Miscellaneous Writing
Thoughts on Learning from LLMs
A few thoughts on my frustrations learning about general topics from LLMs. This essay touches on how generative AI shifts the comparative value of creation and curation.
Random Graph Games
This is an expository essay on my favorite theorem in mathematics. I'm not going to tell you what the theorem is here; that would ruin the surprise. Instead, I am going to invite you to play a game with me...
Preventing a flash war: Countering the risk of AI-driven escalation on the battlefield
This post on the Penn Center for Ethics and the Rule of Law blog, coauthored with Lieutenant Colonel Kyle Brown and Professor Jesse Hamilton, draws a parallel between the 2010 Flash Crash in financial markets and the emerging risks of escalation in AI-enabled warfare.
We argue that human oversight alone may be insufficient to prevent rapid, unpredictable escalation and propose battlefield “circuit breakers” as a regulatory safeguard. Drawing on lessons from finance and national security strategy, the piece outlines how the United States could lead in developing international mechanisms to prevent unintended AI-driven conflict.
Nicotine Supplementation and Thesis Writing
This essay recounts my self-experiment of using nicotine lozenges to overcome motivational paralysis while writing my PhD thesis. Demoralized after restarting research late in graduate school, I struggled to write despite my desire to finish my degree. I devised a regimen linking nicotine intake exclusively to thesis work, drawing on a half-remembered study about nicotine and habit formation. My results were striking: productivity surged, emotional resistance waned, and withdrawal proved trivial.
In retrospect, there were methodological flaws, and I have some ethical qualms about sharing this piece of writing. However, I believe that nicotine supplementation greatly improved my capacity for sustained focused labor, and I would not have completed my dissertation without it. I simply had to share.
Proofs as Rhetoric
In Proofs as Rhetoric, I argue that mathematical proofs, as we write and communicate them, are not purely formal deductions but acts of persuasion—rhetorical constructions aimed at convincing an audience that a result is true. Drawing on classical rhetorical concepts like logos, ethos, and pathos, I explore how proofs depend not only on logical soundness, but also on the credibility of the author and the emotional resonance of the argument.
These ideas have practical consequences for both research and teaching. In research, we must tailor our proofs to the expectations and norms of different mathematical communities. In the classroom, recognizing the rhetorical nature of student proofs changes how we assess clarity, style, and understanding.
Ultimately, I advocate for a broader recognition of rhetoric as a central component of mathematical practice and pedagogy.
(An earlier version of this essay appeared on Machine Appreciation.)
A Dialogue on a Classic Interview Puzzle
A mathematical dialogue that revisits a well-known envelope puzzle, probing the assumptions behind a strategy that seems to beat random guessing. What begins as a clever probabilistic trick unravels under scrutiny, touching on free will, canonical null-set structures, and the purpose of "real-world framing" in math puzzles.
I would eventually like to update this piece and submit it to a recreational mathematics journal like The Mathematical Intelligencer, but for now, it lives here.
(An earlier version of this essay appeared on Machine Appreciation.)
Stop Having Opinions
Social media and mass media have radically expanded not just the range of acceptable opinions (the Overton Window), but also the number of topics on which we're expected to have a view. Call it the Overton Breadth. We're expected to have opinions on everything from housing policy to international trade deals to local school board decisions halfway across the country. If you don't have an opinion on these matters, it's interpreted as a character flaw.
This essay explores how the pressure to opine on every issue leads to social tension, cognitive overload, and worse policy outcomes. It's a case for epistemic humility; we should recognize our limitations to deeply understand and analyze all of the issues du jour. Most of us are wrong about most things. Why pretend otherwise?
The Continuum Hypothesis is True in Practice
Most mathematicians know the standard story: the Continuum Hypothesis (CH) was the first major open question in set theory, placed at the top of Hilbert's famous list in 1900, and ultimately shown to be independent of the ZFC axioms. Is it true? Is it false? Dealer's choice. While historically correct, the standard story is incomplete (as so many tales involving Gödel are). For nearly a century, numerous mathematicians believed CH was true and tried to prove it. One of the most ambitious efforts was the so-called Perfect Set Program.
This essay traces the mathematical and historical arc of the Perfect Set Program–a failed attempt to prove CH by showing that every uncountable subset of the reals contains a perfect subset. While it failed to prove CH, of course, it led to beautiful results and a fascinating partial truth: in most natural settings, uncountable sets of reals do contain perfect subsets. CH may be independent of set theory, but the program left us with the conclusion that it's “true in practice.”
Machine Appreciation
My old group blog with some friends in the Penn math department. Some of the posts here are excellent, some are fun and silly, some are a little too ambitious, and some are largely unfinished. Several of my blog posts evolved into more polished essays on this site.