About the Work
Science and Method is a collection of essays and lectures by the French mathematician and philosopher of science Henri Poincaré, assembled in the early twentieth century. The work reflects Poincaré’s broad intellectual range, touching on mathematics, physics, astronomy, and the philosophy of science. The volume is framed by a preface from Bertrand Russell, who praises Poincaré as the most eminent scientist of his generation and characterizes his philosophical writings as “untrammelled reflections” possessing “freshness of actual experience.” Russell notes that Poincaré’s work supplies a growing need for intelligible accounts of science’s philosophic outcome, particularly in an era of specialization and revolutionary progress in physics.
The book’s introduction, written by Poincaré himself, establishes the scientific method as consisting of observation and experiment, while acknowledging the practical necessity of selecting facts due to the scientist’s finite time. Poincaré addresses a range of topics including errors of observation, the relativity of space, definitions in education, and the state of mechanics, which he suggests “seem to be on the point of undergoing a complete revolution.” The work also touches on astronomy’s limitations regarding experimental method and sketches the history of French geodesy. The volume concludes with reflections on the Milky Way and French geodesy, followed by general conclusions about the nature of scientific inquiry.
Context
The book emerges from a period of profound transformation in physics and mathematics. Poincaré writes at a moment when the eighteenth-century conception of “laws of nature” is being displaced by the “working hypothesis,” as Russell notes in his preface. The discovery of radium and cathode rays, the development of electron theory, and the challenges to Newtonian mechanics all figure prominently in the work. Poincaré engages with the latest experimental results of his day, including Kaufmann’s experiments on radium rays, Michelson’s interferometer experiments, and Fizeau’s measurements of light velocity in moving media.
The intellectual context includes the debate over the foundations of mathematics, with Poincaré critiquing the efforts of logisticians such as Russell, Couturat, Hilbert, Peano, Whitehead, and Burali-Forti to reduce mathematics to logic. He also engages with Cantor’s theory of transfinite numbers and the antinomies that arose from it. The work reflects the growing specialization of science, which Poincaré sees as both a threat and an opportunity, noting that unexpected concurrences between different parts of science drive advancement. The book also responds to philosophical questions about the nature of space, time, and chance, engaging with the legacy of Kant and the debate between empiricism and rationalism.
Structure of the Book
The book is organized into four main books. Book I, “The Scientist and Science,” contains chapters on the selection of facts and the future of mathematics, along with discussions of mathematical discovery and chance. Book II, “Mathematical Reasoning,” addresses the relativity of space, mathematical definitions and education, and the relationship between mathematics and logic. Book III, “The New Mechanics,” examines the implications of radium and electron theory for classical dynamics, including discussions of Lorentz’s theory, the principle of relativity, and the nature of mass. Book IV, “Astronomical Science,” contains reflections on the Milky Way and the history of French geodesy.
The work opens with Russell’s preface and Poincaré’s introduction, which together frame the collection’s concerns. The first book establishes the methodological foundations, arguing for the necessity of fact selection and exploring the future directions of mathematical research. The second book delves into the nature of space, the psychology of mathematical discovery, and the contested relationship between mathematics and logic. The third book addresses the revolutionary developments in physics, particularly the challenges posed by radium and electrons to Newtonian mechanics. The fourth book applies these considerations to astronomical questions and concludes with the history of French geodesy, followed by Poincaré’s general conclusions about scientific method.
Detailed Section-by-Section Summary
Preface and Introduction
Bertrand Russell’s preface praises Poincaré as the most eminent scientist of his generation, whose philosophical writings possess a “freshness of actual experience” rather than the dryness of professional philosophy. Russell argues that the revolutionary progress in physics has displaced the eighteenth-century conception of “laws of nature” with the “working hypothesis,” making a philosophy of science increasingly necessary. He quotes Poincaré’s dictum that doubting everything or believing everything are equally convenient solutions, both dispensing with reflection. While Russell criticizes Poincaré’s objections to mathematical logic as based on misconception, he praises his books for supplying intelligible accounts of science’s philosophic outcome.
Poincaré’s introduction establishes that scientific method consists of observation and experiment, but since the scientist lacks infinite time, he must select facts—a question confronting physicist, historian, and mathematician alike. He addresses errors of observation that partially compensate through chance, the relativity of space, and the “deceptive illusion” that mathematics reduces to formal logic. He notes mechanics seems on the point of complete revolution, and sketches French geodesy’s history to show “what time and trouble are involved in the conquest of a single new decimal.”
The Selection of Facts
Poincaré cites Tolstoi’s view that “Science for Science’s sake” is absurd since facts are infinite, arguing that selection must be guided by utility. He rejects both a “greedy and narrow plutocracy” and a “virtuous unaspiring democracy,” maintaining that scientists believe in a “hierarchy of facts.” Citing Mach, he notes that disinterested “fools who died poor” preceded practical triumphs and “saved their successors the trouble of thinking.” The most interesting facts are those with a chance of recurring—simple facts, since chance can mingle but cannot unmingle. He observes that sociology has “the greatest number of methods and the least results,” and that once a rule is established, exceptions become important. The scientist studies nature because of its “intellectual beauty” arising from “the harmonious order of its parts,” and economy of thought is both practical and beautiful, like “the graceful caryatids of the Erechtheum.” He suggests natural selection may explain why the Greeks, loving intellectual beauty, triumphed over barbarians loving “garish colours and the blatant noise of the drum.”
The Future of Mathematics
Poincaré proposes studying mathematics’ history to foresee its future, comparing this to extrapolation. Past “prophets of ill” claimed all solvable problems were solved, yet the meaning of “solution” has repeatedly been extended—from rule and compass for Greeks, to extraction of radicals, to algebraic functions. Mathematics will develop in all directions, but this could become alarming as riches become embarrassing. The mathematician creates his own facts through combinations; an isolated combination is valueless unless it takes place in a class of analogous combinations, becoming a law. “The invention of a new word will often be sufficient to bring out the relation, and the word will be creative.” He cites Mach’s multiplication table example: recording that 6 times 7 are 42 saves millions from repeating the operation.
Value, Elegance, and Exactness in Mathematics
A fact’s importance is measured by the “return” it gives—the amount of thought it enables us to economize. In physics, facts embedded in general laws yield large returns by enabling prediction of many others; the same holds in mathematics. A difficult calculation gains nothing unless it helps foresee analogous results, but if the “groping” reveals a profound analogy between the problem and a broader class, one acquires not merely a result but “a new force.” A new result must unite previously scattered, seemingly foreign elements, introducing order where disorder reigned. Mathematicians value elegance not from dilettantism but because harmony, symmetry, and unity enable clear comprehension of whole and parts, revealing analogies and generalizations. Elegance may arise from surprise at unlooked-for associations or from contrast between simple means and complex problems. Since the mid-nineteenth century, mathematicians increasingly seek absolute exactness, though taken too literally this implies no mathematics existed before 1820—an exaggeration. However, if future demonstrations follow the exactness model, works become exceedingly long, losing harmony.
Terms, Language, and the Two Directions of Mathematics
A well-chosen term like “uniformity of convergence” made whole classes of long reasonings unnecessary. Mathematics is “the art of giving the same name to different things”—things similar in form can be “run in the same mould.” Well-chosen terms remove exceptions, which are pernicious because they conceal laws; the term “energy” in physics created a law by eliminating exceptions, while “group” and “invariable” revealed the essence of many reasonings. A new transformation draws ten or twenty theorems from one. Mathematics borders philosophy and physics, with mathematicians advancing in opposite directions: one side reflects on itself, studying postulates, unusual geometries, and strange functions, but the main forces must direct toward nature.
Qualitative Solutions and Specialization
Engineers need not integrals in finite terms but the general behavior of functions or specific figures. Formerly equations were solved only by finite known functions, impossible in about ninety-nine cases out of a hundred. One should solve qualitatively—knowing the curve’s general form—then use infinite converging series. Today such solutions fail when convergence is too slow or terms lack law. Problems are now “more or less solved” by series of varying convergence and harmony; imperfect solutions may lead to better ones, and slow convergence may only demonstrate possibility, which engineers find absurd but mathematicians value for future generations. As science grows, specialization threatens progress; unexpected concurrences between parts drive advancement, and congresses at Heidelberg and Rome remedy this by putting mathematicians in touch.
Arithmetic, Algebra, and Geometry
Arithmetic progresses slower than algebra and analysis because continuity fails—every whole number is individual, an exception. Arithmetic should model itself on algebra; transcendental numbers parallel transcendental functions, and the theory of congruents parallels algebraic equations. Completing this parallelism will advance indeterminate analysis. The theory of algebraic equations continues attracting attention; algebra needs “interesting combinations,” and a new indeterminate analysis will treat polynomials as unknowns. Geometry adds value through well-formed language and grouping, setting new problems analysis wouldn’t pose. The senses assist intellect but fail beyond three dimensions; geometry of more than three dimensions is qualitative, not merely quantitative. Geometry of Position studies relations of position eliminating magnitudes, remaining true even in crude imitation. Riemann’s work demonstrates its importance; constructing it in higher spaces will enable seeing into hyperspace.
Cantorism and Postulates
Cantor introduced a new method of considering mathematical infinity, defining per genus supremum rather than by construction—horrifying minds like Hermitte’s. Paradoxes reminiscent of Zeno of Elea and the Megara school appeared, and remedies were sought. Poincaré holds that only entities definable in finite words should be introduced. Hilbert enumerated axioms and postulates; classification work remains for philosophers.
Mathematical Discovery and the Unconscious
The genesis of discovery interests psychology since the mind borrows least from the exterior world. An enquiry by L’Enseignement Mathématique confirmed Poincaré’s conclusions, though without unanimity. If mathematics invokes only logic accepted by all well-formed minds, why are many impervious? Error arises because between a proposition’s appearance as conclusion and later use as premise, time passes; one forgets its meaning and substitutes a different proposition. Memory failure causes mechanical misapplication. Mathematical aptitude isn’t mere memory or attention—Gauss was exceptional in being both genius and calculator; Poincaré himself cannot add without mistakes and would be a bad chess player. Memory is guided by the “general trend of the argument”; a demonstration is syllogisms in order, and order matters more than elements.
Among the combinations chosen by the discoverer, the most fruitful are often those formed from elements borrowed from widely separated domains, though most incongruous combinations are fruitless. Discovery is selection, but unfruitful combinations do not present themselves to the discoverer’s mind; everything happens as if the discoverer were a secondary examiner interrogating candidates already declared eligible after a preliminary test. Poincaré recounts his personal recollections of writing his first treatise on Fuchsian functions. For a fortnight he attempted to prove there could not be any function analogous to what he later called Fuchsian functions, working an hour or two daily without result. One night, after taking black coffee contrary to his custom, he could not sleep; a host of ideas surged in his head until two coalesced to form a stable combination. By morning he had established the existence of one class of Fuchsian functions derived from the hypergeometric series, needing only a few hours to verify.
He then deliberately sought to represent these functions by the quotient of two series, guided by analogy with elliptical functions, forming what he called Theta-Fuchsian series. Leaving Caen for a geological conference, at Coutances, as he put his foot on the step of a break, the idea came to him that the transformations used to define Fuchsian functions were identical with those of non-Euclidian geometry. He felt absolute certainty at once, verifying later at Caen. Studying arithmetical questions without apparent result, he went to the seaside. Walking on the cliff, the idea came with conciseness, suddenness, and immediate certainty that arithmetical transformations of indefinite ternary quadratic forms are identical with those of non-Euclidian geometry. Returning to Caen, he deduced consequences: there are Fuchsian groups other than those corresponding with the hypergeometric series, and consequently other Fuchsian functions. He laid siege systematically, but one difficulty held out. Leaving for Mont-Valerien for army service, while crossing the street, the solution came all at once. After service, he composed his definitive treatise at a sitting.
These sudden illuminations indicate previous unconscious work. Conscious work, interrupted and followed by rest, proves fruitful because rest was occupied with unconscious work; the revelation may come during conscious work, which performs only the unlocking process. Unconscious work is not fruitful unless preceded and followed by conscious work. The second period is necessary to verify results; the feeling of absolute certainty can deceive, notably with ideas coming in bed in semi-somnolent condition. The subliminal ego plays an important part in mathematical discovery, but it is not purely automatic; it has discernment, tact, lightness of touch, can select and divine, perhaps better than the conscious ego. Alternatively, all combinations are formed automatically, but only interesting ones cross into consciousness. The privileged unconscious phenomena are those which most deeply affect our sensibility. Mathematical beauty, harmony of numbers and forms, and geometric elegance constitute a real aesthetic feeling. The useful combinations are precisely the most beautiful, charming the special sensibility all mathematicians know. The subliminal ego blindly forms many combinations; almost all are without interest, without action on aesthetic sensibility, so consciousness never knows them. A few are harmonious, useful, and beautiful, affecting the geometrician’s sensibility, which directs attention upon them. This aesthetic sensibility plays the part of the delicate sieve; the man without it will never be a real discoverer.
The conscious ego is strictly limited; the subliminal ego’s limitations are unknown, but it cannot form all possible combinations. The explanation may lie in the period of preliminary conscious work. Poincaré offers a crude comparison: future elements of combinations resemble Epicurus’s hooked atoms. When the mind is in complete repose, atoms are immovable, attached to the wall. During unconscious work, some are detached and set in motion, ploughing through space like a swarm of gnats or gaseous molecules in kinetic theory. Their collisions produce new combinations. Preliminary conscious work liberates some atoms, detaching them from the wall; after agitation, they continue to circulate freely. The will did not select them at random but in pursuit of a definite aim; the liberated atoms are those from which we may reasonably expect the desired solution. Only combinations including at least one deliberately selected atom have any chance of being formed. Unconscious work never supplies ready-made the result of a lengthy calculation; all we can hope from inspirations is points of departure. Calculations must be made in the second period of conscious work, demanding discipline, attention, will, and consciousness. In the subliminal ego reigns liberty, the absence of discipline and disorder born of chance, which permits unexpected couplings. Poincaré spoke of a night of excitement when he worked as though in spite of himself; in such cases we assist at our own unconscious work, which becomes partly perceptible to the over-excited consciousness.
The Nature of Chance
Bertrand asks at the beginning of his “Calculus of Probabilities”: “How can we venture to speak of the laws of chance? Is not chance the antithesis of all law?” Probability is the opposite of certainty, what we are ignorant of. The ancients distinguished phenomena obeying harmonious laws from those attributed to chance, which could not be predicted because not subject to any law; chance had a precise, objective meaning. But we have become complete determinists; every phenomenon has a cause, and an infinitely powerful mind could have foreseen it. For such a being, chance would have no meaning. Chance is only the measure of our ignorance; fortuitous phenomena are those whose laws we are ignorant of.
The Relativity of Space
A deformed mirror image preserves relations; deformation is only perceived because the real world and our undeformed body exist alongside. If our body were deformed identically, the deformation would be unascertainable. Two universes where each object in one has an image in the other are indistinguishable to their respective inhabitants; if a window opened between them, each would call the other’s geometry a grotesque image, and no one will ever know which is right. Space is “amorphous”; things in it give it form. We have no direct intuition of distance—a distance could become a thousand times greater overnight without our perceiving it. One part of space is not absolutely equal to another; it is relative to inhabitants. We lack direct intuition of magnitude; we only relate magnitude to measuring instruments. Our body is the instrument, serving as a system of axes of co-ordinates. Judging that two objects occupy the same place means they hold the same relative position to our body—not absolute space. “Same position” means the same extension of the arm reaches both. This is a system of “parries”: the same parry answers several blows; the same blow may be parried several ways. Objects occupying the same point have nothing in common except that the same parry defends against them.
The telegraph-wire model illustrates this: centripetal wires (warnings) and centrifugal wires (remedies) connected via a central exchange; several centripetal wires can act on one centrifugal wire, and vice versa. This “distribution board” is our whole geometry; intuition of straight line and distance is consciousness of these associations and their “imperious character.” These associations are old, racial conquests produced by natural selection; without them organism defence was impossible. A frog with its head cut off rubs acid off its skin with the nearest foot; if that foot is cut, the other foot removes it—a “double parry.” This multiplicity and co-ordination is space. The “evidence of the truths of geometry” is merely repugnance at breaking very old, successful habits.
Restricted space (within arm’s reach) requires memory to extend. A sea-polype fixed to the ground would have no space beyond its tentacles. We perform “long-distance parries”—a complexus of successive sensations requiring memory. A point is defined by the succession of movements to reach it from an initial body position; axes attach to that initial position, which is arbitrarily chosen. Memory can go back more or less, yielding indeterminateness—this is relativity. Absolute space no longer exists; only space relative to an initial body position. A being fixed to the ground would have relative space but not be conscious of it, since his axes never change; we, who can choose among several systems, are conscious of relativity. Restricted space is not homogeneous (some points cost great effort). Extended space appears homogeneous: movements from different positions accompanied by the same muscular sensations define equivalent points. Homogeneity and relativity are the same thing. Great space (lodging the universe) is reached by an act of imagination: imagining a giant reaching planets in steps, or a Lilliputian self on a miniature world—impossible without prior restricted and extended space.
Why Three Dimensions?
Referring to the distribution board with dangers and remedies connected, Poincaré notes this is not a description of the nervous system but psychological association between series of sensations. The fundamental law states: if two dangers are both associated with one remedy, and one of them with a second remedy, then the other danger is generally associated with that second remedy. If this were not true, contradiction would arise in defining a point. If rigorously true, space would be discontinuous, with discrete, separated categories, no order, no three dimensions. In reality, a remedy answers two dangers if the point corresponding to it is sufficiently close to both; another remedy may be near one but not the other. This yields sequences where each term is associated with neighbours but not distant terms. Categories overlap, making space continuous. Order is no longer arbitrary, and experience shows this order as a “three-circuit distribution board”—hence three dimensions. Three dimensions is a property of our distribution board, residing in human intelligence; destroying some associations could give a fourth dimension. The exterior world counts: the connexion between a danger and a remedy exists because the remedy effectively defends against the danger—a fact exterior to us. The board adapted to a world with natural solids moving by laws of unvarying solids. Three-dimensional language is easiest for our world, and beings with a four-dimensional board might not be able to live and defend themselves in our world.
Two Kinds of Mathematical Minds
Poincaré observes that even children incapable of becoming mathematicians must still be taught mathematics. He notes a fundamental division among mathematicians: logicians like Weierstrass and intuitionists like Riemann. Students show the same division, with some preferring “analysis” and others “geometry.” Poincaré argues that changing this natural disposition is useless and undesirable, as both types are needed for the advancement of science, and one cannot declare whether Weierstrass or Riemann should be preferred.
Definitions and Understanding
“Understanding” has multiple meanings, Poincaré explains. Some seek images, while others combine empty, purely intelligible forms. Fractions exemplify this: primary schools define them by cutting up apples or pies, while higher institutions define a fraction as two whole numbers separated by a horizontal line with conventions for operations. This suits students already familiar with fractions through apple-cutting. Poincaré admires Hilbert’s “Grundlagen der Geometrie,” which begins with “three systems of THINGS”—points, straight lines, planes—whose nature we need not know, but he wouldn’t recommend it to schoolboys.
Fourth-Grade Circle Lesson
A teacher dictates: “A circle is the position of the points in a plane which are the same distance from an interior point called the centre.” Neither good nor bad pupils understand. When the teacher draws a circle, pupils think “a circle is a round” would have sufficed. The teacher is right—the pupils’ definition has no demonstrative value—but pupils must be made to see they don’t understand what they think they understand.
Historical Evolution
Books from fifty years ago lack exactness. They assumed continuous functions cannot change sign without passing through zero; now we prove it. They assumed calculus rules apply to incommensurable numbers; now we prove it. Intuition cannot give exactness or certainty—it wrongly teaches that every continuous function has a derivative. Exactness required introducing exactness into definitions. The vague idea of continuity resolved into a system of inequalities on whole numbers. Mathematics has been “arithmetized.”
Loss of Objectivity
Mathematics gained exactness but lost objectivity by withdrawing from reality. Obstacles moved to the frontier. We reject empirical elements, preserve a priori ones; one property serves as definition, but we must prove this property belongs to real objects. Logic breeds monsters—weird functions with no continuity or continuity without derivatives. These strange functions are logically most general. Formerly functions served practical ends; now they show ancestors’ reasonings at fault.
Intuition’s Necessity
Pure logic cannot give the view of the whole; intuition must. The continuous function begins as a chalk line, is purified into inequalities, the “centering” removed like an arch’s support—but the instructor must recall the original image or the pupil won’t understand why inequalities were scaffolded.
Educational Principles
Educators must make children pass through all stages their ancestors passed. Pupils imagine they know fractions, continuity, curved areas; if told they don’t understand, they’ll think mathematics is arbitrary subtleties or become like Greek sophists. Later, ripened minds will welcome demonstrations. The principal aim is developing faculties, especially intuition—through it the mathematical world touches the real world. Engineers need to see aspects quickly, not split hairs. Future teachers need first principles but also intuition. Pure geometricians need intuition for discovery: “by logic that we prove, but by intuition that we discover.” Logic doesn’t tell which road leads to the desired end.
Written Exercises
At the École Polytechnique, insisting on written compositions would exclude pupils who know their subject but can’t apply it; Poincaré prefers those who understand thoroughly.
Definitions and Axioms
We cannot prove or define everything; intuition must be drawn upon. Every definition implies an axiom asserting existence of the defined object. Definitions are constructions of simpler notions; we must explain why elements were assembled thus, what need it fills, what familiar object is its rough image. The choice of name must be explained through analogies. Justification should precede and prepare the statement, led up to by particular examples. The definition must distinguish the object from neighbouring objects.
Arithmetic
Whole numbers need no definition. Addition cannot be logically defined—we must stop somewhere; we start with concrete examples. Subtraction is the inverse of addition, but begin with examples showing the relation. Multiplication: show a problem solved by adding equal numbers, then note multiplication is quicker. Division: inverse of multiplication, beginning with sharing examples. Fractions: expound proportion theory first, with rule-of-three problems and fractional data; use geometrical figures. After defining fraction multiplication, justify by demonstrating commutativity, associativity, distributivity. Arithmetic discovered fractions through geometry’s requirements.
Geometry
The straight line—the common definition “shortest path” doesn’t satisfy Poincaré. Start with the ruler; verification by revolving it is the true definition (a straight line is an axis of rotation); verification by sliding gives another property. Shortest path is a theorem too advanced for secondary education; show a verified ruler applied to a taut thread. Multiply axioms when needed. Uncle Sarcey said theatre audiences accept initial postulates but are inexorable on logic once the curtain rises—same in mathematics. Circle: start with compass; pupils recognize the curve; note constant distance between points, one fixed, one movable—leading naturally to logical definition. Plane: implies an axiom; use a drawing-board with movable ruler applied constantly (three degrees of freedom); compare with cylinder and cone (two degrees); three drawing-boards sliding while in contact; two boards applicable to a third are applicable to each other.
Can Mathematics Be Reduced to Logic?
Poincaré poses the opening question: can mathematics be reduced to logic without appealing to principles peculiar to itself? A school of ardent logicians claims yes, using a special sign-only language understood only by initiates.
Cantor’s Work
Poincaré reviews Cantor’s introduction of actual infinity—a quantity regarded as having already passed all limits, unlike the philosophical “becoming” infinity previously used. Cantor asked whether there are more points in space than whole numbers. He termed these quantities transfinite cardinal numbers and also imagined transfinite ordinal numbers.
The Logicians’ Program
Some mathematicians now make the theory of finite numbers depend on Cantor’s transfinite cardinals, hoping to prove all arithmetic and algebra without principles foreign to logic. Poincaré calls this contrary to healthy psychology and doubts its accuracy.
Cantorian Antinomies
These logicians have arrived at contradictory results—the Cantorian antinomies—but persist, modifying rules without assurance new contradictions won’t appear. Poincaré compares them to the Lernaean hydra, too numerous to defeat.
Couturat’s Claims
M. Couturat’s “Les Principes des Mathématiques” analyzes recent works by Mr. Russell and Signor Peano. Couturat claims they settled the Leibnitz-Kant dispute, showing no a priori synthetic judgment exists and mathematics is entirely reducible to logic.
Hilbert’s Formalism
Poincaré quotes Hilbert’s purely formal geometry—points, lines, planes as unknown things. Poincaré doesn’t blame Hilbert, who needed mechanical form to enumerate axioms completely.
Two Conditions for Disguised Definitions
First, one must demonstrate postulates involve no contradiction; otherwise they remain axioms. Second, when a word is defined and later used, one must verify the implicit meaning matches the explicit definition.
Burali-Forti’s Treatise
Signor Burali-Forti’s “Una Questione sui Numeri transfiniti” presented the first antinomy of transfinite numbers. Poincaré notes Burali-Forti defines 1 with a formula containing both “1” and “One”—a suspected petitio principii.
Russell’s Logic
Mr. Bertrand Russell subordinates class logic to proposition logic, studies conjunctions if, and, or, not, and concludes a false proposition involves all others. Russell introduces undefinable words and undemonstrable principles—which Poincaré calls appeals to intuition.
Peano’s Axioms
Couturat presents Peano’s five axioms as disguised definitions, but Poincaré argues the demonstration of non-contradiction requires the very principle of complete induction being justified—a circularity.
The Circle of Logic and Arithmetic
Common logical principles already imply arithmetical notions like “whole” and “number,” creating a circle; thus, logic and arithmetic must be developed simultaneously to avoid paradox. Poincaré states that this criticism of Hilbert applies equally to Russell’s logic, which treats logic as anterior to arithmetic, whereas Hilbert treats them as simultaneous.
Hilbert’s Symbols
Hilbert introduces the object “1” as a mere symbol, but then uses “two, three, or several times repeated,” which introduces the notion of number, a petitio principii Hilbert later tries to patch up. Hilbert introduces two objects, “1” and “=”, forms all combinations, and divides them into classes of entities and nonentities, with affirmative propositions assigning to entities and negative ones to nonentities.
Russell and Hilbert Compared
A key difference: for Russell, a variable x is absolutely indeterminate; for Hilbert, it represents only combinations of already-defined objects. Hilbert states that indeterminates in axioms represent only objects already acquired, and axioms must be re-proven when new objects are added. Russell’s view is one of comprehension (starting with the general idea of entity); Hilbert’s is one of extension (only combinations of known objects).
Hilbert’s Axioms
Hilbert introduces two axioms (every quantity equals itself; operations on identical quantities give identical results), treating them as postulates defining the symbol “=”, but must show they don’t lead to contradiction. Poincaré notes Hilbert uses the reasoning of Section III, apparently without recognizing he is making complete induction. Poincaré finds the end of Hilbert’s treatise enigmatical and contradictory, saying Hilbert breaks down in demonstrating that the definition of whole number by the axiom of complete induction involves no contradiction, just as Russell and M. Couturat broke down.
Geometry and Induction
On geometry, M. Couturat says complete induction doesn’t intrude, but Poincaré notes it intrudes at page 114 of Mr. Halsted’s “Rational Geometry.” Geometry, once intuition’s domain, is now where logisticians triumph, but the fundamental theorem—that axioms don’t involve contradiction—cannot be demonstrated without induction. Hilbert demonstrates this via analysis, arithmetic, and induction; any other demonstration would still rely on this principle since the possible consequences are infinite.
The Principle of Induction
The conclusion: the principle of induction cannot be the disguised definition of the whole number. Poincaré compares three “truths”: the principle of complete induction, Euclid’s postulate, and the physical law that phosphorus melts at 44° centigrade. He admits the second is a disguised definition (of the straight line) but not the first or third. A definition is acceptable only if shown non-contradictory; this is impossible for the first, demonstrated by Hilbert for the second, and clear for the third, but the third concerns physical existence, not absence of contradiction. Also, in applications, the principle of induction presents itself with two non-identical but equivalent definitions of finite whole number: one obtainable by successive additions, and another as that about which we can reason by recurrence. These are equivalent only by an a priori synthetic judgment, not by purely logical processes. For the straight line, there is only one expressible definition; non-Euclidian straight lines are circles orthogonal to a sphere, and calling Euclidian lines “straight” is by definition. For phosphorus, the true definition would be “this piece of matter that I see before me in this bottle.”
The Phosphorus Example Clarified
Poincaré clarifies his phosphorus example: he meant “all bodies which possess such and such properties in finite number (namely, the properties of phosphorus given in chemistry books, with the exception of its melting-point) melt at 44° centigrade.” He contrasts the straight line and phosphorus: if light doesn’t satisfy Euclid’s postulate, we could conclude light is not rectilineal, but it would be foolish to adopt the former definition (that straight line is the trajectory of light) because light deviates from other properties of the straight line and is subject to change. If phosphorus melts at 43.9°, it would be foolish to conclude it’s not true phosphorus, since we can’t change a substance’s name with each new decimal.
The Verdict on Russell and Hilbert
Summing up, Russell and Hilbert have made great efforts with original, profound, often true views, but they have not settled the controversy between Kant and Leibnitz nor destroyed the Kantian theory of mathematics.
The Last Efforts of the Logisticians
Logisticians have transformed logistic; Russell modified his views. Poincaré questions whether logistic rules are fruitful and infallible, and whether they can demonstrate the principle of complete induction without intuition. M. Couturat has “most childish illusions” about fruitfulness, claiming logistic lends “stilts and wings” to discovery, but Peano, despite his fine work (e.g., his curve filling a whole area), hasn’t flown in ten years. Poincaré finds logistic gives “shackles,” not conciseness—it takes 27 equations to establish that 1 is a number. Logistic forces step-by-step advance; it’s surer but not more expeditious—”leading-strings,” not wings. Rules must be followed blindly, so they must be infallible; logisticians can’t say “we make mistakes” for them mistakes are death. The logisticians applied their rules and fell into contradiction, so they’re altering the rules and “sacrificing the notion of class.” Russell attempts to reconcile contradictions by “restricting or even sacrificing the notion of class.”
The Demonstrations of the Principle of Induction
Poincaré examines demonstrations of the induction principle by Whitehead and Burali-Forti. He defines a “recurrent class” as every class of numbers including zero, and including n+1 if it includes n. An “inductive number” is every number forming part of all recurrent classes. For this definition to be “predicative,” “all recurrent classes” must mean only those whose definitions don’t contain the notion of inductive number—otherwise a vicious circle results. Whitehead failed to take this precaution, making his argument vicious and illegitimate, even if it happens to reach a true conclusion. A definition containing a vicious circle defines nothing; a “non-predicative class” isn’t empty but has uncertain boundaries.
Burali-Forti’s Demonstration
Burali-Forti’s demonstration requires two postulates: first, that at least one infinite class always exists; second, a formula stating that the number of combinations formable with several objects is smaller than the number of those objects. The first postulate is no more evident than the principle to be demonstrated; the second is untrue, as Whitehead showed.
Zermelo’s Axiom
Zermelo’s axiom holds that from any aggregate (or each aggregate in an aggregate of aggregates), one can select an element at random, even with infinitely many aggregates. Borel rejected it; others accepted it. Russell pronounces no opinion. Poincaré offers a picturesque example: with as many pairs of boots as whole numbers, the number of boots equals the number of pairs only if right and left boots are distinguishable—otherwise Zermelo’s axiom is needed to select randomly.
Conclusions on Logistic
A demonstration based on Analytical Logic reduces to tautology when definitions are substituted; logic remains barren unless fertilized by intuition. Logisticians’ definitions are non-predicative, containing hidden vicious circles, so their arguments cannot melt into identities—logistic engenders antinomies. Belief in actual infinity births these definitions; the word “all” requires actual infinity for precise meaning. Formal logic requires immutable classifications; indefinite objects may force classification changes, exposing antinomies. There is no actual infinity; Cantorians forgot this. Russellian logistic requires actual infinity, unlike Hilbertian logistic. Russell is preparing to change everything—the old Logistic is dead, with zigzag and no-classes theories disputing succession.
Mechanics and Radium
Poincaré questions whether Dynamics’ principles face abandonment after radium’s discovery. He recalls principles: inertia (A); acceleration proportional to force divided by mass, with mass constant (B); forces arising from other material points, depending on relative positions and velocities (C); equality of action and reaction (D). These principles hold only for low velocities. Cathode rays and radium rays achieve velocities a thousand times greater than Mercury’s. Radium emits α, β, γ rays; Poincaré discusses β rays, analogous to cathode rays. Crookes proposed molecular bombardment (emission theory); Hertz proposed ether undulations. Facts favored Crookes: cathode rays carry negative charge, are deviated by magnetic and electric fields. Measurements yield velocities of 6,000–20,000 miles per second; the charge-to-mass ratio is about a thousand times that of hydrogen ions. Wiechert’s experiments confirmed the theory’s magnitude. β rays of radium reach 60,000–120,000 miles per second.
Longitudinal and Transversal Mass
Self-induction acts as inertia opposing current variation. Cathode rays, as convection currents, produce induction effects (Rowland proved convection currents produce magnetic effects; Cremien and Pender demonstrated induction effects). Corpuscles possess double inertia: actual plus apparent electro-magnetic inertia. Abraham’s theoretical work shows fictitious mass varies with velocity; total longitudinal and transversal masses differ and depend on velocity. Kaufmann’s experiments on radium rays determined the relation between velocity and charge-to-total-mass ratio. Assuming charge and actual mass are constant across corpuscles, Kaufmann compared Abraham’s law with experiment: the actual mass is nil. Mass appears to be purely electro-magnetic, increasing with velocity, becoming infinite at light’s velocity; transversal mass no longer equals longitudinal mass; principle B fails.
Canal-Rays
Goldstein’s canal-rays (Kanalstrahlen) carry positive electricity, emitted behind a pierced cathode. Radium emits similar, absorbable α rays. Measurements show lower velocity and lower charge-to-mass ratio; positive corpuscles are larger if charges are equal and opposite. These corpuscles are named electrons (though the name now applies only to negative corpuscles).
Lorentz’s Theory
Matter consists entirely of electrons with enormous charges, appearing neutral because opposite charges balance—a solar system model with a large positive electron and negative planetary electrons. Electrons are immersed in ether, identical everywhere. Luminous waves perturb electrons, which react on the ether, explaining refraction, dispersion, double refraction, absorption, and light emission. In metals, movable electrons circulate freely like gas molecules, producing currents and incandescence when deflected at surfaces. In dielectrics, electrons oscillate about fixed positions, explaining non-conductivity, transparency, and refraction. The theory accounts for known facts and predicted Zeeman’s phenomenon.
Mechanical Consequences
Two hypotheses: (1) positive electrons have actual mass, negative electrons lack it, mechanics unaffected; (2) neutral atoms don’t exist, all electrons lack actual mass—mass becomes purely fictitious, no longer constant, mechanics upset. The difference in mass might be explained by positive electrons being much smaller “holes in the ether.” Kaufmann’s method can’t decide using canal rays.
The Ether Question and the Principle of Relativity
The search for Earth’s motion through the ether took an alternative form: comparing the apparent positions of a star through a telescope filled with air versus one filled with water, exploiting the different velocities of light in each medium. The results were negative, showing that the laws of reflection and refraction appear unaltered by Earth’s motion. Two explanations were offered. The first, Hertz’s theory, proposed that the ether is entirely displaced by moving bodies, so that all instruments and the ether move together. The second, Fresnel’s view, held that the ether is absolutely at rest, with refringent mediums only partially carrying it along. Lorentz refined this, positing a stationary ether with electrons alone moving. Fizeau’s experiment, comparing light velocity in moving water and air, confirmed Fresnel’s partial displacement, and Michelson’s repetition of the experiment yielded the same result, leading to the rejection of Hertz’s theory.
Experience thus showed that only relative velocities of material bodies can be disclosed, a principle of relativity. Lorentz’s concept of “local time” explained how two observers moving together, setting their watches by optical signals, would each find a systematic error—a time proper to their place—that compensates perfectly when the square of the aberration is neglected. However, Michelson’s experiment, which involved rays traversing different distances, could no longer neglect this square, yet still returned a negative result. To explain this, Lorentz and Fitz-Gerald hypothesized that bodies in motion contract in their direction of travel by about one part in 200 million, while their perpendicular dimensions remain unchanged. This contraction is undetectable by measuring instruments, as the yard-measures themselves contract equally. A sphere at rest becomes a flattened ellipsoid in motion, but observers, being deformed themselves, believe it is still a sphere. The law of contraction is precisely chosen so that the wave surfaces of light appear as elongated ellipsoids, with the source’s actual position as the focus, making the compensation exact.
The Principle of Reaction and the New Dynamics
Under Lorentz’s theory, the principle of reaction—that action and reaction are equal and opposite—fails for electrons alone, since reaction cannot be simultaneous. A Hertz excitator at a parabolic mirror’s focus would recoil like a cannon firing, but it would be firing energy, not matter. This idea is supported by Maxwell-Bartholi pressures, which explain comet tails as small particles repelled by sunlight. The radiometer, which initially turned the wrong way, only succeeded with a better vacuum and non-blackened plates. Hertz’s theory would have given perfect compensation, but Lorentz’s gives imperfect compensation, nil in space. Since Fizeau’s experiment rejects Hertz’s theory, Lorentz’s must be adopted, and the principle of reaction must be abandoned.
This new relativity has profound consequences. The Lorentz-Fitz-Gerald contraction must extend to electrons themselves, which are spherical at rest but become flattened ellipsoids in motion. Abraham, however, considered electrons as spherical and undeformable. For perfect compensation, two conditions are required: first, that positive electrons have no real mass, only fictitious electromagnetic mass, and second, that all forces are electromagnetic in origin or vary with velocity accordingly. This implies that “there is no more matter,” and gravitation itself must be explained electromagnetically or modified. The argument can be presented without assuming deformation, treating electrons as material points whose mass varies as if they were deformed. Kaufmann’s experiment tested the law of mass variation to decide between Abraham’s and Lorentz’s theories. His first attempts were insufficiently accurate, but repeated with more precautions, they seemed to confirm Abraham. A footnote reports that Bucherer, repeating the experiment with new precautions, obtained results confirming Lorentz’s views. Poincaré, however, questions the measurement of the electrostatic field, wondering if it is truly uniform between the condenser armatures.
The Principle of Inertia and the Wave of Acceleration
The principle of inertia remains true in the new Dynamics: an isolated electron moves rectilinearly and uniformly, though Lindemann raised objections. An electron in the ether behaves like a body in an ideal fluid, with a wake accompanying it and energy increasing its inertia. In Lorentz’s hypothesis, vis viva is not proportional to the square of velocity, and mass, momentum, and vis viva become infinite at the velocity of light, so no body can exceed it. The apparent paradox of adding velocities—120,000 plus 120,000 miles per second equaling 240,000—vanishes when using Lorentz’s local time method. Langevin named two disturbances: the wave of velocity, the wake from uniform motion, and the wave of acceleration, light-like waves from acceleration. Rectilineal uniform motion conserves energy, but acceleration dissipates energy as light waves. These effects are negligible in ordinary mechanics, celestial motions, and radium rays, which are quasi-stationary. Cases with great acceleration include oscillating electrons in incandescent gases, gas absorbing light, Hertz’s excitator radiating waves, electrons deflected at incandescent metal surfaces, and cathode rays striking the anticathode to produce Röntgen rays.
Gravitation
Mass has two definitions: as a coefficient of inertia and as a coefficient of attraction. Whether attraction increases with velocity like inertia does is undecidable. In Lorentz’s hypothesis, electrons of similar sign repel and opposite signs attract. Material molecules are like solar systems with zero net charge, so total electric action should be nil—yet molecules attract gravitationally. Two hypotheses arise: either gravitation is a separate force, or attraction between positive and negative charges exceeds repulsion between like charges, as in Franklin’s hypothesis. The question remains unresolved in motion.
The Milky Way
The flattened form of the Milky Way invites three hypotheses. First, a provisional equilibrium from collisions: stars acquire perpendicular velocities and escape the plane, making the system tend toward a spherical form, the only equilibrium figure for an isolated gaseous mass. Second, common rotation could cause flattening, as with Earth and Jupiter. The Milky Way’s density is 10^11 times lower than the Sun’s, so a revolution velocity 10^11 times slower would be equivalent for flattening. A velocity one-thirtieth of a second of arc per century would be rapid, almost too rapid for stable equilibrium. Individual motions would appear uniformly distributed, teaching nothing about rotation since we rotate with the system. Spiral nebulae, if foreign Milky Ways, could reveal rotation, though they are at immense distances. Fixed stars disclose Earth’s diurnal rotation despite their distance, but the Milky Way’s rotation is absolutely slow, requiring thousands of years of observation.
Third, Stratonoff’s recent work suggests the Milky Way is itself a spiral nebula. Irregular nebulae, like Orion’s, have discontinuous spectra and are not composed of stars, their distribution depending on the Milky Way. Spiral nebulae, however, are independent and composed of stars—other Milky Ways. They show rotation: all spiral radii curve in the same direction, the advancing wing hanging back on the pivot. They resemble gas in permanent motion with internal currents. If the central nucleus rotates too rapidly, centrifugal force prevails at the equator, and stars escape forming divergent currents, their angular velocity diminishing as radius increases. Alternatively, stars lose velocity, stop, and attraction brings them back, creating centripetal currents in the first rank and centrifugal in the second. A permanent status establishes as the advancing wing’s attraction retards the pivot and vice versa. Swarms concentrate into radii because existing swarms attract emerging stars, accentuating inequalities. With rotation, four curved radii intersecting at 90° may form stable equilibrium if rotation is sufficiently rapid.
The age question remains: statistical equilibrium requires many collisions, implying great antiquity. But equilibrium is not ultimate, as gases in motion are viscous and velocities expend. The Milky Way loses stars occasionally, like atmospheric molecules escaping, limiting its duration. Calculations give enormous ages, conflicting with physicists’ estimate that suns last only about fifty million years. Either stars reached the adult period simultaneously while matter was dark, or visible stars are a minority versus extinct or future-luminous ones—contradicting the absence of considerable dark matter. Poincaré ends with this difficulty unresolved.
French Geodesy
Geodesy provides the framework for maps, essential for public works, but its higher purpose is understanding nature’s unity. Maupertuis and La Condamine’s eighteenth-century expeditions tested Newton’s flattening theory against Cassini’s elongation theory. M. Faye found dense rocks beneath oceans and empty spaces beneath continents. France’s contributions include the French Academy sending Maupertuis and Clairaut to the Arctic circle and Bouguer and La Condamine to the Andes. Voltaire first praised Maupertuis extravagantly, then attacked him via Dr. Akakia. His lines “Vous avez confirmé dans des lieux pleins d’ennui / Ce que Newton connut sans sortir de chez lui” are unjust—theory and experience are equally indispensable.
Delambre and Méchain measured an arc from Dunkirk to Barcelona during revolutionary turmoil. Delambre faced suspicious municipalities and missing steeples, with white linen signals mistaken for counter-revolutionary standards. Méchain in Spain faced hostile peasants fearing diabolical instruments. His second expedition to the Balearic Isles failed, and he died after applying for recall. Arago and Biot completed the work; Arago was imprisoned, read of his own execution, escaped to Algiers, was captured by a Spanish privateer, released after the Dey threatened war over lions destined for Napoleon, crossed Kabylia on foot, and preserved his observations and instruments. General Berrier joined Spain and Africa via four peaks. Signals from Nice to Corsica will measure light velocity. Colonel Defforges’ pendulum determines gravity precisely. The geographical department of the army now directs French geodesy. Captains Maurain and Lacombe surveyed Quito, and General Alfaro of Ecuador called them “los hombres de hierro.” Lieutenant-Colonel Bourgeois commanded the definitive mission despite climate difficulties at 13,000 feet.
General Conclusions
Scientists must select facts. Some facts teach nothing beyond themselves; others reveal new laws. This classification is relative to our mind’s frailty—complex facts exceed our comprehension, while simple facts, or those where chance produces compensation, yield large returns. Mathematical and physical discovery both rise from fact to law. The mathematician’s mind works through three forms: the inventive creator, the unconscious geometrician constructing instinctive space notions, and the school youth learning first principles. Intuition and generalization are essential throughout. Mathematical reasoning is genuine induction, proceeding from particular to universal; attempts to reduce it to logic have failed.
Main Arguments
Poincaré advances several central arguments throughout the work. First, he argues that the scientific method requires selection among facts, since the scientist lacks infinite time. This selection is guided by utility, but also by the beauty of intellectual harmony. He rejects both the view that “Science for Science’s sake” is absurd and the notion that only immediately practical results matter, citing the disinterested “fools who died poor” whose work preceded practical triumphs.
Second, Poincaré argues that mathematics will continue to develop in all directions, and that its future can be foreseen by studying its history. He contends that the meaning of “solution” has repeatedly been extended throughout mathematical history, and that this process will continue. He emphasizes the importance of analogy, well-chosen terminology, and the economy of thought, arguing that a fact’s importance is measured by the “return” it gives in enabling us to economize thought.
Third, Poincaré presents a theory of mathematical discovery based on his own experience. He argues that sudden illuminations indicate previous unconscious work, and that the subliminal ego plays an important part in mathematical discovery. He describes the process as one of selection among combinations, guided by an aesthetic sensibility that recognizes mathematical beauty. The useful combinations are precisely the most beautiful, and the man without this aesthetic sense “will never be a real discoverer.”
Fourth, Poincaré argues against the reduction of mathematics to logic. He contends that the logisticians’ program fails because their definitions are non-predicative, containing hidden vicious circles, and because the principle of complete induction cannot be demonstrated without appealing to intuition. He concludes that logic remains barren unless fertilized by intuition, and that the attempts of Russell and Hilbert have not settled the controversy between Kant and Leibnitz.
Fifth, Poincaré argues that the principles of Newtonian mechanics are facing abandonment in light of radium’s discovery. He examines the evidence from cathode rays and radium rays, which achieve velocities far greater than those previously studied, and discusses how mass appears to be purely electro-magnetic, increasing with velocity and becoming infinite at light’s velocity. He concludes that the principle of reaction must be abandoned under Lorentz’s theory, and that “there is no more matter” in the traditional sense.
Key Concepts
The selection of facts: Poincaré argues that scientists must choose among the infinite facts available to them. The most interesting facts are those with “a chance of recurring”—simple facts, since chance can mingle but “cannot unmingle.” He introduces the hierarchy of facts and the importance of exceptions once a rule is established.
Economy of thought: Drawing on Mach, Poincaré argues that science aims to economize thought. A fact’s importance is measured by the amount of thought it enables us to economize. The multiplication table example illustrates this: recording that 6 times 7 are 42 saves millions from repeating the operation.
The creative word: Poincaré argues that “the invention of a new word will often be sufficient to bring out the relation, and the word will be creative.” Well-chosen terms like “uniformity of convergence,” “group,” and “invariable” revealed the essence of many reasonings and made whole classes of long reasonings unnecessary.
The subliminal ego: Poincaré’s theory of mathematical discovery posits an unconscious mind that forms combinations during periods of rest. The subliminal ego has discernment and can select and divine, perhaps better than the conscious ego. He compares future elements of combinations to Epicurus’s hooked atoms, which are detached and set in motion during unconscious work.
The distribution board: Poincaré’s model of space perception compares the nervous system to a telegraph exchange with centripetal wires (warnings) and centrifugal wires (remedies). This “distribution board” is our whole geometry; intuition of straight line and distance is consciousness of these associations and their “imperious character.”
The relativity of space: Poincaré argues that absolute space no longer exists; only space relative to an initial body position. Space is “amorphous”; things in it give it form. We have no direct intuition of distance, and one part of space is not absolutely equal to another.
The principle of complete induction: Poincaré examines demonstrations of this principle by Whitehead and Burali-Forti, arguing that they fail due to vicious circles. He concludes that the principle cannot be the disguised definition of the whole number and requires an a priori synthetic judgment.
Longitudinal and transversal mass: In the new mechanics, Poincaré discusses how mass varies with velocity. Abraham’s theoretical work shows that fictitious mass varies with velocity, and that total longitudinal and transversal masses differ and depend on velocity. Kaufmann’s experiments suggested that actual mass is nil and mass is purely electro-magnetic.
Local time: Lorentz’s concept of local time explains how two observers moving together would find systematic errors when setting watches by optical signals. Each watch marks “local time” proper to its place.
The Lorentz-Fitz-Gerald contraction: To explain Michelson’s negative result, Lorentz and Fitz-Gerald hypothesized that bodies in transposition contract in the direction of motion. Measuring instruments cannot disclose this contraction since yard-measures contract equally.
Themes
A recurring theme is the relationship between intuition and logic in mathematical and scientific reasoning. Poincaré consistently argues that pure logic is insufficient for discovery, and that intuition plays an indispensable role. He states that “by logic that we prove, but by intuition that we discover.” This theme runs through his critique of the logisticians, his theory of mathematical discovery, and his educational philosophy. The book suggests that the attempt to reduce mathematics to logic alone is not merely mistaken but dangerous, as it leads to antinomies and barren formalism.
Another theme is the aesthetic dimension of science. Poincaré argues that scientists study nature because it is beautiful—”intellectual beauty” arising from “the harmonious order of its parts.” This aesthetic sensibility is not a luxury but a practical necessity, since it guides the selection of fruitful combinations in mathematical discovery. The useful combinations are precisely the most beautiful, and the aesthetic sense serves as a “delicate sieve” that filters out unfruitful possibilities. Poincaré connects this to economy of thought, suggesting that aesthetic satisfaction and intellectual economy are linked.
The theme of revolutionary change in physics pervades the third book. Poincaré presents the discoveries of radium and electrons as threatening to overturn the fundamental principles of mechanics. He writes at a moment when the old certainties are crumbling, and the work captures the excitement and anxiety of this transition. The principles of inertia, conservation of mass, and equality of action and reaction all come under scrutiny, and Poincaré suggests that “mechanics seem to be on the point of undergoing a complete revolution.”
The theme of education and the transmission of scientific knowledge also appears prominently. Poincaré discusses the challenges of teaching mathematics to students who lack mathematical aptitude, the importance of intuition in education, and the need to make students pass through all stages their ancestors passed. He criticizes overly formal approaches to definition and advocates for beginning with concrete examples before introducing abstract definitions. The work suggests that education must cultivate both logical rigor and intuitive understanding.
The theme of chance and determinism runs through the discussion of probability. Poincaré notes that the ancients gave chance a precise, objective meaning, but that modern determinism has made chance merely “the measure of our ignorance.” Fortuitous phenomena are those whose laws we are ignorant of. This theme connects to his discussion of the selection of facts, where chance plays a role in compensation of errors.
The Author’s Conclusions
Poincaré concludes that scientists must select facts, and that this selection is guided by the return they give in economizing thought. Some facts teach nothing beyond themselves, while others reveal new laws. This classification is relative to our mind’s frailty—complex facts exceed our comprehension, while simple facts yield large returns. Mathematical and physical discovery both rise from fact to law.
He concludes that the mathematician’s mind works through three forms: the inventive creator, the unconscious geometrician constructing instinctive space notions, and the school youth learning first principles. Intuition and generalization are essential throughout. Mathematical reasoning is genuine induction, proceeding from particular to universal, and attempts to reduce it to logic have failed.
On the new mechanics, Poincaré concludes that the principle of reaction must be abandoned under Lorentz’s theory, and that mass appears to be purely electro-magnetic. He notes that Kaufmann’s experiments supported Abraham’s theory, though Bucherer’s later results confirmed Lorentz’s views. The principle of inertia remains true in the new dynamics, but no body can exceed light’s velocity. He leaves the question of gravitation’s relationship to these new theories undecided.
On the Milky Way, Poincaré examines three hypotheses for its flattened form—provisional equilibrium from collisions, common rotation, and Stratonoff’s suggestion that it is itself a spiral nebula. He finds difficulties with each, particularly the conflict between the enormous ages implied by statistical equilibrium and physicists’ estimate that suns last only about fifty million years. He ends with this difficulty unresolved.
The book closes with Poincaré’s reflections on French geodesy, celebrating the contributions of Maupertuis, La Condamine, Delambre, Méchain, Arago, and others who measured the Earth’s shape and size under difficult conditions. He emphasizes that theory and experience are equally indispensable, and that the history of geodesy shows “what time and trouble are involved in the conquest of a single new decimal.”
Significance
Science and Method captures a pivotal moment in the history of science, when the foundations of physics and mathematics were being questioned and transformed. Poincaré’s work is significant for its synthesis of technical scientific content with philosophical reflection, written by a practitioner of the highest rank. His theory of mathematical discovery, based on his own experience with Fuchsian functions, has become a classic account of the role of unconscious processes in creative work.
The book’s critique of the logisticians’ program to reduce mathematics to logic is historically important, representing a major alternative to the views of Russell, Hilbert, and others. Poincaré’s insistence on the role of intuition and his identification of vicious circles in non-predicative definitions anticipated later developments in the foundations of mathematics.
The discussion of the new mechanics documents the transition from Newtonian physics to the relativistic and quantum theories that would follow. Poincaré’s engagement with Lorentz’s theory, the principle of relativity, and the concept of local time places him at the forefront of these developments. His conclusion that “there is no more matter” in the traditional sense captures the radical implications of electron theory.
The work also stands as a contribution to the philosophy of science, arguing for the importance of selection, economy of thought, and aesthetic sensibility in scientific practice. Poincaré’s reflections on the relativity of space, the nature of chance, and the relationship between intuition and logic continue to be relevant to contemporary debates. The book’s educational philosophy, emphasizing the cultivation of intuition alongside logical rigor, remains influential in discussions of mathematics pedagogy.
