Miscellanea

i was never naturally good at mathematics. reading and writing came easily to me as a child, while mathematics seemed to require some faculty that everyone else had acquired without telling me. when my class learned multiplication in grade school, i remained behind for something like a month because i simply could not comprehend it.

eventually i did. i remember the satisfaction of understanding it much better than i remember the confusion itself. mathematics began to feel less like an aptitude i had failed to inherit and more like a puzzle whose rules could be learned, although sometimes embarrassingly slowly. i discovered that i liked quantification itself: the possibility that some enormous or apparently disorderly thing could be translated into relationships, that the mess of the world might submit, however partially, to a few symbols on a page. at the same time, i loved the stars with none of this sophistication at all.

at my grandparents' house i would sit on the balcony and stare at the sky for hours. one particularly clear night i recorded something like thirty minutes of myself talking about it. i was very young and have no idea what i managed to discuss for that long. i wish i still had the video, mostly for endearment purposes, but also because i would like to meet that particular intensity again: a child with thirty uninterrupted minutes of things to say about the sky, before she knew enough to be embarrassed by how little she knew. before i knew the name of the profession, i said i wanted to be "a mathematician but for space." i had found the desire before i had the vocabulary for it.

the attraction eventually shifted from things that could be answered to things that apparently could not. i would learn enough about some problem to formulate a question and then discover, with a mixture of frustration and delight, that there was no settled answer. sometimes nobody knew; sometimes several incompatible explanations survived; sometimes the question itself turned out to have been badly posed. i loved that uncertainty had structure, that there were better and worse ways not to know something. every answer seemed capable of exposing another question behind it. this could have been discouraging. instead it made the universe feel inexhaustible. i wanted to keep going.

i became preoccupied with the big bang and white holes. the latter interested me because certain solutions of general relativity permit something resembling the temporal reverse of a black hole: a region from which things may emerge but which they cannot enter. speculative connections between white holes and cosmology made the idea irresistible to me. there was an apparent symmetry to it that appealed to a fairly primitive intuition i had about physics: if one solution permits an input with no recoverable output, what becomes of the reversed solution? whether nature has any obligation to satisfy that intuition is, of course, another question.

general relativity was full of offenses against intuition like this. space and time ceased to be the passive stage on which physics occurred and became participants in it. gravity could be understood geometrically. sufficiently extreme distributions of matter could produce objects from which even light could not return. none of this becomes less strange merely because the mathematics is capable of describing it.

gravity also becomes peculiar on sufficiently large scales. it is long-range and, unlike electromagnetism, has no opposite gravitational charge available to screen it away. a sufficiently large self-gravitating system therefore does not necessarily behave according to the thermodynamic intuitions developed from ordinary matter in a box. gravitating systems clump. stranger still, there are circumstances in which losing energy can make such a system hotter. negative specific heat sounds almost like a contradiction until the gravitational dynamics are taken seriously.

i have always liked physical results of this kind, where the absurdity turns out to belong not to nature but to an intuition we forgot we were assuming. quantum mechanics provided these in abundance. the double-slit experiment particularly bothered me. ordinary language seems almost deliberately inadequate to describe what is happening; familiar categories like particle and wave become unreliable if treated too literally, while the experimental result remains perfectly indifferent to the discomfort of our description. eventually this altered what i wanted from understanding. i did not need the universe to become intuitive. i wanted it to become intelligible, and there was something almost intoxicating about discovering that these were not the same demand.

i first found the name for what i wanted to become when i was sixteen, at a museum planetarium. the presentation discussed dark matter (see dark matter). i found it beautiful that something could be indispensable to our account of the universe while remaining inaccessible to direct observation in the ordinary sense. its presence is inferred from gravitational effects; the visible universe does not behave correctly without something else there.

there is something almost philosophical about knowing an object first through the consequences of its absence. dark matter is not visible, but the universe keeps betraying the fact that something is missing from the visible account. i had already become fascinated by the ways the universe seems to resist ordinary distinctions like finite and infinite, something i have written about elsewhere in relation to black holes, and dark matter belonged to the same intellectual territory. the universe was not simply large; it was incomplete as an object of knowledge. what we did not know was not necessarily waiting at some remote edge of existence. some of it was apparently holding galaxies together.

this was around the time i learned the word astrophysicist. suddenly there was a profession attached to the thing i had been doing in miniature for years: looking upward, learning just enough to make the questions worse, and wanting desperately to know more. i don't remember ever making a particularly measured decision about it. i found something i loved and allowed it to swallow me whole.

there is a minor contradiction in how i arrived here. mathematics first appealed to me because it made things quantifiable. astrophysics became important to me because it revealed the limits of that satisfaction. the equations can be extraordinarily precise while the ontology remains unsettled. we can know with considerable confidence that an effect exists without knowing what produces it. a successful mathematical description does not always amount to an explanation, and precision does not abolish mystery. sometimes it only tells us exactly where the mystery begins.

this is probably also why i have never been particularly fond of anthropic reasoning when it is allowed to masquerade as an explanation. obviously any universe containing observers must have conditions compatible with the existence of observers. the statement is difficult to dispute. i am less convinced by what is sometimes made of it.

selection effects matter, and our existence necessarily constrains the universe we are capable of observing. but there seems to me an important difference between identifying the conditions under which an observation is possible and explaining why those conditions obtain. "we could not observe otherwise" is not quite the same proposition as "this is why the universe is this way." i am much more interested in the possibility of a mechanism.

this is one reason cosmology appeals to me. its most interesting problems frequently make the boundary between physics and philosophy difficult to maintain. eventually a question about what the universe does becomes a question about what would constitute an explanation of what the universe does. physics can tell us that a model works extraordinarily well and leave intact the more irritating question of what we mean when we say that we understand it.

actually studying astrophysics has made the relationship less romantic in some ways and much more intimate in others. loving a subject from a distance is easy. it asks almost nothing of you. loving it while it assigns you problem sets, humiliates you on exams, consumes afternoons you wanted for something else, and repeatedly exposes the limits of your own understanding is a different relationship entirely. sometimes i resent how much of myself i have given it. then i encounter some new problem, some result that should not make sense and does, some question nobody has answered satisfactorily, and immediately i want to give it more.

i am still not naturally intuitive at mathematics or physics. i often need a derivation written without omitted steps, or need to approach the same argument several times before its structure becomes apparent. there are students who seem capable of looking at a problem and immediately seeing the appropriate mathematical machinery. i generally cannot. under the artificial pressure of an exam or a semester this difference becomes consequential. if something takes me twice as long to understand, understanding it eventually does not give me twice as many hours in the week.

this is frustrating in a way i don't particularly want to romanticize. difficulty is not automatically virtuous. sometimes taking longer simply means taking longer, and sometimes being confused is just being confused. but neither is confusion proof of incapacity. i wish i had understood the difference much earlier.

i also have an unfortunate tendency to become impatient with classes when the material is far removed from whatever question currently interests me. this is not an especially defensible objection to having to learn things. astrophysics is a discipline rather than a collection of my favorite cosmological problems, and expertise requires spending an enormous amount of time on foundations before being entitled to say anything interesting about their limits.

the difficulty has also changed what i think i might want to do with the subject. i have become increasingly interested in teaching and science communication. i don't believe intellectual seriousness requires unnecessary opacity. almost any scientific idea can be given an accessible point of entry without pretending that the idea itself is simple. simplification and explanation are not the same operation. the best explanation does not remove the difficulty from an idea; it gives somebody somewhere to stand while they encounter it.

i am admittedly capable of being terrible at this in conversation. i skip intermediate steps, arrive at a conclusion that makes perfect sense inside my own head, and then discover that nobody else was given the intervening argument. probably an adhd thing. having to reconstruct so many mathematical ideas deliberately, however, has made me unusually conscious of how quickly an explanation can lose somebody.

there is something unfortunate about the way mathematics is often encountered as a test of identity. a child struggles with it and concludes not merely that she does not understand multiplication yet, but that she is not a "math person." difficulty becomes evidence of incompatibility. a temporary state of not knowing is promoted into a permanent fact about the kind of mind she possesses. i could very easily have made that inference about myself.

instead, i was behind everyone else until i understood multiplication, and then multiplication was fun.

i suspect there are people who would love physics if they were given enough time to get past the humiliation of not understanding it immediately. perhaps there are people who would look at dark matter the way i did at sixteen if somebody first convinced them that confusion was not evidence that they had no business thinking about it. there are subjects people abandon because they dislike them, and subjects they abandon because they have been made to feel ridiculous for loving something they do not yet understand. i am much more bothered by the second possibility.

i would like to be one of the people who makes that possibility less likely. i know what it is like to love a subject before feeling entitled to it, and to want something badly enough that being bad at it hurts. perhaps this is another form of yearning: not simply wanting to know, but wanting to become capable of knowing.

the practical realities of studying astrophysics have complicated my relationship with it without making the attachment any smaller. if anything, knowing more has made the object of my affection expand. every piece of understanding seems to expose the perimeter of something larger. i still want to know why an apparently sensible physical intuition fails; i still find it beautiful that we can infer something we cannot see because the universe behaves incorrectly in its absence; i still love the instant when something incomprehensible becomes comprehensible without becoming any less strange. i could spend my whole life doing this and there would still be more universe left over.

after all of the terminology and mathematics and embarrassment and wonder, i have apparently only found increasingly elaborate ways to express the same desire i had before i knew what to call it: i still want to be a mathematician for space.

last edited: 13 september 2026