The mathematics of a better world

Numbers, proofs, logic. 

As far back as I can remember, I have loved mathematics. Growing up in a family of scientists in Barcelona, I gained a deep appreciation for mathematics from a young age when my father would challenge me to find the next term of sequences at the dinner table. I always found math so beautiful and elegant — like a painting or a poem, or the classical music I loved playing on the piano and violin. The golden ratio, fractals, Euler’s identity… everything seemed to carry a sense of perfect symmetry and harmony.

But what fascinated me most about math was not its beauty. It was the fact that it is the only discipline that uncovers truths through logic alone. It’s not a question of empirical validity. It’s not a matter of opinion. If a statement is true, you can prove it through reasoning, with just pen and paper. The real world can be full of prejudice, conflicts, and imperfections, but, as a kid, whether alone on my couch or with friends, I could immerse myself in this captivating world of reason and truth, where everything always seemed to work perfectly.

Years later, at age 17, I left home for the first time and flew across the Atlantic for the chance to study mathematics at Harvard University. I had only been to the United States once before, when the Catalan science foundation Joves i Ciència sent me to MIT to participate in a summer research program.

During my first year at Harvard, I happened to walk into the first lecture of the course “Introduction to Algorithms,” and that day, we learned how to multiply two numbers faster than using grade-school multiplication. This is how I discovered the field of theoretical computer science, and I was immediately hooked. Just as we can prove things about geometric objects, I learned that we can prove things about algorithms. We can prove how fast they run and how much memory they need. We can even prove that some problems, such as Alan Turing’s Halting Problem, cannot be solved by any algorithm at all.

However, the world outside of my lectures and textbooks was changing rapidly. Algorithms were no longer being used just to invert matrices or to find the shortest path in a graph. They were increasingly being used to make decisions about people, deciding who received—and who did not receive—job opportunities, loans, insurance, healthcare, education, and social services. Algorithms had entered the criminal justice system, where their assessments could influence decisions about bail, sentencing, and parole. Algorithms were also making choices for us, deciding what products we buy, what ads we see, what songs we listen to, what shows we watch, even whom we date. 

And in many cases, they were failing us. Opaque algorithms were making biased, unreliable, and discriminatory decisions, causing real harm to real people. One scholar even referred to them as “weapons of math destruction” in a famous book.

Learning about these failures stunned me. I felt the collision between my two worlds: the world of numbers and the world of people. If algorithms are all logic and rigor, what went wrong?

That’s when I realized that even the most logical of fields are never free of value judgments. We choose what problems to solve, what to optimize for, and which questions are worth asking in the first place. For example, algorithms are typically designed to optimize for global accuracy. Yet optimizing solely at a global level fails to ensure accuracy across subgroups of the population, potentially causing substantial and unanticipated harm to those communities. Choosing to optimize globally, without also optimizing locally for subgroups, is itself a value choice.

Once I recognized the subjective nature of research, my mathematical refuge of logic and objectivity did not come crashing down. Instead, it opened up to the real world.

I realized that I did not have to leave behind the things I loved—proofs, abstractions, theorems—to make a meaningful impact in the world. In fact, I could use these exact tools and bring mathematical precision to help make algorithms and AI benefit, not harm, society.

This realization ultimately inspired me to pursue a PhD and now forms the focus of my research at Stanford. I am part of a growing research community that uses mathematics to provide provable answers to questions at the intersection of algorithms and society, including how to ensure that predictions work well across different groups, how to protect people’s privacy while learning from their data, how to allocate opportunities and resources fairly, and how to use large language models to surface representative viewpoints, among many other topics. Our community is creating new theorems, new textbooks (and a forthcoming book by my PhD supervisor, Professor Omer Reingold, Equal under the Algorithm: A People’s Guide to Fair Computation), new university courses, and new conferences, all of which have profoundly deepened our understanding of algorithms and computer science as a whole. We have even managed to use these new tools from algorithmic fairness to improve central results in “classical” computer science from back in the 1990s, which predate all of these efforts.

Perhaps most importantly, my journey through theoretical computer science has taught me that, as scientists, we can improve the world not only through policy and politics, but also from within our own disciplines. We can—and should—use fundamental research to address society’s most pressing challenges. After all, the boundaries of a scientific field are not set in stone; they are drawn and redrawn by the questions we choose to ask.

Numbers, proofs, logic. The concepts that captivated me as a child now enable me to improve the impact of algorithms on people’s lives. I now see that even those of us who work in the world of abstraction are not actually standing outside of the real world.

We are actively shaping it.

Sílvia Casacuberta Puig (2025 cohort), from Barcelona, Spain, is pursuing a PhD in computer science at Stanford School of Engineering. She graduated summa cum laude and Phi Beta Kappa from Harvard University with a double bachelor’s degree in mathematics and computer science, a minor in philosophy, and a concurrent master’s degree in computer science. She researches the societal impacts of algorithms through the lenses of theoretical computer science. Her undergraduate thesis won the Captain Jonathan Fay Prize, awarded to the three best theses of Harvard College’s graduating class, and was featured in Quanta magazine. She then attended the University of Oxford as a Global Rhodes Scholar, pursuing a master’s degree in computer science. Sílvia has conducted research at the Max Planck Institute for Intelligent Systems, ETH Zurich, IBM Research, Imperial College London, and MIT. She has received multiple teaching awards, a CRA Outstanding Undergraduate Research Award, and a Stanford Graduate Fellowship.

Knight-Hennessy scholars represent a vast array of cultures, perspectives, and experiences. While we as an organization are committed to elevating their voices, the views expressed are those of the scholars, and not necessarily those of KHS.