Book: La Science et la Méthode (Science and Method)

Overview
La Science et la Méthode collects Henri Poincaré's reflections on how science and mathematics progress, why certain methods succeed, and what guides discovery. The essays blend philosophical analysis with concrete reflections on mathematical practice, arguing that scientific reasoning is neither a simple application of logic nor a blind accumulation of facts. Poincaré treats creativity, intuition, and formal reasoning as complementary elements that together shape scientific advances.

Method and Heuristics
Poincaré sets out practical rules that scientists often unconsciously follow: reduce problems to simpler cases, exploit analogy, vary hypotheses, and seek invariants. These heuristics are not formal algorithms but educated habits that guide fruitful exploration. He emphasizes the provisional and experimental character of methods, which evolve as new problems and tools emerge, and underscores the importance of choosing fruitful questions over rote application of procedures.

Creativity and the Unconscious
Central to Poincaré's account is the creative unconscious. He insists that many mathematical discoveries arise from a period of subconscious work in which the mind reorganizes elements until a sudden insight appears. Conscious reasoning verifies and refines the result, but the initial leap often escapes deliberative control. Poincaré describes the interplay between disciplined preparation and unexpected flashes of invention, arguing that both systematic practice and openness to intuition are essential for discovery.

Role of Hypotheses
Hypotheses are presented as heuristic tools that structure thought and guide experiment rather than as immutable truths. Poincaré treats them as provisional constructs chosen for their explanatory power and convenience. He cautions against dogmatic attachment to particular hypotheses and stresses that empirical testing, mathematical consistency, and simplicity determine their retention or rejection.

Conventionalism and Geometry
A striking theme is the conventional nature of certain foundational choices, especially in geometry and the principles of mechanics. Poincaré argues that axioms such as the nature of space are not compelled purely by experience but reflect conventions selected for their simplicity and usefulness. This "conventionalism" does not deny empirical constraint but locates scientific freedom in selecting frameworks that best organize phenomena with minimal arbitrariness.

Mathematical Style and Rigor
Mathematical practice combines rigorous proof with aesthetic judgment. Poincaré defends rigorous demonstration while recognizing that exposition, choice of notation, and the intuitive framing of problems deeply influence discovery and comprehension. Elegance and simplicity are not mere ornaments but serve as practical guides to fruitful directions and clearer proofs, aiding both invention and communication.

Experiment, Induction, and Theory Choice
Empirical evidence and logical argument play complementary roles. Induction supplies generalizations from observed cases, while deductive reasoning draws out their consequences. Poincaré emphasizes the pragmatic criteria for theory choice: predictive power, simplicity, coherence with established knowledge, and the capacity to unify disparate facts. Scientific theories are judged by how well they organize experience and open new avenues for research.

Impact and Legacy
Poincaré's blend of philosophical subtlety and intimate knowledge of mathematical practice shaped later debates about scientific methodology, creativity, and the nature of mathematical truth. His insistence on the heuristic role of hypotheses, the importance of intuition, and the conventional elements of foundational choices influenced both philosophers of science and working scientists. The essays remain a lively account of how reason and imagination combine to drive scientific progress.