Word Gems: Mathematics: Why Study Math?

WHY YOU SHOULD CHOOSE MATH IN HIGH SCHOOL

by Espen Andersen, Associate Professor, Norwegian School of Management and Associate Editor, Ubiquity

[The following article was written for Aftenposten, a large Norwegian newspaper. The article encourages students to choose math as a major subject in high school - not just in preparation for higher education but because having math up to maximum high school level is important in all walks of life. Note: This translation is slightly changed to have meaning outside a Norwegian context.]

A recurring problem in most rich societies is that students in general do not take enough math - despite high availability of relatively well-paid jobs in fields that demand math, such as engineering, statistics, teaching and technology. Students see math as hard, boring and irrelevant, and do not respond (at least not sufficiently) to motivational factors such as easier admission to higher education or interesting and important work.

It seems to me we need to be much more direct in our attempts to get students to learn hard sciences in general and math in particular. Hence, addressed to current and future high school students, here are 12 reasons to choose lots of math in high school:

Choose math because it makes you smarter. Math is to learning what endurance and strength training is to sports: the basis that enables you to excel in the specialty of your choice. You cannot become a major sports star without being strong and having good cardiovascular ability. You cannot become a star within your job or excel in your profession unless you can think smart and critically -- and math will help you do that.

Choose math because you will make more money. Winners of American Idol and other "celebrities" may make money, but only a tiny number of people have enough celebrity to make money, and most of them get stale after a few years. Then it is back to school, or to less rewarding careers ("Would you like fries with that?"). If you skip auditions and the sports channels and instead do your homework -- especially math -- you can go on to get an education that will get you a well-paid job. Much more than what pop singers and sports stars make -- perhaps not right away, but certainly if you look at averages and calculate it over a lifetime.

Choose math because you will lose less money. When hordes of idiots throw their money at pyramid schemes, it is partially because they don't know enough math. Specifically, if you know a little bit about statistics and interest calculations, you can look through economic lies and wishful thinking. With some knowledge of hard sciences you will probably feel better, too, because you will avoid spending your money and your hopes on alternative medicine, crystals, magnets and other swindles -- simply because you know they don't work.

Choose math to get an easier time at college and university. Yes, it is hard work to learn math properly while in high school. But when it is time for college or university, you can skip reading pages and pages of boring, over-explaining college texts. Instead, you can look at a chart or a formula, and understand how things relate to each other. Math is a language, shorter and more effective than other languages. If you know math, you can work smarter, not harder.

Choose math because you will live in a global world. In a global world, you will compete for the interesting jobs against people from the whole world -- and the smart kids in Eastern Europe, India and China regard math and other "hard" sciences as a ticket out of poverty and social degradation. Why not do as they do -- get knowledge that makes you viable all over the world, not just in your home country?

Choose math because you will live in a world of constant change. New technology and new ways of doing things change daily life and work more and more. If you have learned math, you can learn how and why things work, and avoid scraping by through your career, supported by Post-It Notes and Help files -- scared to death of accidentally pressing the wrong key and running into something unfamiliar.

Choose math because it doesn't close any doors. If you don't choose math in high school, you close the door to interesting studies and careers. You might not think those options interesting now, but what if you change your mind? Besides, math is most easily learned as a young person, whereas social sciences, history, art and philosophy benefit from a little maturing -- and some math.

Choose math because it is interesting in itself. Too many people - including teachers - will tell you that math is hard and boring. But what do they know? You don't ask your grandmother what kind of game-playing machine you should get, and you don't ask your parents for help in sending a text message. Why ask a teacher -- who perhaps got a C in basic math and still made it through to his or her teaching certificate -- whether math is hard? If you do the work and stick it out, you will find that math is fun, exciting, and intellectually elegant.

Choose math because you will meet it more and more in the future. Math becomes more and more important in all areas of work and scholarship. Future journalists and politicians will talk less and analyze more. Future police officers and military personnel will use more and more complicated technology. Future nurses and teachers will have to relate to numbers and technology every day. Future car mechanics and carpenters will use chip-optimization and stress analysis as much as monkey wrenches and hammers. There will be more math at work, so you will need more math at school.

Choose math so you can get through, not just into, college. If you cherry-pick the easy stuff in high school, you might come through with a certificate that makes you eligible for a college education. Having a piece of paper is nice, but don't for a second think this makes you ready for college. You will notice this as soon as you enter college and have to take remedial math programs, with ensuing stress and difficulty, just to have any kind of idea what the professor is talking about.

Choose math because it is creative.* Many think math only has to do with logical deduction and somehow is in opposition to creativity. The truth is that math can be a supremely creative force if only the knowledge is used right, not least as a tool for problem solving during your career. A good knowledge of math in combination with other knowledge makes you more creative than others.

Choose math because it is cool. You have permission to be smart, you have permission to do what your peers do not. Choose math so you don't have to, for the rest of your life, talk about how math is "hard" or "cold". Choose math so you don't have to joke away your inability to do simple calculations or lack of understanding of what you are doing. Besides, math will get you a job in the cool companies, those that need brains.

You don't have to become a mathematician (or an engineer) because you choose math in high school. But it helps to chose math if you want to be smart, think critically, understand how and why things relate to each other, and to argue effectively and convincingly.

Math is a sharp knife for cutting through thorny problems. If you want a sharp knife in your mental tool chest - choose math!

*This point was added by Jon Holtan, a mathematician who works with the insurance company If.

Source: Ubiquity Volume 7, Issue 11 (March 21, - March 27, 2006) www.acm.org/ubiquity

Mathematics reveals hidden patterns that help us understand the world around us. Now much more than arithmetic and geometry, mathematics today is a diverse discipline that deals with data, measurements, and observations from science; with inference, deduction, and proof; and with mathematical models of natural phenomena, of human behavior, and social systems.

The process of "doing" mathematics is far more than just calculation or deduction; it involves observation of patterns, testing of conjectures, and estimation of results. As a practical matter, mathematics is a science of pattern and order. Its domain is not molecules or cells, but numbers, chance, form, algorithms, and change. As a science of abstract objects, mathematics relies on logic rather than on observation as its standard of truth, yet employs observation, simulation, and even experimentation as means of discovering truth.

The special role of mathematics in education is a consequence of its universal applicability. The results of mathematics--theorems and theories--are both significant and useful; the best results are also elegant and deep. Through its theorems, mathematics offers science both a foundation of truth and a standard of certainty. In addition to theorems and theories, mathematics offers distinctive modes of thought which are both versatile and powerful, including modeling, abstraction, optimization, logical analysis, inference from data, and use of symbols.

Mathematics, as a major intellectual tradition, is a subject appreciated as much for its beauty as for its power. The enduring qualities of such abstract concepts as symmetry, proof, and change have been developed through 3,000 years of intellectual effort. Like language, religion, and music, mathematics is a universal part of human culture.

Career Opportunities

The study of mathematics can lead to a variety of exciting professional careers. Basic research, engineering, finance, business, and government service are among the opportunities open to those with mathematical training. Moreover, with the increasing importance of basic science and information technology, prospects for careers in the mathematical sciences are very good. Mathematical analysis and computational modeling are important for solving some of the most pressing problems of our time - new energy resources, climate change, risk management, epidemiology, to name a few. We must strive to maintain our technological edge; mathematical skills will be crucial to this effort.

Some more specific business positions include portfolio analysis, design studies, statistical analysis, computer simulation, software design and testing, and other areas of operations research. There are extensive opportunities for mathematics in finance, the actuarial fields, and economic forecasting.

Many laboratories, both government and private, maintain independent research staffs that include mathematicians. Their work often deals with the development of new technology, including research in basic physics and software development, as well as applied mathematics. Numerical simulation, such as weather and climate forecasting, depends heavily on the use of supercomputers.

Another major reason for studying mathematics is to pursue a career in teaching. Studies of mathematics scores in the United States have indicated a crisis in the making, as mathematical education fails to meet minimum standards. At the university level, science and engineering departments may encounter a growing gap between supply and demand of faculty. Dwindling numbers of students are entering into Ph.D. study while there will be a bumper crop of faculty retirees in the next twenty years. These trends are beginning to be reversed. Rising public concern has already begun to drive teachers' salaries upward, and there is renewed interest in teaching as a career among college graduates.

Practical considerations aside, there is the pleasure of learning, applying, and creating mathematics. Real world issues pose problems that can be studied by formulating and analyzing mathematical models. In some cases applications may lead to new mathematics, and a new branch of the science is born. In other cases abstract theory finds unexpected practical purpose. Working on research problems is exciting; solving difficult problems successfully is, for many, satisfaction enough.

While a career in mathematics can be very attractive, it takes time to acquire the necessary skills, particularly for basic research at the Ph.D. level. Graduate study is essential for most fields. The undergraduate course sequence provides a foundation upon which more advanced mathematics will be built. In graduate study, one or two further years of coursework completes this basic training. Thereafter, more specialized courses, often at the frontiers of research, are taken. Applied mathematics students will take courses in various application areas to acquire experience in modeling the real world, and to learn how mathematics can help with problems from the physical and biological sciences, and in finance.

The breadth and depth of work will depend on the degree level. With an M.S. degree, the student is prepared for many jobs in government, business, and industry; with the Ph.D. degree the choices are wider. Many Ph.D. mathematicians join the faculty of a university or four-year college, where they not only teach but also conduct research and publish their results in scholarly journals and books. Others take post-doctoral positions at various laboratories around the world, where work of interest to them is being done. Still others pursue careers in corporate research and management. With either an M.S. or a Ph.D., starting salaries are significantly higher than those of graduates with bachelor's degrees.

At both the M.S. and Ph.D. levels, graduate study in mathematics develops a number of important skills for solving problems suggested either by mathematics or by real world questions. Foremost is the ability to break complex issues into smaller, more manageable problems, until a model is reached which can be thoroughly studied and understood. Applied mathematics develops the art of extracting quantitative models from problems of physics, biology, engineering and economics. This ability comes from experience, such as that acquired gradually from examples studied in graduate courses.

An undergraduate student wishing to enter graduate study in mathematics should first satisfy the basic undergraduate requirements. The most essential courses are the calculus sequence (often three one-term courses and a course in advanced calculus) and a course in linear algebra. Courses in probability, statistics, and an introduction to computer science are also useful. Courses in algebra and topology can provide an introduction to more abstract mathematics. Students interested in applied mathematics will probably want to consider taking core courses from another department, such as physics, chemistry or biology. Introductory courses in ordinary and partial differential equations are useful. It is desirable to master at least one computer language.

If you like math and the challenges it offers, why not study it at university? The mathematical sciences are developing rapidly, providing many career opportunities and challenges both in traditional fields and in new exciting areas where mathematics and statistics are essential tools. If you decide to pursue mathematical studies at university, you will be choosing not only a fascinating and prestigious subject, but also a program which will give you excellent prospects for a rewarding career.

Here are just a few of the fields that our recent mathematics and statistics graduates have found employment:

Mathematics is also an excellent preparation for further study in various fields. Many of our students have continued in such areas as:

Mathematics courses are preparation for business calculations, for handling your money sensibly and for courses in sciences, engineering and technology. You should view mathematics as an opportunity to strengthen your thinking skills.

In mathematics courses you should not only meet calculations to do but also the chains and threads of reason and persuasion which justify them and links them together. Understanding and following the rules and patterns of mathematics, practices and nurtures an ability to think and reason well. Mathematics provides a neutral territory for the practice of rule and pattern-based reason and logic. The opinions and views you meet in daily life say and care little about what mathematical conclusions should be.

If you find yourself in a course which gives formulas and numbers to use in them but does not expect you to use algebra, you are wasting your time. Your time would be better spent studying algebra, and then taking a more advanced course that respects your intelligence. Similarly in college, if you find a course which gives you formulas and numbers to use in them and also talks at length about rates of change without expecting you to understand calculus, then a calculus course would be of better use of your time.

Students ask: Why do I need to know how to calculate the square root of a number? What is the why calculate square root - beyond (or behind) it is used in math formulas? Are square roots really needed in life outside just math studies?

Here's an idea of how you as a teacher/parent can at the same time show students one important real-life application of square root AND let them ponder where math is needed AND hopefully pique their interest into math problems in general. This lesson idea will work best when you've taught the square root concept but not yet touched on Pythagorean theorem.

Then, beyond this simple example, square root as a CONCEPT is needed to understand other math concepts and to study math further. Studying math is like building a block wall or building: you need the blocks on the lower part to build on, and if you leave holes in your building, you can't build on the hole.

So, you can mention to your students that the concept of square root is a prerequisite and ties in with many many other math concepts:

there are probably more examples that I didn't think of for right now. If you do, please send them in!

The answer to that is that in many, many fields and occupations that require higher education like computer science, electronics, engineering, medicine (doctors), trade and commerce analysts, ALL scientists, etc. In short, if someone is even considering higher education, they should study algebra.

Algebra also develops thinking skills and problem solving skills. And, it is needed to take your SAT test.

Then you can admit to your student(s) that yes, square roots are not needed in every single occupation, those of mostly manual labor for example. That is no big secret; kids know that. But try to ask them that do they know for sure what they are going to become as adults? Most kids in middle school are not sure at all. If they are not sure, they better study all school subjects and learn all they can so that when they finally have some idea, they won't be stopped because of not having studied and understood square roots.

And, even if they think they know what they want to be, how many times have young people changed their mind?? Even we as adults don't necessarily know what kind of job or career changes are awaiting us. In times past, you could pretty well bank on either becoming a housewife (girl), or continuing in your father's occupation (boy). In today's world this is not so. Kids have more freedom in choosing - but the other side is that young people need to study more to get a good solid basic education. Sometimes young people just need an adult to tell them about these things: they don't know all about their future so they need to keep studying, even math.

Why Do We Study Calculus?
or,
a brief look at some of the history of mathematics

The question I am asked most often is, "why do we study this?" (or its variant, "will this be on the exam?"). Though some students will eventually use integrals and derivatives in their work in physics, chemistry, or economics, most will never use epsilons and deltas. Applied mathematicians may use a theorem such as "the limit of the product is the product of the limits"; we only need epsilons and deltas to prove such theorems. If the applied mathematician takes the attitude that "I trust the pure mathematicians who say they have proved this theorem," then the applied mathematician does not need to study epsilons and deltas at all.

But calculus is not a just vocational training course. In part, students should study calculus for the same reasons that they study Darwin, Marx, Voltaire, or Dostoyevsky: These ideas are a basic part of our culture; these ideas have shaped how we perceive the world and how we perceive our place in the world. To understand how that is true of calculus, we must put calculus into a historical perspective; we must contrast the world before calculus with the world after calculus. (Probably we should put more history into our calculus courses. There is a growing movement among mathematics teachers to do precisely that.)

The earliest mathematics was perhaps the arithmetic of commerce: If I am willing to trade 3 of my goats for one of your cows, how many goats will 4 cows cost me? The ancient Greeks did a great deal of clever thinking, but very few experiments; this led to some errors. For instance, Aristotle observed that a rock falls faster than a feather, and concluded that heavier objects fall faster than lighter objects. Aristotle's views persisted for centuries, until the discovery of air resistance.

A much more dramatic tale comes with astronomy, which studied and tried to predict things that were out of human reach and apparently beyond human control. "Fear not this dark and cold; warm times will come again. The seasons are a cycle. The time from the beginning of one planting season to the beginning of the next planting season is almost 13 cycles of the moon -- almost 13 cycles of the blood of fertility." The gods who lived in the heavens were cruel and arbitrary -- too much rain or too little rain could mean famine.

The earth was the center of the universe. Each day, the sun rose in the east and set in the west. Each night, the constellations of stars rose in the east and set in the west. The stars were fixed in position, relative to each other, except for a handful of "wanderers," or "planets"; their motions were much more complicated. Astrologers kept careful records of the motions of the planets, so as to predict their future motions and (hopefully) their effects on humans.

In 1543 Copernicus published his observations that the motions of the planets could be explained more simply by assuming that the planets move around the sun, rather than around the earth -- and that the earth moves around the sun too; it is just another planet. This makes the planets' orbits approximately circular. The church did not like this idea, which made earth less important and detracted from the idea of humans as God's special creation.

During the years 1580-1597, Brahe and his assistant Kepler made many accurate observations of the planets. Based on these observations, in 1596 Kepler published his refinement of Copernicus's ideas. Kepler showed that the movements of the planets are described more accurately by ellipses, rather than circles. Kepler gave three "laws" that described, very simply and accurately, many aspects of planetary motion:

The few people who understood geometry could see that Kepler had uncovered some very basic truths.

In 1609 Galileo took a "spyglass" -- a popular toy of the time -- and used it as a telescope to observe the heavens. He discovered many celestial bodies that could not be seen with the naked eye. The moons of Jupiter clearly went around Jupiter; this gave very clear and simple evidence supporting Copernicus's idea that not everything goes around the earth. The church punished Galileo, but his ideas, once released to the world, could not be halted.

Galileo is also credited with realizing that, aside from air resistance, light objects drop just as fast as heavy ones. He made careful measurements of times that it took balls of different sizes to roll down ramps. There is a story that Galileo dropped objects of different sizes off the Leaning Tower of Pisa, but it is not clear that this really happened. However, we can easily run a "thought-experiment" to see what would happen in such a drop. If we describe things in the right way, we can figure out the results:

Drop 3 identical 10-pound weights off the tower; all three will hit the ground simultaneously. Now try it again, but first connect two of the three weights with a short piece of thread; this has no effect, and the three weights still hit the ground simultaneously. Now try it again, but instead of thread, use superglue; the three weights will still hit the ground simultaneously. But if the superglue has dried, we see that we no longer have three 10-pound weights; rather, we have a 10-pound weight and a 20-pound weight.

Some of the most rudimentary ideas of calculus had been around for centuries, but it took Newton and Leibniz to put the ideas together. Independently of each other, around the same time, those two men discovered the Fundamental Theorem of Calculus, which states that integrals (areas) are the same thing as antiderivatives. Though Newton and Leibniz generally share credit for "inventing" calculus, Newton went much further in its applications. A derivative is a rate of change, and everything in the world changes as time passes, so derivatives can be very useful. In 1687 Newton published his "three laws of motion," now known as "Newtonian mechanics"; these laws became the basis of physics.

To explain planetary motion, Newton's basic laws must be combined with his law of gravitation:

Newton's laws were simpler and more intuitive as Kepler's, but they yielded Kepler's laws as corollaries. Newton's universe is sometimes described as a "clockwork universe," predictable and perhaps even deterministic. We can predict how billiard balls will move after a collision. In principle we can predict everything else in the same fashion; a planet acts a little like a billiard ball. (In practice, most motions are a bit more complicated. For our everyday experiences here on the surface of the earth, it may be more appropriate to think of an atom as a billiard ball. Most things are made of trillions of trillions of trillions of atoms, so I can't predict how you will behave. But Newton could predict how the planets would behave.)

Suddenly the complicated movements of the heavens were revealed as consequences of very simple mathematical principles. This gave humans new confidence in their ability to understand -- and ultimately, to control -- the world around them. No longer were they mere subjects of incomprehensible forces. The works of Kepler and Newton changed not just astronomy, but the way that people viewed their relation to the universe. A new age began, commonly known as the "Age of Enlightenment"; philosophers such as Voltaire and Rousseau wrote about the power of reason. Surely this new viewpoint contributed to

Newton's laws of motion only described; they did not explain. Newton described how much gravity there is, with mathematical preciseness, but he did not explain what causes gravity. Are there some sort of "invisible wires" connecting each two objects in the universe and pulling them toward each other? Apparently not. How gravity works is understood a little better nowadays, but Newton had no understanding of it whatsoever. Thus, when Newton formulated his law of gravity, he was also implicitly formulating a new principle of epistemology (i.e., of how we know things): we do not need to have a complete explanation of something, in order to have useful (predictive) information about it. That principle revolutionized science and technology.

That principle can be seen in the calculus itself. Newton and Leibniz knew how to correctly give the derivatives of most common functions, but they did not have a precise definition of "derivative"; they could not actually prove the theorems that they were using. Their descriptions were not explanations. They explained a derivative as a quotient of two infinitesimals (i.e., infinitely small but nonzero numbers). This explanation didn't really make much sense to mathematicians of that time; but it was clear that the computational methods of Newton and Leibniz were getting the right answers, regardless of their explanations. Over the next couple of hundred years, other mathematicians -- particularly Weierstrass and Cauchy -- provided better explanations (epsilons and deltas) for those same computational methods.

(It may be interesting to note that, in 1960, logician Abraham Robinson finally found a way to make sense of infinitesimals. This led to a new branch of mathematics, called nonstandard analysis. Its devotees claim that it gives better intuition for calculus, differential equations, and related subjects; it yields the same kinds of insights that Newton and Leibniz originally had in mind. A college calculus book based on this approach was published by Keisler in 1986. However, it did not catch on. The nonstandard analysis approach is too radically different from the more traditional epsilon-delta approach, and perhaps the nonstandard analysis approach is also too abstract for today's college freshmen.)

Yet another chapter is still unfolding in the interplay between mathematics and astronomy. In recent years, astronauts have been able to see, from outer space, that the earth is indeed round, like a ball. Of course, even before the advent of space travel, mathematicians and geographers were able to determine that fact, by careful measurements on the ground. But the radius of the earth is large (4000 miles), and so the curvature of the two-dimensional surface is too slight to be evident to a casual observer. In an analogous fashion, our entire universe, which we perceive as three-dimensional, may have a slight curvature; this question was raised a couple of hundred years ago when Gauss and Riemann came to understand non-Euclidean geometries. The curvature of the physical universe is too slight to be detected by any instruments we have yet devised. Astronomers hope to detect it, and deduce the shape of the universe, with more powerful telescopes that are being built even now.

Human understanding of the universe has gradually increased over the centuries. One of the most dramatic events was in the late 19th century, when Georg Cantor "tamed" infinity and took it away from the theologians, making it a secular concept with its own arithmetic. We may still have a use for theologians, since we do not yet fully understand the human spirit; but infinity is no longer a good metaphor for that which transcends our everyday experience.

Cantor was studying the convergence of Fourier series and was led to consider the relative sizes of certain infinite subsets of the real line. Earlier mathematicians had been bewildered by the fact that an infinite set could have "the same number of elements" as some proper subset. For instance, there is a one-to-one correspondence between the natural numbers

But this did not stop Cantor. He said that two sets "have the same cardinality" if there exists a one-to-one correspondence between them; for instance, the two sets above have the same cardinality. He showed that it is possible to arrange the rational numbers into a table (for simplicity, we'll consider just the positive rational numbers):

1/1	1/2	1/3	1/4	...
2/1	2/2	2/3	2/4	...
3/1	3/2	3/3	3/4	...
4/1	4/2	4/3	4/4	...
...	...	...	...	...

This shows that the set of all ordered pairs of positive integers is countable -- i.e., it can be arranged into a list; it has the same cardinality as the set of positive integers. Now, run through the list, crossing out any fraction that is a repetition of a previous fraction (e.g., 2/2 is a repetition of 1/1). This leaves a slightly "shorter" (but still infinite) list

containing each positive rational number exactly once. Thus the set of positive rational numbers is countable. A similar argument with a slightly more complicated diagram shows that the set of all rational numbers is also countable. However, by a different argument (not given here), Cantor showed that the real numbers cannot be put into a list -- thus the real numbers are uncountable. Cantor showed that there are even bigger sets (e.g., the set of all subsets of the reals); in fact, there are infinitely many different infinities.

As proof techniques improved, gradually mathematics became more rigorous, more reliable, more certain. Today our standards of rigor are extremely high, and we perceive mathematics as a collection of "immortal truths," arrived at by pure reason, not even dependent on physical observations. We have developed a mathematical language which permits us to formulate each step in our reasoning with complete certainty; then the conclusion is certain as well. However, it must be admitted that modern mathematics has become detached from the physical world. As Einstein said,

As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.

For instance, use a pencil to draw a line segment on a piece of paper, perhaps an inch long. Label one end of it "0" and the other end of it "1," and label a few more points in between. The line segment represents the interval [0,1], which (at least, in our minds) has uncountably many members. But in what sense does that uncountable set exist? There are only finitely many graphite molecules marking the paper, and there are only finitely many (or perhaps countably many) atoms in the entire physical universe in which we live. An uncountable set of points is easy to imagine mathematically, but it does not exist anywhere in the physical universe. Is it merely a figment of our imagination?

It may be our imagination, but "merely" is not the right word. Our purely mental number system has proved useful for practical purposes in the real world. It has provided our best explanation so far for numerical quantities. That explanation has made possible radio, television, and many other technological achievements --- even a journey from the earth to the moon and back again. Evidently we are doing something right; mathematics cannot be dismissed as a mere dream.

The "Age of Enlightenment" may have reached its greatest heights in the early 20th century, when Hilbert tried to put all of mathematics on a firm and formal foundation. That age may have ended in the 1930's, when Gödel showed that Hilbert's program cannot be carried out; Gödel discovered that even the language of mathematics has certain inherent limitations. Gödel proved that, in a sense, some things cannot be proved. Even a mathematician must accept some things on faith or learn to live with uncertainty.

Some of the ideas developed in this essay are based on the book Mathematics: The Loss of Certainty, by Morris Kline. I enjoyed reading that book very much, but I should mention that I disagreed with its ending. Kline suggests that Gödel's discovery has led to a general disillusionment with mathematics, a disillusionment that has spread through our culture (just as Newton's successes spread earlier). I disagree with Kline's pessimism. Mathematics may have some limitations, but in our human experience we seldom bump into those limitations. Gödel's theorem in no way invalidates Newton, Cantor, or the moon trip. Mathematics remains a miraculous device for seeing the world more clearly.

One question which is raised by many people is, "Why should I study mathematics?". The question is usually asked from a perspective that there is probably no good and desirable reason for the speaker to study mathematics, but he will tolerate the minimum required because he has to, and then get on to more valuable and important things.

I readily acknowledge that there are many math classes which are drudgery and a general waste of time, and that many people have had experiences with mathematics which give them good reason to hold a distaste for the discipline. However, it is my hope that I may provide readers with an insight that there is something more to mathematics, and that this something more may be worthwhile.

Let's begin by looking at the reasons that the reader may already have come across for why he should study mathematics:

< >There are certain basic computational skills that are needed in life. People should be able to figure out whether a 24-pack of their favorite soda for $3.89 is a better or worse deal than a 12-pack for $1.99.It builds character. I suffered through mathematics for such-and-such many years. So should you.

The answer which many nonmathematicians may have is something along the lines of, "Mathematics, at its heart, is about learning and using formulas and things like that. In gradeschool, you learn the formulas and methods to add, subtract, multiply, and divide; then in middle school and high school it is on to bigger and better formulas, like the formula for the slope of a line passing through two points. Then in college, if your discipline unfortunately requires a little mathematics (such as the social sciences requiring statistics), you learn formulas that are even more complicated and harder to remember. The deeper you go into mathematics, the more formulas and rote methods you have to learn, and the worse it gets."

The best response I can think of to that question is to respond by analogy, and my response is along the following lines:

A child is in school will be taught various grammatical rules, sentence diagramming, and so on. These will be drilled and studied for quite a long while, and it must be said that this is not the most interesting of areas to study.

An English teacher who is asked, "Is this what your discipline is really about?", will almost certainly answer, "No!". Perhaps the English student is proficient in grammar, but that's not what English is about. English is about literature--about stories, about ideas, about characters, about plots, about poetic description, about philosophy, about theology, about thinking, about life. Grammar is not studied so that people can suffer through learning more pointless grammar; grammar is studied to provide students with a basic foundation from which they will be able to use the English language. It is a little drudgery which is worked through so that students may behold an object of great beauty.

This is the function of the formulas and rules of mathematics. Not rules and formulas so that the student is prepared for more rules and formulas, but rules and formulas which are studied so that the student can go past them to see what mathematics is really about.

And what is mathematics really about? Before I give a full answer, let me say that it is something like what English is about.

The one real glimpse that someone who has been through high school may have had of mathematics is in the study of geometry. There are a few things about high school geometry that I would like to point out:

Imagine a dream where there are many pillars--some low, some high--all of which are too high to step up to, and all of which are wide enough to stand upon.

Now imagine someone dreaming this dream. That person looks at one of the pillars and asks, "Has anyone been on top of that pillar?" Then one of the Inhabitants of his dream answers, "No, nobody has been on top of that pillar." Then the person looks at another of the pillars, which has a set of stairs next to it, and asks, "Has anyone been on top of that pillar over there?". The answer is, "Yes, someone has, and has left behind a set of steps. You may take those steps and climb up on top of the pillar yourself, if you wish."

And this person continues, and sees more pillars. Some of them stand alone, too high to step up to, and nobody has been to those. Others have had someone on top, and there is always a set of steps which the person left behind, by which he may climb up personally. And the steps go every which way--some go straight up, some go one way and then another, some seem to almost go sideways. Some are very strange. Some pillars have more than one set of steps. But all of them lead up to the top of the pillar.

The person dreaming may well have the impression that one gets atop a pillar by laying down one step, then another, then another, until one has assembled steps that reach to the top of the pillar. And, indeed, it is possible to climb the steps up to the pillars that others have gone to first.

And the person sees what really happens when the guide becomes very excited and says, "Look over there! There is a great athlete who is going to attempt a pillar that nobody has ever been atop!"

And the athlete runs, and jumps, and sails through the air, and lands on top of the pillar.

And when the athlete lands, there appears a set of stairs around the pillar. The athlete climbs up and down the stairs a few times to tidy them up for other people, but the stairs were produced, not by laying down slabs of stone one atop another, but by jumping.

Then the guide explained to the dreamer that the athlete had learned to jump not only by looking at the steps that others had left, but by jumping to other pillars that already had steps, instead of using the steps.

The pillars that stand with steps leading up to them are mathematical facts that have been proven.

The steps are the steps of proofs, the little assertions. As some of the steps are bizarre, so are some proofs. As some pillars have more than one path of steps, so some facts have more than one known proof.

The leap is a flash of intuition, by which the mathematician knows which of many steps will take him where he wants to go.

As the steps appeared when the leap was made, so the proof appears when the flash of intuition comes. The athlete then tidied up the steps, as the mathematician writes down and clarifies the proof, but the proof comes from jumping, not from building one step on another.

Finally, the athlete became an athlete not only by climbing up and down existing steps, but also by jumping up to pillars that already had steps--one becomes skilled at making intuitive leaps, not only by learning existing proofs, but also by solving already proven problems as if there were no proof to read.

As one philosophy major commented to me, "Mathematicians do proofs, but they don't use them."

That flash of insight is the flash of inspiration that artists work under, and in this sense a mathematician is very similar to an artist. (What do a mathematician and an artist have in common? Both are pursuing beauty, to start with...)

This character of mathematics that is captured in geometry is true to geometry, but the actual form that it takes is largely irrelevant. Other branches of mathematics, properly taught, could accomplish just the same purpose, and for that matter could just as well replace geometry. Two other disciplines which draw heavily on applied mathematics, namely computer science and physics, have essentially the same strong points. I would hold no objections, for that matter, if high school geometry classes were replaced by strategy games like chess and go.

Mathematics is about puzzle solving; I would refer the reader to works such as Raymond Smullyan's The Lady or the Tiger? and Colin Adams's The Knot Book: an Elementary Introduction to the Mathematical theory of Knots. There are many people to whom mathematics is a recreation, consisting of the pleasure of solving puzzles. If mathematics is approached as memorizing incomprehensible formulas and hoping to have the good luck to guess the right formula at the right time, it will be a chore and a torture. If it is instead approached as puzzle solving, the activity will yield unexpected pleasure.

My father has a doctorate in physics and teaches computer science. He has said, more than once, that he would like for all of his students to take physics before taking his classes. There is a very important and simple reason for this. It is not because he wants his students to program physics simulators, or because there is any direct application of the mathematics in physics to the computer science he teaches. There isn't. It is because of the problem solving, the manner of thinking. It is because someone who has learned how to think in a way that is effective in physics, will be able to think in a way that is effective in computer science.

This applies to other disciplines as well. Ancient Greek philosophers, and medieval European theologians, made the study of geometry a prerequisite to the study of their respective disciplines. It was not because the constructions or theorems would be directly useful in making claims about the nature of God. Like physics and computer science, there was no direct application. But in order to study geometry, one had to be able to think rigorously, analytically, critically, logically, and abstractly.

Thinking logically and abstractly is an important discipline in life and in other academic disciplines that consist of thinking--it has been said that if you can do mathematics, you can do almost anything. The main reason mathematics is valuable to the non-mathematician is as a form of weight lifting for the mind. Even when the knowledge has no application, the finesse that's learned can be useful.

To the non-mathematician, mathematics is a valuable discipline which offers practice in how to think well--both analytic thought and problem solving. Mathematics classes will most profitably be approached, not as "What is the formula I have to memorize," but with ideas such as those enumerated here. The nonmathematician who approaches a mathematics class as an opportunity for disciplined thought and problem solving will do better, profit more, and maybe, just maybe, enjoy the course.

It is my the hope that this essay have provided the nonmathematician with an inkling of why it is profitable for people who aren't going to be mathematicians to still 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'?", 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 knots to indignation at perceived frivolity and uselessness, as if research must either be directly responsible for new technology and wonder drugs 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 crossword puzzles, word games, and other "frivolous" intellectual games which don't directly produce new technology or solve world hunger. Many people intentionally spend time solving 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 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 key 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.

Similarly, 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; likewise, 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, and 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 trivial 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 survive in a very specific environment, 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 a theorem has been proven, 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 salesmen, 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.

PROBLEM SOLVING. Working on a problem, and finally figuring out a solution is very satisfying. There is a feeling of accomplishment to have triumphed over an obstacle by the power of your mind, as well as the pleasure of any creative endeavor. One attraction that is particular to mathematics is the sense of certainty: when you have a solution, you know it is a solution. In some ways, this is an aspect shared with certain kinds of puzzles, which most mathematicians love. (Click here for a few favorite puzzles on another part of this website.)

PATTERNS and STRUCTURE. Mathematics is all about patterns and order. The way that ideas fit together and unexpected connections emerge make the subject both fascinating and aesthetically compelling. Achieving a new level of understanding often has the same emotional impact as a deeply moving piece of music or looking out over a breath-taking vista.

PRECISION of THOUGHT. Mathematics is distinguished by the precision with which concepts can be articulated, and the certainty with which arguments can be advanced. This attribute accounts in large measure for the near absence of controversy in mathematics. With rare exceptions, mathematicians do not argue with each other about what is true. This is a marked contrast with about every other field of thought.

Everyone wants to be understood, and communication is a fundamental human impulse. Knowing that you can communicate just what you mean, that you will be understood clearly, and that there will be consensus in the entire community of mathematicians is very gratifying. Similarly, you can have confidence that your arguments will be judged on their merits, and that when you are right the community will recognize and acknowledge that. Indeed, mathematics seems like one of the few subjects in which you can really convince a determined skeptic simply by the inevitability of your arguments.

APPLICABILITY. Mathematics has proven incredibly useful in just about every field of human endeavor. It is used in science, in business, and in industry. It is used in all branches of government to help describe and manage our complex society. It has even been used as a way to understand and develop art and music. Many mathematicians get great satisfaction from helping solve problems with real significance in life.

HISTORY and PHILOSOPHY. The development and evolution of mathematical thought are inextricably tied up with the history and philosophy of western culture. Many of the greatest mathematicians have also been philosophers, including Descartes, Leibniz, and Russell, to name just a few. The tradition of viewing mathematics as fundamental in human thought dates to Aristotle and Plato, and is respected by modern philosophers like Kant. Many mathematicians are interested in both mathematics and philosophy, and find similar satisfactions from understanding both subjects.

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Why Do We Study Calculus? or, a brief look at some of the history of mathematics

Why Do We Study Calculus?
or,
a brief look at some of the history of mathematics