Book: La Science et l'Hypothèse (Science and Hypothesis)

Overview
Henri Poincaré presents a spirited examination of how mathematics and physics rest on choices that are not purely empirical but partly conventional. He questions the status of axioms, the nature of mathematical truth, and the meaning of geometry when confronted with physical measurement. The tone blends philosophical reflection, mathematical insight, and critique of naive empiricism, inviting readers to rethink the relationship between theory and observation.
Poincaré emphasizes that scientific ideas often serve as frameworks for organizing experience rather than literal descriptions of reality. He treats hypotheses not as immutable truths but as tools whose value lies in simplicity, convenience, and explanatory power.

Mathematics, Logic, and the Status of Axioms
Poincaré argues that mathematics is synthetic and inventive: its axioms are not discovered facts about the external world but conventions chosen to enable powerful deductions. He defends the autonomy of mathematical thought, showing that logical deduction operates within systems built from primitive assumptions whose selection is guided by intuition and practical considerations.
He distinguishes between definitions, theorems, and conventions, and insists that mathematical axioms cannot be reduced simply to empirical generalizations. Logic supplies the structure for mathematical reasoning, but the initial postulates acquire their meaning through the way they organize mathematical practice and applications.

Conventionalism and Geometry
A central thesis is conventionalism about geometry: geometrical axioms are not forced by experience because different geometric frameworks can be empirically equivalent once physical laws are adjusted. Poincaré illustrates that measurements interpreted as evidence for Euclidean or non-Euclidean geometry depend on choices about the underlying physical assumptions, such as the behavior of measuring rods and light.
He shows that geometry is a language for describing spatial relations and that adopting one geometry over another often amounts to a decision of convenience. The choice is influenced by simplicity, familiarity, and how naturally it meshes with other established principles, rather than by an absolute experimental verdict.

Hypotheses and the Method of Physics
Poincaré treats scientific hypotheses as provisional instruments that should be judged by their empirical adequacy and heuristic fruitfulness. He emphasizes the role of creativity in proposing hypotheses and the pragmatic criteria, such as simplicity and coherence, that guide theory choice when empirical data alone are underdetermined.
He explores how physical laws are underpinned by conventions and metaphysical assumptions, and he warns against elevating hypotheses to metaphysical certainties. Scientific progress arises from continually testing, refining, or replacing hypotheses in light of their explanatory scope and the stability of predictions.

Limits of Empirical Knowledge
Poincaré underscores inherent limits to what observation can decide, stressing that empirical evidence may be compatible with multiple theoretical frameworks. He introduces an early form of underdetermination: measurements cannot uniquely determine the form of foundational principles when auxiliary hypotheses can be adjusted.
This recognition leads to a tempered epistemology that acknowledges the provisional and sometimes pragmatic character of scientific knowledge. Certainty in the strictest sense is unattainable, but well-chosen conventions and robust hypotheses provide reliable, if fallible, guides for inquiry.

Legacy and Continuing Relevance
Poincaré's perspectives anticipated key debates in philosophy of science and informed later thinkers about the conventional aspects of scientific frameworks. His blend of mathematical rigor and philosophical subtlety influenced discussions on the foundations of geometry, the nature of scientific explanation, and the role of conventions in theory formation.
The argument that choice and pragmatism enter at the foundations of science remains influential, providing a caution against simplistic realism and a reminder that scientific theories are judged not only by how they map onto observations but by how coherently and economically they structure understanding.