Error: I'm afraid this is the first I've heard of a "comments" flavoured Blosxom. Try dropping the "/+comments" bit from the end of the URL.

Wed, 31 Jan 2007

Sum and Difference

"Democracy is the worst form of government except for all those others that have been tried."
– Winston Churchill

Churchill's quote really drives to the heart of the matter. It makes the point, in a somewhat sarcastic way, that the problems with democracy are known and acknowledged, and yet inevitable. Implicit in his sarcasm is the sober warning that the troubles associated with democracy are not a ready excuse for dismissing it when what it says appears inconvenient.

As I explained last month, I think this is sage advice. Even if we disagree with the consensus, we usually don't "get into a fight over it" as Rich commented. We go along. Most times, this is to everyone's benefit. In fact, quite often, we may look back and realize we were wearing the big yellow shoes, orange hair, and rubber nose when we disagreed with the group. In hindsight we may be relieved that we didn't get our way.

And although Churchill's joke makes it seem that he believes democracy was discovered by trial and error, I'm sure that such a brilliant man realized something deeper was going on.

Anthropomorphizing a consensus may lead to category errors, as Shiloh warned, but I believe we are largely stuck with those errors. We are stuck because we may not have the capacity to "get outside" our perception of consensus. We appreciate aggregates with built-in, pattern matching capabilities. These sum-and-difference-recognizing brain functions operate behind the curtain that separates the conscious from the unconscious. After all, we, ourselves are an emergent property of a consensus of neurons. How our own emergence happens has many aspects that may always remain baffling to us. It's no surprise, therefore, that there are baffling aspects to any analysis of what any aggregate "thinks" or "feels" or "wants".

Like most great concepts, the idea of consensus – or a common perceived agreement pattern in an aggregate – has an opposite that is also a great concept. The opposite of consensus is dissension – the opposite of our facile perception of aggregates is our equally facile perception of differences.

Much of this may be truly mysterious, but there are definitely aspects of consensus that are well understood. Going further, I would claim that the concept is innate in human thinking. It is a concept that pervades reason, emotion, reverence, morality. We can't unrope from this cognitive mountaineering crew even if we want to. Denying that consensus in perception is like denying the law of excluded middle permeates logic. We risk babbling nonsense if we don't demand that a proposition be either true, or untrue, but not both at the same time. Denying our innate perception of aggregates and differences leads to the same sort of madness.

Perhaps I was unclear last month in my exposition of this central point. From the tone of the comments, there seemed to be some confusion about what I was getting at. I think my writing was unfortunately ambiguous in places, and not a little muddled in others. I was not arguing for or against a particular consensus on a particular political issue. Nor am I questioning the merits of a particular political system. Rather, I'm arguing strongly against categorical dismissals of the primality of consensus in human endeavor, as in Crichton's: "if it's consensus, it isn't science." Taken seriously (rather than as the rhetorical argumentation it surely is) a statement like Crichton's is utter claptrap. Consensus and aggregation concepts, from statistical hypothesis testing to peer review, are the very flesh and blood of science. You can't excise consensus from the scientific process without killing science.

A secondary point I was attempting to make last month – that I also seemed to botch – is that consensus and decision making (whether or not it's called small "d" democracy) is equally inextricable. Decisions simply cannot be made without consensus in some form, some place. Even in anarchy, a consensus of the neurons in one brain must cooperate to propel a suicide bomber toward his bloody task.

In the hope of getting my argument back on track, let me give several examples illustrating how we grok consensus, and its opposite, at the deepest level. I will trace these examples through various ancient and modern theoretical directions, showing the well known ways that consensus succeeds in each of them, and the well known ways that it fails.

I think it's best to begin with phenomenology. When anyone looks at a string of characters like this

  000111111111111111111111111111111

We also effortlessly perceive in the above string that all the "0" digits stand together in a tidy group. Should we look at a string like this

  001011111111111111111111111111111

  111110111111111111111011111101111

Consensus, in essence, relates to the idea that a summation has an independent meaning from the things summed; this is the idea that adding together things results in a new thing. Buckminster Fuller called it synergy: the triangle is more than just three lines. In fact, the lines weren't even three till they were perceived in that triangular group – now were they even lines till our brain aggregates the consensus of sensation.

Oddly, consensus is more often associated with stasis and the inability to change than the idea that bringing together disparate entities creates change in the form of an emergence of a new entity. Mankind's first recorded expression of the static concept of consensus dates back at least to 450BC and the Greek Philosopher Zeno of Elea – possibly the world's first conservative – who doubted that change was possible. In the well known "Zeno's paradox" of the tortoise and Achilles, Zeno doubts that any summation of small changes can add up to a substantive disturbance of the status quo. The logic of the argument can seem irrefutable (especially to those with low levels of serotonin), but reality refutes Zeno trivially: swift Achilles passes the slow tortoise easily and reliably. Change is possible.

Such a static view is the opposite of what another equally influential Greek claimed a hundred years earlier. Heraclitus – possibly the world's first liberal – claimed that consensus was illusion and the world is in a constant (sic) state of change – an incessant flux. We can never step into the same river twice. Heraclitus doubts that any summation of differences can converge to something constant and unchanging. Here the logic is inherently self contradictory (how can change be constant), but evidence for the truth of the Heraclitus flux is manifold in nature. Change is everywhere; who hasn't felt overwhelmed by it, as if the idea that anything can endure must be an illusion.

Both of these opposing viewpoints make a certain kind of sense and we still hear them used every day, applied by all sorts of people in all sorts of contexts. The views of Zeno and Heraclitus represent a fundamental dichotomy – maybe even the only dichotomy: all other conceptual divisions being mere variations on this theme of sum and difference. The list is long: sum and difference, peace and war, plenum and void, cooperation and conflict, harmony and dissonance....

But the fact that these ideas were invented 2500 years ago and live on in myriad raw forms today does not mean they have never been refined, amplified, and delicately reworked into greater sophistication. We should not dismiss something because it is a descendant of these ancient ideas. Rather, we should appreciate how deep these ideas run.

It took about two thousand years for mankind to arrive at what may be our best strict understanding thus far of what Zeno and Heraclitus were driving at. The Integral and Differential Calculus – the study of aggregation and partition in advanced mathematics – was developed by Leibniz and Newton in order to formalize and quantify what was till then a merely qualitative understanding of these two intertwined concepts of sums and differences. Thanks to the calculus, we now know in very rich detail exactly how small changes add up to something substantial, and when they don't. Conversely, we know when aggregates resist change, and when they don't.

The processes studied in calculus tend to be strong, strict convergences to particular outcomes. Adding together many small bits in a particular way reaches a constant quantitative sum strictly, certainly, and with few caveats or side-conditions. Do it a different way, and it doesn't reach a constant sum, with equally strict conditions. For example, in Zeno's paradox we are lead to doubt that 1/2 + 1/4 + 1/8 + 1/16 + ... adds up to 1, but in calculus we learn the certainty of how this convergence can be rigorously and absolutely true for this process of aggregation and a wide collection of similar processes. We also learn that 1/2 + 1/3 + 1/4 + 1/5 + ... does not reach a static conclusion. And we learn detailed rules about how to tell these cases apart.

But not all aggregation processes are so clear cut. In the science of probability and statistics, the concept of convergence in the aggregate, or the so called "regression to the mean" is a far weaker and tenuous concept. Probability retains all the formality and rigor of mathematics, but now the results don't have the same kind of certainty. For example, if we flip a fair coin N times, the total number of heads will be about N/2, but there is no ironclad guarantee of this when conducting an experiment to "verify" the truth of it. We might get N heads; we might get none; we might get something in between. There's no way to say, a priori, what we will get, and yet we can still make predictions that will be true "most" of the time – or at least, we have developed cognitive tools that give us a pragmatic way to bet on soft probabilistic convergence that tends to keep us alive as a species.

Related to soft probabilistic convergence are results called limit theorems. These theorems say how the consensus (the sum) of many independent experiments is distributed, statistically. The most well known of these is called The Central Limit Theorem, which states that sums of "most" random processes will be approximately normally (Gaussian) distributed.

As humans we somehow "get" limit theorems intuitively and I believe that this understanding is the basis for our direct understanding of the behavior of averages over a wide gamut of applications – from the "average American voter" to the average depth of the river. Could it be that our ancestors survived better guiding their actions by perceived averages and induction. They learned that they could grab the most berries off a bush by groping them together in a hand at what seems the densest fruiting area without considering exactly which or how many individual berries they were grabbing.

Even with all calculus shows us about the quantitative nature of change and constancy, and softer versions of these concepts from probability, it is still not the whole story. There remain many important aspects of the arguments of Zeno and Heraclitus that slip past our best rigorous tools. These aspects, resistant to quantitative or even logical analysis concern the mystery of human perception. Calculus and probability comprises much of what logic and reason can tell us, but it misses some other elements. There is something very messy about the perception of consensus and the perception of change – something that may have evolved into our brains through the eons of selection based on our success at staying alive and reproducing by means of our skills at perception.

In the beginning of the 20th century psychologists such as Max Wertheimer proposed a theory of perception that had aggregate objects perceived within an environment according to the consensus of their elements. The component elements of an object form a Gestalt, a unified concept or pattern that is primary, defining the parts of which it was composed, rather than the other way around. The individual elements are nothing until the Gestalt "emerges".

Gestalt theories of perception systematize this magic emergence with the additional concepts of reification, invariance, and multistability. In reification, we recognize that human perception adds something that is not present before the emergence; it's a creative process. The property of invariance possessed by human perception allows us to persistently see the same object even when the sense data is distorted in various ways.

Finally, mustistability acknowledges that perception is fundamentally ambiguous – that distinctly different gestalt aggregates may emerge from the same collection of sense fragments. Dwelling on the implications of multistability can be scary stuff. Multistability may be an inevitable consequence of the non-linear perceptive flash of insight. We wouldn't have our ability to perceive true pattern out of seemingly unrelated entities without a chance, hopefully small, of seeing something that isn't actually there, or seeing the wrong thing.

The concepts of game theory provide a different and very illuminating view of the strategic aspects of consensus and dissent in our lives. You may recall the classic Prisoner's Dilemma game where two criminals are interrogated separately and rewarded or punished based on how they behave in cooperation or in conflict. Studies of this basic game, and other similar games have shown how tempered cooperation and altruism can still result from players committed to selfish individual advantages. This may explain how society, with its tendency to be guided mostly by consensus, yet showing occasional bursts of conflict, might have evolved.

Prisoner dilemma games usually preclude in their rules any form of direct communication between the players. Collusion in such a vulgar way is not allowed. Yet success always emerges best from the strategies that implicitly communicate. Like bidding conventions in the game of bridge, patterns of conflict and cooperation form a larger pattern that is recognized by the players as information being communicated. It's not like partners in bridge actually show each other their cards, but after a round or two of bidding, they each have a good idea what the other holds. Bystanders may just see the surface dance between consensus and conflict, but there is a deeper process happening as the players reach consensus through that dance.

With advances in electronics, high speed data communication first became widespread in the 1950 and 60s. Along with this technology, the science of information theory developed. Communication engineers wanted to know the most efficient way to communicate information over a telephone line or radio link, and information theory explained how.

One central result in information theory is the concept of an optimal matched filter. If Alice wants to communicate a letter from some alphabet to her friend Bob, she may send a picture of it over some communications link. For simplicity, let's assume that link is the Postal Service. When Bob receives the picture, it is folded, spindled, and mutilated as a result of its rough journey through the mails. Bob wants to determine what letter it is with the best chance of success and the least chance of making an error, confusing a corrupted "O" with a "Q" perhaps. Information theory tells him that his best strategy for success is to compare the received character with each character in the alphabet. To do the comparison, he should divide up the received character and a potential candidate into tiny pixels, as small as possible, and compare each pixel by pixel, forming a tally where the pixels agree in color. The tallies for each candidate are then compared, and the letter with the largest "vote" total of matches is "elected".

Although I have oversimplified some of the practical details of the matched filtering process, this process (with the details) has been proved to be the best that can be done in a general, common context. The similarity to elections and democracy is striking. It's also interesting to note that the chance of catastrophic error always exists, but can be made vanishingly small with sufficiently sophisticated voting schemes.

Induction puzzles also relate to game and information theory and the emergence of consensus through implicit communication (with a vanishingly small yet inevitable risk of catastrophic failure). The classic induction puzzle is the three hat problem.

Blindfolded, you are brought into a room with two other players. Either a black hat or a white hat is placed on each player's head. When the blindfolds are removed simultaneously, the object is to be the first to determine the color of your own hat, which you cannot see directly, although you can see the colors on the other players' heads. Participants are not allowed to communicate, but are obliged to make some predetermined sign to the others based on what they see (e.g. raise their hand if they see a white hat).

In these puzzles, a winning player needs to make assumptions about the reasoning ability of the other players and "bootstrap" a way to a solution by observing the actions and inactions by the others and thereby eliminating possibilities that one would expect the other players would have been able to solve by that time.

What's most interesting about these puzzles in the context of our discussion of consensus is how a consensus in the actions of others guides successful individual action without any firm logical basis for that action. It works well most of the time – stunningly well – but it still is a tower of cards. Induction strategies can fail, and fail in an embarrassingly catastrophic way.

Take, for example, the Saddam Hussein paradox. It is announced that Saddam will be hanged sometime in the next week, and the execution will be televised, but for security reasons, the exact day of the execution will not be revealed in advance. The media pundits conclude from this announcement that Saddam will never be hanged. After all, he cannot be hung on the last day of the week because, having lived till then, the date of his hanging will be thereby known. The last day thus eliminated, he cannot be hanged on the next to last day either, as, again the day would be known in advance. Following this process of elimination to its "logical" conclusion, the pundits decide that Saddam can never be hung. Of course, he is hung on the second day of the week and the everyone is totally surprised, as was the intended security plan all along.

Generalizations of the three hat puzzle with a large number of hats leads to even more facinating results. Once again, implicit communication turns out to be the best strategy. In fact, one can apply results from information coding theory to these puzzles to develop "near perfect" strategies that achieve asymptotic certainty of correct hat guessing for everyone, but retain an ever so small chance of going catastrophically wrong, resulting in everyone being stumped.

These have been just a few examples of how the concept of consensus is intertwined with many different theoretical disciplines. Our perception of aggregates succeeds almost certainly and almost always, but has an inherent risk of failure. That is not to say, however, that cooperation and consensus are "the worst form of government, except for everything else". Rather, it shows how unboundedly good it can be – how it can benefit everyone – without being stuck in a Zeno-like stasis with no possibility of change. Our modern theoretical view of the inevitable yin and yang of consensus might not wholly please the Enlightenment thinkers that originally argued for consensus-based decision making, nor might it please the logical positivists that attempted to nail down every loose end with the tools of reason, but I personally believe that this richer understanding of consensus will lead mankind to a better understanding of ourselves.