Science and Hypothesis asks how much of physical science is genuine knowledge of the world and how much is the free work of the mind. Poincaré opens against two easy positions. To the casual observer, scientific truth is unassailable and its logic infallible. To the disillusioned, once the role of hypothesis is noticed, the whole edifice looks ready to fall at a breath. He calls both attitudes superficial, since to doubt everything or to believe everything alike spares one the labour of reflection. His own method is to examine the role of hypothesis carefully and to distinguish its several kinds: some are verifiable and, once confirmed, become fertile truths; some merely help fix our ideas; and some are hypotheses only in appearance, being definitions or conventions in disguise.
The first part turns to number and magnitude. Poincaré asks how mathematics can be both rigorous and genuinely informative rather than a vast tautology, and locates its creative power in reasoning by recurrence, a kind of mathematical induction that lets a single argument cover an infinity of cases. The notions of continuous magnitude and the measuring of quantity, he argues, are not simply read off from nature but constructed by the mind to organise experience.
The second part, on space, contains the book's most famous argument. Geometry rests on axioms that are neither self-evident truths nor experimental facts. Euclid's parallel postulate cannot be proved, and the work of Lobatschewsky, Bolyai, and Riemann shows that consistent non-Euclidean geometries are possible. Poincaré concludes that the axioms of geometry are definitions in disguise. To ask whether Euclidean geometry is true has no meaning; one geometry can only be more convenient than another, and Euclidean geometry stays in use because it is the simplest and fits the behaviour of ordinary solids. Experiment guides this choice without imposing it.
The third part carries the same analysis into mechanics. Treatises blur what is experiment, what is reasoning, what is convention, and what is hypothesis. There is no absolute space and no absolute time; to say that two periods are equal has meaning only by convention, and even the law of inertia is partly a disguised definition. Mechanics is built on principles that began in experiment but have hardened into conventions we keep because they are convenient, not because new experiments could never touch them. Energy and thermodynamics receive a similar reading.
The fourth part, on Nature, treats physics directly. Experiment is the sole source of truth, yet bare facts are not science: science is built of facts as a house is built of stones, but a heap of stones is not a house, so the physicist must generalise and exhibit foresight. Good hypotheses are fruitful and testable, and even a hypothesis later discarded has usually served by exposing real relations. Because prediction is only probable, Poincaré pauses over the calculus of probabilities and the role of our belief in simplicity. Reviewing optics and electro-dynamics, he shows that as theories are overturned their equations and relations persist, which is his ground for holding that science grasps the relations between things even when its images of the things change.