Making Sense of Sets, in Theory and Life

Set theory has applications in the real world, from bars to train schedules

ILLUSTRATION: TOMASZ WALENTZ

10 COMMENTS

By 

Eugenia Cheng

Aug. 30, 2018 11:43 a.m. ET

Mathematics often helps us to think about issues that don’t seem mathematical. One area that has surprisingly far-reaching applications is the theory of sets. Sets are one of the most basic objects in mathematics, since we almost always have a collection of things we are trying to study.

But interesting sets are more than a random bunch of objects, just as a house is more than a pile of bricks. Mathematical structure arises from relationships, such as addition and multiplication, distance and closeness, or—in the case of sets—ordering.

An ordered set is based on a defined relationship among its objects. For example, when people line up at a store’s checkout counter, they are usually arrayed in an unambiguous order, based on who arrived first, second and so on. The people form an ordered set. By contrast, at many bars, customers have to crowd around until a bartender notices them. This is not an ordered set.

In mathematics, we often think about comparing sets using functions, which pair up the objects in one set with the objects in another. In a train timetable, for example, departure times are paired with the corresponding arrival times. When applying functions to ordered sets, it is revealing to ask whether or not the order is “preserved.” If a local train leaves earlier than an express train for the same destination but arrives later, the function does not preserve the order of the set.

When functions are not order-preserving in mathematics, it causes structural breakdown. In life, it can cause antagonism: At a bar, if people are not served in the order they arrived, they feel miffed. A more serious example is the late Sen. John McCain, who declined to accept early release as a prisoner of war unless those captured before him were released.

Trouble also can arise when one set of people is ordered according to two different rules. At a workplace, for instance, the employees can be arranged by age and also by the seniority of their positions. When the order is not preserved in moving from one set to the other—when, say, a young person is the superior of an older person—tensions may ensue.

Or consider a single set of people arranged by both wealth and some ostensible marker of privilege. Arguments can arise when the function comparing them does not preserve order—for example, if a white person is alleged to be more privileged than a nonwhite person, even though the nonwhite person is richer.

Orderings of sets are seldom so straightforward, of course. We can stack our Tupperwaredishes neatly if they fit inside each other, but some won’t fit if they are different shapes. In mathematics this is called a partially ordered set, in which some things are allowed to be “incomparable.” If we have multiple criteria for ordering a set, we might find one object higher up according to one rule, and another object higher up according to the other.

Thus, if you are looking to buy an apartment, you might find one that is in better condition on the inside although another has a much better location. This can cause indecision if you don’t know how to rank the criteria themselves in order of importance. You could make a strict hierarchy if you disregard all but one of the criteria, but it would be rather short-sighted to buy an apartment based on one criterion alone.

Partially ordered sets are more difficult to deal with but are also more common. They are just one of the many types of subtle mathematical relationships that have much to teach us about navigating the complexities of our own lives.

—Dr. Cheng’s new book, “The Art of Logic in an Illogical World,” will be published by Basic Books on Sept. 11.

Paradoxes, From Your Coffee to Calculus

• IDEAS 
•  EVERYDAY MATH

Puzzles that point out the limits of logical thinking, help improve it and otherwise deal with contradictions in life.

ILLUSTRATION: TOMASZ WALENTA

13 COMMENTS

By 

Eugenia Cheng

July 25, 2018 11:34 a.m. ET

If you wear glasses you’ve probably experienced the problem of being unable to find your glasses because you’re not wearing your glasses. There’s also the conundrum of needing to drink coffee before you’re capable of making coffee in the morning. More seriously, there’s the question of whether tolerant people should be tolerant of intolerance. These are all types of paradox arising from self-reference, as in the sentence “I am lying.”

Mathematicians and philosophers have long studied paradoxes. In 1901, Bertrand Russell discovered a famous paradox that shook the foundations of mathematics. It is often described informally by imagining a man in a town who is a barber. He shaves every man in the town who does not shave himself and nobody else. This causes a looped-up contradiction if we start wondering whether or not the barber shaves himself: If he doesn’t, then he does. And if he does, then he doesn’t. It’s a paradox.

This might sound pedantic and contrived, but the mathematical study of paradoxes is important for several reasons. The first is that the presence of paradoxes alerts us to the limits of our own logical thinking and points us in the direction of improving it. Russell’s paradox was formally stated in terms of sets and caused mathematicians to realize that a naive or intuitive definition of a set as “a collection of things” is not logically sound when self-referential sets become involved. Because mathematics is based on making sound logical deductions, the logic of the initial definitions is crucial. Paradoxes tell us that greater care is sometimes needed in setting out these premises.

Thinking about how to resolve paradoxes has helped mathematicians make progress throughout history. More than 2,000 years ago, the Greek philosopher Zeno pondered a range of paradoxes having to do with motion. One of them said: In order to travel from A to B, you must first cover half the distance, then half the remaining distance, then half the remaining distance and so on forever. We need to cover an infinite number of distances, but we only have finite time, so we can never arrive.

Here the conclusion is blatantly untrue: We arrive at places every day. To resolve this paradox, what was needed was a new theory of adding up infinitely many infinitely small numbers. This launched the field of calculus, which analyzes things that change continuously. And calculus, of course, would become a crucial tool of science, engineering and economics. Thus the apparently arcane study of ancient paradoxes led to the mathematical foundation of much of modern life, whether or not technology users are aware of the math involved.

MORE IN EVERYDAY MATH

• How Numbers Can Check ThemselvesJune 20, 2018
• The Best Way to Think About Probabilities May 17, 2018

Understanding how mathematicians avoid paradoxes in abstract cases can help us to work out how to deal with similar contradictions in life. For the paradox of the barber, we can simply say, “Oh well, no such barber can exist.” But Russell’s paradox is formally stated in terms of sets rather than barbers, and the paradox comes from considering the set containing “every set not containing itself.” Instead of just saying this set can’t exist, we say it exists but at a different level from ordinary sets. It is a way of breaking the self-referential loop.

This has helped me work out how to break the loop problems in life as well. I’ve made it easier to find my glasses without wearing my glasses, for instance, by always storing them in the same place. For my coffee, I’ve broken the loop by setting up my espresso machine the night before so that in the morning I just have to turn it on. In the case of tolerance, I make a distinction between people themselves and their ideas about other people, so that I can be tolerant of individuals who are intolerant but don’t have to accept their ideas.

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Paradoxes are not just useful puzzles in mathematics. They can point the way to thinking more clearly about resolving complex situations in everyday life.

Simple Rules for Escaping Infinite Decisions

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• IDEAS 
•  EVERYDAY MATH

Are life’s endless menus exhausting you? Escape the burdens of the ‘axiom of choice’

ILLUSTRATION: TOMASZ WALENTA

By 

Eugenia Cheng

March 8, 2018 10:38 a.m. ET

3 COMMENTS

Sometimes I stare at a restaurant menu with no idea how to choose my dinner. It might be because the menu has too many appealing options, but often it’s because I have decision fatigue. This is when you make too many decisions in one day and then find it hard to make any more. It doesn’t matter how important those decisions are. It could be deciding what shoes to wear or whether to buy a house. Each decision takes a toll.

There is a piece of mathematics dealing with this problem called the axiom of choice. It holds that, given any collection of sets, with each one containing at least one object, it is possible to select exactly one object from each set—even if the collection of sets is infinite. The axiom of choice asserts, in other words, that we can make an infinite number of individual choices.

In mathematics, an axiom is a basic truth that we assume without proof. In fact, we can’t prove it. If we could prove it, it would be called a theorem. Although mathematics is all about proving things with rigorous logic, we have to start with some assumptions because it is not possible to prove anything from nothing.

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The axiom of choice can be expressed in terms of what mathematicians call a choice function. Given a collection of choices from, say, an infinite set of restaurant menus, a choice function selects one meal from each. Because menus are so varied, however, that could be quite a burden of choice!

Some choice functions, however, can let us escape the axiom of choice—by making an infinite number of choices unnecessary. Faced with an infinite number of menus, our choice function could be: “Always pick the least expensive item” or “Always pick the least caloric.” We would make one global choice and repeat it infinitely instead of making an infinite number of individual choices.

The same holds for simpler choices, but as with the menus, our ability to escape the axiom of choice depends on the sort of objects we are choosing. Consider the difference, in terms of organizing your closet, between sets that consist of pairs of socks or pairs of shoes. For the socks, you have to choose which of the pair to tidy first (assuming the socks have no features distinguishing left from right). With the shoes, you can just make one choice at the start: to put away, say, the left shoe first every time. Again, you’re able to make the same choice an infinite number of times.

MORE IN EVERYDAY MATH

• A Thinking Person’s Guide to the Rubik’s Cube February 2, 2018
• Holiday Sweets and the Math of Exponentials January 4, 2018

In mathematics as in real life, dealing with the prospect of infinite choices is daunting, but we can avoid it by making one global choice instead. The problems arise when the choices can’t be easily ranked or when they’re not distinguishable from one another.

Another way to define a choice function that escapes the burden of infinite choices is to use an algorithm. This doesn’t make one global choice but instead provides a scheme for choosing, without your having to make any decisions along the way. For example, I could choose to walk the same route to work every day, or I could use an algorithm that says I will take the most direct route in the city grid until I hit a red light at a crosswalk, at which point I will turn and take the most direct route from my new position. The first way removes all choices but might be boring; the second provides some variation but without my having to make any decisions.

The axiom of choice is a powerful tool, but efficient mathematicians prefer to avoid using it where possible, saving it only for problems where it’s really necessary. We can likewise avoid decision fatigue in daily life by using global choices or algorithms, thus saving our decision-making energy for more important choices.