Check the units

"Is a human life worth more than the economy?" sounds profound and is dimensionally confused. Lives and dollars are different kinds of thing — different units — and the sentence quietly demands an exchange rate it never states. You cannot answer it as written any more than you can answer whether three kilograms is longer than two hours. The grandeur is a symptom: the question has skipped the step where it would have to name the number it is secretly using.

Physicists never make this mistake for long, because a wrong equation announces itself. The units don't match, and you can see it before you compute anything — a term in metres cannot be added to a term in seconds, and the moment one is, the whole line is void. The same discipline, turned on arguments, catches a huge class of nonsense: comparing incommensurables, adding what can't be added, and the missing conversion factor hiding inside a confident claim. Dimensional analysis is a bullshit detector you can run before you know any of the facts.

Every quantity carries a dimension — the kind of thing it is: length L, time T, mass M, money $, people N, happiness U. Add a quantity, then an operator, then another. The checker tracks dimensions through the algebra honestly: × and ÷ always work and make new compound units; + and − demand identical units, and flag the moment they don't.

Ordinary claims carry implicit units too. The auditor parses each for the dimensions it's really comparing, then flags the verdict: a category error (comparing genuinely different dimensions), a hidden exchange rate (a comparison that only works once you name a conversion), or a legitimate rate (unlike units combined honestly by ÷). Reveal each to see the conversion the claim is smuggling.

Before precision, before checking a single number, ask whether the result is in the units the question asked for. A speed should come out in L·T⁻¹; a cost-per-life in $ per person. Pick a question, build a rough estimate, and the bridge tells you whether your combination lands in the right dimension — the first sanity check on any Fermi guess.

The bridge doesn't care if your numbers are good. It cares that money ÷ people gives dollars-per-person while money × people gives the meaningless $·N — same inputs, and only one can possibly answer the question. A wrong-dimensioned estimate is wrong before it is even inaccurate.

Dimensional consistency is a strange kind of test: it can only ever say maybe. It cannot certify that a formula is right — plenty of dimensionally perfect equations are physically false — but it can certify, absolutely and instantly, that a dimensionally wrong one is meaningless. That asymmetry is the whole value. Passing the check is cheap and proves little; failing it is fatal and proves everything. So you run the cheap test first, and only spend attention on claims that survive it.

The mechanics are just algebra, but the algebra encodes a rule about kinds. Multiplication and division are generative: they take unlike quantities and make new, legitimate ones — distance over time becomes speed, dollars over people becomes a per-capita. Addition and subtraction are conservative: they demand that both sides already be the same kind of thing. You can multiply anything by anything, but you can only add like to like. A claim that adds unlike units hasn't made an error of degree; it has failed to be a claim at all — it is not even wrong.

1. Build a category error and watch it flag. In the units checker, load happiness = money + friends, or make your own: any two different-coloured units joined by +. The offending operator turns red. Now swap the + for a × or ÷ and watch the violation vanish as a new compound unit appears — the same quantities, suddenly well-formed.
2. Audit the sample claims for hidden exchange rates. In the auditor, reveal "that's a lot of money to save one life" and "the risk isn't worth the reward." Both look like statements; both are actually asking for a number — an exchange rate — they decline to state. Then reveal "productivity per capita" to see what an honest comparison looks like.
3. Check a Fermi estimate's output units. Pick "what does it cost to save one life?", then try money × people and money ÷ people. Only one lands in the units the question asked for — and you knew which before estimating any number at all.

A surprising number of "profound" disputes secretly demand a conversion factor between incommensurables — a life in dollars, justice in growth, art in utility, a species in jobs. Dimensional analysis does not set the rate; nothing here can tell you what a life is worth. What it does is force you to admit that a rate is needed and to state it. And stating it dissolves a great deal of fake depth. "You can't put a price on a life" is often not a moral position but a refusal to look at the price you are already paying — every guardrail not built, every drug not subsidised, reveals a finite number.

The move clarifies real tradeoffs at the same time as it deflates false ones. Once the exchange rate is on the table, disagreement becomes tractable: we can argue about whether a statistical life should be valued at two million or ten, which is a hard, honest, finite argument — instead of trading grand incommensurable slogans that were never comparable to begin with. The question that felt bottomless was bottomless only because it kept its units hidden. Name them, and it acquires a floor.

Once you have the check, you read quantitative rhetoric differently. Watch for the classic unit-crimes. Comparing a rate to a level — "the deficit is rising while the debt is huge" treats a flow and a stock as rivals, when the honest question is how $/T relates to $ (see stocks and flows). Comparing a per-capita to a total — country A spends more in total, country B more per person, and both are quoted as if they settled the same argument. And the missing multiplication: quoting a probability against a payoff without ever multiplying them into an expected value, so a tiny risk of catastrophe and a huge risk of nothing get the same rhetorical weight.

"Check the units" is portable because it needs no expertise in the subject. You do not have to know climate science, or epidemiology, or finance, to notice that a sentence has added a rate to a total or compared a dollar figure to a probability. The category error lives in the grammar of the claim, not in its facts — which is exactly why you can catch it first, and why it travels from physics into any argument that dares to use numbers.