Science and Hypothesis

Overview

Two hundred and fifty pages, published in 1902, assembled from essays Poincare had written for philosophical reviews over the previous decade and rewritten into a continuous book. It was a bestseller, translated everywhere, and read by people who had never opened a mathematics text.

Poincare was then the most eminent mathematician alive. He had created algebraic topology, transformed the theory of differential equations, found the first example of what would later be called chaos in the three-body problem, and come within a hair of special relativity. The authority behind the book is not that of a philosopher commenting on science from outside.

The structure is in four parts: number and magnitude, space, force, and nature. The first three are the philosophical argument. The fourth is a tour of the physics of 1900, still valuable as a record of what the world looked like immediately before it changed, though the specifics are dated in a way the rest is not.

The prose is unusually good. Poincare writes in short paragraphs, poses each question as a question, and never hides behind formalism. Almost no mathematics is required.

The argument

Part I: number. He starts with the question of what makes mathematical reasoning productive. If mathematics were purely deductive it would be an immense tautology, every theorem contained in the axioms, and it is hard to see why anyone would need to prove anything or how a mathematician could be surprised.

His answer is reasoning by recurrence, mathematical induction: prove it for one, prove that if it holds for n it holds for n plus one, and you have proved it for an infinity of cases in two steps. Poincare argues this cannot come from logic, because logic can only unpack what is already stated, and cannot come from experience, because no finite set of observations reaches an infinity. It must therefore be a synthetic a priori judgement, an intuition of the indefinite repetition of an act once it is possible. This is the one place he agrees with Kant, and he is emphatic about it.

The magnitude chapters discuss the mathematical continuum and argue that it too is a construction, made because the physical continuum is intolerably contradictory: we can find A equal to B, B equal to C, and A distinguishable from C, and mathematics invents a continuum without that defect.

Part II: space, and the conventionalist thesis. The centre of the book.

Poincare begins with non-Euclidean geometry, which by 1900 was seventy years old and still widely regarded as a curiosity. He explains Lobachevsky’s geometry, where many parallels pass through a point, and Riemann’s, where there are none, and stresses that both are perfectly consistent. He gives his own model: imagine a world inside a sphere where temperature falls to absolute zero at the boundary and all bodies contract in proportion, so that the inhabitants can never reach the edge, find their world infinite, and develop Lobachevskian geometry as the natural description. Nothing is wrong with them.

Then the argument. Geometric axioms are not a priori, because their denials are consistent and imaginable. They are not experimental, because experiments are performed on physical bodies, not on the abstract figures geometry is about, and no measurement of a physical triangle tests a mathematical one.

What remains is convention. We choose Euclidean geometry, and Poincare thinks we always will, because it is the simplest and because the group of its motions is the simplest group, not because it is true. One geometry cannot be truer than another, only more convenient.

The strongest version comes when he considers the obvious objection: could astronomers not simply measure the parallax of distant stars and find out? Poincare answers no, because what the measurement tests is the whole package of geometry and optics together. Confronted with a discrepancy we may always keep Euclidean geometry and change our account of how light travels. Any geometry can be held true given suitable adjustments elsewhere, which is why the question is not empirical.

He also argues that the space of geometry is not the space of our senses, which is neither continuous nor infinite nor three-dimensional in the required way, and reconstructs the idea of space from the correlation between our muscular sensations and our perceptions, which is a piece of psychology as much as philosophy.

Part III: force. The same treatment applied to mechanics, and the result is a careful sorting.

The law of inertia is not a priori and not established by experiment, since nobody has ever observed a body free of all forces. The principle of the equality of action and reaction, the conservation of energy, the definition of mass: each is examined and found to be at bottom a definition dressed as a discovery, adopted because it organises experience well.

But Poincare is not a sceptic and takes care to say so. Conventions are not arbitrary. They are chosen under the pressure of experimental facts, and a convention that made the description of the world grotesquely complicated would be abandoned. Freedom in the choice is not freedom from constraint.

Part IV: nature. Optics, electrodynamics, the ether, Lorentz’s theory, and a chapter on the calculus of probabilities. He asks of each hypothesis whether it is doing real work or merely hiding a gap, and his treatment of the ether is striking: he says its existence is a matter of indifference, since only the relations between phenomena are accessible to us, and it will one day be discarded as useless.

He was three years from Einstein’s paper and closer to it than anyone else.

How to read it

Read Part II first if you read nothing else. Chapters 3 to 5 contain the whole argument for conventionalism and they are about forty pages.

Do not skip his model of the disc world. It is the clearest thing ever written about what it would be like to inhabit a non-Euclidean space, and once you have followed it the thesis stops sounding paradoxical.

Hold the distinction between convention and arbitrariness firmly. Poincare is constantly misread as saying that science is a matter of taste. He says the opposite: the choice is free but the pressure of facts is real, and some conventions are enormously more convenient than others.

Note what he got wrong, and why it is interesting. Poincare insisted we would never abandon Euclidean geometry because it is the simplest, and thirteen years later general relativity abandoned it, because keeping it would have required physics so contorted that the whole package was less simple. That is his own criterion used against his prediction, and it is the best test of his framework.

Read the inertia chapter beside Newton’s Principia and the induction chapter beside Kant’s first Critique. Both are direct arguments with those books.

And read Part IV as history. The physics is superseded, but watching one of the two or three best-informed people in the world assess the state of the field in 1900 tells you what a moment before a revolution feels like from inside.

Questions it raises

• If no experiment can decide between geometries, what exactly is being chosen when we choose one?
• Poincare says a convention is free but guided. Where is the line between a definition and a discovery?
• Mathematical induction is his one synthetic a priori. Is it, or can it be recovered from logic?
• Any theory can be saved by adjusting something else. Does that make refutation impossible in principle?
• General relativity abandoned Euclidean geometry on grounds of convenience. Does that vindicate Poincare or refute him?
• He called the ether a matter of indifference because only relations are knowable. How far does that principle extend?