Why Study Mathematics?

As a researcher of mathematics, people I meet are frequently curious about what my work is about. Whenever I'm asked to explain my research, I invariably get two questions. The first, of course, is "What exactly is 'low-dimensional topology' or 'non-associative algebra'?", and the second is inevitably "...and what are the practical applications of your research?", betraying the common attitude that 'if it doesn't lead immediately to fabulous new technology or wonder cures for deadly diseases, it ain't worth our time or money.' Similarly, my students in various math classes often demand to know "When are we going to use this in our daily lives?" or "Why do we need to know this?". It is this latter question that I intend to address here.

As with many questions, the answers to these questions depend on who's asking and their motivation for asking. Students may ask the question as a way of complaining about course material they don't like, e.g., "I don't see how this will ever be useful to me, so I don't think I should have to learn it," or they may be genuinely curious about why the course content is what it is and why the course is required for their degree. When I'm asked the question by non-mathematicians about the applications of my research, the question's tone can vary from a curiosity about why someone would spend time studying things like knots to indignation at perceived frivolity and uselessness, as if research must either be directly responsible for new technology or amount to hard-earned tax dollars being wasted on frivolous nonsense.

I find this attitude quite curious, given that millions of people every day spend countless hours of leisure time doing sudoku, crossword puzzles, word games, and other "frivolous" intellectual games which don't directly produce new technology or solve world hunger. People intentionally spend time solving these and other puzzles, many of which are simply thinly disguised math problems, for no other reward than the pleasure of solving the puzzles. No one attacks such puzzle-solving as a waste of time, because it's clear that the goal is simply the enjoyment of solving the puzzle.

It is this very same enjoyment of solving puzzles that drives mathematicians to solve mathematical problems.We study mathematical problems because we're curious about the problems and we want to know the answers; it's fun to find new things out. The great French mathematician Pierre de Fermat, whose famous 'last theorem' problem was recently solved and who contributed a number of important theorems to the field of number theory, was in fact a lawyer by day who solved mathematics problems as a way of relaxing. It is his hobby we remember him for, some 300 years later, not his profession.

One difference between mathematics problems and jigsaw puzzles is that some mathematical puzzles turn out to have consequences in the real world. Indeed, many of the puzzles mathematicians solve arise from efforts to understand how features of the world work, though certainly not every problem arises this way. Riemann's generalization of Euclid's geometry, for instance, turns out to be exactly the language Einstein and others needed to explain gravitation. Mathematics is humanity's best tool for trying to describe how the world works; indeed, the unreasonable effectiveness of mathematics at explaining the world is a bit of a mystery.

This unreasonable effectiveness of mathematics is both a asset and a curse. It's clearly an asset in that it generates interest in the subject and funding for research from private companies and government agencies willing to pay to have key problems solved. On the other hand, it's a curse because it leads people to misunderstand the reason for doing mathematics in the first place; because some math turns out to be just what we need to solve certain problems, suddenly people expect all math to have obvious immediate applications. It's sort of like a musical group that starts composing and performing music out of love for their art form, but then has a hit song which sells well -- people then get the idea that music is about making money, and if the group's next album doesn't sell as well as their first hit, suddenly they're considered laughable failures, even though to the group it was never about the money.

A record that doesn't sell well initially may become a big retro hit in later years, and records can have big success within smaller music circles without necessarily topping the mainstream charts. Similarly, mathematical concepts that are not successfully applied to real-world problems immediately after their conception are frequently found to have important applications later on as the subject matures or as new developments in the other sciences call for new mathematical approaches. Some mathematical ideas may be useful for very specialized problems without necessarily having a broad range of applications.

In any case, the study of mathematics motivated by sheer intellectual curiosity is known as 'pure mathematics,' while mathematics studied as means for solving practical problems rather than as an end in itself is known as 'applied mathematics.' Even within the mathematics community, not everyone agrees on which problems are interesting; sometimes problems which are not interesting on their own are studied in order to solve other problems which we do find inherently interesting. This can even lead to some pure mathematicians criticizing other pure mathematicians for wasting time on ideas without applications!

The moral is that what counts as an 'application' depends entirely on what you're interested in. A topologist might study group theory as a way of developing methods for distinguishing topological spaces, while an algebraist might study group theory because she's interested in groups themselves.

In particular, for most non-mathematicians, mathematics is at best a means to some practical end, whether optimizing costs, calculating the trajectory of a spacecraft, or just finishing a degree and getting a job. Anyone interested in a technical career will need to be familiar with some level of mathematics, and the cumulative nature of mathematics makes it inevitable that students will end up studying some topics that don't directly contribute to the mathematics they'll use in their career. Nevertheless, the fact that a student can't see an immediate application for a particular topic does not imply that the student will never use that topic; in many cases, the techniques ones studies in early classes are used to construct the more advanced and powerful machinery one uses later. For example, integration and power series techniques that students see as 'pointless' in calculus become valuable tools for solving differential equations, which are the natural language in which most of the practical problems in the real world are expressed.

This example brings up another reason why students who view mathematics as a means to the end of solving practical problems often don't see the point of many math classes, namely the simple fact that most of us vastly underestimate the complexity of the world we live in. Most of the quantities in which we have a natural interest, such as what the temperature will be tomorrow or what the Dow will be at closing, are functions of (that is, they depend on) many variables, each of which is changing over time. The equations which model such complex quantities are called partial differential equations, and to solve them requires a good understanding of topics such as trigonometry, multi-variable calculus, and linear algebra, at a minimum. Even so, it is not hard to write down partial differential equations which no one (yet) knows how to solve.

Therefore, when we write 'word problems' in early classes such as algebra, trig, and even calculus, the problems frequently sound phony and contrived, and in large part this is because they are phony and contrived; we have to make many unrealistic simplifying assumptions in order to make the problems actually solvable with the limited mathematics available to students at these early stages. Nevertheless, we feel compelled to try to come up with such problems in order to convince the students that the techniques they are learning are in fact useful, when the real reason for studying the techniques is that they are building blocks for future techniques which are more useful.

Does this mean that students must either enjoy studying mathematics for it own sake or devote several years of their lives studying course after course of mathematics in order to derive any benefit from taking math classes? Absolutely not! There is a far greater benefit from studying mathematics that one can begin seeing a payoff from almost immediately, assuming that the student is willing to actually take the subject of mathematics seriously and put some time into studying the topics presented in the course. What is this payoff?

Consider a baby playing with blocks. The infant struggles to grasp the object, and through hours of trial and error learns to use its hands in order to achieve the desired effect. Are we to conclude that the only benefit the child gets from successfully placing the square block in the square hole is the satisfaction of a goal achieved and a slightly more organized playpen?

Of course not. The real progress the baby has made is in developing motor skills, hand-eye coordination, and spatial intuition. These skills are more general and have much greater utility than the task for which they were initially developed. Similarly, the greatest benefit that comes from successfully studying mathematics is the ability to think carefully. Indeed, it is precisely because students of mathematics do not realize that solving the problems they're working on requires a new way of thinking that these problems seem so difficult.

I've found that around 70% of the mathematical topics that I've studied have a point of view from which the topic seems very natural and intuitive, so that the definitions and theorems say exactly what they "should" say, and the topic becomes quite simple. Of the remaining 30%, I'm not sure (and never can be sure) how many of the topics I've simply not found the right way to think about and how many are Just Plain Hard.

It may sound like intellectual snobbery to say that learning to think carefully requires studying mathematics, but the simple fact is that human mind is not built to understand the way the world works; rather, it's built to stay alive in the plains and savannas, and thus is prone to all sorts of faulty thinking: self-deception is the key to lying convincingly, which can be a useful survival strategy. Wishful thinking, confirmation bias, and myriad other forms of incorrect reasoning are what come naturally to human beings, and learning to avoid these common errors takes real work. Like the baby putting the square block into the square hole, solving mathematical problems requires learning to think in a new way, culminating in the form of reasoning known as mathematical rigor, which is so powerful that it is the only human activity which leads to undeniable absolute objective truth. Once you've seen the proof of a theorem, to deny the theorem is to misunderstand what it says.

Mathematical thinking is the basis of all of the sciences. You cannot be a scientist without learning mathematical thinking. Even scientists who don't make use of more advanced mathematical techniques need to be able to deduce which predictions follow from their hypotheses and which do not. Engineers who don't learn how to think like a mathematician leave themselves and their clients open to errors which can prove to be costly or fatal. Even if you have no intention of going into a technical field, mathematical thinking is the key to solving problems, to avoiding being taken in by scam artists, to making successful decisions. You don't have to be Sherlock Holmes to benefit from making careful observations and sound deductions. Only if you know the difference between valid and invalid reasoning can you spot the errors, intentional or not, in the reasoning of salespeople, lawyers, and politicians who want your money and votes. Careful thinking is necessary to distinguish fact from fiction, which is a skill we all require if we're going to live in a democracy where citizens sit on juries and elect leaders.

Ultimately, this is why we should all study mathematics, even if we never end up using the particulars of algebra or calculus. Like the baby's block problem, studying these subjects sharpens the mind and develops deductive reasoning skills which are far more important than the problems that we use to learn the skills. Moreover, these skills are a kind of intellectual technology which our ancestors have fought hard to create and to pass on to us, skills which are not part of our natural environment and which come to us only after hard work and study.

Copyright © 2004-2008 Sam Nelson

Permission to reprint without modification for academic use granted (but please email me anyway!)