= Solution
\b[Explicit reference to laws is not necessary for every good <scientific explanation>. Laws remain indispensable in many explanations, but their occurrence alone is not sufficient either.] The issue turns partly on whether a law means a contingent regularity of nature, any supporting generalization, or even a mathematical necessity.
The <deductive-nomological model> offers a strong positive case. Laws and circumstances entail the phenomenon, and at least one law is essential to the derivation. For example, <Newton's second law> and an inverse-square <gravitational force>, with appropriate initial conditions, yield predictions for an idealized two-body <orbit>. A universal relation unifies cases that otherwise look like unrelated observations. <Carl Hempel> and <Paul Oppenheim> explicitly develop adequacy conditions for such deductive explanations; their 1948 paper should not be read as a finished treatment of every statistical explanation.<Studies in the Logic of Explanation by Carl Hempel and Paul Oppenheim>[section 3, the deductive schema and conditions of adequacy]{ref}
However, derivability does not always give the right explanatory direction. In a simplified <geometrical optics> model, a vertical pole of height $h$ on level ground casts a shadow of length $\ell=h/\tan\theta$ at solar elevation $\theta$. Given $h$ and $\theta$, one derives $\ell$; given $\ell$ and $\theta$, one derives $h$. The same geometrical relations enter both derivations. Yet the shadow does not normally explain why the pole was built to that height. The reverse calculation is a way to measure its height. The difference concerns which features produce or constrain which others, rather than deductive validity alone.
<Mechanistic explanation> is another challenge to a compulsory law-based format. A clock stopped because debris blocked a particular wheel, thereby preventing the mechanism from advancing. Inspecting the obstruction, removing it and observing restored operation can answer the causal question with considerable precision. The explanation describes organization and a relevant difference-making event without stating a universal law. It is more informative about this failure than the observation that every component obeyed <Newtonian mechanics>.
A defender of the <covering-law model> can reply that the causal account relies implicitly on regularities: debris that offers no resistance would not stop the wheel. This is a fair warning against treating a bare chronology as an explanation. But it does not establish that explanatory understanding requires identifying exceptionless fundamental laws, or that replacing the account with a complete microscopic derivation would improve the answer. Stable, tested relations in the actual operating regime may be sufficient.
An explanation can also turn on a mathematical constraint. Suppose an engineer asks why a connected network with four odd-degree junctions cannot be inspected by a route using every segment exactly once. Represent junctions as <vertices> and segments as <edges> of an <undirected graph>. Every intermediate visit pairs an incoming and an outgoing <edge>, so only the two endpoints of an open <trail> can have odd <vertex degree>. Four such <vertices> therefore exclude an <Eulerian trail>. The impossibility persists if segment lengths, construction materials or walking speeds change while the connections remain fixed. This is the obstruction in the <Seven Bridges of Königsberg>.<Solutio problematis ad geometriam situs pertinentis by Leonhard Euler>[sections 6, 9 and 20: connectivity data, the four-odd-region obstruction and the general rule]{ref}
The physical application requires that the <graph> represent the allowed routes correctly. Once that is granted, the explanatory necessity comes from <parity>, rather than a contingent <physical law>. Calling every mathematical constraint a law would preserve the original claim only by greatly widening its meaning. A critic may restrict “scientific” to causal explanations, but that restriction needs justification when applied mathematics answers a concrete engineering question.
Thus laws are powerful explanatory resources, and good explanations normally support relevant what-if judgements. Those judgements can be grounded in laws, mechanisms or mathematical constraints. Requiring explicit reference to laws imposes an unnecessarily narrow format; requiring some disciplined account of dependence is much more defensible.
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