For about a decade, I watched mathematics lectures I could not follow.
Not partially follow. Not follow with effort. Simply not follow: the symbols moved across the screen and I understood, at best, the English words connecting them. I watched Schuller explain fiber bundles. I watched lectures on Lie groups, representation theory, algebraic topology. I took notes I couldn't read back meaningfully a week later.
Anyone watching would have said I was wasting my time.
I wasn't.
What I was doing, without knowing I was doing it, was learning what mathematics feels like before learning what it says. Every lecture I didn't understand was teaching me something I couldn't have gotten from a textbook I could follow: the texture of mathematical thinking. I was learning taste before I had the technical vocabulary to exercise it.
There is a word for this in agriculture. You prepare soil before you plant. The preparation looks like nothing; you are just turning earth, adding things that will not be visible in the harvest. Someone walking past the field would see no progress. But the harvest depends on it entirely.
Mathematical maturity, I think, has a soil phase. Most people skip it or are rushed through it. Coursework moves too fast, problem sets demand execution before intuition, examinations reward procedure over understanding. You learn to do mathematics before you learn to see it.
I did it backwards. I saw it, or began to see it, for years before I could do it. The doing is coming now, relatively quickly, because the ground was already prepared.
Last week I opened Serre's Linear Representations of Finite Groups, a graduate text terse even by mathematical standards, and found it readable. Not easy. Readable. A few years ago it would have been impenetrable.
Something shifted. Not a single thing I learned, but a consolidation of many things. The σ-algebras and measurable functions I was working through in Durrett. The group theory I have been building toward for years. The probability and maximum entropy I absorbed from Kardar. Individually these were hard. Together they are beginning to feel like a language.
The shift felt sudden. It wasn't. It was ten years arriving at once.
I think about this when people ask how to learn mathematics outside of institutions. The usual advice is to find good textbooks, do the problems, seek out communities. All of that is true. But there is something prior to all of it that almost nobody talks about: you have to spend time inside mathematics that is too hard for you, not to extract information from it, but to let it form you.
The confusion is not failure. The confusion is the soil.
I also think about timing. I was fortunate that AI arrived when it did, late enough that I had already laid the ground, early enough that it can help me build on it. If I had access to something that could explain everything clearly during the years I was watching lectures without understanding, I would have been tempted to resolve every confusion immediately. I would have learned to execute before I learned to see.
The not-understanding was doing something the understanding couldn't have done. It was teaching me what I was reaching for before I had hands to reach with.
I am not a mathematician yet in any formal sense. I am a molecular/computational biologist who has spent a decade making herself into something harder to categorize. The mathematics is still being consolidated. The technical fluency still sometimes lags the structural intuition.
But I can read Serre now. I am working through Durrett. I can ask questions about proofs that go directly to what the proof is actually doing, not just whether it is correct.
That came from the years of not understanding. I am certain of it.
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