Review: Natalie Angier on Eugenia Cheng

Welcome to Book Post, your subscription-based book review service. This week we are very pleased to bring storied science journalist Natalie Angier on the lively and inventive science explainer Eugenia Cheng, and the question—can logic help us sort out our our differences?


Eugenia Cheng uses the example of the Battenberg Cake to to analyze the relationships among sets of alternatives.  Yum.

Among the many pleasures supplied by Eugenia Cheng’s third book, The Art of Logic in an Illogical World, is relief from a dilemma I have long struggled with, and perhaps you have too. It runs like this: for me, it is important that the rights I can claim as an adult citizen of the United States should be extended to all other American adults. If I can marry the person of my choice, for instance, then others should be allowed to do so, too. Of course, this is not a position universally shared. Even after the Obergefell decision by the U.S. Supreme Court in 2015, many Americans remain opposed to marriage between people of the same sex and even feel justified refusing to bake cakes or arrange flowers for such weddings. I find that position insupportable. But while I may object to those who aim to overturn Obergefell—or who oppose the building of a mosque in a neighborhood with many churches, for that matter, or who fly large Confederate flags outside a public park—I’ve often worried that my righteous indignation is not quite right. If I’m so tolerant, shouldn’t my call for inclusiveness extend to those who, by the looks of it, seek actively or effectively to exclude others?

According to Cheng, a mathematician and scientist in residence at the School of the Art Institute of Chicago and one of the clearest, liveliest, and most essential math popularizers of our time, the tools of logic can be applied to help illuminate many of the impasses of the day, including the frustrating dilemma above. She compares the problem to a paradox famously articulated by the mathematician and philosopher Bertrand Russell in 1901. Imagine a town in which all the men shave, and a male barber shaves all men who don’t shave themselves. Who, then, shaves this town barber? “Formally, the paradox is stated in terms of sets,” Cheng writes. “It says, consider the set S of all sets that are not members of themselves. Is this set a member of itself? If it is, then it isn’t. And if it isn’t, then it is. It’s a paradox.” The paradox may seem unresolvable, Cheng continues, but the mistake lies in assuming that all sets operate on the same level. They do not, and in fact they should not. “As long as we have different levels, we can declare that a set of all sets is on a different level” than other sets, she says, “and this prevents us making statements that loop back on themselves.”  The idea that “if I’m tolerant then I must embrace intolerance” is a subtle form of Russell’s paradox, she says, which can be resolved by adjusting the scope, or level, of the set. We can define tolerant, for example, to mean “tolerant of all things that do not hurt other people,” just as the free speech clause in the Bill of Rights can be narrowed to exclude dangerous speech like yelling “fire” in a crowded theater. 

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But Cheng does not settle for defining away a problem; she shows how the application of familiar logical operations can help inform our response to it. Everyone knows that, in English, a double negative statement equals a positive statement, as in, “I don’t see any reason why it wouldn’t be Russia.” Similarly, in math, a negative number times a negative number makes a positive number, as does a positive times a positive; while a negative number times a positive number, or a positive times a negative, yields a negative result.

How does this set of operations apply to our tolerance/intolerance conundrum? If you think of tolerance as a positive figure and intolerance as a negative one, here’s what you come up with when combining the two: tolerance of tolerance gives you tolerance; intolerance of intolerance also resolves as tolerance; but both intolerance of tolerance and tolerance of intolerance results in more intolerance. If your ultimate goal is tolerance, you not only are relieved of the obligation to add intolerance to your circle of inclusion, you logically should speak up against intolerance whenever possible.

Cheng emphasizes that logical thinking is by no means the solution to all our ills, but it can make an excellent starting point. Just as math helps us see patterns through the process of sequential abstraction—one banana plus two apples becomes 1 plus 2 becomes a plus b—so an argument can be clarified by stripping it down to its fundamental components and the elements generalized and compared. Taking that axiomatic approach, debates over seemingly unrelated topics like subsidized health care, affirmative action programs, and the handling of sexual harassment claims can be seen as variations on a theme, all three stemming from one basic question: are we more comfortable with false positives, or with false negatives? Would we rather err on the side of too much health coverage, while possibly paying for people who don’t need the help; would we rather seek out minority candidates to hire and risk bringing in the occasional unprepared individual; would we rather take all claims of sexual harassment seriously enough to investigate, even if that means some accusations will prove unfair or overblown? These are social policies set toward an acceptance of false positive results. Conversely, we can take a minimalist approach and end up with a lot of false negatives: cancers left undetected, talented members of minorities unhired, and sexual harassment accepted unless there is overwhelming physical evidence. Cheng acknowledges that a basic preference for one style over another, generous vs. stringent, undertreat or overtreat, is ultimately subjective, and that is not a bad thing. After a problem has been reduced to its axiomatic core, we must still seek guidance from our emotions: Star Trek’s logical, green-blooded Mr. Spock, steered by his half-human heart.


Natalie Angier has been a science columnist for The New York Times since 2007. She is the author of four books, most recently The Canon: A Whirligig Tour of the Beautiful Basics of Science.


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