A simple explanation of how differential gears work and why they are necessary.
“I have never done anything like others,” Benoit Mandelbrot (1924-2010) once said.
That statement is proven time and time again in his autobiography: The Fractalist.
Mandelbrot is independent almost to a fault, his book an interesting memoir from the man who revitalized visual geometry, and whose ideas about fractals have changed how we look at physics, engineering, arts, medicine, finance, and biology.
Nearly all common patterns in nature are rough. They have aspects that are exquisitely irregular and fragmented—not merely more elaborate than the marvelous ancient geometry of Euclid but of massively greater complexity. For centuries, the very idea of measuring roughness was an idle dream. This is one of the dreams to which I have devoted my entire scientific life.
Let me introduce myself. A scientific warrior of sorts, and an old man now, I have written a great deal but never acquired a predictable audience. So, in this memoir, please allow me to tell you who I think I am and how I came to labor for so many years on the first-ever theory of roughness and was rewarded by watching it transform itself into an aspect of a theory of beauty.
Mandelbrot was full of insight.
What shape is a mountain, a coastline, a river or a dividing line between two river watersheds? … Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.
While that sounds obvious it wasn’t at the time. In showing that triangles, squares, and circles are more prevalent in textbooks than reality, he brought to life the discipline now known as fractal geometry, a general theory of “roughness.”
Mandelbrot was fascinating, in part, because he never stayed in one place very long.
An acquaintance of mine was a forceful dean at a major university. One day, as our paths crossed in a busy corridor, he stopped to make a comment I never forgot: “You are doing very well, yet you are taking a lonely and hard path. You keep running from field to field, leading an unpredictable life, never settling down to enjoy what you have accomplished. A rolling stone gathers no moss, and—behind your back—people call you completely crazy. But I don’t think you are crazy at all, and you must continue what you are doing. For a thinking person, the most serious mental illness is not being sure of who you are. This is a problem you do not suffer from. You never need to reinvent yourself to fit changes in circumstances; you just move on. In that respect, you are the sanest person among us.”
Quietly, I responded that I was not running from field to field, but rather working on a theory of roughness. I was not a man with a big hammer to whom every problem looked like a nail. Were his words meant to compliment or merely to reassure? I soon found out: he was promoting me for a major award.
Is mental health compatible with being possessed by barely contained restlessness? In Dante’s Divine Comedy, the deceased sentenced to eternal searching are pushed to the deepest level of the Inferno. But for me, an eternal search across countless scientific fields beyond obvious connection managed to add up to a happy life. A rolling stone perhaps, but not an unresponsive one. Overactive and self-motivated, I loved to roll along, stopping to listen and preach in lay monasteries of all kinds—some splendid and proud, others forsaken and out of the way.
He had a different way of looking at things. For example, he saw math problems as geometry.
I would raise my hand and describe my findings: “Monsieur, I see an obvious geometric solution.” I quickly grasped the most abstract problem that the teacher could contrive. And then — with no effort, conscious search, or delay — I continued along a path that somehow avoided every difficulty…. I managed to be examined on the basis of speed and good taste in, first, translating algebra back into geometry, and then thinking in terms of geometric shapes. My analytic skills remained so-so, but that did not matter — the hard work was done by geometry, and it sufficed to fill in short calculations that even I could manage.
Ultimately, The Fractalist is proof that “force of character and independence” can take some to great heights.
Continuing with our recent exploration of sleep, I came across this passage by sleep researcher Rosalind Cartwright in The Twenty-four Hour Mind: The Role of Sleep and Dreaming in Our Emotional Lives, on the role of dreams:
Despite differences in terminology, all the contemporary theories of dreaming have a common thread—they all emphasize that dreams are not about prosaic themes, not about reading, writing, and arithmetic, but about emotion, or what psychologists refer to as affect. What is carried forward from waking hours into sleep are recent experiences that have an emotional component, often those that were negative in tone but not noticed at the time or not fully resolved. One proposed purpose of dreaming, of what dreaming accomplishes (known as the mood regulatory function of dreams theory) is that dreaming modulates disturbances in emotion, regulating those that are troublesome. My research, as well as that of other investigators in this country and abroad, supports this theory. Studies show that negative mood is down-regulated overnight. How this is accomplished has had less attention.
I propose that when some disturbing waking experience is reactivated in sleep and carried forward into REM, where it is matched by similarity in feeling to earlier memories, a network of older associations is stimulated and is displayed as a sequence of compound images that we experience as dreams. This melding of new and old memory fragments modifies the network of emotional self-defining memories, and thus updates the organizational picture we hold of ‘who I am and what is good for me and what is not.’ In this way, dreaming diffuses the emotional charge of the event and so prepares the sleeper to wake ready to see things in a more positive light, to make a fresh start. This does not always happen over a single night; sometimes a big reorganization of the emotional perspective of our self-concept must be made — from wife to widow or married to single, say, and this may take many nights. We must look for dream changes within the night and over time across nights to detect whether a productive change is under way. In very broad strokes, this is the definition of the mood-regulatory function of dreaming, one basic to the new model of the twenty-four hour mind I am proposing.
(*Update*) While not on dreams, this passage on circadian rhythms from Zoobiquity: What Animals Can Teach Us About Health and the Science of Healing, fits very much into our recent look at sleep:
Studies have shown that disrupting circadian rhythms by even one hour during the switch to daylight saving time may increase depression, traffic accidents, and heart attacks. These rhythms affect consumption and metabolism in animals—it is hard to imagine that they aren’t also playing a role in human appetites as well. Controlling environmental light with lamps, TVs, and computers gives us incredible flexibility and productivity. But it interrupts daily and yearly cycles that were billions of years in the making and are shared by countless creatures on our planet.
Michael Mauboussin commenting on Daniel Kahneman:
When asked which was his favorite paper of all-time, Daniel Kahneman pointed to “On the Psychology of Prediction,” which he co-authored with Amos Tversky in 1973. Tversky and Kahneman basically said that there are three things to consider in order to make an effective prediction: the base rate, the individual case, and how to weight the two. In luck-skill language, if luck is dominant you should place most weight on the base rate, and if skill is dominant then you should place most weight on the individual case. And the activities in between get weightings that are a blend.
In fact, there is a concept called the “shrinkage factor” that tells you how much you should revert past outcomes to the mean in order to make a good prediction. A shrinkage factor of 1 means that the next outcome will be the same as the last outcome and indicates all skill, and a factor of 0 means the best guess for the next outcome is the average. Almost everything interesting in life is in between these extremes.
To make this more concrete, consider batting average and on-base percentage, two statistics from baseball. Luck plays a larger role in determining batting average than it does in determining on-base percentage. So if you want to predict a player’s performance (holding skill constant for a moment), you need a shrinkage factor closer to 0 for batting average than for on-base percentage.
I’d like to add one more point that is not analytical but rather psychological. There is a part of the left hemisphere of your brain that is dedicated to sorting out causality. It takes in information and creates a cohesive narrative. It is so good at this function that neuroscientists call it the “interpreter.”
Now no one has a problem with the suggestion that future outcomes combine skill and luck. But once something has occurred, our minds quickly and naturally create a narrative to explain the outcome. Since the interpreter is about finding causality, it doesn’t do a good job of recognizing luck. Once something has occurred, our minds start to believe it was inevitable. This leads to what psychologists call “creeping determinism” – the sense that we knew all along what was going to happen. So while the single most important concept is knowing where you are on the luck-skill continuum, a related point is that your mind will not do a good job of recognizing luck for what it is.
Mauboussin is the author of The Success Equation: Untangling Skill and Luck in Business, Sports, and Investing.
A great passage from Erik Hollnagel‘s Barriers And Accident Prevention on the search for causes:
Whenever an accident happens there is a natural concern to find out in detail exactly what happened and to determine the causes of it. Indeed, whenever the result of an action or event falls significantly short of what was expected, or whenever something unexpected happens, people try to find an explanation for it. This trait of human nature is so strong that we try to find causes even when they do not exist, such as in the case of misleading or spurious correlations. For a number of reasons humans seem to be extremely reluctant to accept that something can happen by chance. One very good reason is that we have created a way of living that depends heavily on the use of technology, and that technological systems are built to function in a deterministic, hence reliable manner. If therefore something fails, we are fully justified in trying to find the reason for it. A second reason is that our whole understanding of the world is based on the assumption of specific relations between causes and effects, as amply illustrated by the Laws of Physics. (Even in quantum physics there are assumptions of more fundamental relations that are deterministic.) A third reason is that most humans find it very uncomfortable when they do not know what to expect, i.e., when things happen in an unpredictable manner. This creates a sense of being out of control, something that is never desirable since – from an evolutionary perspective – it means that the chances of survival are reduced.
This was described by Friedrich Nietzsche when he wrote:
[T]o trace something unknown back to something known is alleviating, soothing, gratifying and gives moreover a feeling of power. Danger, disquiet, anxiety attend the unknown – the first instinct is to eliminate these distressing states. First principle: any explanation is better than none … The cause-creating drive is thus conditioned and excited by the feeling of fear.
A well-known example of this is provided by the phenomenon called the gambler’s fallacy. The name refers to the fact that gamblers often seem to believe that a long row of events of one type increases the probability of the complementary event. Thus if a series of ‘red’ events occur on a roulette wheel, the gambler’s fallacy lead people to believe that the probability of ‘black’ increases. … Rather than accepting that the underlying mechanism may be random people invent all kinds of explanations to reduce the uncertainty of future events.
“The decision to reject one paradigm is always simultaneously the decision to accept another, and the judgment leading to that decision involves the comparison of both paradigms with nature and with each other.”
The progress of science is commonly perceived of as a continuous, incremental advance, where new discoveries add to the existing body of scientific knowledge. This view of scientific progress, however, is challenged by the physicist and philosopher of science Thomas Kuhn, in his book The Structure of Scientific Revolutions. Kuhn argues that the history of science tells a different story, one where science proceeds with a series of revolutions interrupting normal incremental progress.
“A prevailing theory or paradigm is not overthrown by the accumulation of contrary evidence,” Richard Zeckhauser wrote, “but rather by a new paradigm that, for whatever reasons, begins to be accepted by scientists.”
Between scientific revolutions, old ideas and beliefs persist. These form the barriers of resistance to alternative explanations.
Zeckhauser continues “In this view, scientific scholars are subject to status quo persistence. Far from being objective decoders of the empirical evidence, scientists have decided preferences about the scientific beliefs they hold. From a psychological perspective, this preference for beliefs can be seen as a reaction to the tensions caused by cognitive dissonance. ”
* * *
Gary Taubes posted an excellent blog post discussing how paradigm shifts come about in science. He wrote:
…as Kuhn explained in The Structure of Scientific Revolutions, his seminal thesis on paradigm shifts, the people who invariably do manage to shift scientific paradigms are “either very young or very new to the field whose paradigm they change… for obviously these are the men [or women, of course] who, being little committed by prior practice to the traditional rules of normal science, are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them.”
So when a shift does happen, it’s almost invariably the case that an outsider or a newcomer, at least, is going to be the one who pulls it off. This is one thing that makes this endeavor of figuring out who’s right or what’s right such a tricky one. Insiders are highly unlikely to shift a paradigm and history tells us they won’t do it. And if outsiders or newcomers take on the task, they not only suffer from the charge that they lack credentials and so credibility, but their work de facto implies that they know something that the insiders don’t – hence, the idiocy implication.
…This leads to a second major problem with making these assessments – who’s right or what’s right. As Kuhn explained, shifting a paradigm includes not just providing a solution to the outstanding problems in the field, but a rethinking of the questions that are asked, the observations that are considered and how those observations are interpreted, and even the technologies that are used to answer the questions. In fact, often the problems that the new paradigm solves, the questions it answers, are not the problems and the questions that practitioners living in the old paradigm would have recognized as useful.
“Paradigms provide scientists not only with a map but also with some of the direction essential for map-making,” wrote Kuhn. “In learning a paradigm the scientist acquires theory, methods, and standards together, usually in an inextricable mixture. Therefore, when paradigms change, there are usually significant shifts in the criteria determining the legitimacy both of problems and of proposed solutions.”
As a result, Kuhn said, researchers on different sides of conflicting paradigms can barely discuss their differences in any meaningful way: “They will inevitably talk through each other when debating the relative merits of their respective paradigms. In the partially circular arguments that regularly result, each paradigm will be shown to satisfy more or less the criteria that it dictates for itself and to fall short of a few of those dictated by its opponent.”
But Taubes’ explanation wasn’t enough to satisfy my curiosity.
The Structure of Scientific Revolutions
To learn more on how paradigm shifts happen, I purchased Kuhn’s book, The Structure of Scientific Revolutions, and started to investigate.
“The decision to reject one paradigm is always simultaneously the decision to accept another, and the judgment leading to that decision involves the comparison of both paradigms with nature and with each other.”
Anomalies are not all bad.
Yet any scientist who pauses to examine and refute every anomaly will seldom get any work done.
…during the sixty years after Newton’s original computation, the predicted motion of the moon’s perigee remained only half of that observed. As Europe’s best mathematical physicists continued to wrestle unsuccessfully with the well-known discrepancy, there were occasional proposals for a modification of Newton’s inverse square law. But no one took these proposals very seriously, and in practice this patience with a major anomaly proved justified. Clairaut in 1750 was able to show that only the mathematics of the application had been wrong and that Newtonian theory could stand as before. … persistent and recognized anomaly does not always induce crisis. … It follows that if an anomaly is to evoke crisis, it must usually be more than just an anomaly.
So what makes an anomaly worth the effort of investigation?
To that question Kuhn responds, “there is probably no fully general answer.” Einstein knew how to sift the essential from the non-essential better than most.
When the anomaly comes to be recognized as more than another puzzle of science the transition, or revolution, has begun.
The anomaly itself now comes to be more generally recognized as such by the profession. More and more attention is devoted to it by more and more of the field’s most eminent men. If it still continues to resist, as it usually does not, many of them may come to view its resolution as the subject matter of their discipline. …
Early attacks on the anomaly will have followed the paradigm rules closely. As time passes and scrutiny increases, more of the attacks will start to diverge from the existing paradigm. It is “through this proliferation of divergent articulations,” Kuhn argues, “the rules of normal science become increasing blurred.
Though there still is a paradigm, few practitioners prove to be entirely agreed about what it is. Even formally standard solutions of solved problems are called into question.”
Einstein explained this transition, which is the structure of scientific revolutions, best. He said: “It was as if the ground had been pulled out from under one, with no firm foundation to be seen anywhere, upon which one could have built.”
All scientific crises begin with the blurring of a paradigm.
In this respect research during crisis very much resembles research during the pre-paradigm period, except that in the former the locus of difference is both smaller and more clearly defined. And all crises close in one of three ways. Sometimes normal science ultimately proves able to handle the crisis—provoking problem despite the despair of those who have seen it as the end of an existing paradigm. On other occasions the problem resists even apparently radical new approaches. Then scientists may conclude that no solution will be forthcoming in the present state of their field. The problem is labelled and set aside for a future generation with more developed tools. Or, finally, the case that will most concern us here, a crisis may end up with the emergence of a new candidate for paradigm and with the ensuing battle over its acceptance.
But this isn’t easy.
The transition from a paradigm in crisis to a new one from which a new tradition of normal science can emerge is far from a cumulative process, one achieved by an articulation or extension of the old paradigm. Rather it is a reconstruction of the field from new fundamentals, a reconstruction that changes some of the field’s most elementary theoretical generalizations as well as many of its paradigm methods and applications.
Who solves these problems? Do the men and women who have invested a large portion of their lives in a field or theory suddenly confront evidence and change their mind? Sadly, no.
Almost always the men who achieve these fundamental inventions of a new paradigm have been either very young, or very new to the field whose paradigm they change. And perhaps that point need not have been made explicit, for obviously these are men who, being little committed by prior practice to the traditional rules of normal science, are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them.
Therefore, when paradigms change, there are usually significant shifts in the criteria determining the legitimacy both of problems and of proposed solutions.
That observation returns us to the point from which this section began, for it provides our first explicit indication of why the choice between competing paradigms regularly raises questions that cannot be resolved by the criteria of normal science. To the extent, as significant as it is incomplete, that two scientific schools disagree about what is a problem and what is a solution, they will inevitably talk through each other when debating the relative merits of their respective paradigms. In the partially circular arguments that regularly result, each paradigm will be shown to satisfy more or less the criteria that it dictates for itself and to fall short of a few of those dictated by its opponent. There are other reasons, too, for the incompleteness of logical contact that consistently characterizes paradigm debates. For example, since no paradigm ever solves all the problems it defines and since no two paradigms leave all the same problems unsolved, paradigm debates always involve the question: Which problems is it more significant to have solved? Like the issue of competing standards, that questions of values can be answered only in terms of criteria that lie outside of normal science altogether.
Many years ago Max Planck offered this insight: “A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.”
If you’re interested in learning more about how paradigm shifts happen, read The Structure of Scientific Revolutions.