Have you ever gone to the phone to call a friend only to have your friend ring you first? What are the odds of that? Not high, to be sure, but the sum of all probabilities equals one. Given enough opportunities, outlier anomalies — even seeming miracles — will occasionally happen.

Let us define a miracle as an event with million-to-one odds of occurring (intuitively, that seems rare enough to earn the moniker). Let us also assign a number of one bit per second to the data that flow into our senses as we go about our day and assume that we are awake for 12 hours a day. We get 43,200 bits of data a day, or 1.296 million a month. Even assuming that 99.999 percent of these bits are totally meaningless (and so we filter them out or forget them entirely), that still leaves 1.3 “miracles” a month, or 15.5 miracles a year

Thanks to our confirmation bias, in which we look for and find confirmatory evidence for what we already believe and ignore or discount disconfirming evidence, we will remember only those few astonishing coincidences and forget the vast sea of meaningless data.

We can employ a similar back-of-the-envelope calculation to explain death premonition dreams. The average person has about five dreams a night, or 1,825 dreams a year. If we remember only a tenth of our dreams, then we recall 182.5 dreams a year. There are 300 million Americans, who thus produce 54.7 billion remembered dreams a year. Sociologists tell us that each of us knows about 150 people fairly well, thus producing a network social grid of 45 billion personal relationship connections. With an annual death rate of 2.4 million Americans, it is inevitable that some of those 54.7 billion remembered dreams will be about some of these 2.4 million deaths among the 300 million Americans and their 45 billion relationship connections. In fact, it would be a miracle if some death premonition dreams did not happen to come true!

These examples show the power of probabilistic thinking to override our intuitive sense of numbers, or what I call “folk numeracy,” in parallel with my previous columns on “folk science” (August 2006) and “folk medicine” (August 2008) and with my book on “folk economics” (The Mind of the Market). Folk numeracy is our natural tendency to misperceive and miscalculate probabilities, to think anecdotally instead of statistically, and to focus on and remember short-term trends and small-number runs. We notice a short stretch of cool days and ignore the long-term global-warming trend. We note with consternation the recent downturn in the housing and stock markets, forgetting the half-century upward-pointing trend line. Sawtooth data trend lines, in fact, are exemplary of folk numeracy: our senses are geared to focus on each tooth’s up or down angle, whereas the overall direction of the blade is nearly invisible to us.

The reason that our folk intuitions so often get it wrong is that we evolved in what evolutionary biologist Richard Dawkins calls “Middle World” — a land midway between short and long, small and large, slow and fast, young and old. Out of personal preference, I call it “Middle Land.” In the Middle Land of space, our senses evolved for perceiving objects of middling size — between, say, grains of sand and mountain ranges. We are not equipped to perceive atoms and germs, on one end of the scale, or galaxies and expanding universes, on the other end. In the Middle Land of speed, we can detect objects moving at a walking or running pace, but the glacially slow movement of continents (and glaciers) and the mind-bogglingly fast speed of light are imperceptible. Our Middle Land timescales range from the psychological “now” of three seconds in duration (according to Harvard University psychologist Stephen Pinker) to the few decades of a human lifetime, far too short to witness evolution, continental drift or long-term environmental changes. Our Middle Land folk numeracy leads us to pay attention to and remember short-term trends, meaningful coincidences and personal anecdotes.

Next month, in Part 2, we will consider how randomness rules our lives through the metaphor of “the drunkard’s walk,” well elucidated by physicist Leonard Mlodinow of the California Institute of Technology in his new book of the same title.

In belles lettres the witty literary slight has evolved into a genre because, as 20th-century trial lawyer Louis Nizer noted, “A graceful taunt is worth a thousand insults.” To wit, from high culture, Mark Twain: “I didn’t attend the funeral, but I sent a nice letter saying I approved of it.” Winston Churchill: “He has all the virtues I dislike and none of the vices I admire.” And from pop culture, Groucho Marx: “I’ve had a perfectly wonderful evening. But this wasn’t it.” Scientists are no slouches when it comes to pitching invectives at colleagues. Achieving almost canonical status as the ne plus ultra put-down is theoretical physicist Wolfgang Pauli’s reported harsh critique of a paper: “This isn’t right. It’s not even wrong.” I call this Pauli’s proverb.

Columbia University mathematician Peter Woit recently employed Pauli’s proverb in his book title, a critique of string theory called Not Even Wrong (Basic Books, 2006). String theory, Woit argues, is not only based on nontestable hypotheses, it depends far too much on the aesthetic nature of its mathematics and the eminence of its proponents. In science, if an idea is not falsifiable, it is not that it is wrong, it is that we cannot determine if it is wrong, and thus it is not even wrong.

Not even wrong. What could be worse? Being wronger than wrong, or what I call Asimov’s axiom, well stated in his book The Relativity of Wrong (Doubleday, 1988): “When people thought the earth was flat, they were wrong. When people thought the earth was spherical, they were wrong. But if you think that thinking the earth is spherical is just as wrong as thinking the earth is flat, then your view is wronger than both of them put together.”

Asimov’s axiom holds that science is cumulative and progressive, building on the mistakes of the past, and that even though scientists are often wrong, their wrongness attenuates with continued data collection and theory building. Satellite measurements, for instance, have shown precisely how the earth’s shape differs from a perfect sphere.

The view that all wrong theories are equal implies that no theory is better than any other. This is the theory of the “strong” social construction of science, which holds that science is inextricably bound to the social, political, economic, religious and ideological predilections of a culture, particularly of those individuals in power. Scientists are knowledge capitalists who produce scientific papers that report the results of experiments conducted to test (and usually support) the hegemonic theories that reinforce the status quo.

In some extreme cases, this theory that culture shapes the way science is conducted is right. In the mid-19th century, physicians discovered that slaves suffered from drapetomania, or the uncontrollable urge to escape from slavery, and dysaethesia aethiopica, or the tendency to be disobedient. In the late 19th and early 20th centuries, scientific measurements of racial differences in cognitive abilities found that blacks were inferior to whites. In the mid- 20th century, psychiatrists discovered evidence that allowed them to classify homosexuality as a disease. And until recently, women were considered inherently inferior in science classrooms and corporate boardrooms.

Such egregious examples, however, do not negate the extraordinary ability of science to elucidate the natural and social worlds. Reality exists, and science is the best tool yet employed to discover and describe that reality. The theory of evolution, even though it is the subject of vigorous debates about the tempo and mode of life’s history, is vastly superior to the theory of creation, which is not even wrong (in Pauli’s sense). As evolutionary biologist Richard Dawkins observed on this dispute: “When two opposite points of view are expressed with equal intensity, the truth does not necessarily lie exactly halfway between them. It is possible for one side to be simply wrong.”

Simply wrong. When people thought that science was unbiased and unbound by culture, they were simply wrong. On the other hand, when people thought that science was completely socially constructed, they were simply wrong. But if you believe that thinking science is unbiased is just as wrong as thinking that science is socially constructed, then your view is not even wronger than wrong.

I had long wanted to meet Edward R. Tufte — the man the New York Times called “the da Vinci of data” because of his concisely written and artfully illustrated books on the visual display of data — and invite him to speak at the Skeptics Society science lecture series that I host at the California Institute of Technology. Tufte is one of the world’s leading experts on a core tool of skepticism: how to see through information obfuscation.

But how could we afford someone of his stature? “My honorarium,” he told me, “is to see Feynman’s van.”

Richard Feynman, the late Caltech physicist, is famous for working on the atomic bomb, winning a Nobel Prize in Physics, cracking safes, playing drums and driving a 1975 Dodge Maxivan adorned with squiggly lines on the side panels. Most people who saw it gazed in puzzlement, but once in a while someone would ask the driver why he had Feynman diagrams all over his van, only to be told, “Because I’m Richard Feynman!”

Feynman diagrams are simplified visual representations of the very complex world of quantum electrodynamics (QED), in which particles of light called photons are depicted by wavy lines, negatively charged electrons are depicted by straight or curved nonwavy lines, and line junctions show electrons emitting or absorbing a photon. In the diagram on the back door of the van, seen in the photograph above with Tufte, time flows from bottom to top. The pair of electrons (the straight lines) are moving toward each other. When the left-hand electron emits a photon (wavy-line junction), that negatively charged particle is deflected outward left; the right-hand electron reabsorbs the photon, causing it to deflect outward right.

Feynman diagrams are the embodiment of what Tufte teaches about analytical design: “Good displays of data help to reveal knowledge relevant to understanding mechanism, process and dynamics, cause and effect.” We see the unthinkable and think the unseeable. “Visual representations of evidence should be governed by principles of reasoning about quantitative evidence. Clear and precise seeing becomes as one with clear and precise thinking.”

The master of clear and precise thinking meets the master of clear and precise seeing in what I call the Feynman-Tufte Principle: a visual display of data should be simple enough to fit on the side of a van.

As Tufte poignantly demonstrated in his analysis of the space shuttle Challenger disaster, despite the 13 charts prepared for NASA by Thiokol (the makers of the solid-rocket booster that blew up), they failed to communicate the link between cool temperature and O-ring damage on earlier flights. The loss of the Columbia, Tufte believes, was directly related to “a PowerPoint festival of bureaucratic hyperrationalism” in which a single slide contained six different levels of hierarchy (chapters and subheads), thereby obfuscating the conclusion that damage to the left wing might have been significant. In his 1970 classic work The Feynman Lectures on Physics, Feynman covered all of physics — from celestial mechanics to quantum electrodynamics — with only two levels of hierarchy.

Tufte codified the design process into six principles: “(1) documenting the sources and characteristics of the data, (2) insistently enforcing appropriate comparisons, (3) demonstrating mechanisms of cause and effect, (4) expressing those mechanisms quantitatively, (5) recognizing the inherently multivariate nature of analytic problems, (6) inspecting and evaluating alternative explanations.” In brief, “information displays should be documentary, comparative, causal and explanatory, quantified, multivariate, exploratory, skeptical.”

Skeptical. How fitting for this column, opus 50 for me, because when I asked Tufte to summarize the goal of his work, he said, “Simple design, intense content.” Because we all need a mark at which to aim (one meaning of “skeptic”), “simple design, intense content” is a sound objective for this series.

Consider the following quotes, written by authors of recently self-published books purporting to revolutionize science:

“This book is the culmination of nearly twenty years of work that I have done to develop that new kind of science. I had never expected it would take anything like as long, but I have discovered vastly more than I ever thought possible, and in fact what I have done now touches almost every existing area of science, and quite a bit besides … I have come to view [my discovery] as one of the more important single discoveries in the whole history of theoretical science.”

“The development of this work has been a completely solitary effort during the past thirty years. As you will realize as you read through this book, these ideas had to be developed by an outsider. They are such a complete reversal of contemporary thinking that it would have been very difficult for any one part of this integrated theoretical system to be developed within the rigid structure of institutional science.”

Both authors worked in isolation for years. Both produced remarkably self-consistent theories and make equally extravagant claims about overturning the foundations of physics in particular and science in general. Both shunned the traditional route of submitting their work to peer-reviewed scientific journals and instead chose to take their ideas straight to the public. And both texts are filled with self-produced diagrams and illustrations alleging to reveal the fundamental structures of nature.

There is one distinct difference between the two authors: one was featured in Time, Newsweek and Wired, and his book was reviewed in the New York Times. The other has been largely ignored, apart from a few exhibits at art museums. Their bios help to clarify these dissimilar receptions.

One of the authors earned his Ph.D. in physics at age 20 at the California Institute of Technology, where Richard Feynman called him “astonishing,” and he was the youngest to ever win a prestigious MacArthur “genius award.” He founded an institute for the study of complexity at a major university, then quit to start his own software company, where he produced a wildly successful computer program used by millions of scientists and engineers. The other author has been an abalone diver, gold miner, filmmaker, cave digger, repairman, inventor and owner-operator of a trailer park. Can you guess the names of the authors and which author penned which quote?

The first quote comes from Stephen Wolfram, the Caltech whiz and author of A New Kind of Science, in which the fundamental structure of the universe and everything in it is reduced to computational rules and algorithms that produce complexity in the form of cellular automata. The second comes from James Carter, the abalone diver and author of The Other Theory of Physics, proffering a “circlon” theory of the universe, wherein all matter is founded on hollow, ring-shaped tubes that link everything together.

Whether Wolfram is correct remains to be seen, but eventually we will find out because his ideas will be tested in the competitive marketplace of science. We may never know the veracity of Carter’s ideas. Why? Because, like it or not, in science, as in most human intellectual endeavors, who is doing the saying matters as much as what is being said, at least in terms of getting an initial hearing.

Science is, in this sense, conservative and sometimes elitist. It has to be in order to survive in a surfeit of would-be revolutionaries. For every Stephen Wolfram there are 100 James Carters. There needs to be some screening process whereby truly revolutionary ideas are weeded out from ersatz ones. Enter the skeptics. We are interested in the James Carters of the world because in the interstices between science and pseudoscience, the next great revolution may arise. Although most of these ideas will land on the junk heap, you never know until you look at them closely.