Math and Language
Saturday, March 18th
Get ready for SUMS 2023: Math and Language!
What are the languages we use to understand mathematics? What math can we use to understand language? Join us as we think about the intersections of math and language!
Emily Riehl (Johns Hopkins), Philip Ording (Sarah Lawrence),
Rob Lewis (Brown), and Danny Calegari (University of Chicago).
Email any questions to sums [@] brown [dot] edu.
Morning Plenary Talks
On the art of giving the same name to different things
Emily Riehl, Johns Hopkins University
9am - 10am
Mathematics has developed an increasingly “higher dimensional” point of view of when different things deserve the same name, relaxing the traditional logical notion of equality to isomorphism (from Greek isos “equal” and morphe “form” or “shape”) and equivalence (from Latin aequus “equal” and valere “be well, be worth”). In practice, mathematicians tend to become more flexible in determining when different things deserve the same name as those things become more complicated. Unfortunately, these pervasive notions of sameness no longer satisfy the indiscernibility of identicals — the assertion that if two objects are the same they must share the same properties — essentially because the traditional set theoretical foundations of mathematics make it too easy to formulate “evil” statements. However, in a new proposed foundation system there are common rules that govern the meaning of identity for mathematical objects of any type that allow one to “transport” information along any identification. Moreover, as a consequence of a new "univalence" axiom, these identity types are faithful to the meanings of sameness that have emerged from centuries of mathematical practice.
Emily Riehl's homepage
Photo Credits: Marshall Clarke
Philip Ording, Sarah Lawrence College
10am - 11am
Drawing on literary, historical, philosophical, and mathematical texts, this talk will trace a number of problems and creative possibilities for writing that ensue from the sometimes friendly, sometimes fraught relationship between mathematics and language.
Philip Ording's homepage
Illustration by Anna Higgie
Afternoon Plenary Talks
The Formal Language of Mathematics
Rob Lewis, Brown University
3:30pm - 4:30pm
Sometimes it's clear that a mathematical statement is well-formed, that a definition is precise, or that a proof is correct. Sometimes the opposite is clear. And sometimes nothing is clear either way! The formalist perspective on mathematics, dating back to Hilbert and beyond, maintains that there is a rigorous logical grammar underlying all of these things. So goes the old story: humans don't write in this grammar directly, but they could, if they really tried. Modern pieces of software called proof assistants make the formalist dream feasible in practice. In a proof assistant, math looks a little like programming, but instead of running your code the computer checks your proofs. We'll talk about some of the challenges in developing a language for this, and discuss some examples of what has been achieved using these tools.
Rob Lewis' homepage
Danny Calegari, University of Chicago
4:30pm - 5:30pm
Names are boxes. Names are metaphors. Names are tools. Names are scaffolds. Names are stories. Naming well or ill has an enormous impact on the practice of mathematics. I will discuss some examples.
Danny Calegari's homepage