Subtitled: The Power of Mathematical Thinking.
Third post (after this and then this) about this fascinating book, an examination of several basic principles (linearity, inference, expectation, regression, and existence) and how they apply to every-day, real world situations, situations that are often misunderstood by ordinary “common sense”. The author, a one-time child genius, is a professor of mathematics at the University of Wisconsin-Madison, and has written for Slate, Wired, and other publications, including an occasional column for Slate.
The third of five parts is about “Expectation”. He discusses how to calculate the ‘expected value’ of a single lottery ticket — which is not the same as the value you expect of any given ticket (which is 0). Should you play? Yes, when the jackpot is high enough that the expected value is higher than the cost of the ticket. Famous case of a Massachusetts game, WinFall, in which a bunch of MIT students figures out the odds, bought huge numbers of tickets when the odds favored it, and won. And kept winning — Massachusetts authorities didn’t care, as long as they were making money — until a newspaper uncovered the story, the state shut down that game.
(Ch12) If you never miss a plane, you’re spending too much time in airports. This balance of cost vs inconvenience applies to other areas of life, e.g. occasional reports of government waste, such as paying benefits to dead people. The cost of eliminating *all* waste outweighs the benefits of never making such mistakes. (That such incidents of government waste, no matter how rare, get media coverage, exaggerates the issue and hides the true cost/benefit analysis.)
Same logic applies to the famous “Pascal’s wager” about belief in god; the flaw in his argument is that he doesn’t consider other options, e.g. the existence of a god who damns Christians and favors others.
This discussion leads to the idea of ‘utils’, how to evaluate cost vs benefit in subjective ways, e.g. a thousand dollars is worth more to someone who has no money than to someone who already has a thousand dollars. Different people have different util curves; some people work only until they have enough money, then stop.
(Ch13) WinFall involved different strategies, which correspond to the idea of finite geometries, e.g. the Fano plane, which has just seven points and seven lines or curves, each with three points. And these correspond to winning strategies for choosing lottery numbers.
This carries over into the redundancy codes invented for transmitting signals to satellites, and analogous patterns in natural language, and the problem of packing spheres into the least possible volume.
Lotteries? Despite the odds, people play them anyway, because of some concept of ‘fun’ that is independent of those expected values. Just as people start businesses, despite the odds against.
Part IV is about “Regression”, (Ch14) beginning with a 1930s study about successful businesses that discovered that the most and least successful businesses didn’t stay that way; they ‘regressed to the mean’. This wasn’t a discovery about human nature; it was a discovery about statistics, and explains why second novels aren’t as good as first ones, and why football players perform worse in the second year after they are signed.
(Ch15) The idea of ‘scatter’ charts, their patterns, and early ideas of profiling criminals by compiling data about their head size, finger length, and so on — “bertillonage”, which eventually gave way to fingerprinting.
(Ch16) Correlation: you can find correlations between virtually any two variables. As everyone knows, that doesn’t imply causation.
Even in the 1950s, strong correlation was seen between smoking and lung cancer, but some analysts warned about drawing conclusions about causation. (Really. Maybe, e.g., having early stages of cancer, such as symptoms like a slight chronic inflammation, prompt a desire for the relief and comfort of smoking…)
Author makes a crucial point: It’s not always wrong to be wrong. The detections of correlations like that between smoking and lung cancer lead to public healthy policies that are sometimes mistaken, and have to be changed. But if you wait for absolute certainly before any such policy, you’ll never get anywhere, and people will die while you’re waiting for perfect evidence. “If you never give advice until you’re sure it’s right, you’re not giving enough advice.”
Part V is about “Existence”. These chapters address ideas that are relatively more familiar to me.
Ch17: “There Is No Such Thing as Public Opinion”. This addresses the paradoxes of opinion polls, how, depending on how questions are asked, contradictory results appear. E.g., Americans want a smaller government, but when asked which government programs they would cut, more people want to increase spending on programs than cut them (education, health, defense, etc.).
The central issue is that each voter’s stance is rational, but in aggregate, they’re nonsensical. (A prominent example: a binary poll about Obamacare shows that most are opposed; but a more nuanced poll shows that more approve or *want it expanded* than those who disapprove of it altogether.)
This leads to a discussion about voting methods — a topic I’m familiar with, given the science fiction community’s nerdish obsession with fairness in voting, in the elaborate ‘Australian ballot’ procedures applied to the Hugo Awards. (This idea has trickled down to some examples of ‘instant-runoff’ voting in some state elections.)
As has been revealed before, there is no voting system that cannot lead to counter-intuitive results. Example: a three-way presidential election; if most people prefer candidate A, or candidate B, yet most of them rate candidate C second, candidate C might well win. (There are examples.)
Ch18: “Out of Nothing I have Created a Strange New Universe”. The idea of what is *true* vs what is *right* via rules and procedures exists in many fields. The most fundamental is in math, the struggle for two thousand years to try to deduce Euclid’s fifth axiom (the one about parallel lines) from the first four. The breakthrough came in the 19th century, when several mathematicians realized that it could not—in fact, alternatives to Euclid’s fifth would be equally valid logically, and would describe different geometries! E.g., that of a sphere. This was an astonishing result; thus the chapter title’s quote.
This leads to a discussion about whether mathematical expressions *mean* or whether they should be *defined*. The parallel is in law, where issues are *defined* by the results of the voting process, never mind what voters might have meant. The famous example: the 2000 election, where Justice Scalia’s policy of deferring to the process reigned.
The problem with this is that such procedures never admit new evidence. [Not in this book, but recently in the news: cases where someone was convicted to Death Row, and despite DNA evidence of innocence, are left there, because after all the *procedures were followed*, and later evidence doesn’t matter.]
The champion of formalism in math was David Hilbert, who famously, in 1900, put forth a list of 23 great problems for math to solve. His intent was to build mathematics from the ground up, using precise formalism, and assuming that no contradictory results could emerge.
That assumption wasn’t true, as Kurt Gödel demonstrated in 1931 — his famous “incompleteness” theorem.
Conclusion: How to Be Right
Author discusses a summer job with a researcher who *wanted an answer*, never mind qualifications; quotes FDR and John Ashbery; and discusses reactions to Nate Silver, how people misunderstand his predictions of percent chance of voting results.
Conclusions? Avoid precision; it’s misleading. People are more tolerant of contradictions than computers are (citing Captain Kirk’s numerous defeats of computers through logical paradoxes). If you have an idea, try to prove your theorem by day, and disprove it by night. Apply that idea in all areas of life, and it will force you to confront the reasons why you believe what you believe.
What’s true is that the sensation of mathematical understanding – of suddenly knowing what’s going on, with total certainty, all the way to the bottom — is a special thing, attainable in few if any other places in life. You feel you’ve reached into the universe’s guts and put your hand on the wire. It’s hard to describe to people who haven’t experienced it.
…To do mathematics is to be, at once, touched by fire and bound by reason. This is no contradiction. Logic forms a narrow channel through which intuition flows with vastly augmented force.