The only such proofs that warrant the name are those in mathematics and formal logic. Each logical step has to come with a logically sufficient justification. That way each logical step comes with binary certainty. Then the final result itself follows with binary certainty. It is as if the prover had multiplied the number 1 by itself for each step in the proof. The result is still the number 1. That is why the final result warrants a declaration of QED. That is also why the process comes to an unequivocal halt if the prover cannot justify a step. Any act of faith or guesswork or cutting corners will destroy the proof and its demand for binary certainty.
The catch is that we can really only prove tautologies.
The great binary truths of mathematics are still logically equivalent to the tautology 1 = 1 or Green is green. This differs from the factual statements we make about the real world—statements such as “Pine needles are green” or “Chlorophyll molecules reflect green light.” These factual statements are approximations. They are technically vague or fuzzy. And they often come juxtaposed with probabilistic uncertainty: “Pine needles are green with high probability.” Note that this last statement involves triple uncertainty. There is first the vagueness of green pine needles because there is no bright line between greenness and non-greenness—it is a matter of degree. There is second only a probability whether pine needles have the vague property of greenness. And there is last the magnitude of the probability itself. The magnitude is the vague or fuzzy descriptor “high,” because here, too, there is no bright line between high probability and not-high probability.
No one has ever produced a statement of fact that has the same 100 percent binary truth status as a mathematical theorem. Even the most accurate energy predictions of quantum mechanics hold only out to a few decimal places. Binary truth would require getting it right out to infinitely many decimal places.
Most scientists know this and rightly sweat it. The logical premises of a math model only approximately match the world the model purports to model. It is not at all clear how such grounding mismatches propagate through to the model’s predictions. Each infected inferential step tends to degrade the confidence of the conclusion, as if multiplying fractions less than one. Modern statistics can appeal to confidence bounds if there are enough samples and if the samples sufficiently approximate the binary assumptions of the model. That at least makes us pay in the coin of data for an increase in certainty.
It is a big step down from such imperfect scientific inference to the approximate syllogistic reasoning of the law. There the disputant insists that similar premises must lead to similar conclusions. But this similarity involves its own approximate pattern matching of inherently vague patterns of causal conduct or hidden mental states such as intent or foreseeability. The judge’s final ruling of “granted” or “denied” resolves the issue in practice. But it is technically a non sequitur. The product of any numbers between zero and one is again always less than one. So the confidence of the conclusion can only fall as the steps in the deductive chain increase. The rap of the gavel is no substitute for proof.
Such approximate reasoning may be as close as we can come to a QED moment when using natural language. The everyday arguments that buzz in our brains hit far humbler logical highs. That is precisely why we all need to prove something at least once—to experience at least one true QED moment. Those rare but godlike tastes of ideal certainty can help keep us from mistaking it for anything else.
Objects of Understanding and Communication
Richard Saul Wurman
Architect; cartographer; founder, TED Conference; author, 33: Understanding Change & the Change in Understanding
The waking dream I have for my toolkit is one filled with objects of understanding and communication.
The tools in my toolkit respond to me. They nod when I talk, give me evidence of me, and suggest secondary and tertiary journeys that extend my curiosities.
This toolkit is woven of threads of ignorance and stitches of questions that invite knowledge in.
In this weave are maps and patterns with enough stitches to allow me to make the choice, as I wish, to add a tiny drop of superglue.
I want an iPhone/iPad/iMac that nods.
The first movies archived stage shows. The iPad and Kindle archive magazines, newspapers, and books.
I want a new modality with which I can converse at differing levels of complexity, in different languages, and which understands the nuance of my questions.
I want help flying through my waking dreams connecting the threads of these epiphanies.
I believe we are at this cusp.
A first toe in the warm bath of this new modality.
Life As a Side Effect
Carl Zimmer
Journalist; author, The Tangled Bank: An Introduction to Evolution; blogger, The Loom
It’s been more than 150 years since Charles Darwin published the Origin of Species, but we still have trouble appreciating the simple, brilliant insight at its core. That is, life’s diversity does not exist because it is necessary for living things. Birds did not get wings so that they could fly. We do not have eyes so that we can read. Instead, eyes, wings, and the rest of life’s wonder have come about as a side effect of life itself. Living things struggle to survive, they reproduce, and they don’t do a perfect job of replicating themselves. Evolution spins off that loop, like heat coming off an engine. We’re so used to seeing agents behind everything that we struggle to recognize life as a side effect. I think everyone would do well to overcome that urge to see agents where there are none. It would even help us to understand why we are so eager to see agents in the first place.
The Veeck Effect
Gregory Cochran
Adjunct professor of anthropology, University of Utah; coauthor (with Henry Harpending), The 10,000-Year Explosion: How Civilization Accelerated Human Evolution
There’s an invidious rhetorical strategy we’ve all seen, and I’m afraid that most of us have inflicted it on others as well. I call it the Veeck effect (of the first kind*)—it occurs whenever someone adjusts the standards of evidence in order to favor a preferred outcome.
Why Veeck? Bill Veeck was a flamboyant baseball owner and promoter. In his autobiography, Veeck—As in Wreck, he described installing a flexible fence in the right field of the Milwaukee Brewers. At first he put the fence up only when facing a team full of power hitters, but eventually he took it to the limit, putting the fence up when the visitors were at bat and taking it down when his team was.
The history of science is littered with flexible fences. The phlogiston theory predicted that phlogiston would be released when magnesium burned. It looked bad for that theory when experiments showed that burning magnesium became heavier—but its supporters happily explained that phlogiston had negative weight.
Consider Johannes Kepler. He came up with the idea that the distances of the six (known) planets could be explained by nesting the five Platonic solids. It almost worked for Earth, Mars, and Venus but clearly failed for Jupiter. He dismissed the trouble with Jupiter, saying, “Nobody will wonder at it, considering the great distance.” The theory certainly wouldn’t have worked with any extra planets, but fortunately for Kepler’s peace of mind, Uranus was discovered well after his death.
The Veeckian urge is strong in every field, but it truly flourishes in the human and historical sciences, where the definitive experiments that would quash such nonsense are often impossible, impractical, or illegal. Nowhere is this tendency stronger than among cultural anthropologists, who at times seem to have no raison d’être other than refurbishing the reputations of cannibals.
Sometimes this has meant denying a particular case of cannibalism—for example, among the Anasazi of the American Southwest. Evidence there has piled up and up. Archaeologists have found piles of human bones with muscles scraped off, split open for marrow, polished by stirring in pots. They have even foun
d human feces with traces of digested human tissue. But that’s not good enough. For one thing, this implication of ancient cannibalism among the Anasazi is offensive to their Pueblo descendants, and that somehow trumps mounds of bloody evidence. You would think that the same principle would cause cultural anthropologists to embrace the face-saving falsehoods of other ethnic groups—didn’t the South really secede over the tariff? But that doesn’t seem to happen.
Some anthropologists have carried the effort further, denying that any culture was ever cannibalistic. They don’t just deny Anasazi archaeology; they deny every kind of evidence, from archaeology to historical accounts, even reports from people alive today. When Álvaro de Mendaña discovered the Solomon Islands, he reported that a friendly chieftain threw a feast and offered him a quarter of a boy. Made up, surely. The conquistadors described the Aztecs as a cannibal kingdom. Can’t be right, even if the archaeology supports it. When Papuans in Port Moresby volunteered to have a picnic in the morgue—to attract tourists, of course—they were just showing public spirit.
The Quaternary mass extinction, which wiped out much of the world’s megafauna, offers paleontologists a chance to crank up their own fences. The large marsupials, flightless birds, and reptiles of Australia disappeared shortly after humans arrived, about fifty thousand years ago. The large mammals of North and South America disappeared about ten thousand years ago—again, just after humans showed up. Moas disappeared within two centuries after Polynesian colonization in New Zealand, while giant flightless birds and lemurs disappeared from Madagascar shortly after humans arrived. What does this pattern suggest as the cause? Why, climate change, of course. Couldn’t be human hunters—that’s unpossible!
The Veeck effect is even more common in everyday life than it is in science. It’s just that we expect more from scientists. But scientific examples are clear-cut, easy to see, and understanding the strategy helps you avoid succumbing to it.
Whenever some administration official says that absence of evidence is not evidence of absence, whenever a psychiatrist argues that Freudian psychotherapy works for some people, even if proven useless on average, Bill Veeck’s spirit goes marching on.
Supervenience!
Joshua Greene
Cognitive neuroscientist and philosopher, Harvard University
There’s a lot of stuff in the world: trees, cars, galaxies, benzene, the Baths of Caracalla, your pancreas, Ottawa, ennui, Walter Mondale. How does it all fit together? In a word . . . supervenience. (Verb form: to supervene.) Supervenience is a shorthand abstraction, native to Anglo-American philosophy, that provides a general framework for thinking about how everything relates to everything else.
The technical definition of supervenience is somewhat awkward. Supervenience is a relationship between two sets of properties. Call them Set A and Set B. The Set A properties supervene on the Set B properties if and only if no two things can differ in their A properties without also differing in their B properties.*
This definition, while admirably precise, makes it hard to see what supervenience is really about, which is the relationships among different levels of reality. Take, for example, a computer screen displaying a picture. At a high level, at the level of images, a screen may depict an image of a dog sitting in a rowboat, curled up next to a life vest. The screen’s content can also be described as an arrangement of pixels, a set of locations and corresponding colors. The image supervenes on the pixels. This is because a screen’s image-level properties (its dogginess, its rowboatness) cannot differ from another screen’s image-level properties unless the two screens also differ in their pixel-level properties.
The pixels and the image are, in a very real sense, the same thing. But—and this is key—their relationship is asymmetrical. The image supervenes on the pixels, but the pixels do not supervene on the image. This is because screens can differ in their pixel-level properties without differing in their image-level properties. For example, the same image may be displayed at two different sizes or resolutions. And if you knock out a few pixels, it’s still the same image. (Changing a few pixels will not protect you from charges of copyright infringement.) Perhaps the easiest way to think about the asymmetry of supervenience is in terms of what determines what. Determining the pixels completely determines the image, but determining the image does not completely determine the pixels.
The concept of supervenience deserves wider currency because it allows us to think clearly about many things, not just about images and pixels. Supervenience explains, for example, why physics is the most fundamental science and why the things that physicists study are the most fundamental things. To many people, this sounds like a value judgment, but it’s not, or need not be. Physics is fundamental because everything in the universe, from your pancreas to Ottawa, supervenes on physical stuff. (Or so “physicalists” like me claim.) If there is a universe physically identical to ours, it would include a pancreas just like yours and an Ottawa just like Canada’s.
Supervenience is especially helpful when grappling with three contentious and closely related issues: (1) the relationship between science and the humanities, (2) the relationship between the mind and brain, and (3) the relationship between facts and values.
Humanists sometimes perceive science as imperialistic, as aspiring to take over the humanities, to “reduce” everything to electrons, genes, numbers, neurons—and thus to “explain away” all the things that make life worth living. Such thoughts are accompanied by disdain or fear, depending on how credible such ambitions are taken to be. Scientists, for their part, are indeed sometimes imperious, dismissing humanists and their pursuits as childish and unworthy of respect. Supervenience can help us think about how science and the humanities fit together, why science is sometimes perceived as encroaching on the humanist’s territory, and the extent to which such perceptions are and are not valid.
It would seem that humanists and scientists study different things. Humanists are concerned with things like love, revenge, beauty, cruelty, and our evolving conceptions of such matters. Scientists study things like electrons and nucleotides. But sometimes it seems like scientists are getting greedy. Physicists aspire to construct a complete physical theory, often called a “Theory of Everything” (TOE). If humanists and scientists study different things, and if physics covers everything, then what is left for the humanists? (Or, for that matter, nonphysicists?)
There is a sense in which a TOE really is a TOE, and there is a sense in which it’s not. A TOE is a complete theory of everything upon which everything else supervenes. If two worlds are physically identical, then they are also humanistically identical, containing exactly the same love, revenge, beauty, cruelty, and conceptions thereof. But that does not mean that a TOE puts all other theorizing out of business—not by a long shot. A TOE won’t tell you anything interesting about Macbeth or the Boxer Rebellion.
Perhaps the threat from physics was never all that serious. Today, the real threat, if there is one, is from the behavioral sciences, especially the sciences that connect the kind of “hard” science we all studied in high school to humanistic concerns. In my opinion, three sciences stand out in this regard: behavioral genetics, evolutionary psychology, and cognitive neuroscience. I study moral judgment, a classically humanistic topic. I do this in part by scanning people’s brains while they make moral judgments. More recently I’ve started looking at genes, and my work is guided by evolutionary thinking. My work assumes that the mind supervenes on the brain, and I attempt to explain human values—for example, the tension between individual rights and the greater good—in terms of competing neural systems.
I can tell you from personal experience that this kind of work makes some humanists uncomfortable. During the discussion following a talk I gave at Harvard’s Humanities Center, a prominent professor declared that my talk—not any particular conclusion I’d drawn, but the whole approach—made him physically ill. The subject matter of the humaniti
es has always supervened on the subject matter of the physical sciences, but in the past a humanist could comfortably ignore the subvening physical details, much as an admirer of a picture can ignore the pixel-level details. Is that still true? Perhaps it is. Perhaps it depends on one’s interests. In any case, it’s nothing to be worried sick about.
The Culture Cycle
Hazel Rose Markus and Alana Conner
Hazel Rose Markus is the Davis-Brack Professor in the Behavioral Sciences at Stanford University and coauthor (with Paula M. L. Moya), Doing Race: 21 Essays for the 21st Century. Alana Conner is a science writer, social psychologist, and curator, The Tech Museum, San Jose, California.
Pundits invoke culture to explain all manner of tragedies and triumphs: why a disturbed young man opens fire on a politician; why African-American children struggle in school; why the United States can’t establish democracy in Iraq; why Asian factories build better cars. A quick click through a single morning’s media yields the following catch: gun culture, Twitter culture, ethical culture, Arizona culture, always-on culture, winner-take-all culture, culture of violence, culture of fear, culture of sustainability, culture of corporate greed.
Yet no one explains what, exactly, culture is, how it works, or how to change it for the better.
A cognitive tool that fills this gap is the culture cycle, a tool that not only describes how culture works but also prescribes how to make lasting change. The culture cycle is the iterative recursive process whereby people create cultures to which they later adapt, and cultures shape people so that they act in ways that perpetuate the cultures.
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