Mathematicians From The United States

"Any bit of nonsense can be computerized—astrology, biorhythms, the I Ching—but that doesn’t make the nonsense any more valid."

"I remember thinking of mathematics as a kind of omnipotent protector. You could prove things to people and they would have to believe you whether they liked you or not."

"I’ve argued that the set of standard questions journalists ask and readers want answers should be enlarged. Besides Who, What, Where, When, Why, and How, it should include How many? How likely? What fraction? How does the quantity compare with other quantities? What is its rate of growth, and how does that compare? What about the self-referential aspects of the story? Is there an appropriate degree of complexity in it? Are we looking at the right categories and relations? How much of the story is independent of its reporting? Are we especially vulnerable to the availability error or to anchoring effects? If statistics are presented, how were they obtained? How confident can we be of them? Were they derived from a random sample or from a collection of anecdotes? Does the correlation suggest a causal relationship, or is it merely a coincidence? And do we understand how the people and various pieces of an organization reported upon are connected? What is known about the dynamics of the whole system? Are they stable or do they seem sensitive to tiny perturbations? Are there other ways to tally any figures presented? Do such figures measure what they purport to measure? Is the precision recounted meaningful?"

"There surely is something to these terms, but too often they’re the result of minds intent on discovering meaning where there is only probability."

"Those who can make you believe absurdities can make you commit atrocities."

"To follow foolish precedents, and wink with both eyes, is easier than to think."

"It’s always healthy to recognize facts."

"Almost any bunkum has some partial validity, and we regularly read into the confusing mess what we want to see."

"All art, in fact, has these two aspects: its content and its frame (or setting), which sets it apart from nonart and which says of itself, “This is not an everyday sort of communication. This is unreal.”"

"Innumerate people characteristically have a strong tendency to personalize—to be misled by their own experiences, or by the media’s focus on individuals and drama."

"A tendency to drastically underestimate the frequency of coincidences is a prime characteristic of innumerates, who generally accord great significance to correspondences of all sorts while attributing too little significance to quite conclusive but less flashy statistical evidence."

"Humor, since it depends on so many emotional, social, and intellectual facets of human beings, is particularly immune to computer simulation."

"The necessity of this psychic stepping back (or up) to the metalevel is probably what is meant when people say that a sense of perspective is needed for an appreciation of humor. It also explains why dogmatists, idealogues, and others with one-track minds are often notoriously humorless."

"Appreciating humor—even recognizing it—requires human skills of the highest order (level?); no computer comes close to having them."

"After all, one must have some grasp of logic even to recognize a non sequitur."

"The moral, again, is that some unlikely event is likely to occur, whereas it’s much less likely that a particular one will...The paradoxical conclusion is that it would be very unlikely for unlikely events not to occur. If you don’t specify a predicted event precisely, there are an indeterminate number of ways for an event of that general kind to take place."

"Disproving a claim that something exists is often quite difficult, and this difficulty is often mistaken for evidence that the claim is true...Presented as I am periodically with these and other fantastical claims, I sometimes feel a little like a formally dressed teetotaler at a drunken orgy for reiterating that not being able to conclusively refute the claims does not constitute evidence for them."

"Define God in a sufficiently nebulous way as beauty, love, mysterious complexity, or the ethereal taste of strawberry shortcake, and most atheists become theists. Still, although one can pose as Humpty Dumpty and aver, “When I use a word, it means just what I choose it to mean, neither more nor less,” others needn’t play along."

"While not a panacea, candidly recognizing the absence of any good logical arguments for God’s existence, giving up on divine allies and advocates as well as taskmasters and tormentors, and prizing a humane, reasonable, and brave outlook just might help move this world a bit closer to a heaven on earth."

"Together the, two ingredients—a perceived incongruity with a point and an appropriate emotional climate—seem to be both necessary and sufficient for humor."

"There’s always enough random success to justify almost anything to someone who wants to believe."

"Two dangers threaten the world—order and disorder."

"When the law’s on your side, pound the law. When the facts are on your side, pound the facts. And when neither is on your side, pound the table."

"Gullible citizens are a demagogue’s dream."

"One can and should debate whether the tests in question are appropriate for the purposes at hand, but one shouldn’t be surprised when normal curves behave normally."

"In general, any differences between two groups will always be greatly accentuated at the extremes."

"Even the most superficial of a newspaper reveals an important aspect of human psychology: our preoccupation with the short term."

"Rigid distinctions between the deep and the shallow are generally themselves quite superficial."

"There is no such thing as free lunch, and even if there were, there’d be no guarantee against indigestion."

"Too often, this concern for the big picture is simply obscurantist and is put forward by people who prefer vagueness and mystery to (partial) answers. Vagueness is at times necessary and mystery is never in short supply, but I don’t think they’re anything to worship. Genuine science and mathematical precision are more intriguing than are the “facts” published in supermarket tabloids or a romantic innumeracy which fosters credulity, stunts skepticism, and dulls one to real imponderables."

"Correlation and causation are two quite different words, and the innumerate are more prone to mistake them than most."

"It’s time to let the secret out: mathematics is not primarily a matter of plugging numbers into formulas and performing rote computations. It is a way of thinking and questioning that may be unfamiliar to many of us, but is available to almost all of us."

"If we’re not keenly aware of the choices we’re making, we’re not likely to work for better ones."

"Our two most basic political ideals—liberty and equality—are, in their purest forms, incompatible. Complete liberty results in inequality, and mandatory equality leads to a loss of liberty."

"Having been involved in a couple of lawsuits as an expert probability witness and having observed that a prudent skepticism is often less prized than an indefensible certainty, I turned down preliminary requests from both sides to testify."

"The fashion pages have always puzzled me. In my smugly ignorant view, the articles appear to be so full of fluff and nonsense as to make the astrology columns seem insightful by comparison."

"You can only predict things after they’ve happened."

"There are, of course, innumerable abuses, countless possible misinterpretations, and depressingly many biased studies, but, done right, the process works; it yields knowledge."

"Always be smart; seldom be certain."

"There is a fine line between public expressions of faith and aggressive declarations thereof, and religious tolerance is inversely proportional to the latter."

"Bad things happen periodically, and they’re going to happen to somebody. Why not you?"

"Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of axiomatic approaches. However, there are internal problems with real or quaternionic quantum theory. Here we argue that these problems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the "three-fold way". ... This three-fold classification sheds light on the physics of time reversal symmetry, and it already plays an important role in particle physics."

"The conformal invariance of the Yang-Mills equations in four dimensions greatly facilitates the study of the temporal asymptotic behavior of their solutions."

"The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on. The complex numbers are a slightly flashier but still respectable younger brother: not ordered, but algebraically complete. The quaternions, being noncommutative, are the eccentric cousin who is shunned at important family gatherings. But the octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative."

"Besides their possible role in physics, the octonions are important because they tie together some algebraic structures that otherwise appear as isolated and inexplicable exceptions."

"The most popular approach to quantum gravity is string theory. Despite decades of hard work by many very smart people, it's far from clear that this theory is successful. It's made no predictions that have been confirmed by experiment. In fact, it's made few predictions that we have any hope of testing anytime soon! Finding certain sorts of particles at the big new particle accelerator near Geneva would count as partial confirmation, but string theory says very little about the details of what we should expect. In fact, thanks to the vast "landscape" of string theory models that researchers are uncovering, it keeps getting harder to squeeze specific predictions out of this theory."

"When the Löwenheim-Skolem theorem is applied to particular formal systems, we obtain as special cases: Every group, field, ordered field, etc., has a countable subsystem of the same type. A more spectacular result follows from applying the theorem to set theory (a system which we shall later formalize): There is a countable collection of sets, such that if restrict the membership relation to these sets alone, they form a model for set theory (more precisely all the true statements of set theory are true in this model). In particular, within this model which we may denote by M, there must be an uncountable set. This paradox, that a countable model can contain an uncountable set, is explained by noting that to say a set is uncountable merely asserts the nonexistence of a one-one mapping of the set with the set of integers. The "uncountable" set in M set actually has only countably many members in M, but there is no one-one correspondence \underline {within} M of this set with the set of integers."

"In 1963 P. J. Cohen completed Gödel's linguistic attack on set theory by introducing the immensely valuable, syntactic notion of forcing, and by using it to demonstrate that the axioms of set theory were not powerful enough to prove Cantor's continuum hypothesis. Thus the presently existing axioms of set theory leave the most celebrated hypothesis of set theory shrouded in uncertainty."

"To him mathematics was a unified subject that one could master broadly. He had a deep understanding of most areas, and he taught advanced courses in logic, analysis, differential equations, algebra, topology, Lie theory, and number theory on a regular basis. He felt that good mathematics should be easy to understand and that it is always based on simple ideas once you get to the bottom of the issue. This attitude extended to a strong belief that the well-recognized unsolved problems in mathematics are the heart of the subject and have clear and transparent solutions once the right new ideas and viewpoints are found. This belief gave him the courage to work on notoriously difficult problems throughout his career."

"The object of mathematics is to discover "true" theorems. We shall use the term "valid" to describe statements formed according to certain rules and then shall discuss how this notion compares with the intuitive idea of "true"."