Hi I’m A. I have bipolar disorder type 1. I’ve also been trying to teach myself math. I’m working on a book regarding my experiences learning math with an impaired brain (ie while I was in psychosis). I have liberally used math metaphors I barely understand and was hoping someone could sanity check them for me. Here’s the relevant post:
Later, when she has language for it again, Maya will think of her mind that spring as a graph. Nodes for fields: genetics, group theory, linguistics, topology, whatever she happened to touch that week. Edges for the connections she drew between them, by hand, in real time, the way you'd wire a circuit. Most of the time the graph is sparse and that sparseness is the whole game. A real edge is a bet. It says these two things are related in a way that could, in principle, turn out to be false, and the not-yet-falsified ones are what she has built a career on.
That spring the graph does not stay sparse. It fills in. Every field starts reaching for every other field, DNA to group theory to linguistics to gravity, until there's an edge running between any two nodes she can name, and a graph like that, a complete graph, carries no information at all. Not because it's wrong exactly. Because it can no longer be wrong. When everything is connected to everything, there is nothing left for a connection to distinguish, and nothing left for a mind to be surprised by, and surprise, it turns out, was the thing doing all the work.
She won't know any of this while it's happening. While it's happening it just feels like speed.
2n² − 4n + 31 is prime.
She writes it in the notebook without a qualifier, the way you write something you have already checked. It has the shape of a fact. It sits on the page the same way the differential geometry notes sit two pages later, in the same hand, the same ink, the same unhurried certainty: let M be a manifold and p ∈ M. Nothing on the page marks one as different from the other. Why would it. From inside, they arrive by the same process. She thinks, therefore it holds.
It is three in the morning, or it is eleven in the morning, the distinction has stopped doing any work for her. What matters is that the notebook is open and something in her is moving very fast and very cleanly, the way a river moves fast and clean right before a drop it can't see yet.
She is mapping the genome onto a tetrahedron. One base, one vertex. The length of each edge is the distance between two bases. AATCCGC becomes a shape, and the shape can be embedded in a 3-sphere, and the embedding links to the Ising model, and she writes this down with an arrow, information links to Ising, as though the arrow itself were doing the proving. On the same page, a triangle labeled with entropy: entropy for gene distribution across cells, entropy for gene distribution within a cell, cell state as a set of gene abundances within a certain range. It reads like the start of something real. It reads like the start of everything she has ever actually built, which is exactly the problem, because it is not building on anything. It is borrowing the posture of rigor and spending it somewhere rigor was never earned.
She turns the page and she is inside topos theory. How to organize mathematical knowledge using topoi and adjoint functors, concrete to abstract, structured sets to theories to categories of theories. This part is correct. This part could sit in a textbook. And then, without any seam she can find later, without any hinge she can point to and say here, this is where it turned, the same page starts explaining morphemes. A sentence is a set of morphemes. Morphemes are literally about relative locations of positive and negative charges. She underlines literally. She does not mean it as a metaphor and at three in the morning she is incapable of noticing that she does not mean it as a metaphor.
There is a place on another page where the pen just goes in circles. Not a doodle. Not decoration. A place where a thought is moving faster than the hand can turn it into language, so the hand does the only thing left to do, which is press the same small loop into the paper again and again until there's a black knot sitting in the middle of an otherwise empty half page. She comes back to that knot more than once across these weeks. She never writes anything next to it. There is nothing to write, she thinks. That's what it looks like when the thought outruns the words for it, and the body just registers that something enormous is happening, roughly here, roughly now, without being able to say what.
What the DNA-and-topos-theory page feels like from inside is not easy to name. Not "delusional," not quite. Not "brilliant" either, though both are true in their own domains at other points in this stretch. It feels like fluency. It feels like the specific, unmistakable sensation of a formalism clicking into place, the same sensation she's had a hundred times before over real problems, except this time the formalism is clicking into a lock that doesn't exist. The feeling doesn't come with a watermark. It doesn't dim itself, doesn't hedge itself, doesn't announce itself as unreal. It arrives with the exact same weight as the manifold notes two pages over, notes that are, as it happens, just right.
Here is the only thing from that stretch that can actually be verified, months later, the notebook open on her lap: 2n² − 4n + 31 is not always prime. Try n = 31. She didn't check n = 31. She didn't check anything. She had stopped needing to.
This is the difference, she will come to think, and it is not the difference between sense and nonsense. The prime claim is a real claim. It says something specific enough to be wrong, and it is wrong, which makes it the most honest artifact in the whole notebook: proof that some thread of her was still trying to run a test. Morphemes are literally about relative locations of charges is not that. It was never a claim that could have failed. It was an edge added to a graph that had already stopped being able to tell the difference between a pattern and a coincidence, because it had run out of room to be surprised by either.
There's a fact from information theory she'll come back to, much later, on an ordinary afternoon with nothing at stake: the most compressed a sequence can ever be is exactly the point at which it becomes indistinguishable from noise. A message with real structure, real pattern, can always be squeezed smaller, because the pattern is redundancy, and redundancy is exactly what compression removes. Squeeze all the way, past every regularity, and what's left has no structure left to exploit. It looks random. Not because nothing is happening, but because everything that could be predicted already has been, and prediction was the only thing separating signal from static in the first place. A maximally compressed sequence and a maximally random one are, to any observer without the key, the same object. That spring, her mind was doing something like this in reverse and calling it compression anyway. Every field folded into every other field, DNA into topology into linguistics into gravity, each fold presented as a discovery, a redundancy finally noticed and removed. But a real compression shrinks toward a shorter description of the same thing. Hers didn't shrink. It just kept adding edges until the graph was so dense it read exactly like noise, and noise, examined closely enough by a mind convinced of its own rigor, will always look like it's on the verge of resolving into a pattern. That is what she mistook for being close to something. She wasn't close to a pattern. She was standing at the point where pattern and its total absence stop being distinguishable from each other, and the notebook, if you read it end to end, is a record of a mind that could no longer tell which side of that point it was on
