a hidden assumption
A hidden assumption is a fact a proof leans on without ever stating it as a definition, axiom, or previously proved theorem — usually because it looks too obvious to question, or because the picture seems to make it true. It is the most dangerous flaw in a deductive argument precisely because it hides: the chain of reasoning looks airtight, yet one of its links is invisible. Hunting for hidden assumptions is the core discipline of rigorous foundations, and the history of geometry is largely the history of finding and naming them.
Euclid's Elements is the classic hunting ground, and the assumptions there fall into recognizable kinds. Continuity assumptions: that two circles which appear to cross actually share a point (used in the very first proposition), with nothing in the postulates guaranteeing it. Order or betweenness assumptions: that a line entering a triangle must leave it — the content later captured by Pasch's axiom — and that a point read off the diagram as 'inside' really is between two others. Diagram-driven steps generally: concluding a configuration is impossible because you cannot draw it, or that points lie in a certain arrangement because that is how the figure happens to be sketched. Each of these felt self-evident and each turned out to need an explicit axiom.
The cure is exactly Hilbert's programme: make every such tacit fact into a stated axiom (Pasch's axiom for that triangle-crossing, the continuity axioms for circle intersections) so that no step relies on inspecting a figure. The deeper lesson generalizes far beyond geometry: a picture is a wonderful guide to intuition but a treacherous source of proof, because a diagram can quietly smuggle in special features — a particular order of points, an assumed intersection — that the general claim must not depend on. Rigour is, in the end, the practice of refusing to believe anything you have not been given permission to assume.
A famous fake 'proof' that every triangle is isosceles works entirely by drawing the figure with a point in a position that, for a non-isosceles triangle, actually falls outside the triangle — the picture lies. The argument has no arithmetic error; its only flaw is a hidden assumption about where a point lands, read off a misleading diagram.
A diagram can smuggle in a fact the postulates never grant — that smuggled fact is the hidden assumption.
A hidden assumption is not a calculation mistake; it is a missing axiom. The fix is never to compute more carefully but to state the assumption explicitly — or discard the step that needed it. This is why proofs should not depend on reading a figure.