More on this book
Kindle Notes & Highlights
Read between
September 21, 2020 - February 18, 2021
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.
SECONDARY, COLLEGE, AND GRADUATE EDUCATION
In addition to some understanding of algebra, geometry, and analytic geometry, high school students should be exposed to some of the most important ideas of so-called finite mathematics. Combinatorics (which studies various ways of counting the permutations and combinations of objects), graph theory (which studies networks of lines and vertices and the phenomena which can be modeled by such), game theory (the mathematical analysis of games of all sorts), and especially probability, are increasingly important.
The students who do major in mathematics in college, taking the basic courses in differential equations, advanced calculus, abstract algebra, linear algebra, topology, logic, probability and statistics, real and complex analysis, etc., have a large number of options, not only in mathematics and computer science but in an increasing variety of fields which utilize mathematics. Even when companies recruit for jobs that have nothing to do with mathematics, they often encourage math majors to apply, since they know that analytical skills will serve anyone well, whatever the job.
Given this sorry fact, it’s perhaps not surprising that few educated people will admit to being completely unacquainted with the names Shakespeare, Dante, or Goethe, yet most will openly confess their ignorance of Gauss, Euler, or Laplace, in some sense their mathematical analogues. (Newton doesn’t count, since he’s much more famous for his contributions to physics than for his invention of calculus.)
Martin Gardner, Douglas Hofstadter, and Raymond Smullyan
INNUMERACY AND THE TENDENCY TO PERSONALIZE
Too many people, in my opinion, maintain a “Why me?” attitude toward their misfortunes. You needn’t be a mathematician to realize that there’s something wrong statistically if most people do this.
Bad things happen periodically, and they’re going to happen to somebody. Why not you?
THE UBIQUITY OF FILTERING AND COINCIDENCE
Remember that rarity in itself leads to publicity, making rare events appear commonplace.
Our innate desire for meaning and pattern can lead us astray if we don’t remind ourselves of the ubiquity of coincidence, a ubiquity which is the consequence of our tendency to filter out the banal and impersonal, of our increasingly convoluted world, and, as some of the earlier examples showed, of the unexpected frequency of many kinds of coincidence.
The sequel to a great movie is usually not as good as the original. The reason may not be the greed of the movie industry in cashing in on the first film’s popularity, but simply another instance of regression to the mean. A great season by a baseball player in his prime will likely be followed by a less impressive season. The same can be said of the novel after the best-seller, the album that follows the gold record, or the proverbial sophomore jinx. Regression to the mean is a widespread phenomenon, with instances just about everywhere you look.
DECISIONS AND FRAMING QUESTIONS
If we have no direct evidence or theoretical support for a story, we find that detail and vividness vary inversely with likelihood; the more vivid details there are to a story, the less likely the story is to be true.
In their fascinating book Judgement under Uncertainty, Tversky and Kahneman describe a different variety of the seemingly irrational innumeracy that characterizes many of our most critical decisions. They ask people the following question: Imagine you are a general surrounded by an overwhelming enemy force which will wipe out your 600-man army unless you take one of two available escape routes. Your intelligence officers explain that if you take the first route you will save 200 soldiers, whereas if you take the second route the probability is 1⁄3 that all 600 will make it, and 2⁄3 that none
...more
MATH ANXIETY
The truism that one learns how to read by reading and how to write by writing extends to solving mathematical problems (and even to constructing mathematical proofs).
ROMANTIC MISCONCEPTIONS
That said, objections to being identified for special purposes by number (social security, credit cards, etc.) seem silly. If anything, a number in these contexts enhances individuality; no two people have the same credit-card number, for example, whereas many have similar names or personality traits or socioeconomic profiles. (I personally use my middle name—John Allen Paulos—to keep the masses from confusing me with the Pope.)
Many people believe that determining the truth of any mathematical statement is merely a matter of mechanically plugging into some algorithm or recipe which will eventually yield a yes or a no answer, and that given a reasonable collection of basic axioms, every mathematical statement is either provable or unprovable. Mathematics in this view is cut-and-dried and calls for nothing so much as mastery of the requisite algorithms, and unlimited patience. The Austrian–American logician Kurt Gödel brilliantly refuted these facile assumptions by demonstrating that any system of mathematics, no
...more
DIGRESSION: A LOGARITHMIC SAFETY INDEX
Furthermore, since perceptions tend to become realities, the natural tendency of the mass media to accentuate the anomalous, combined with an innumerate society’s taste for such extremes, could conceivably have quite dire consequences.
5
Statistics, Trade-Offs, a...
This highlight has been truncated due to consecutive passage length restrictions.
PRIORITIES—INDIVIDUAL VS. SOCIETAL
This chapter will concentrate on the harmful social effects of innumeracy, with particular emphasis on the conflict between society and the individual.
LAISSEZ-FAIRE: ADAM SMITH OR THOMAS HOBBES
If the members of a particular “society” never behave cooperatively, their lives are likely to be, in Thomas Hobbes’s words, “solitary, poor, nasty, brutish and short.”
BIRTHDAYS, DEATH DAYS, AND ESP
Descriptive statistics, the oldest part of the subject and the part with which people are most familiar, is at times (though not always) a dreary discipline, with its ceaseless droning about percentiles, averages, and standard deviations. The theoretically more interesting field of inferential statistics uses probability theory to make predictions, to estimate important characteristics of a population, and to test the validity of hypotheses.
The latter notion—statistical testing of hypotheses—is simple in principle. You make an assumption (often forbiddingly termed the null hypothesis), design and perform an experiment, then calculate to see if the results of the experiment are sufficiently probable, given the assumption. If they aren’t, you throw out the assumption, sometimes provisionally accepting an alternative hypothesis. In this sense, statistics is to probability as engineering is to physics—an applied science based on a more intellectually stimulating foundational discipline.
TYPE I AND TYPE II ERRORS: FROM POLITICS TO PASCAL’S WAGER
A Type I error occurs when a true hypothesis is rejected, and a Type II error occurs when a false hypothesis is accepted.
When punishment is being meted out, the stereotypical conservative is more concerned with avoiding Type I errors (the deserving or guilty not receiving their due), whereas the stereotypical liberal worries more about avoiding Type II errors (the undeserving or innocent receiving undue punishment).
POLLING WITH CONFIDENCE
OBTAINING PERSONAL INFORMATION
TWO THEORETICAL RESULTS
The central limit theorem states that under a wide variety of circumstances this will always be the case—averages and sums of nonnormally distributed quantities will nevertheless themselves have a normal distribution.
CORRELATION AND CAUSATION
BREAST CANCER, MUGGINGS, AND WAGES: SIMPLE STATISTICAL MISTAKES
ODDS AND ADDENDA
There’s a strong human tendency to want everything, and to deny that trade-offs are usually necessary.
Because of their positions, politicians are often more tempted than most to indulge in this magical thinking.
Close
Statistical tests and confidence intervals, the difference between cause and correlation, conditional probability, independence, and the multiplication principle, the art of estimating and the design of experiments, the notion of expected value and of a probability distribution, as well as the most common examples and counter-examples of all of the above, should be much more widely known.
Probability, like logic, is not just for mathematicians anymore. It permeates our lives.
The desire to arouse a sense of numerical proportion and an appreciation for the irreducibly probabilistic nature of life—this, rather than anger, was the primary motivation for the book.
Also by John Allen Paulos
ABOUT THE AUTHOR
