Math as instrument: lessons from functional programming

Speaker: Spencer Breiner (NIST) Abstract: There is a sharp break in the mathematics curriculum, somewhere between vector calculus and abstract algebra, where the default tone of mathematical texts shifts quite abruptly. Before the break, we learn specific algorithms to solve specific problems, gaining an operational understanding of numbers and how they behave along the way. Mathematics after the break focuses on theory, usually in definition-theorem-proof style, and while much is gained in terms of clarity and structured thought, the shift from primarily instrumental to primarily theoretical instruction often undervalues operational knowledge and examples. The central theme of this talk is that society at large and mathematicians in particular could benefit from a less theoretical, more operational “user’s manual” for the toolbox of modern (20ᵗʰ century) mathematics. The theory of monads, developed for pure mathematics and imported into functional programming (FP), provides an interesting case study. Comparing the introductory materials prepared by the two communities, three observations stand out: (i) FP introductions are more instrumental than theoretical–when and how should we use monads in a computational model–with substantially lower prerequisites than their mathematical counterparts. (ii) Computation provides an operational basis for learning, with theory-driven guard rails (e.g., typechecking) that are baked in rather than (explicitly) learned. (iii) Instrumental and theoretical knowledge are only weak substitutes (in either direction), with a particular tension between context and generality. Theory is hard, and computational support for instrumental learning is an underexploited route to expand interest in and utility of modern mathematics. Talk at the AMS Special Session on "Compassion as a Mathematical Practice" at the 2026 Joint Mathematics Meetings in Washington, DC. Browse the rest of the videos from this session in the following playlist:    • AMS Special Session on "Compassion as a Ma...