A mathematical theory begins from axioms: statements accepted without proof, from which theorems are derived by valid inference. Theorems are only as secure as the axioms they use. The classic illustration is Euclid's parallel postulate. Replacing it with alternatives does not produce contradictions; it produces different, internally consistent geometries (hyperbolic and elliptic) in which, for example, the angles of a triangle no longer sum to 180°. A theorem's dependence on its axioms can be made explicit: if you know which axioms a proof uses, you know exactly what must be accepted for the theorem to hold.
A debate case is a chain of inference from premises. Some premises are stated; many are hidden. Hidden premises function like unexamined axioms: if an opponent rejects one, every conclusion built on it falls. Good case construction lists the premises each claim depends on, separates empirical premises (which evidence can test) from normative ones (value commitments), and anticipates which premise the other side will attack. Good rebuttal identifies the single premise on which the most conclusions depend.
Consider the Proposition in the zoning motion. Claim: relaxing restrictive zoning will make housing more affordable. Hidden premises: (1) zoning is a binding constraint, meaning developers would build more if allowed; (2) the additional homes are built where demand is high; (3) added supply lowers prices or slows their growth more than it raises land values nearby; (4) affordability improvements reach lower-income renters, not only higher-income buyers. Reject premise (1) for a given city (because, say, construction costs or interest rates are the binding constraint) and the chain stops at the first step. In the interactive, switch premises off and see which conclusions survive.
Conclusions that still follow
Mathematical axioms are chosen; debate premises are claims about the world or about values, and many can be tested or argued for. Unlike axioms, empirical premises can be partly true, true in some places and not others, or true with some probability. Rejecting an axiom changes a mathematical system without making it wrong; rejecting a factual premise may simply make an argument false. The analogy teaches the structure of dependence, not that premises are arbitrary.