John Allen Paulos

American mathematician

John Allen Paulos (born July 4, 1945) is an American professor of mathematics at Temple University in Philadelphia, Pennsylvania. He is a prominent figure as a writer and speaker on mathematics and the importance of mathematical literacy. Paulos writes about many subjects, especially of the dangers of mathematical innumeracy; that is, the layperson's misconceptions about numbers, probability, and logic.

Paulos in 2015


All page numbers from the 1982 paperback edition published by The University of Chicago Press ISBN 0-226-65025-1
  • Together the, two ingredients—a perceived incongruity with a point and an appropriate emotional climate—seem to be both necessary and sufficient for humor.
    • Chapter 1, “Mathematics and Humor” (p. 10)
  • After all, one must have some grasp of logic even to recognize a non sequitur.
    • Chapter 2, “Axioms, Levels, and Iteration” (p. 19)
  • 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.
    • Chapter 2, “Axioms, Levels, and Iteration” (p. 27)
  • Appreciating humor—even recognizing it—requires human skills of the highest order (level?); no computer comes close to having them.
    • Chapter 3, “Self-Reference and Paradox” (p. 50)
  • Humor, since it depends on so many emotional, social, and intellectual facets of human beings, is particularly immune to computer simulation.
    • Chapter 3, “Self-Reference and Paradox” (p. 51)
  • 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.”
    • Chapter 3, “Self-Reference and Paradox” (p. 53)
All page numbers from the 1990 trade paperback edition published by Vintage Books ISBN 0-679-72601-2
  • 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.
    • Introduction (p. 6)
  • 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.
    • Chapter 2, “Probability and Coincidence” (p. 35)
  • 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.
    • Chapter 2, “Probability and Coincidence” (pp. 37-38; ellipsis represents elision of examples)
  • There’s always enough random success to justify almost anything to someone who wants to believe.
    • Chapter 2, “Probability and Coincidence” (p. 44)
  • 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.
    • Chapter 2, “Probability and Coincidence” (p. 62)
  • To follow foolish precedents, and wink with both eyes, is easier than to think.
  • Any bit of nonsense can be computerized—astrology, biorhythms, the I Ching—but that doesn’t make the nonsense any more valid.
    • Chapter 3, “Pseudoscience” (p. 68)
  • 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.
    • Chapter 3, “Pseudoscience” (pp. 95-96; ellipsis represents elision of new age examples)
  • 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.
    • Chapter 4, “Whence Innumeracy?” (p. 99)
  • Bad things happen periodically, and they’re going to happen to somebody. Why not you?
    • Chapter 4, “Whence Innumeracy?” (p. 110)
  • 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.
    • Chapter 4, “Whence Innumeracy?” (pp. 126-127)
  • There is no such thing as free lunch, and even if there were, there’d be no guarantee against indigestion.
    • Chapter 5, “Statistics, Trade-Offs, and Society” (p. 147)
  • Correlation and causation are two quite different words, and the innumerate are more prone to mistake them than most.
    • Chapter 5, “Statistics, Trade-Offs, and Society” (p. 159)
  • If we’re not keenly aware of the choices we’re making, we’re not likely to work for better ones.
    • Chapter 5, “Statistics, Trade-Offs, and Society” (p. 176)
All page numbers from the April 1996 first trade paperback edition published by Anchor Books ISBN 0-385-48254-X
The chapters in the book are not numbered, and are numbered here for ease of reference
  • 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.
    • Introduction (p. 3)
  • You can only predict things after they’ve happened.
    • Section 1, “Politics, Economics, and the Nation” Introduction (p. 7; quoting Eugene Ionesco)
  • 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.
    • Section 1, “Politics, Economics, and the Nation” Introduction (pp. 7-8)
  • 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.
    • Chapter 9, “Fraud Alleged in Pennsylvania Senate Race” (p. 45)
  • In general, any differences between two groups will always be greatly accentuated at the extremes.
    • Section 2, “Local, Social, and Business Issues” Chapter 11, “Company Charged with Ethnic Bias in Hiring” (p. 60)
  • 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.
    • Section 2, “Local, Social, and Business Issues” Chapter 11, “Company Charged with Ethnic Bias in Hiring” (p. 61)
  • Even the most superficial of a newspaper reveals an important aspect of human psychology: our preoccupation with the short term.
    • Chapter 21, “Researchers Look to Local News for Trends” (p. 96)
  • 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.
    • Section 3, “Lifestyle, Spin, and Soft News” Chapter 23, “Tsongkerclintkinbro Wins” (p. 106)
  • Two dangers threaten the world—order and disorder.
    • Chapter 27, “Special Investigator Says Full Story Not Told” (p. 123; quoting Paul Valéry)
  • Gullible citizens are a demagogue’s dream.
    • Section 4, “Science, Medicine, and the Environment” Chapter 38 “More Dismal Math Scores for U.S. Students” (p. 165)
  • Rigid distinctions between the deep and the shallow are generally themselves quite superficial.
    • Section 5, “Food , Book Reviews, Sports, Obituaries” Introduction (p. 169)
  • 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.
    • Chapter 40, “Top Designs for the Busy Working Woman” (p. 173)
  • There are, of course, innumerable abuses, countless possible misinterpretations, and depressingly many biased studies, but, done right, the process works; it yields knowledge.
    • Chapter 42, “New Survey Reveal Changing Attitudes” (p. 180)
  • There is a fine line between public expressions of faith and aggressive declarations thereof, and religious tolerance is inversely proportional to the latter.
    • Chapter 50, “Which Way Mecca?” (p. 198)
  • Always be smart; seldom be certain.
    • Conclusion (p. 201)
  • 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?
    • Conclusion (p. 201)
  • Almost any bunkum has some partial validity, and we regularly read into the confusing mess what we want to see.
    • Conclusion (p. 202)
All page numbers from the first trade paperback edition published in 2009 by Hill and Wang ISBN 978-0-8090-5918-8
  • Those who can make you believe absurdities can make you commit atrocities.
  • 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.
  • For the record, natural selection is a highly nonrandom process that acts on the genetic variation produced by random mutation and genetic drift and results in those organisms with more adaptive traits differentially surviving and reproducing.
    • Part 1 “Four Classical Arguments”, Chapter 2 “The Argument from Design (and Some Creationist Calculations)” (p. 19)
  • You would think that the obvious irreligious objection would come to almost anyone’s mind when reading a religious tome or holy book. What if you don’t believe the holy book’s presuppositions and narrative claims and simply ask for independent argument or evidence for God’s existence? What if you’re not persuaded by the argument that God exists because His assertion that He exists and discussion of His various exploits appear in this book about Him that believers say He inspired?
    • Part 2 “Four Subjective Arguments”, Chapter 2 “The Argument from Prophecy (and the Bible Codes)” (p. 63)
  • Claiming that a holy book’s claims are undeniable because the book itself claims them to be is convincing only to the convinced.
    • Part 2 “Four Subjective Arguments”, Chapter 2 “The Argument from Prophecy (and the Bible Codes)” (p. 64)
  • Confirmation of a person’s unreliable statement by another unreliable person makes the statement even less reliable.
    • Part 2 “Four Subjective Arguments”, Chapter 2 “The Argument from Prophecy (and the Bible Codes)” (p. 65)
  • The whole weight of science is the prima facie evidence against a miracle having occurred.
    • Part 2 “Four Subjective Arguments”, Chapter 5 “The Argument from Interventions (and Miracles, Prayers, and Witnesses)” (p. 88)
  • It’s become somewhat fashionable to say that religion and science are growing together and are no longer incompatible. This convergence is, in my opinion, illusory. In fact, I don’t believe that any attempt to combine these very disparate bodies of ideas can succeed intellectually.
    • Part 2 “Four Subjective Arguments”, Chapter 5 “The Argument from Interventions (and Miracles, Prayers, and Witnesses)” (pp. 88-89)
  • The universe acts on us, we adapt to it, and the notions that we develop as a result, including the mathematical ones, are in a sense taught us by the universe. Evolution has selected those of our ancestors (both human and not) whose behavior and thought were consistent with the workings of the universe.
    • Part 3 “Four Psycho-Mathematical Arguments”, Chapter 4 “The Universality Argument (and the Relevance of Morality and Mathematics)” (p. 131)
  • The connections among morality, prudence, and religion are complicated and beyond my concerns here. I would like to counter, however, the claim regularly made by religious people that atheists and agnostics are somehow less moral or law-abiding than they. There is absolutely no evidence for this, and I suspect whatever average distance there is along the nebulous dimension of morality has the opposite algebraic sign.
    • Part 3 “Four Psycho-Mathematical Arguments”, Chapter 5 “The Gambling Argument (and Emotions from Prudence to Fear)” (p. 139)
  • It’s always healthy to recognize facts.
    • Part 3 “Four Psycho-Mathematical Arguments”, Chapter 6 “Atheists, Agnostics, and “Brights”” (p. 146)
  • 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.
    • Part 3 “Four Psycho-Mathematical Arguments”, Chapter 6 “Atheists, Agnostics, and “Brights”” (p. 149)
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