The figures in this group fall into two sets. The statistical paradoxes show data telling opposite stories depending on how it is cut, and they are among the most practically important entries in the ledger. The sayings, some ancient and some from working scientists, record what it feels like to pursue knowledge and how easily the pursuit goes wrong.
When the numbers reverse
Simpson's Paradox is the central case: a trend present in every subgroup can reverse when the groups are combined. Its best-known example came from Berkeley's 1973 admissions, where women appeared to be admitted at lower rates overall while most departments admitted them at equal or higher rates, because women applied more often to the most competitive departments. Berkson's Paradox shows selection into a sample creating correlations that do not exist in the population. The Birth-Weight Paradox and the Obesity Paradox are medical versions, in which conditioning on a diagnosis makes a harmful factor look protective. Lord's Paradox shows two defensible analyses of the same data reaching opposite conclusions.
What knowing feels like
The sayings address the pursuit itself. Plato's Cave, the oldest, describes people mistaking shadows for reality and resisting the one who tells them otherwise. The Procrustean Bed names the habit of forcing observations to fit the model. Polanyi's Paradox, that we know more than we can tell, explains why expertise resists being written down. Planck's Principle is the most pessimistic: a new scientific truth wins not by convincing its opponents but because they eventually die.
Several sayings come from working scientists and are wry rather than grand. Twyman's Law, that any figure that looks interesting is probably wrong, is a data analyst's first instinct. Orgel's Second Rule, that evolution is cleverer than you are, warns biologists against assuming a feature has no function. Asimov's Corollary observes that when the public rallies to an idea distinguished experts have dismissed, the experts are probably right after all, and the Law of the Infinite Cornucopia notes that any doctrine one wants to believe can find a supply of arguments.
Where it applies
The statistical paradoxes are essential in medicine, social policy and any field that compares groups. Simpson's paradox appears in hospital league tables, school comparisons and hiring statistics, wherever groups differ in their mix of easy and hard cases. Berkson's paradox appears in any study that recruits from hospitals, clinics or selective programmes. Twyman's law is the right reflex for anyone reading a dashboard, and Polanyi's paradox explains why automation has advanced fastest where tasks can be spelled out and slowest where they rely on tacit skill.
Where it is argued over
The paradoxes are settled as mathematics; the disputes are about interpretation, and in particular which way of cutting the data answers the question asked. Lord's paradox remained contested for decades until causal inference methods clarified that the two analyses answer different questions. The obesity paradox carries a contested flag because researchers disagree about whether it is an artefact of selection or reflects something real. Planck's principle has been tested: a 2019 study found that research fields did open up to newcomers after the deaths of prominent scientists, which offered partial support.
Why it is worth knowing
The paradoxes teach the most useful single lesson in statistics: a number means nothing until you know how the groups behind it were formed. The sayings add the human side, that knowledge is resisted, tacit and easily bent to fit what people want. Together they explain why this ledger reports how each claim was established as well as what it says.
The entries behind this essay: Figure in Science & Epistemology. Every subject has its essays on the essays page.