On the other hand, cooperation enables specialization and expertise which are needed for innovation. The tool-maker cooperates with the farmer so better tools can be made more quickly, enhancing the farmer's productivity and the artisan's welfare. Conflicts arise over what constitutes progress. Stem cell research, genetic engineering, nuclear power technology: progress or plague? Cooperative collective decision making enables the constructive resolution of these value-based conflicts.

Stress and leisure. Challenge, necessity and stress all motivate innovation. If you have no problems, you are unlikely to be looking for solutions. On the other hand, the leisure to think and tinker is a great source of innovation. Subsistence societies have no resources for invention. In assessing the implications of industrial efficiency, Bertrand Russell praised idleness in 1932, writing: "In a world where no one is compelled to work more than four hours a day, every person possessed of scientific curiosity will be able to indulge it, and every painter will be able to paint without starving ...." Stress is magnified by the unknown consequences of the stressor, while leisure is possible only in the absence of fear.

New replaces Old. Yin and yang are complementary opposites that dynamically interact. In Hegel's dialectic, tension between contradictions is resolved by synthesis. Human history is written by the victors, who sometimes hardly mention those swept into Trotsky's "dustbin of history". "In the evening resides weeping; in the morning: joy." (Psalm 30:6). Change and stability; conflict and cooperation; stress and leisure.

No progress without innovation; no innovation without discovery; no discovery without the unknown; no unknown without fear. There is no progress without pain.

Robustness and Locke's Wingless Gentleman

Our ancestors have made decisions under uncertainty ever since they had to stand and fight or run away, eat this root or that berry, sleep in this cave or under that bush. Our species is distinguished by the extent of deliberate thought preceding decision. Nonetheless, the ability to decide in the face of the unknown was born from primal necessity. Betting is one of the oldest ways of deciding under uncertainty. But you bet you that 'bet' is a subtler concept than one might think.

We all know what it means to make a bet, but just to make sure let's quote the Oxford English Dictionary: "To stake or wager (a sum of money, etc.) in support of an affirmation or on the issue of a forecast." The word has been around for quite a while. Shakespeare used the verb in 1600: "Iohn a Gaunt loued him well, and betted much money on his head." (Henry IV, Pt. 2 iii. ii. 44). Drayton used the noun in 1627 (and he wasn't the first): "For a long while it was an euen bet ... Whether proud Warwick, or the Queene should win."

An even bet is a 50-50 chance, an equal probability of each outcome. But betting is not always a matter of chance. Sometimes the meaning is just the opposite. According to the OED 'You bet' or 'You bet you' are slang expressions meaning 'be assured, certainly'. For instance: "'Can you handle this outfit?' 'You bet,' said the scout." (D.L.Sayers, Lord Peter Views Body, iv. 68). Mark Twain wrote "'I'll get you there on time' - and you bet you he did, too." (Roughing It, xx. 152).

So 'bet' is one of those words whose meaning stretches from one idea all the way to its opposite. Drayton's "even bet" between Warwick and the Queen means that he has no idea who will win. In contrast, Twain's "you bet you" is a statement of certainty. In Twain's or Sayers' usage, it's as though uncertainty combines with moral conviction to produce a definite resolution. This is a dialectic in which doubt and determination form decisiveness.

John Locke may have had something like this in mind when he wrote:

"If we will disbelieve everything, because we cannot certainly know all things; we shall do muchwhat as wisely as he, who would not use his legs, but sit still and perish, because he had no wings to fly." (An Essay Concerning Human Understanding, 1706, I.i.5)

The absurdity of Locke's wingless gentleman starving in his chair leads us to believe, and to act, despite our doubts. The moral imperative of survival sweeps aside the paralysis of uncertainty. The consequence of unabated doubt - paralysis - induces doubt's opposite: decisiveness.

But rational creatures must have some method for reasoning around their uncertainties. Locke does not intend for us to simply ignore our ignorance. But if we have no way to place bets - if the odds simply are unknown - then what are we to do? We cannot "sit still and perish".

This is where the strategy of robustness comes in.

'Robust' means 'Strong and hardy; sturdy; healthy'. By implication, something that is robust is 'not easily damaged or broken, resilient'. A statistical test is robust if it yields 'approximately correct results despite the falsity of certain of the assumptions underlying it' or despite errors in the data. (OED)

A decision is robust if its outcome is satisfactory despite error in the information and understanding which justified or motivated the decision. A robust decision is resilient to surprise, immune to ignorance.

It is no coincidence that the colloquial use of the word 'bet' includes concepts of both chance and certainty. A good bet can tolerate large deviation from certainty, large error of information. A good bet is robust to surprise. 'You bet you' does not mean that the world is certain. It means that the outcome is certain to be acceptable, regardless of how the world turns out. The scout will handle the outfit even if there is a rogue in the ranks; Twain will get there on time despite snags and surprises. A good bet is robust to the unknown. You bet you!

An extended and more formal discussion of these issues can be found elsewhere.