The World of Mathematics

Poincaré, The True Father of Chaos …

Jules Henri Poincare

Henri Poincaré was one of the last universal mathematicians. He made contributions to the theory of complex functions, to number theory, algebraic and differential geometry, and to many branches of applied mathematics, including celestial mechanics.

He invented the notion of an abstract dynamical system in order to attack the question of the stability of the solar system, and, in the course of this study, invented the field of topology. He believed that all physical laws should be “invariant under the Lorentz group.” This insight, which expresses the hidden symmetries of Maxwell’s equations, leads logically to Einstein’s theory of special relativity when applied to Newtonian mechanics.

Henri Poincaré can be said to have been the originator of algebraic topology and of the theory of analytic functions of several complex variables. Poincaré entered the Ecole Polytechnique in 1873 and continued his studies, as a student of Charles Hermite,at the Ecole des Mines, from which he received his doctorate in mathematics in 1879. He was appointed to a chair of mathematical physics at the Sorbonne in 1881, a position he held until his death. Before the age of 30 he developed the concept of automorphic functions which he used to solve second order linear differential equations with algebraic coefficients.

His Analysis situs, published in 1895, is an early systematic treatment of topology. Poincaré can be said to have been the originator of algebraic topology and of the theory of analytic functions of several complex variables. He also worked in algebraic geometry and made a major contribution to number theory with work on Diophantine equations. In applied mathematics he studied optics, electricity, telegraphy, capillarity, elasticity, thermodynamics, potential theory, quantum theory, theory of relativity and cosmology. He is often described as the last universalist in mathematics.

In the field of celestial mechanics he studied the three-body-problem, and theories of light and electromagnetic waves. He is acknowledged as a co-discoverer, with Albert Einstein and Hendrik Lorentz, of the special theory of relativity.

His major works include Les Méthods nouvelle de la méchanique celeste in three volumes published between 1892 and 1899 and Lecons de mecanique celeste (1905). In the first of these he aimed to completely characterise all motions of mechanical systems. He invoked an analogy with fluid flow. He also showed that previous series expansions used in studying the 3-body problem were convergent, but not in general uniformly convergent, so putting in doubt the stability proofs of Lagrange and Laplace.

He also wrote many popular scientific articles including Science and Hypothesis (1901), Science and Method (1908), and The Value of Science (1904). A quote from Poincaré is particularly relevant to this collection on the history of mathematics. In 1908 he wrote The true method of foreseeing the future of mathematics is to study its history and its actual state.

The Poincaré conjecture is one of the most baffling and challenging unsolved problems in algebraic topology. Homotopy theory reduces topological questions to algebra by associating with topological spaces various groups which are algebraic invariants. Poincaré introduced the fundamental group to distinguish different categories of two-dimensional surfaces. He was able to show that any 2-dimensional surface having the same fundamental group as the two-dimensional surface of a sphere is topologically equivalent to a sphere. He conjectured that the result held for 3-dimensional manifolds and this was later extended to higher dimensions.

Surprisingly proofs are known for the equivalent of Poincaré’s conjecture for all dimensions strictly greater than 3. No complete classification scheme for 3-manifolds is known, so, there is no list of possible manifolds that can be checked to verify that they all have different homotopy groups. Poincaré was first to consider the possibility of chaos in a deterministic system, in his work on planetary orbits.

“A very small cause which escapes our notice determines a considerable effect that we cannot fail to see, and then we say that the effect is due to chance. If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of the same universe at a succeeding moment. But even if it were the case that the natural laws had nolonger any secret for us, we could still know the situation approximately. If that enabled us to predict the succeeding situation with the same approximation, that is all we require, and we should say that the phenomenon had been predicted, that it is governed by the laws. But is not always so; it may happen that small differences in the initial conditions produce very great ones in the finalphenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible…”. (Poincaré)

Little interest was shown in his work until the modern study of chaotic dynamics began in 1963.Thus Poincaré can truly be said to be is the Father of Chaos.


Poincaré quotes:

“Mathematicians are born, not made.”

“I entered an omnibus to go to some place or other. At that moment when I put my foot on the step the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with non-Euclidean geometry.”

“In the old days when people invented a new function they had something useful in mind. Now, they invent them deliberately just to invalidate our ancestors’ reasoning, and that is all they are ever going to get out of them.”

“How is an error possible in mathematics? A sane mind should not be guilty of a logical fallacy, yet there are very fine minds incapable of following mathematical demonstrations. Need we add that mathematicians themselves are not infallible?”

“Point set topology is a disease from which the human race will soon recover.”

“Mathematics is the art of giving the same name to different things. [As opposed to the quotation: Poetry is the art of giving different names to the same thing].”

“Later generations will regard Mengenlehre (set theory) as a disease from which one has recovered. [Whether or not he actually said this is a matter of debate amongst historians of mathematics.]”

“What is it indeed that gives us the feeling of elegance in a solution, in a demonstration? It is the harmony of the diverse parts, their symmetry, their happy balance; in a word it is all that introduces order, all that gives unity, that permits us to see clearly and to comprehend at once both the ensemble and the details.”

“Thus, be it understood, to demonstrate a theorem, it is neither necessary nor even advantageous to know what it means. The geometer might be replaced by the “logic piano” imagined by Stanley Jevons; or, if you choose, a machine might be imagined where the assumptions were put in at one end, while the theorems came out at the other, like the legendary Chicago machine where the pigs go in alive and come out transformed into hams and sausages. No more than these machines need the mathematician know what he does.”

“Talk with M. Hermite. He never evokes a concrete image, yet you soon perceive that the more abstract entities are to him like living creatures.”

“Science is built up with facts, as a house is with stones. But a collection of facts is no more a science than a heap of stones is a house.”

“A scientist worthy of his name, about all a mathematician, experiences in his work the same impression as an artist; his pleasure is as great and of the same nature.”

“The mathematical facts worthy of being studied are those which, by their analogy with other facts, are capable of leading us to the knowledge of a physical law. They reveal the kinship between other facts, long known, but wrongly believed to be strangers to one another.”

“Mathematicians do not study objects, but relations between objects. Thus, they are free to replace some objects by others so long as the relations remain unchanged. Content to them is irrelevant: they are interested in form only.”

“Thought is only a flash between two long nights, but this flash is everything.”

“The mind uses its faculty for creativity only when experience forces it to do so.”

“Mathematical discoveries, small or great are never born of spontaneous generation They always presuppose a soil seeded with preliminary knowledge and well prepared by labour, both conscious and subconscious.”

“Absolute space, that is to say, the mark to which it would be necessary to refer the earth to know whether it really moves, has no objective existence…. The two propositions: “The earth turns round” and “it is more convenient to suppose the earth turns round” have the same meaning; there is nothing more in the one than in the other.”

” …by natural selection our mind has adapted itself to the conditions of the external world. It has adopted the geometry most advantageous to the species or, in other words, the most convenient. Geometry is not true, it is advantageous.”

“One would have to have completely forgotten the history of science so as to not remember that the desire to know nature has had the most constant and the happiest influence on the development of mathematics.”

“Les faits ne parlent pas.” “Facts do not speak.”

“If one looks at the different problems of the integral calculus which arise naturally when one wishes to go deep into the different parts of physics, it is impossible not to be struck by the analogies existing. Whether it be electrostatics or electrodynamics, the propogation of heat, optics, elasticity, or hydrodynamics, we are led always to differential equations of the same family.”

“Science is built up of facts, as a house is built of stones; but an accumulationof facts is no more a science than a heap of stones is a house.”

“To doubt everything, or, to believe everything, are two equally convenient solutions; both dispense with the necessity of reflection.”

“Ideas rose in clouds; I felt them collide until pairs interlocked, so to speak, making a stable combination.”

“Invention consists in avoiding the constructing of useless contraptions and in constructing the useful combinations which are in infinite minority. To invent is to discern, to choose.”

“Thus, be it understood, to demonstrate a theorem, it is neither necessary nor even advantageous to know what it means …”


Biographies
Kurt Gödel

Douglas Hofstadter author of Gödel, Escher, Bach: an Eternal Golden Braid; and Guggenheim Fellow, 1980-81.

Omar Khayyam, poet and mathematician

Jules Henri Poincare a mathematician, physicist, and philosopher of science.

Tutorials
Introduction to Green’s Functions
The Equichordal Point Problem