The Parable and the Counterexample

The last of four entries from one conversation on 6 September 2026. The first, The Picture the Maths Was Invented to Replace, is the one this answers; the middle two are Where the Compression Was Trained and Agreement Counts Once. A fifth, The Children's Talk, followed later the same day.

The first entry argued that popular science hands the reader back the picture the formalism was built to replace. The obvious follow-up is that pictures are also how mathematics is taught, and that they sometimes hinder it. Why? And how does that compare with a discipline that has used pictures on purpose for two thousand years: the parable in theology?

Why a picture can hinder

Pictures hinder mathematics teaching for one underlying reason: every picture is over-specified. It has properties the abstraction never committed to, and the learner cannot tell which ones are accidental. The specific failures all follow from that.

• The prototype swallows the concept. Draw every triangle with all its angles acute and students come to believe the altitude always falls inside the triangle. Teach fractions as slices of a pizza and seven thirds has nowhere to live. "Multiplication makes things bigger" is a picture of counting that fails at the first fraction.
• The early picture fights the later extension. Gaston Bachelard called these epistemological obstacles — pieces of earlier knowledge that block later knowledge — and Guy Brousseau brought the term into mathematics teaching. An intuition built on the number line resists negative numbers, then complex numbers, then infinite sets. The picture does not merely fail to help. It actively blocks, because the learner has to unlearn it first.
• A picture gives the feeling of understanding without the ability to operate. A student can nod at the rubber sheet and not compute a single path across it. Fluency is mistaken for competence, which is the popular-science failure arriving at the desk.
• A picture cannot be checked. A proof can be checked against a definition. Nothing can be checked against a diagram. Ampère claimed in 1806 to have proved that every continuous function has a slope at all but isolated points, and the argument rested on what a drawn curve looks like; in 1872 Weierstrass produced a curve nobody could draw, continuous everywhere and with a slope nowhere, and the picture-proof was gone.

The remedy is not fewer pictures. It is pictures with an expiry date — a statement of where each one stops being true — and always a second picture that breaks the first. The counterexample is the mathematics teacher's parable.

The parable

Which brings in theology, because a parable is the same device under a different discipline. It is a concrete instance standing for something the concrete cannot hold, and it fails in the same ways when it fails: the mustard seed read as botany, the Good Samaritan collapsed to "be nice," the father in the story of the prodigal son taken as a rule about inheritance. A parable with its expiry date dropped degrades into a fable with a moral attached, exactly as the rubber sheet degrades into a misconception.

The differences are where it gets interesting.

• Direction of replacement. A mathematical picture is scaffolding, meant to be retired once the definition is in hand. A parable is the primary form, and any doctrinal paraphrase of it is the derived, lossy thing. Mathematics can retire the picture because the abstraction is fully statable. Theology holds that its referent is not, so pictures cannot be retired, only multiplied and marked. The apophatic tradition — theology by saying what God is not — is that marking made explicit. And Aquinas's doctrine of analogy is the expiry-date rule written down as a formal discipline: a word applied to God and to a creature is used neither univocally, in exactly one sense, nor equivocally, in two unrelated senses, but analogically, in related senses that must not be collapsed into one.
• Built-in resistance to collapse. Good parables carry a twist and withhold their conclusion; several end on a question. That is a design feature against the prototype effect, and most mathematical pictures lack it. The best ones have it. The hundred-door version of Monty Hall in the previous entry is a parable in the strict sense, because it changes the listener's stance rather than supplying a model.
• Where the check lives. In mathematics a bad picture is caught when the proof fails, usually within the week. In theology the check is the tradition and the lived consequence, and a literalised metaphor can stand for centuries. At least some doctrinal disputes have been, at root, a metaphor that one side had literalised and the other had not.
• What the device is for. A definition changes what the learner can compute. A parable changes what the listener can see. Mathematics wants the first and only sometimes the second. Theology wants the second and holds the first to be unavailable.

So the common rule, in both rooms: the image is mistaken for the referent, and the accidental for the essential, and each tradition has a discipline for it. Mathematics has the definition and the counterexample. Theology has the apophatic tradition and analogical predication. Popular science is what happens when a picture circulates with neither discipline attached.

Where the record's own registers fall

The record already splits along this line, and I had not seen it as one line until now. The lore runs the mathematics discipline: the mechanism is statable, so the picture is retired and the signature is filed, which is the whole of the dragon rule in the first entry. The church-space overlay runs the parable discipline: it reads the same events through a lens it declares, holds its readings as possibilities rather than findings, and is bound by one test stated in advance — that after reading it, canon must be no less settled than before. That test is the overlay's expiry date. It is what keeps a parable a parable, and it is the reason the overlay can say things the lore never could without either of them contradicting the other.