Mathematical Pluralism

Multiplicity in the math of AI, Systems, and Society.

Last week, I attended a great conference hosted by the Emory Center for Ethics on AI, Systems, and Society. Organized by Alex Tolbert, the conference brought together a unique mix of philosophers and computer scientists interested in understanding the foundations of algorithmic fairness and automating decision-making. It gave me plenty to think about.

My main takeaway is a reinforced commitment to mathematical pluralism. Analytic tools from computer science, philosophy, mathematics, and statistics are inherently local in their validity, and we should embrace a Cartwrightian dappled view of epistemology rather than an imperialist approach to understanding and decision making. The keywords in decision making—be they probable, casual, or rational—do not have universal mathematical definitions. They mean different things at different times to different people, and these meanings can be simultaneously valid.

A pathology of formal methods is taking terms that everyone uses and turning them into impenetrable jargon that only professors can deploy. It is absurd to think that only mathematicians can understand what words like likely, probable, causal, or rational mean. Creating analytic boxes around these terms invents mathematics. It does not establish conceptual grounding. Indeed, it creates mathematical probability, mathematical causation, and mathematical rationality. These are locally valid but deeply limited modeling frameworks.

Let’s take mathematical causality as an example. A general scientific theory of causality that covers all aspects of causation is a pipe dream. But then what is “causal inference?” The academic discipline amounts to browbeating people with p-values derived from real or imaginary randomized trials. Not every statement about worldly causation needs a DAG or identification strategy. Indeed, we should flip it around: there is a very narrow set of problems for which the effect size of interventions can be estimated from differences in means of a real or hypothetical randomized trial. This limits us to a very specific subset of what Aristotle called “efficient causes.” It would be better to understand the boundaries of this set of causes rather than to try to convince people that this is the only path to understanding causation.

Most causal inference adherents admit limits but hope they can patch their methodology with appropriate addenda. But what if they can’t? What if there are some causes that can’t be RCTed? Issa Kohler-Hausmann and Lily Hu have highlighted these limitations in social settings for years. Perhaps their most famous example is in understanding resume audit studies. In a famous paper “Are Emily and Greg More Employable than Lakisha and Jamal?” by Marianne Bertrand and Sendhil Mullainathan ran a randomized controlled trial where they submitted resumes with “black sounding” and “white sounding” names at random and found “black sounding” names yielded fewer job offers than “white sounding.” Was this intervening on race? No. It was intervening on name. What is the meaning of intervening on name on a resume? A “white resume” means something unique, and changing the leading name creates discordance. A recruiter might take pause reading a resume of a person named “Lakisha” who lists membership in the Daughters of the American Revolution. You can’t just toggle a name and think you have toggled “race.”

But of course, in law and policy, we can argue that racialization causes discrimination. We don’t need to use the Neyman-Rubin potential outcomes framework to do this. Legal and political reasoning doesn’t need to rely on backfitting a probabilistic data generating process onto reality. We can push for the world we want without mathematics.

Similarly, I’m perplexed by the value of modeling people as utility maximizers. We have known almost from the beginning that people don’t follow this model. Von Neumann and Morgenstern formalized utility maximization in 1944. Flood and Dresher proved it unpredictive of human behavior in 1950. Few theories in social science have been so thoroughly invalidated while remaining so enticing to so many people. When you build a model of utility maximizing agents and extract a public policy from that, why should I care? The model is not even wrong. How could it be useful?

I could make a case for utility maximization in designing voting or allocation systems that are transparent. Mathematical instrumental rationality can help design certain fair gaming scenarios, but such rules don’t help predict how people will behave. Similarly, mathematical causal inference can give regulatory guidance for drug approval. It’s likely not helpful in understanding political interventions. Mathematical statistical forecasting can help tabulate cost-benefit analyses of policies for large-scale human systems. Maybe it doesn’t give good guidelines for how to treat individuals.

My argument is not that statistico-computational methods are useless in the human-facing sciences. It is that such methods have sweet spots, and we need a methodological pluralism to make sense of our world and to imagine a better one. Our current academic models are allergic to such pluralism but shouldn’t be.

Pluralism is vital because the nexus of analytic philosophy, mathematical statistics, computer science, and econometrics have no way of reasoning about people. People are not computers! And yet these communities refuse to admit it.

But this leads to the obvious question of what computer scientists should do if they want to engage with social issues. Algorithms cause harm, and we need to consider their impacts on people. If our products are causing social problems, we should be able to help provide solutions.

To remedy these harms, what if we’re best served as technical stewards? Our role is as communicators—though not as authorities—on what our systems can and cannot do. We can partner with ethicists tasked with understanding the human-machine interface and its legal, ethical, and societal implications. We can work with other disciplines to craft policies, rules, and interventions that shape the world in our values.

At the Emory conference, Leonard Harris convincingly argued for pluralistic accounts of the tragedy of racism, not leaning on any particular methodology as fully explanatory. Multiple analytical narratives bring multiple truths and potential paths out. Mathematical pluralism is necessary as our analytic philosophical storytelling always has limits. Qualitative and quantitative are not axes. Multiple truths do exist and need to exist. It would be nice to hear that there’s one simple trick to making sense of the world. But there is no trick, and multiple, seemingly contradictory truths hold at once.