Conclusions

We have completed our journey, our tour around various aspects of mathematical life. We looked at the beginnings of mathematical life for children and students. Then we studied some of its special features as a unique subculture of modern society. We saw its ability, on the one hand, to provide its devotees with solace and refuge; and we saw, on the other hand, its dangers in permitting them isolation and eccentricity, which have in some rare cases extended to utter madness. We looked next at some of the glue that holds the mathematical community together, in a chapter on friendships, a chapter on communities, and a chapter on aging. Then our last two chapters turned to teaching, learning, and schooling.

Such a broad and inclusive look at mathematical life is hardly intended to prove a theory or preach a moral. But we can point to a few important issues. One is an obvious but often overlooked psychological fact: mathematical work, like every other kind of deeply engaged intellectual or artistic work, is deeply emotional. It relies on intense motivation; it brings with it elation and disappointment, happiness and grief.

Some feelings about clarity and certainty, and the pursuit of answers to long unresolved problems, are specific to mathematics. Other emotions are shared across disciplines. These include the pleasures of mentoring, the challenge of teaching, and the rewards of participating in a caring community, as well as the discomfort of competition for prizes and fame. The joy of discovery, an emotion that mathematicians share with scientists and artists, was thus celebrated by Paul Halmos: “The joy of suddenly learning a former secret and the joy of suddenly discovering a hitherto unknown truth are the same to me—both have the flash of enlightenment, the almost incredibly enhanced vision, and the ecstasy and euphoria of released tension.”¹ But mathematicians are particularly vulnerable to a sense of inadequacy in a profession that remembers and honors so many of its most illustrious contributors.

The applied mathematician Fern Hunt has said, “No matter how good you actually are, there is definitely somebody who can run rings around you. . . . This and the fact that mathematics is a field a lot of people have trouble with causes a great deal of anxiety both within and outside the profession.”²

We have met Lipman Bers before in this book, as a beloved mentor at the Courant Institute and as a leader among mathematicians struggling against abuses of human rights He expressed some of these emotions in an interview.³

Question: “When you say that mathematics is a very cruel profession, do you mean because the standards are so high?”

Bers: “The standards are high, and you never know whether you will be able to hack it. First you are afraid that you won’t be able to understand your professors. Then you are afraid that you won’t be able to write a thesis.”

Bers was then asked whether, in spite of doubts, one does at last realize that one has succeeded. He answered: “If you have done something, yes. Nothing can compare with this pleasure! But then you start worrying—will you be able to do it again?”⁴

We have repeatedly mentioned the appeal of clarity and elegance that many future mathematicians find alluring. The usefulness of mathematics to physics, engineering, biology, and other disciplines also is a great motivation and satisfaction for future mathematicians.

But ambiguity, contradiction, and paradox are also inherent in mathematics. Life is ambiguous and contradictory. Mathematics is part of life. In so far as a philosophy of mathematics describes the total mathematical situation—process as well as content—naturally, it is also bound to be ambiguous. As William Byers writes, “Logic moves in one direction, the direction of clarity, coherence, and structure. Ambiguity moves in the other direction, that of fluidity, openness, and release. Mathematics moves back and forth between these two poles. . . . It is the interaction between these different aspects that gives mathematics this power.”⁵

Mathematical culture includes not only known results and theorems but also open problems. These challenges call forth some of the feelings Bers identified—doubts and questions, as well as the pleasure of reaching a solution. Problems may be ambiguous. Working on them requires one to live with the tension of uncertainty. Mathematicians cherish stories about one of their heroes’ arduous journey to resolve some long-standing conjecture. Such stories are told and retold as part of mathematical history and culture.

One powerful theme in this book has been the need for balance between single-minded absorption, on the one hand, and intellectual and emotional breadth, on the other. Part of the fascination of mathematics is its clarity, aesthetic appeal, and precision. But total immersion in these aspects can lead to a teaching style devoid of humor, lightness of touch, or compassion. It can even endanger someone who is vulnerable to obsession. For many mathematicians, a counterweight to immersion in their intellectual work is provided by love of friends and family or by joy in music or in nature. We think of the chamber music players of Göttingen and New York, and of the many hikers, swimmers, tennis players, collectors of butterflies or minerals, and lovers of music or poetry.

The second conclusion to this work is that despite its appearance of being individual and solitary, mathematical life is social as well as emotional. Every bit of mathematical work, whether problem solving, theory building, or practical application, takes its meaning and value from its interest and relevance to the mathematical community and to the larger society. This recognition goes counter to the stereotype of mathematics as an extreme academic ivory tower, as a sort of closed subculture disconnected from the concerns of socially oriented scholars or the public at large. By looking at controversies related to race, gender, age, and prize competitions, we have seen mathematical life entangled with the challenges and conflicts of contemporary culture. At the same time, immersion in mathematics has offered a temporary haven from war, persecution, and injustice.

Because it is social, there is always an ethical aspect to mathematical life. What you do affects others and can be helpful or harmful. In general, as part of a school or a university or even a corporation or a bureaucracy, a mathematician like anyone else may be competitive or cooperative, constructive or destructive, helpful or harmful. Moreover, since mathematics is integrally connected—financially, politically, and ideologically—to the larger society, the role a mathematician plays in his or her own professional community—for or against freedom of thought, social advancement, and human welfare—is subject to the same ethical judgments that would apply in any other realm of social life.

Applied mathematicians, who collaborate with physicists, biologists, or engineers, are judged by the usefulness of their work in real-world terms. Will they contribute to sophisticated methods of destruction? Or will their work be used for the benefit of humankind?

But the special activities of mathematicians are teaching and research. For a teacher, the main question is, What do you do for your students? Do you help to overcome the alienation so many young people feel when confronted with this rigorous discipline? Do you share with them your passion for the beauty of this discipline? Do you share your own turmoil when a solution is evasive?

As part of an ethical approach to the teaching of mathematics we have argued against using it as a filter—using it to decide who may get into graduate or professional programs. Mathematical knowledge is relevant for engineering. But for fields such as medicine, there are more appropriate ways to choose future doctors.

What are ethical concerns for mathematical researchers? Bill Thurston writes that the goal of mathematical research is to advance mathematics—not just to pile up theorems and proofs attached to your name. Do you try to make it possible for others as well as yourself to make big discoveries?

For those like G. H. Hardy who see themselves above all as artists, it is appropriate to be evaluated in the way one evaluates a composer or a novelist. Do you stay with the multitude? Or do you follow your own vision where it leads, even away from the most popular and acceptable trend? Do you go for the easy product for a guaranteed payoff without too much time and struggle? Or do you take on the most demanding project of which you are capable?

Mathematics is part of the broad tapestry of human thought. Like other parts of art and science, it is a search for pattern, harmony, proportion, and application. It offers dangers and frustrations, unreasonable and impossible demands. It also offers intense and memorable pleasures and satisfactions. Frustrations and satisfactions, dangers and pleasures, all are part of a deeply demanding and rewarding way of life.

Bibliography

Albers, D. J., Alexanderson, G. L., & Reid, C. (Eds.) (1990). More mathematical people: Profiles and interviews. Boston: Birkhäuser.

Byers, W. (2007). How mathematicians think: Using ambiguity, contradiction, and paradox to create. Princeton, N.J.: Princeton University Press.

Halmos, P. R. (1985). I want to be a mathematician. New York: MAA Spectrum, Springer-Verlag.

Henrion, C. (1997). Women in Mathematics: The addition of difference. Bloomington, Ind.: Indiana University Press, p. 226.

Thurston, W. (2005). On proof and progress in mathematics. In R. Hersh (Ed.). Unconventional essays on the nature of mathematics. New York: Springer.