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The useful mathematics hiding inside everyday wisdom.
There’s an old saying that every story has three sides: yours, mine, and the truth.
I’ve always liked it. Two people can watch the same event, come away with incompatible accounts, and be completely sincere. The saying gives that uncomfortable fact a nice little ending: somewhere beyond their disagreement sits the truth, waiting patiently for a third person to report it.
But who is that person? And what makes their account more reliable?
That’s the question that sent me back to a collection of aphorisms I’d been gathering. Some sounded like advice your grandfather might give. Others sounded like things a statistician would say after a third coffee. A few turned out to be compact versions of genuinely powerful ideas from mathematics, science, and philosophy.
I began to suspect that the appeal of a good aphorism isn’t just that it’s memorable. It’s that, sometimes, a whole way of thinking has been compressed into a sentence.
There’s a famous maxim, often associated with psychologist Kurt Lewin, that nothing is as practical as a good theory. The phrasing is older than Lewin, as Arthur Bedeian’s history of the quotation shows, but the thought is worth keeping. A useful theory changes the questions we ask before we make a decision.
That’s what this series is about.
One distinction matters from the start. A mathematical theorem follows from stated assumptions. A scientific model must also earn its usefulness against observations. A philosophical principle can clarify reasoning without being a theorem, and a heuristic may be helpful without being universally true. Confusing these categories is an excellent way to turn sound advice into sophisticated nonsense.
We don’t see the world. We see a model of it.
George Box’s line, “All models are wrong, but some are useful,” is easy to misread as an invitation to tolerate sloppy thinking. It isn’t.
A map leaves out almost everything about a city. It can still get you to the station. In fact, including every pipe, crack in the pavement, and passing pigeon would make it a worse navigation tool. A model earns its keep by discarding the right information for the decision at hand.
The real question is not whether the map is identical to the territory. It’s whether the missing details matter for what you’re trying to do.
A financial forecast that ignores a rare event might be good enough for next month’s budget and terrible for deciding whether a company can survive the next decade. A machine-learning model might be excellent on its test distribution and fail when the world changes. A language model can give a fluent account of an event without having the evidence required to distinguish a good explanation from a plausible invention.
None of that is proved by Box’s sentence. It’s an attitude toward modelling, developed in his 1979 discussion of scientific model building. The practical habit it suggests is simple: ask what was left out, and does it matter here?
Even a model that seems to have failed doesn’t tell us, by itself, where the failure lies.
Suppose an experiment produces the wrong result. The hypothesis could be false. Or the instrument could be miscalibrated. Or a background assumption could be false. Or the experiment could have measured something subtly different from what we thought.
This is the insight behind “No hypothesis is ever tested alone,” associated with Pierre Duhem and W. V. O. Quine. Predictions depend on a whole bundle of assumptions, not one isolated proposition. As the Stanford Encyclopedia of Philosophy explains, an unexpected observation tells us that something in the bundle needs attention; it doesn’t identify the guilty premise automatically.
It’s a philosophical thesis about how testing works, not a mathematical licence to rescue any favourite theory forever. You can always invent an excuse for a failed prediction. The hard part is finding an explanation that survives independent scrutiny.
Maybe our third storyteller isn’t standing outside the disagreement after all. Maybe they’re simply holding a different map.
Evidence should change our minds, but not equally
Now imagine someone tells you a coin landed heads ten times in a row. Odd, but possible. If the same person tells you they can predict the next thousand flips of a fair coin, you ought to demand rather better evidence.
“Extraordinary claims require extraordinary evidence” has a rigorous mathematical relative in Bayes’ theorem.
In odds form, it reads:
Posterior odds = Prior odds × Likelihood ratio
The prior odds reflect what you believed before seeing the new observation. The likelihood ratio tells you how much better that observation is explained by one hypothesis than another.
If a claim begins with very low prior odds, modest evidence might not move it very far. The point is not that unfamiliar ideas must be wrong. It’s that evidence should be evaluated in relation to what it would look like under competing explanations.
This is a conditional probability identity, not a theorem assigning objective prior probabilities to every claim in your life. People can reasonably dispute the priors and the likelihood model. What Bayes gives us is the accounting system for those disagreements.
And sometimes the most important evidence is the evidence that never arrives.
The bombers that didn’t come home
During the Second World War, statisticians studying damage to returning aircraft faced a deceptively straightforward question: where should the planes be reinforced?
The visible evidence consisted of aircraft that had been hit and survived. You might naturally put more armour where the surviving planes had the most damage. But Abraham Wald’s analysis turned attention toward something absent from the data: the planes that never returned.
A damaged area that appeared rarely among survivors could be particularly dangerous precisely because aircraft hit there were less likely to make it home. The relevant mathematical work is documented by Mangel and Samaniego’s account of Wald’s aircraft-survivability analysis.
“Look at the planes that didn’t come back.” It’s a memorable way of recognising survivorship bias.
The same mistake appears when we examine only successful founders to discover the habits that produce successful companies, or only profitable investors to identify strategies worth copying. Unless we account for the attempts that failed, success stories may tell us remarkably little about the probability of success.
This isn’t merely a warning about misleading anecdotes. It is a property of selecting observations based on an outcome: the data we can see is conditional on survival.
A useful theory changes the direction of our attention. Instead of asking only what the available data says, we ask what process determined which data was available?
The awkward business of acting on what we measure
So far, we’ve been trying to observe reality. But what if our observations begin changing the very thing we’re observing?
Imagine rewarding a software team for the number of tickets it closes. At first, the number may be a reasonable proxy for completed work. Then people adapt. Large jobs get split into smaller tickets. Easily closed tasks become attractive. Important but ambiguous problems become harder to justify.
The dashboard improves. Whether the product improves is another question.
“When a measure becomes a target, it ceases to be a good measure.” That’s the popular formulation of Goodhart’s law, from Marilyn Strathern’s discussion of audit culture. It is a powerful warning, not a theorem saying every targeted metric instantly becomes useless. The mechanism is that people and institutions adapt to incentives; the connection to the Lucas critique is illuminating, though the two ideas are not identical.
A model of passive observers may stop working when those observers become strategic participants.
And even with an honest metric, improving the most visible part of a system may barely improve the whole.
Suppose a computation spends 90% of its time in a component you can accelerate without limit, while the other 10% remains unchanged. You might expect infinite improvement. You won’t get it.
Amdahl’s law gives the bound:
Speedup = 1 / ((1 − p) + p / s)
Here p is the fraction you can accelerate and s is how much faster that portion becomes. With p = 0.90, even s → infinity leaves you with a maximum tenfold speedup.
The saying “Speeding up what isn’t the bottleneck changes nothing” is slightly too dramatic. It can change something. What the mathematical bound proves is that a fixed, unimproved fraction limits total speedup. The managerial lesson—work on what constrains the whole system—is an application, not the theorem itself.
I like that distinction. It makes the advice more useful, not less. It tells me what I need to measure before deciding where to invest my effort.
Sometimes the easiest way forward is backward
There is one more saying I keep returning to: “Invert, always invert.” Charlie Munger often credited the mathematician Carl Jacobi with the idea, although the exact historical wording is less secure than the retellings suggest; the Mathematical Association of America has examined that attribution.
Want to build a reliable service? First ask how you would make it fail: a single unmonitored dependency, no recovery plan, silent data corruption, credentials living forever. Now work backward.
Want a better scientific argument? Ask what observation would embarrass your preferred explanation. Want a better decision? Ask what would make it regrettable, and whether you can remove that risk.
Inversion isn’t a universal mathematical law. It’s a problem-solving heuristic. But it has a pleasing connection to everything that came before it: examine the assumptions, look for the missing observations, test the alternatives, and find what actually constrains the outcome.
That’s a lot of intellectual machinery packed into three words.
Maybe that’s what makes a theory genuinely practical. Not that it provides the last word, or that it turns judgment into arithmetic, but that it lets you notice something consequential you would otherwise have missed.
And once you’ve noticed it, you can’t quite return to thinking the old way.