Five-minute digest
Science and Hypothesis
Summary
Science and Hypothesis (La Science et l'hypothèse, Paris, 1902) is the first of Henri Poincaré's three books on the foundations of science, written by one of the leading mathematicians of his age for a general educated public. It asks how much of science is fact, how much is reasoning, and how much is choice. Poincaré answers that hypotheses are everywhere in science, but that they are of different kinds: some can be tested and become fruitful truths, some are useful for fixing ideas, and some are only definitions or conventions in disguise. The library text is the English translation by W. J. Greenstreet published in London in 1905, with an introduction by the Cambridge physicist Joseph Larmor. It is an OCR text, and some formulas and words are imperfectly reproduced. Part I, Number and Magnitude, argues that mathematical reasoning is not merely a chain of syllogisms. Its creative power comes from reasoning by recurrence (mathematical induction), which Poincaré treats as a synthetic judgment rooted in the mind itself. He then shows how the idea of a mathematical continuum is built beyond the rough 'physical continuum' of the senses, in which A may seem equal to B and B to C while A differs from C. Part II, Space, describes the non-Euclidean geometries of Lobachevsky and Riemann and asks what the axioms of geometry are. They cannot be truths imposed on us a priori, since consistent alternatives exist, nor experimental facts, since experiments deal with bodies rather than ideal lines. They are conventions, chosen freely but guided by experience, and Euclidean geometry remains the most convenient. Poincaré imagines beings in a heated sphere whose bodies shrink toward the edge, who would naturally adopt a non-Euclidean geometry, and he analyses how visual, tactile and motor sensations give rise to our idea of space. Part III, Force, treats classical mechanics, absolute and relative motion, and energy. The principles of inertia, acceleration and conservation of energy began as experimental laws but have been raised to the rank of principles that experiment can no longer refute. Part IV, Nature, turns to physics proper, where experiment is the source of truth and generalization and hypothesis are indispensable. Poincaré answers those who speak of the 'bankruptcy of science' because theories pass: Fresnel's theory of light was replaced by Maxwell's, yet its equations still state true relations. Chapters on probability, optics and electricity, and electrodynamics (Ampère, Helmholtz, Maxwell, Rowland's experiment) close the book.
Top Lessons
Neither credulous nor sceptical
Poincaré opens by setting aside two easy positions, that science is infallible and that it is a house of cards. Both spare us the work of thinking. The task is to sort hypotheses by kind and see which are tested truths, which are useful aids, and which are conventions.
Geometry as convention
Because consistent non-Euclidean geometries exist, Poincaré argues that Euclid's axioms cannot be necessary truths; because experiments concern physical bodies, not ideal lines, they cannot be experimental facts. They are conventions chosen for convenience, and Euclid's is the simplest and best suited to the behaviour of solid bodies.
Facts and the house of science
Experiment is the only source of new truth, but a heap of facts is not a science. The scientist must choose which facts matter, generalize from them, and foresee. Generalization requires belief in the simplicity and unity of nature, held provisionally.
What survives when theories change
Against talk of the 'bankruptcy of science', Poincaré argues that when one theory replaces another, its equations, which express relations, often remain true. Fresnel's ether was abandoned, but his predictions of optical phenomena still hold. What science knows are relations, not the hidden nature of things.
Verified Quotes
“To doubt everything or to believe everything are two equally convenient solutions ; both dispense with the necessity of reflection.”
“the aim of science is not things themselves, as the dogmatists in their simplicity imagine, but the relations between things; outside those relations there is no reality knowable.”
“Mathematical induction — i.e., proof by recurrence — is, on the contrary, necessarily imposed on us, because it is only the affirmation of a property of the mind itself.”
“One geometry cannot be more true than another; it can only be more convenient.”
“Principles are conventions and definitions in disguise.”
“Experiment is the sole source of truth. It alone can teach us something new ; it alone can give us certainty.”
“Science is built up of facts, as a house is built of stones ; but an accumulation of facts is no more a science than a heap of stones is a house.”
“The true relations between these real objects are the only reality we can attain”
Medical Lens
Not a medical work, but two chapters engage the psychology and physiology of sensation that interested late nineteenth-century psychiatry and psychophysics. Chapter II uses experiments in which weights or lengths that differ slightly cannot be told apart (drawn from the experiments behind Fechner's law, which he says point the opposite way) to describe the contradictory 'physical continuum' of sensation. Chapter IV builds the idea of space from visual sensations, the muscular sensations of eye movement and the tactile and motor sensations of the body, an analysis that belongs to the history of theories of perception.
- Part I, Ch. II, The Physical Continuum
Sensory thresholds in Fechner's experiments: weights of 10 and 11 grams, or 11 and 12, felt as equal though 10 and 12 are distinguished, and what this implies for the idea of magnitude.
- Part II, Ch. IV, Visual Space; Tactile and Motor Space
How retinal images, the muscular sensations of convergence and accommodation, and the body's movements together give rise to representative space.