A graduate student wrote to me recently, after reading something I’d posted about how the value in our work is migrating from execution toward judgment. He’d been circling the same thought: that the future of pure mathematics may be governed less by the practical pressures that shaped the profession for the last century and more by aesthetic sensibility, by taste, by meta-level decisions about what is even worth learning, exploring, or having a machine generate. He noted, correctly, that this isn’t wholly new. Before mathematics was a profession it was a pastime, pursued by people who chose their problems for beauty rather than grant cycles. But it cuts hard against the grain of how academia has actually operated in living memory.
He and his advisor are planning a seminar built on exactly this premise. Rather than grinding through a book theorem by theorem the usual way, they want to read something rich but approachable entirely from the top down. Asking of each idea: why was this interesting? Why were these definitions chosen, these paths of attack and not others? What was the guiding philosophy? The seminar, in other words, is about taste, treated not as an accident of exposure but as something you can deliberately study. And he ended with the question I’ve been turning over some: how do you actually develop taste? Should the tools be part of it or kept out? Some days, he wrote, the progress astonishes him; other days it leaves him distraught.
I don’t have this sorted out either, and I’m still trying to think it through myself. But I’ve come to believe the question is more answerable than it looks, and that the most useful evidence comes from an unexpected place: a chess engine.
Start with what taste even is, because you can’t cultivate a thing you can’t define. I’d resist the romantic story that taste is an ineffable gift you either have or don’t. A more useful definition: mathematical taste is a learned, partly-tacit predictive model of which problems, definitions, and directions will prove fruitful. Where fruitful means generative of more mathematics, connective across areas, and durable over time. Three things fall out of that. Taste is predictive, a bet about the future value of a direction, which is exactly why you can only fully confirm it from the far side, why the worth of a path is visible only once you’ve walked it. It’s partly tacit, which is why it has always been transmitted by apprenticeship and osmosis rather than by a list of rules. And, this is the part we undersell, it’s partly objective. We love to say taste is relative, and at one level it is: I find certain fashionable areas unlovely and won’t spend my life there, and that’s personal. But beneath the personal layer sits a more objective one. A definition that unifies three phenomena previously seen as unrelated is better, more or less regardless of who’s judging. Separating those two layers is the most clarifying move available, because you can teach the objective layer and only cultivate the personal one.
So what are the objective markers? Look at the mathematics people across generations agree is tasteful, and a few features recur. Unification: one idea that explains several things previously thought separate. Inevitability: the sense that a definition is the right one, that once seen it couldn’t have been otherwise. Generativity: it opens more than it closes. Economy: a disproportion between how simple the idea is and how much it yields. Transfer: the perspective migrates to other areas and keeps paying out. What’s useful about this list is that these are all identifiable in a text. A top-down seminar can sharpen its lovely-but-vague questions into something with teeth: which of these markers are present here, and can we reconstruct why this definition was chosen over the plausible alternatives the author silently discarded? That last exercise, recovering the roads not taken, may be the single highest-leverage way to build taste from a book, and it’s one a reader can do entirely alone: try to define the objects yourself, choose your own line of attack, and only then compare against the master’s, feeling exactly where your judgment was cruder.
That solitary exercise is really a borrowed one, and the borrowing points at something larger: taste is studied far more rigorously outside mathematics than within it, and the mechanisms transfer. Consider how connoisseurship is actually trained in art, in wine, in literature. It is never exposure alone. It is exposure plus forced articulation, you are made to say why, aloud, in front of people who can correct you. The poetry workshop, the tasting where you must name what you’re detecting, the art seminar: the shared structure is that a tacit judgment is dragged into words and then contested by a community. That is precisely what a taste seminar is, which is why the instinct behind it is sound. Reading a great book in silence builds less taste than being made to argue, clumsily at first, about why an idea was the right one, and being corrected.
Two more borrowed mechanisms. From wine and music: you don’t develop a palate by consuming one masterpiece repeatedly, but by structured comparison, the great example set beside the merely competent one. Taste is a discrimination, and discriminations sharpen on contrast. So a seminar shouldn’t read only the masterwork; it should sometimes set a tasteful treatment of some material beside a pedestrian one and force the difference into the open. And from the training of copyists: you learn a master’s taste by reproducing their choices until you feel, from the inside, why each was made, the same “redo it yourself before you read the answer” move, now recognizable as a technique with a long pedigree in every aesthetic domain.
Which brings me to chess, the closest analogue we have, and the one that turns all this from philosophy into evidence. Chess is a closed enough world that its version of taste, what players call positional understanding, has actually been measured. And what’s known is directly relevant. Strong players don’t win by calculating more; they perceive better. They see the meaningful pattern faster and prune the space of possibilities by taste before calculating anything. Positional judgment is a pruning mechanism, not a search. That alone reframes what a taste seminar is for: it trains the pruning, the faculty that tells you which of ten questions is worth the calculation in the first place.
Now the part that should interest my correspondent and unsettle us both. Chess is the one domain where we have watched a machine develop taste and then transmit it back to humans. When AlphaZero arrived, grandmasters didn’t describe its play as brute force, they described it as having a style, an aesthetic. It revived a class of positional sacrifices that human taste had long since pruned away as unsound. And then something remarkable happened: top human players studied it and revised their own taste. The machine’s judgment was genuinely novel, and it was genuinely absorbable by humans. That is neither doom nor hype. It’s a third thing, and it’s the most hopeful data point I know, a case where the tool didn’t replace human taste but enlarged it.
So when my correspondent asks whether the tools belong in a taste seminar, I think the honest answer runs against the natural instinct. The natural instinct is to keep them out to protect originality, to develop human taste in isolation so it stays uncontaminated. But the chess evidence suggests the frame is slightly off. The reason to build human judgment first isn’t purity; it’s that you cannot evaluate what a generator produces until your own discrimination is trained, and these tools are least reliable exactly where the seminar is trying to build strength. A language model will produce plausible, competent, tasteless mathematics all day long. That is the most useful teaching material imaginable. Build the discriminator first; then use the fast generator as a foil, have it produce three developments of an idea and rank them by the markers you’ve spent weeks learning to see, and articulate why its default was mediocre. Taste is a discrimination, and the tool is an inexhaustible source of things to discriminate against.
One caution to hold through all of this. The part of taste genuinely at risk is the tacit part, the judgment that used to accrue, for free, from grinding through six weeks in the weeds and coming out knowing the terrain in your body. A seminar that reads about taste from the top down is necessary but not sufficient, because taste also needs the friction of doing: of making a real choice, being wrong in front of someone better, and feeling the correction land. That apprenticeship channel is exactly the one the tools are eroding, which is all the more reason to protect it now that it no longer comes automatically. And there’s a quieter difficulty underneath, one every mentor will recognize: most of us acquired our taste without ever having to say it. Being asked to teach it explicitly means being asked to articulate what we only ever knew in our hands. That’s genuinely hard, and worth admitting at the outset rather than pretending a fluency we don’t have.
I’ll end with the thing I’m least certain about and most convinced matters. The deepest form of taste isn’t the ability to follow the giants, that’s the objective layer, and it’s teachable. The deepest form is the confidence to not follow, on specific, well-chosen occasions: to decline the fashionable turn when your own judgment dissents, to skip the herd’s migration into the technique everyone suddenly loves. I’ve done this a few times, sometimes at a cost, and it’s the part of my own taste I trust most, precisely because it’s the part no advisor handed me. And that’s the point for anyone building this kind of seminar, or sitting in one, or just choosing what to work on next: taste of that kind can’t be taught directly. It can only be permitted, modeled, and grown into. Which may be the real case for reading mathematics from the top down, not that it hands anyone a palate, but that it might give people permission to start trusting the one they’ve been quietly building all along.
The engine learned to sacrifice pieces the masters had written off. Then the masters learned it back. Somewhere in that exchange is the shape of the next few decades, if we’re careful about which half of it is ours to keep.