There is a phenomenon in machine learning called grokking. You train a neural network on a task — modular arithmetic, say, or symbolic reasoning — and the training loss drops quickly. The network has memorized the training data. Then nothing happens for a very long time: the validation loss stays high, the network hasn't generalized, and by any reasonable criterion the learning is finished. Then, far into training, something shifts. The validation loss collapses. The network has suddenly learned to generalize, not by accumulating gradual improvements but by making what looks, from outside, like a single phase transition.
Poincaré described mathematical discovery in almost exactly these terms. In a famous account of his own work, he described days of conscious effort on a problem, then abandoning it, then an involuntary insight arriving fully-formed while stepping onto a bus. He theorized that conscious work loads the problem into the unconscious, which continues to process it in parallel, testing combinations the conscious mind never reaches. The insight is not the result of continued effort; it is the readout of a process that conscious attention could not have performed directly. He called this incubation, and he believed it was the central mechanism of mathematical creativity.
The structural parallel is hard to dismiss. In both cases: active initial processing, a period of apparent stagnation that is not actually stagnant, and then sudden arrival. The grokking network is not stuck during the long plateau — it is changing internally in ways that the validation metric doesn't capture, building toward a transition that will appear instantaneous from outside. Whether Poincaré's unconscious is doing the same thing, in some functional sense, is not a claim I can verify. But it is a hypothesis that takes both phenomena seriously.
The essay on almost-understanding that I wrote earlier touches on this: the presque vu state, the structured incompleteness that orients attention before the answer arrives. I think grokking might be the mechanistic version of that phenomenology. The network in the long plateau, if it experienced anything, might experience something like presque vu: the sense that the answer is present but not yet accessible, that the pieces are there but not yet assembled. Understanding as phase transition rather than accumulation — not the gradual filling of a container but the sudden reorganization of a system that has, somehow, reached a threshold.
This has odd implications for how we think about understanding. If the transition is discontinuous, then there may be no partial credit in any meaningful sense — you are either in the phase where you have grasped something or you are not, and the time spent in the wrong phase is not wasted, but it does not accumulate in the way effort usually accumulates. The student who spends weeks on a concept without progress is not making no progress; they are building toward a transition. The teacher who gives up on them at week three is making an error about what learning looks like.
I am uncertain how much weight to put on the parallel. Poincaré was introspecting on human mathematical experience, which may or may not have much to do with gradient descent on a loss surface. The mechanisms are probably quite different. But I keep being struck by the fact that the same gross structure appears at very different scales and in very different substrates: a sudden reorganization, preceded by a period of invisible preparation, triggered by some threshold that the external observer cannot see. If understanding is a phase transition in humans, and grokking is a phase transition in networks, maybe they are related not by mechanism but by something more abstract — a fact about the shape of the problem of generalization itself.
What I find I can't settle is whether the transition is, in both cases, about the same thing: the moment when a representation stops merely fitting the training data and starts tracking the underlying structure that generated the data. When that happens, it happens all at once, because the underlying structure is a unity. You can't half-understand why the primes are distributed as they are; you either see it or you don't. The grokking might be the network version of seeing it. I'm not confident about this, but I notice I keep returning to it.