Another Sort of Mathematics is a tour through some of the best that has been thought and written about the mysteries of shapes and numbers, including proofs of some of the most beautiful theorems as well as explorations of some of the most famous unsolved problems. The book begins, however, with a dazzling discussion of the nature of mathematics itself, a discipline that finds its source and summit in the true, the good, and the beautiful. In the second chapter, the author discusses the nature of proof and its central role in the art of mathematics.

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I have a theory that has been field-tested on occasion, much to the cha­grin of my children, who are usually present during the field-testing. The theory is that if I ask random people on the street what a biologist does, most of them will be able to offer a fairly accurate description. While they may not understand advanced biological concepts and may not know all of the details of a biologist’s work, most people have a decent mental image of what it means to do biology, even in its various subdisciplines, e.g., cellular biology, field biology, etc. The same thing is true of a variety of careers: history, writing, etc. When I ask people to describe the life of a mathematician, however, I rarely get an answer that captures the essence of this career. I usually get one of three incorrect answers. The first is the image of someone who spends his days figuring out creative ways to multiply large numbers together. We might chuckle about this, but it turns out that it is closer to being correct than the other two answers. It at least has the element of serious play with actual mathematical objects that we saw in our non-standard subtraction algorithms. A second an­swer is a description that more accurately describes an engineer: using math to build structures and solve physical problems. The third answer describes a statistician: someone who collects and analyzes data. Both of these are noble careers, but neither offers an accurate description of a mathematician.

How is it that most people have had thirteen years of mathematics classes from kindergarten through the twelfth grade, and more if they took college classes, and yet have trouble describing what a mathemati­cian does? Of course, I am taking for granted that a mathematician does mathematics. The only possible answer is that in those thirteen years of mathematics classes, most people have not actually done much mathe­matics. This is not to suggest that they have not studied mathematical topics. As we saw, the content of mathematics consists fundamentally of numbers and shapes, and these are certainly covered in elementary, mid­dle, and high school courses. But simply learning facts and skills within a discipline is not the same as learning the discipline itself, and still less is it doing the discipline. Think about learning the names of notes, names of intervals, and even advanced techniques of chord progressions but never experiencing a finished piece of music and still less producing a piece of music either on paper or through the use of an instrument. It could hard­ly be said that one has learned music. Yet this is precisely the case for most people who have survived mathematics curricula over the course of their time as a student. What I posit is that we were all taught computation and application. We were taught to perform operations on numbers, functions, and perhaps even matrices; we were taught to measure certain things about shapes, such as perimeter, area, and volume; and we were taught to apply some of these ideas to “real-world” phenomena.

I will resist the urge to go on a full rant against the push for “real-world problems” in the mathematics textbook industry. I will instead offer three thoughts about applications (word problems). First, learning mathemat­ics is fundamentally about learning mathematics, not about learning to ap­ply it. There is nothing wrong with a word problem, especially for young students, but we must understand its proper place and purpose. When learning what it means to add numbers together, a young student will find it helpful to think about “combining three apples with two apples” and wonder about how many apples there are in total. It is better still if young students have actual, physical objects on their desks to hold and move and use to model addition. Even to young students, however, we do not teach the word problems to show the usefulness of mathematics, as if someday a child might be standing with two apples in one hand and three in another and say, “I wonder what operation I can use to figure out how many apples I have.” The truth is, the child will probably just count the five apples—or better yet, recognize the shape of the number 5—and not worry about the choice of operation, still less about the actual com­putation. The real value of the word problem is that it allows students to contextualize the abstract concept with which they are wrestling, and it is these abstract concepts that form the real stuff of mathematics. We want the students to understand addition itself, not the application about apples, and the word problem helps young students move from something famil­iar to the abstract. In other words, mathematics is not at the service of the word problem; instead, the word problem is at the service of mathematics.

Second, these “real-world problems” are almost never about the real world. Suppose two cars start from the same point and head in two dif­ferent directions, one north and one east. If one car travels at 45 miles per hour and the other car travels at 60 miles per hour, how far apart will they be after six hours? How “real” is this problem? How often have you driven a car at exactly 45 miles per hour for a sustained six hours? How often have you and a friend traveled in two directions that form an exact right angle and maintained that direction the entire trip? How often have you started off with a friend from exactly the same place? The cars would have to be on top of each other! The term “real world” is not quite fitting for this situation. Again, this does not mean students shouldn’t work on these sorts of idealized problems; it simply means that the purpose of a problem like this is not to show the application of mathematical con­cepts. Rather, the purpose is to contextualize the abstract. In this case, the word problem is teaching something about how traveling at different speeds in different directions affects the position of two objects, and it reinforces the importance of the Pythagorean Theorem. Later, in calcu­lus, it will say something about change itself and how related quantities change with respect to one another. The marvel of the fact that mathe­matics can be contextualized lies in its ability to bridge heaven and earth. In other words, the beauty of the problem has more to do with its being a “real problem” than a “real-world problem.”

Third, if mathematics is not first about real-world problems, and if these word problems are not actually “real,” why is there such a push in the textbook industry to include them as upward of 60 percent of the content? The usual answer is that we need to make mathematics “rele­vant” to students. Even given the idealized nature of these problems, the claim is that they demonstrate the usefulness of mathematics for students and thereby motivate them to learn it. We only need to ask ourselves, after thirteen years of mathematics classes that are designed to make mathematics relevant, how many people come out of high school moti­vated by mathematics, let alone deeply in love with it? The reason for this failure is simple. Most of us will never use most of the content we learned in our mathematics classes. Apart from professional mathematicians, mathemat­ics teachers, and perhaps those in a handful of other careers, most people will never even use something as important as the Pythagorean Theorem. This is all to say that the utilitarian push for mathematical application in the pursuit of relevance serves only to present students with mostly irrelevant examples. It fails to do the very thing it claims to do. What does work, what does inspire students, what does cause them to fall in love is to help them to encounter mathematics in all of its purity and beauty. This is not different from other disciplines. The content of mathematics is per­fectly interesting on its own, as are all things true, good, and beautiful. We need only have the eyes to see it and a teacher who can unveil it. It is also not different from falling in love with a person; we do not perform a utilitarian calculation about the other’s relevance to our lives. Instead, we must have an authentic encounter.

Enough about what the art of mathematics is not. Let us turn our at­tention to what it is. First and foremost, mathematics is the study of real things and their properties. Mathematicians look at objects, be they num­bers, shapes, or other more abstract structures, and they look for patterns in these structures. The patterns lead to questions about the universal properties of these structures, which in turn allow the mathematician to pose problems. In solving the problem, the mathematician is producing a logical argument that comes to fruition in the proof. We could therefore say that the art of mathematics is an art of properties, patterns, problems, and proofs. The proof is the end of the mathematical process. It is the formal act of mathematical rhetoric. Remember that rhetoric is the art of bringing oth­ers to truth: it is what prevents the first two arts of language, grammar and logic, from becoming activities merely for the individual. Rhetoric demon­strates that language is an essentially communal activity. It recognizes that truth, upon discovery, demands to be shared. The mathematician shares a discovery and a logical argument, demonstrating it through the mathe­matical proof. It is also the case that proof is the poesis of mathematics, the creative product of the mathematician, in a similar way that the painting is the poesis of the painter, the poem of the poet, and the story of the author.

Papyrus; Hieratic text verso and recto: the "Rhind Mathematical Papyrus" - British Museum

The fact that mathematics ends in an act of rhetoric is why a teacher should insist that a student “show his work.” It is not so that the teacher can “see the thinking” of the student, and still less is it so that the student can “check his work.” Rather, it is because the student has discovered something to be true, and he should now have a desire welling up from within to share this creation with others. If a student solves a problem mentally, the teacher’s response should not simply be “You must show your work.” Instead, the first response should be “Well done, my good and faithful mathematician. Now convince others of the truth you have discovered and put it down in writing for all of posterity!”

For the mathematician, the case is no different. He is a professional “proof writer.” This is why Euclid’s Elements is a marvelous text to use for high school geometry: it is a formal, completed act of mathematics that happens to be accessible at that age. As an original source, it also demon­strates for students that mathematics is a human endeavor, embarked upon by real men and women. I am in good company in recognizing the unique importance of proof, and specifically that found in Euclid, for one’s education. As a young law student and lawyer, Abraham Lincoln carried a copy of Euclid in his satchel. In 1864, the New York Times printed a conversation between Lincoln and the Reverend J. P. Gulliver.  Lincoln remarked on his own education in the law: “In the course of my law-read­ing I constantly came upon the word demonstrate. I thought, at first, that I understood its meaning, but soon became satisfied that I did not.” After perusing several sources and remaining unsatisfied, Lincoln said to himself, “You can never make a lawyer if you do not understand what demonstrate means.” Once he realized this, he said, “I left my situation in Springfield, went home to my father’s house, and staid there till I could give any propositions in the six books of Euclid at sight. I then found out what ‘demonstrate’ means, and went back to my law studies.”

Gulliver echoed Lincoln’s praise for the Elements:

No man can talk well unless he is able first of all to define to himself what he is talking about. Euclid, well studied, would free the world of half its calamities, by banishing half the nonsense which now deludes and curses it. I have often thought that Euclid would be one of the best books to put on the catalogue of the Tract Society, if they could only get people to read it. It would be a means of grace.

Lincoln responded, laughing, “I think so. I vote for Euclid.”

Read more in Another Sort of Mathematics: Selected Proofs Necessary to Acquire a True Education in Mathematics