the second hundred · metaphor 102
Is a marriage, a job, or a friendship more likely to end because it has already lasted a long time — or less? The answer is not a mood. It is the shape of one curve — the hazard rate — and different bonds have opposite shapes.
The hazard rate is the risk that a thing ends right now, given how long it has already lasted. Not the odds it ever ends — the odds it ends this year, conditional on having made it this far. That "given" is the whole trick: it lets the past reach forward and bend the future, and it can bend it either way.
Three shapes divide the world. The dangerous early years: risk is high at the start and falls as the bond proves itself. Wearing out: risk only climbs, and every year survived makes the next one deadlier. The longer it lasts, the safer: risk keeps dropping, so age is armor. Same sentence — "we've lasted this long" — is comfort on one curve and a warning on another. The instrument below lets you flip between them and watch the future invert.
Choose the shape of risk · h(t)
The shape that decides
Watch the second panel: the survival curve only ever falls. Bonds don't un-end, so S(t) — the share still going at time t — slides monotonically toward zero no matter what. Read alone, it flatters a comforting lie: everything decays, so decay must mean the same thing everywhere. It doesn't. The survival curve is downstream bookkeeping. What actually decides your fate is its slope, and the slope is set by the hazard.
The relationship is exact: S(t) = exp(−∫h). Survival is the running exponential of every unit of risk you have accumulated so far. A flat hazard drains survival at a steady rate; a rising hazard makes the curve dive late; a falling hazard makes it plunge early and then flatten into a long, tough tail. Two couples can share the exact same 40% survival at year ten and be on opposite trajectories — one steadying, one accelerating toward the exit. The single number "still together" tells you almost nothing. The shape of the hazard that produced it tells you almost everything.
What to try
Set the shape to wearing out and drag the amber age dial to the right. The bottom-right panel — m(t), the expected years still to come given you have already reached age t — falls as you age. Reaching year twenty doesn't buy you twenty more; it leaves you with fewer than you started with. Now switch to the longer it lasts, the safer and drag again: the same curve climbs. Here reaching year twenty means you expect more road ahead than the day you began — the survivors are the sturdy ones, and you are now one of them.
The dashed memoryless twin is your control. It is a constant-hazard bond with your exact average lifetime, so its expected-remaining-life is a flat line: for it, age genuinely carries no information — a fresh start and a thirty-year veteran face identical odds. Every place your teal m(t) rises above that flat line, lasting has bought you time; every place it dips below, the clock is against you. Try the dangerous early years: expected life rises as you clear the fragile zone, then bends back down as wear-out sets in decades later — comfort first, warning later, from one curve.
The mapping
Sort your attachments by hazard shape and they stop being a mood and start being a forecast. Skills, reputations, canonical books, long friendships tend to run a decreasing hazard — the Lindy pattern, where each year survived is evidence of robustness, and the expected future lengthens with age. Bodies, machines, romances built on novelty, jobs whose thrill was the newness run an increasing hazard — they wear. And a great many real bonds are bathtub-shaped: perilous at the start (the raw first years of a marriage, the probationary months of a job), safe through a long middle, and quietly wearing out at the far end.
This is why "we've lasted this long" is not one sentiment but two opposite ones, and knowing which you are entitled to matters. On a decreasing hazard it is earned comfort: the past is collateral, and endurance predicts more endurance. On an increasing hazard it is a quiet alarm: you have been spending down a fixed reserve, and the length of the record is the size of the bill, not a guarantee against it. The instrument's whole point is that the same words attach to opposite curves — and only the shape can tell you which one you're living inside.
The honest model
The machinery is small and exact. Pick any hazard h(t) ≥ 0. Accumulate it into the cumulative hazard H(t) = ∫h. Survival is S = e−H; the density of ending times is f = h·S (risk now, times the chance of surviving to be exposed to it); and expected remaining life is the area still under survival, normalized by who's left: m(t) = ∫t S ⁄ S(t). A falling hazard forces m(t) to rise; a rising hazard forces it to fall; a flat hazard pins it constant. Those implications are theorems, not opinions — the panels are drawing them, not asserting them.
Here is the one thing the equation cannot see: you. A hazard curve is a population rate — the fraction of many similar bonds that end at each age. Your single marriage is one draw from that population, and a draw is not a rate. The curve says forty percent of couples like yours are still together at year ten; it does not say you are forty percent together. You either last the year or you don't. The model gives you the shape of the odds, honestly and live — and then hands the actual coin to you.
Mathematics ↔ life
| Mathematics | Life |
|---|---|
| hazard rate h(t) | The risk it ends right now, given it has lasted this long — the instantaneous chance of the exit. |
| survival curve S(t) | The chance the bond is still going at time t — the share of couples, jobs, friendships not yet ended. |
| increasing hazard | Wearing out: novelty, bodies, machines. Every year survived makes the next one riskier. |
| decreasing hazard | Getting safer for having lasted: skills, reputations, Lindy bonds. Endurance is evidence of endurance. |
| memoryless (constant h) | Age tells you nothing. The veteran and the newcomer face identical odds; the past buys no forecast. |
| conditional life m(t) | "Given we're still here, how much longer?" — and whether that number grows or shrinks as you age. |
Where the metaphor tears
A hazard curve is estimated across many bonds; it is a rate, and your one relationship is a single draw from it. Knowing that couples like yours have a decreasing hazard tells you the shape of the lottery, not your ticket. The metaphor gives you calibrated odds and then goes silent on the only case you care about.
The instrument assumes you have identified the right shape. In life you rarely have. Your marriage may be Lindy while the average is bathtub, or the reverse; the reference class you belong to is itself a guess, and the wrong reference class hands you a confidently drawn curve that is quietly about somebody else.
Survival curves are silent about their subject; people are not. Believe you are on an increasing hazard and you may act in ways that raise it — vigilance curdling into the resentment that ends things — or, believing you are safe, coast until you aren't. The observed curve reacts to being observed. The map edits the territory it claims only to describe.