
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
'lowdimensional topology' or 'nonassociative 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 nonmathematicians 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 hardearned
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 puzzlesolving 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 realworld 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 nonmathematicians, 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, multivariable 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, handeye 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: selfdeception 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.

