Mathematics, Responsibility, and Joy

Over the past several years, I have been speaking to students from elementary to graduate school about the importance, beauty, and power of mathematics. These conversations are informed by a watershed moment that occurred nine years ago, when I addressed a group of high school math teachers on my 50th birthday. Confronted with the undeniable proof of my aging and wondering about my place in the world now that my children had left home, I changed the topic of my talk from the general importance of math to why it is important that we do math. 

The pursuit of beauty is the fundamental motivation for mathematicians. We are all lured in by some particularly elegant argument or radically new way to think about the world; for instance, the Archimedean proof of the existence of irrational numbers forged my personal love of math. The elegance, rigor, and creativity within the field of mathematics can translate into innovative and effective solutions to physical or social issues.

When we apply math to problems that are posed by industry, government, and society, we expect to change people’s lives. We make financial transactions safer and more efficient; determine health risks within populations; create robust channels of communication; direct more police to select neighborhoods; identify people as potential terrorists; and determine whether a radar return is a flock of birds, a civilian airliner, or an incoming missile. In short, mathematics’ ability to profoundly affect people and communities for both good and ill imposes corresponding moral obligations upon us.

Our first obligation is to predict the impact of our analyses and subsequent decisions on the world. Although the accuracy of these predictions is often limited, the exercise forces us to consider multifarious effects. When submitting a product for approval from the U.S. Food and Drug Administration, for instance, we must describe foreseeable misuse and demonstrate the product’s ability to mitigate such harms. In the absence of mitigation, we need to prove that the product’s benefits offset the possible damages.

If you want to reflect on your career with satisfaction, you should repeat this exercise for every project that you undertake. What is the intended outcome of your work? What are the likely outcomes? How will people actually use your analyses? What technologies and behaviors are you enabling? Do you feel that the benefits of this work outweigh its potential damages?

I began my career by joining the Strategic Defense Initiative (known colloquially as the “Star Wars program”) to protect the U.S. from missile attacks. I took on this work simply because it was both an interesting engineering challenge and an available job. My regrets about this stage of my career do not stem from my participation in the program, but from the fact that I did not make an informed decision about my involvement by considering the arguments for or against the deployment of an imperfect defense in the face of a nuclear attack. In order to look back without regrets, you have to decide whether you are comfortable with the implications of your present work.

Math is pure and absolute, but physical and social systems are chaotic and uncertain. As such, we must be careful not to translate our models’ exact and provable answers into absolute beliefs about machines, natural systems, or societies. Even if we acknowledge the world’s stochastic nature, those who use the models and tools that we create may consider their predictions to be absolute.

Machine translation between languages is a great triumph of applied math; for instance, I could not have imagined Google Translate when I was a student. However, machine translation is still imperfect. In 2020, an Afghan woman’s asylum application was denied in U.S. court because the written submission did not match her initial interviews. This was not a case of the woman attempting to deceive the court, but rather the machine translation changing the personal pronoun “I” to “we” in her written statement [3]. Even good models that are utilized by imperfect systems can cause significant harm.

These imperfect systems are limited by budget or statute and use hard thresholds to determine who will benefit and who will suffer. It is easy to let our algorithms and thresholds disguise our moral choices. For example, a 54-year-old British citizen named Bruce Hardy was undergoing treatment for metastatic kidney cancer in 2008. His doctor recommended a course of Sutent, a new drug that was expected to extend his life for six months at a cost of $54,000. Because British health service policy only valued six months of life at approximately $23,000, the treatment was denied. In the wake of widespread protests, the institute that determined the cost-effectiveness of drugs modified its policy to allow the approval of medications like Sutent [4]. The morally correct position in this case is not obvious, as the same funds that would grant a terminally ill man six more months of life could also fully vaccinate about 50 children [1, 2]. The important lesson is that hard limits and algorithms can obscure the reality of the moral choices at play.

In many cases, the very fact that we are making ethical decisions is opaque. We site a new incinerator near highways and railroads, unaware of its impact on a nearby historic African American community. We segment a city into risk zones for mortgage insurance based on access to emergency services and local crime statistics, not realizing that doing so makes home ownership more inaccessible in a credit-starved neighborhood. Our optimization models incorporate the interests of our employers, the concerns of our coworkers, and our own knowledge and experience. For this reason, meaningful diversity is vital to all workplaces and design processes. One person cannot singlehandedly encompass enough experience to ensure the development of just solutions for all who are affected. I believe that the only way to avoid harming our communities is to guarantee the representation of the entire population throughout every process or project. We need mathematicians who represent all facets of the world, realize the limits of their knowledge, and actively listen to outside concerns and suggestions.

Finally, we must find joy and friendship in our pursuits and interactions. The empathy and humanity of our decisions and analyses depend on our engagement with the people around us, and the variety of experiences, knowledge, and insights that we bring to our analyses improve the creativity and value of our solutions. Math is a particularly beautiful way to describe the world, but art, music, literature, and sport offer equally invaluable insights into the human condition and our place therein. Do not begrudge time spent away from your true work and true calling, as this time allows you to translate the rigor and insight of mathematics into a world and community that becomes better through your presence.

References

[1] Centers for Disease Control and Prevention. (2008, November 25). Archived CDC vaccine price list as of November 25, 2008. Retrieved from https://www.cdc.gov/vaccines/programs/vfc/awardees/vaccine-management/price-list/2008/2008-11-25.html.

[2] Centers for Disease Control and Prevention. (2023, April 27). Child and adolescent immunization schedule by age. Retrieved from https://www.cdc.gov/vaccines/schedules/hcp/imz/child-adolescent.html.

[3] Deck, A. (2023, April 19). AI translation is jeopardizing Afghan asylum claims. Rest of World. Retrieved from https://restofworld.org/2023/ai-translation-errors-afghan-refugees-asylum.

[4] Harris, G. (2008, December 2). British balance benefit vs. cost of latest drugs. The New York Times. Retrieved from https://www.nytimes.com/2008/12/03/health/03nice.html.