i spent about a month being kept behind when my class learned multiplication. the other children seemed to understand it almost immediately, which was particularly embarrassing because i had always found reading easy and assumed that most schoolwork would continue in the same manner. eventually i worked out what was going on and became fascinated with arithmetic. i don't remember the moment it started making sense. i remember wanting to keep doing it afterward, including when there was no homework to finish.

i was already spending a great deal of time looking at the sky. at my grandparents' house i used to sit on the balcony for hours, and once, when i was very young, i recorded approximately thirty minutes of myself talking about astronomy. i would love to know what i was explaining, since i cannot imagine that i had much information at my disposal. i wish i still had the video. i told people i wanted to become "a mathematician but for space," which was my best attempt at describing astrophysics before i knew the profession existed.

for a while i read almost exclusively about the big bang and white holes. the latter are hypothetical regions that appear in certain mathematical extensions of general relativity; unlike black holes, from which light cannot escape beyond the event horizon, nothing can enter a white-hole region. i was interested in how the time-reversed counterpart of a black-hole solution could exist on paper even though there was no evidence that such an object could actually form. i remember searching for articles about whether the big bang might have been a white hole, an idea that appears in some speculative cosmological proposals but is by no means an established description of the early universe. there are plenty of solutions to physical equations that nature has apparently declined to produce.

general relativity was also my introduction to the problem of trying to picture something that is easier to formulate mathematically. mass and energy affect the geometry of spacetime, and the motion of freely falling bodies follows that geometry. the curvature isn't something happening to an ordinary physical surface inside a larger room. i found diagrams of rubber sheets and bowling balls useful until i began asking what, exactly, the sheet was supposed to be bending into. those diagrams have serious limitations, although explaining the actual mathematics to a child would have produced a few more problems.

one result i became interested in later concerns negative heat capacity. a self-gravitating system in approximate virial equilibrium can heat up as it loses energy, because the loss is accompanied by gravitational contraction. you can't apply every intuition developed from an ordinary gas in a container to a large collection of objects bound by their own gravity. this is especially relevant to stellar systems, and it is a good example of why it helps to know which assumptions have gone into a familiar physical rule before deciding that the rule has been violated.

quantum mechanics was harder for me to accept descriptively. the double-slit experiment is routinely introduced with the claim that particles behave like waves, but this can become misleading if you begin imagining the electron as a tiny object deciding which behavior to exhibit. individual detections are localized; the distribution of many detections can produce an interference pattern, provided the experimental arrangement allows it. which-path information affects that pattern. i understood the results long before i could explain the experiment without falling back on some version of the little ball or the water wave, both of which are inadequate. even now i find that the informal explanations require more qualifications than the equations do.

i was sixteen when i attended a planetarium presentation on dark matter. the speaker described how galactic rotation curves, among other observations, indicate more gravitating matter than the visible stars and gas can account for. it was probably the first time i understood how much of astrophysics is concerned with inferring things that have not been directly identified. the measurements are not especially vague: orbital velocities and gravitational lensing can be measured, and theories have to account for them. the question is what produces those effects. i wrote about this separately in a post on dark matter.

i also started reading about the anthropic principle. the observation that any observers must inhabit a universe compatible with their existence is a selection effect, and an important one when comparing cosmological possibilities. it cannot, by itself, tell us what physical process determined the relevant conditions. i kept looking for discussions of the mechanism behind particular cosmological parameters and finding that the literature had not necessarily settled it. some of my questions were probably badly formulated. others still don't have agreed answers.

the actual coursework has involved far less cosmology than all this might suggest. i spend most of my time learning the mathematics needed to approach it, and there are afternoons when i get stuck on a derivation because a textbook has omitted a step that the author evidently regards as too elementary to mention. sometimes it really is elementary and i have simply forgotten it. on exams, knowing roughly what a problem is asking is of limited use if you can't reproduce the argument under time pressure. i'd rather this were not true, but it is.

i sometimes wish introductory courses would spend more time explaining why a particular mathematical method is appropriate instead of demonstrating how to carry it out once that choice has already been made. i can follow a solution and still have trouble identifying the same structure when the numbers or notation change. it makes studying slower. i am trying to improve at working through unfamiliar problems before consulting examples, which is useful and also an excellent way to discover how long an afternoon can be.

science communication interests me for related reasons. an explanation can be technically correct and still leave out the exact step on which somebody's understanding depends. i have done this myself when talking about a subject i know well enough to forget which parts initially confused me. i would like to get better at distinguishing a genuinely necessary simplification from an omission that makes the reader do work the explanation was supposed to do.

right now there is an alarming amount of linear algebra between me and the questions i originally wanted to study. i continue to read about them anyway, usually when i should be doing something else. i still look at the stars when i get the chance, though boston's streetlights are not especially cooperative.

last edited: 31 october 2026