Solve for now

A mathematical adventure through meditation. Ten equations, some borrowed, some bent, some brand new, each with an experiment you can run on your own mind, and an honest note on what the numbers can and can't tell us about sitting still.

By Reid Robison, MD. About 35 minutes, with eleven things to try. No math beyond multiplication is required. Tap any underlined symbol in an equation to see what it means.

The meditation teacher Shinzen Young has a habit I love. He'll be explaining something slippery, like why the same backache wrecks one afternoon and barely registers on another, and then he'll write an equation on the board. Suffering equals pain times resistance. The room changes. People who have nodded politely through twenty minutes of talk about "non-judgmental awareness" suddenly sit up, because an equation makes a claim you can check. If resistance goes to zero, so does the suffering. Does it? Let's find out.¹

Shinzen trained in math and Asian languages before he spent decades in monasteries, and his book The Science of Enlightenment is studded with these little formulas. Some of them are serious and some are winks. In an essay on what mindfulness actually is, he says the ideal definition would be "couched in the language of SI units and mathematical equations," and admits we are "decades, if not centuries" away from that.² Fair enough. But equations are useful long before they're exact. The statistician George Box put it best: all models are wrong, some are useful.³

So here's the plan. We'll walk through ten equations. A few are Shinzen's, and I'll credit him each time. A few belong to the people who invented signal detection theory, learning theory and social psychology. Several are ones I've bent or built to fix something the originals get wrong. Each comes with an exhibit you can play with, and with what the research says. If you collect them all, you'll have a pocket theory of how attention changes experience, and you'll know exactly where that theory runs out.

Before we startCount your breath

Every good experiment needs a baseline. Researchers at the University of Wisconsin looked for an objective way to measure mindfulness, something better than asking people how mindful they feel, and landed on the oldest meditation exercise there is: count your breaths from one to nine, over and over. Mark each out-breath, mark the ninth one differently, and own up when you lose count. People who count more accurately also tend to wander less, and their accuracy improves with training.⁴

Notice what that exercise actually asked of you. To stay on the count, you needed concentration. To feel where one breath ends and the next begins, you needed clarity. And when you lost it, you needed to come back without beating yourself up, which is equanimity. Those three are the backbone of everything that follows.

Equation 1 · ConcentrationThe orbit of attention

When a team at Harvard pinged thousands of people at random moments through a phone app, people's minds were somewhere other than what they were doing 46.9% of the time. They were also less happy while their minds wandered, even when they wandered somewhere pleasant.⁵ Mind-wandering isn't a defect. It's how we plan, rehearse and make sense of things.⁶ But it is the default, and it runs whether we want it to or not.

In 2012, Wendy Hasenkamp put experienced meditators in a scanner and asked them to press a button whenever they caught their mind wandering. What emerged was a cycle: focus, drift, the jolt of noticing, the shift back, focus again. Each phase lit up a different network, from the brain's default-mode "story" system during the drift to the salience network at the moment of catching it.⁷ The cycle has a simple piece of arithmetic in it.

Here's the part that surprised me when I first worked it out. You can raise F in two ways: stay longer, or notice sooner. And noticing sooner is often the bigger lever. If you stay ten seconds and take thirty to notice, you're on target a quarter of the time. Doubling your staying power gets you to 40%. Halving your noticing time gets you to 40% too, and cutting it to five seconds gets you to two-thirds. Meditation researchers call that noticing skill meta-awareness, and it's what the practice trains most directly.⁸

Look at the returns-per-minute number. Every return is a rep. People new to meditation often feel they're failing because they keep drifting off. But you can't do a bicep curl without lowering the weight. Drifting and coming back is the exercise. A session where you wandered off forty times and came back forty times was forty reps.

Equation 2 · ClaritySeeing what's actually there

Shinzen splits clarity into two parts: detection, catching faint or fleeting signals, and discrimination, telling apart things that blur together.² Psychophysicists had a mathematical language for exactly this by the 1960s. Signal detection theory starts from an awkward truth: every perception is a judgment made in noise. Your nervous system is always humming. A faint sensation has to be told apart from the hum, and you can be wrong in two ways, missing something real or "seeing" something that isn't there.⁹

The beauty of d′ is that it separates real sensitivity from bias. Someone who says "yes, I feel it" to everything has lots of hits and lots of false alarms, and a d′ of zero. Real clarity means more hits without more false alarms. Try it.

Does meditation actually sharpen d′? For some senses, yes. In the Shamatha Project, people on a three-month retreat got better at discriminating line lengths flashed on a screen, and kept the gain months later.¹⁰ Back in 1984, Daniel Brown found that people fresh off a mindfulness retreat could detect shorter flashes of light.¹¹ And after three months of intensive insight meditation, people showed a smaller "attentional blink," the half-second after spotting one target during which most of us miss a second one.¹² Shinzen's phrase for why this matters is a good one: what is trackable is tractable.

The honest caveat: it doesn't generalize to everything. Experienced meditators feel more confident about their heartbeat, but they're no better than anyone else at actually detecting it.¹³ Confidence and accuracy are different numbers, and the equation keeps them apart.

Equation 3 · UntanglingWhy overwhelm multiplies

This is my favorite of Shinzen's ideas, and the one I use most with patients. Take any hard moment and look closely and you'll find it's made of strands. Shinzen uses four: mental images (the picture of the disaster), mental talk (the voice narrating it), emotional body sensation (the dread in the gut, the heat in the face) and physical body sensation (the actual ache).² When the strands are tangled together, they feed each other. The picture triggers the voice, which tightens the gut, which makes the picture vivid again.

Shinzen's arithmetic: put each strand at ten out of ten. Tangled, they multiply: 10 × 10 × 10 × 10 = 10,000. Untangled, they merely add: 10 + 10 + 10 + 10 = 40. And with enough concentration to feel one strand at a time, you're only ever holding a single 10. The content didn't change. The relationship to it did.

I love this, but the math has a gap. Real life isn't either fully tangled or fully separate. It's somewhere in between, and it moves. So I built a version with a dial for how tangled things are.

Is any of this measurable? The multiplication isn't; nobody has a meter for strands. But the core claim, that separating and naming the parts of an emotion reduces its grip, has good support. Simply labeling a feeling in words quiets the amygdala's response to it.¹⁴ In a spider-phobia exposure study, people who described their fear out loud ("that ugly spider is terrifying") had less physiological arousal a week later than people who tried to reframe it or distract themselves.¹⁵ And people with finer-grained emotional vocabulary, who can tell "dread" from "irritation" from "grief," cope better under stress and drink and lash out less.¹⁶ That's discrimination doing the work, β coming down.

Equation 4 · The second arrowPain times resistance

About 2,500 years ago, the Buddha described being struck by an arrow. That's pain, and it happens to everyone. Then, he said, the untrained person shoots themselves with a second arrow: the worrying, resenting and fighting that piles onto the first.¹⁷ Shinzen turned that into his most famous equation.¹

It's beautifully simple, and it raises a question: what is R? Shinzen describes it as a multiplier, "two units of actual discomfort multiplied by five units of resistance." Here's a way to give R some structure. Resistance isn't one reaction. It's a reaction to the pain, then a reaction to the reaction ("why is this happening again?"), then a reaction to that. Each echo is some fraction, g, of the one before it. Add them up and you get one of the most useful series in mathematics.

That last line is the interesting part. When g is 0.5, resistance doubles the pain. When g is 0.8, it multiplies it by five, which happens to be Shinzen's example. When g creeps toward 1, the series stops converging and each reaction is as big as the last. Anyone who has had a panic attack knows that place: the fear of the fear of the fear. Clinicians measure something close to g with the Pain Catastrophizing Scale, which asks how much you ruminate on, magnify and feel helpless about pain, and it predicts how much pain hurts better than many physical findings do.¹⁸

The evidence that you can turn g down is surprisingly strong for such a soft-sounding idea. After just four 20-minute training sessions, people meditating in a scanner rated the same painful heat as about 40% less intense and 57% less unpleasant. The unpleasantness dropped more than the intensity, exactly what you'd expect if the second arrow shrinks more than the first.¹⁹ A follow-up blocked the body's opioid system with naloxone and the relief held, so this isn't simply the brain's own morphine.²⁰ Experienced Zen practitioners feel less pain, and their scans show the pain regions working fully while the evaluating, elaborating regions go quiet. They feel the arrow; they skip the story.²¹

Equation 6 · EquanimityThe shape of a feeling

Equanimity is the most misunderstood of the three skills. It doesn't mean feeling less, or not caring. Shinzen defines it by its two opposites: suppression, pushing a feeling down, and identification, grabbing onto it and not letting it pass. Equanimity sits between them. He likes physical analogies: equanimity is like reducing friction in a machine, viscosity in a fluid, or impedance in a circuit.²

The cornstarch joke is worth telling. Some fluids, like paint, get thinner when you stir them. Cornstarch in water does the opposite: stir hard and it locks up solid. Shinzen says equanimity makes you more like paint. When life stirs you, you flow.

The analogy is lovely, but "1 over impedance" is hard to see. So here's a way to look at it that research can actually measure: the shape of an emotional response over time. Every feeling has a rise, a peak and a recovery. Richard Davidson's lab calls this affective chronometry, and in one study, how quickly the amygdala recovered after an upsetting image predicted people's emotional traits better than how strongly it reacted in the first place.²² The peak isn't the problem. The tail is.

Three things to notice. Equanimity has the highest clean peak. You feel the thing fully, which is why it's the opposite of numbness. Suppression flattens the peak but pays later: when people try not to think of a white bear, the bear comes back more often afterward, and habitual suppressors report more negative emotion, not less.^23,24 And grasping holds on, which is easy to see with unpleasant feelings but sneakier with pleasant ones. Grip a lovely moment too tightly and its ending leaves a dip below zero: the craving for more.

That's why Shinzen insists equanimity works on both sides of the ledger. He writes it as purification = (unpleasant × equanimity) + (pleasant × equanimity).² Less suffering from the hard things and deeper fulfillment from the good ones, from the same skill. Researchers have started building scales to measure equanimity on its own, separate from mindfulness in general, because it may be where much of the benefit actually lives.²⁵

Equation 7 · ImpermanenceA walk into one second

Here's where the math gets strange, and where we'll scroll instead of slide. Shinzen describes a sequence that many long-term practitioners recognize. First, experience seems solid. Then you notice things come and go. Then each sensation turns out to shimmer. Eventually, each shimmer turns out to be made of smaller waves, each arising out of nothing and dissolving back into it.² He compares it to Fourier synthesis, the mathematical fact that any shape, however solid it looks, can be built from simple waves stacked together.

Scroll slowly through this one. Each step magnifies the same moment.

Is this physics? No. Your nervous system doesn't literally run a Fourier transform on your shoulder. What the research does support is that perception is built in time, from samples and predictions, and that meditation can change how finely those samples are resolved.¹² One influential theory proposes that deep meditation progressively loosens the brain's predictions, the "thick" model of a solid shoulder, until raw, moment-by-moment input is all that's left.²⁸ I wrote about how priors shape what we see in When the trip doesn't end. It's the same machinery, run in the other direction.

Equation 8 · Putting it togetherMindfulness is a vector

Most questionnaires score mindfulness as one number. Shinzen points out that if mindfulness is really three skills, it's not a number at all. It's a vector: an arrow with a direction as well as a length.²

Thinking in vectors makes two things clear. First, different practices point in different directions. Breath focus mostly builds concentration. Noting builds clarity. Loving-kindness builds something close to equanimity, along with warmth. Researchers have started sorting practices this way too, into attentional, constructive and deconstructive families.²⁹

Second, and more important: direction matters, not just length. Shinzen's suffering ratio, S = D / |M|, says more skill means less suffering. Mostly true. But a long arrow pointing mostly at clarity, with little equanimity behind it, can mean seeing your experience in vivid, high-resolution detail without the capacity to let it be. That's a recipe for being flooded.

This isn't hypothetical. Shinzen is unusually frank about meditation's shadow, including the period some practitioners call the Dark Night, when insight into the instability of self arrives before the equanimity to hold it. He calls it rare, then adds a line I appreciate as a doctor: "rather small ≠ 0."² The research agrees. A systematic review found adverse effects in about 8% of people who meditate.³⁰ In a careful study of standard mindfulness courses, 58% of participants reported at least one unusual or unpleasant experience related to practice, 37% said one interfered with functioning, and about 6% had effects lasting more than a month.³¹ Willoughby Britton's interviews with experienced Western meditators catalogued fear, depersonalization and sleep disruption alongside the bliss.³² Shinzen's advice when too much comes up too fast is exactly what the vector suggests: change the technique, usually toward the equanimity axis, or practice less for a while.

Equation 9 · PracticeHow skill actually grows

Shinzen has an optimistic equation about practice. Each skill has its own reward: concentration has the "taste of focus," clarity the "taste of vividness," equanimity the "taste of purification." As skill sharpens, the reward gets stronger, which motivates more practice, which sharpens skill. "Positive feedback creates an exponential growth curve," he writes.²

Why spoil his exponential? Because people trust what they can see, and nobody's practice looks exponential. Every skill ever studied, from typing to chess to surgery, improves quickly at first and then more slowly.³³ And skills fade with disuse. The feedback loop Shinzen describes is real and lovely, but it lives inside those limits.

What does real data say? In standard mindfulness courses, people are asked to practice 45 minutes a day, six days a week. They actually do about 64% of that, and the amount they practice is linked to how much they benefit, though only modestly (a correlation of 0.26).³⁴ Across dozens of trials, meditation programs improve anxiety, depression and pain by moderate amounts, effect sizes around 0.3 to 0.4, roughly what you'd expect from other active treatments.³⁵ Mindfulness-based cognitive therapy cuts the risk of depressive relapse by about 30% compared with usual care.³⁶ Those are real numbers, and for something free with no side effects for most people, they're good. They are not exponential. A group of researchers wrote a whole paper titled "Mind the hype" for exactly this reason.³⁷

Equation 10 · HabitsUpdating the value of a craving

Sometimes the most powerful thing attention does isn't calming you down. It's correcting your data. In 1972, Robert Rescorla and Allan Wagner wrote down a rule for how animals, people included, learn what to expect from the world.³⁸ It's now the backbone of reinforcement learning, in brains and in computers.

The psychiatrist Judson Brewer saw what this means for addiction. A habit loop runs on a stored value: this will feel good. But if you do the habit on autopilot, you never measure the actual reward, so the stored value never updates. You just keep acting on an old prediction. Brewer asked smokers to smoke mindfully: to notice the smell, the taste, the feeling in their chest. One said it "smells like stinky cheese and tastes like chemicals." That's a large negative prediction error, and V drops. In his trial, 31% of people in mindfulness training were abstinent at four months, compared with 6% in a standard program.³⁹ And mindfulness training weakened the link between craving and actually smoking.^40,41

Set attention to zero and repeat it a hundred times. V never moves. That's the quiet tragedy of autopilot: repetition alone doesn't teach you anything. Then set it to 100% and watch the craving fall to meet reality in a dozen repetitions. No willpower required, just accurate information. Brewer calls this disenchantment, and it feels completely different from forcing yourself to stop.

A bonus equationBehavior is a function of you and where you are

I'll close with an equation that isn't about meditation at all. In 1936, the psychologist Kurt Lewin proposed a formula for all of human behavior: B = f(P, E). Behavior is a function of the person and their environment.⁴² It explains why you're polished at a work dinner and someone else entirely at a family reunion. And it gives two levers for change: change yourself, or change your surroundings. Or both.

Everything in this post has been about that little a. Concentration chooses which part of the environment gets in. Clarity resolves it accurately. Equanimity lets it pass through without sticking. Practice also slowly changes P. But Lewin's original equation keeps us honest. Sometimes E is the problem: an unsafe home, a job that grinds people down, poverty. Mindfulness isn't a substitute for changing those, and critics are right to call out programs that use it to make people quietly put up with things that should change.⁴³

Solve for nowThe equations, for the fridge door

Here's everything we collected, in plain language.

And one practice that uses all of it. It takes three minutes. Concentration first, then clarity, then equanimity, in that order, because each needs a bit of the one before.

Shinzen likes to say the goal of all this is "industrial strength" happiness that doesn't depend on circumstances. I can't promise you that, and neither can any equation. What the math can show you, and what your own two minutes of breath counting probably already did, is that the relationship between you and your experience has variables in it, and some of those variables can move. You can't solve for forever. But you can solve for now, and then do it again.

For a gentler, less mathematical way in, start with Meditation, deconstructed. For the body side of the same story, see Coming home to the body. For a walk instead of a sit, try Signposts inward.

This post is educational and isn't medical advice. Equations here are teaching models, not clinical measurements. If meditation brings up more than you can manage, ease off and talk with a clinician or an experienced teacher. If you're struggling, in the US you can call or text 988 any time.