Sunday, April 8, 2012

Henri Poincare

Jules Henri Poincaré (1854-1912)

Poincaré was an influential French philosopher of science and mathematics,theoretical physicist, distinguished scientist and one of the greatest mathematician.Henri Poincaré , a polymath was called the "last universalist" because he was a great contributor to almost all fields in mathematics, but his work was also broad in physics, philosophy, and psychology. In the foundations of mathematics he argued for conventionalism, against formalism, against logicism, and against Cantor’s treating his new infinite sets as being independent of human thinking. Poincaré stressed the essential role of intuition in a proper constructive foundation for mathematics.He believed that logic was a system of analytic truths, whereas arithmetic was synthetic and a priori, in Kant‘s sense of these terms. Mathematicians can use the methods of logic to check a proof, but they must use intuition to create a proof, he believed.

He maintained that non-Euclidean geometries are just as legitimate as Euclidean geometry, because all geometries are conventions or “disguised” definitions. Although all geometries are about physical space, a choice of one geometry over others is a matter of economy and simplicity, not a matter of finding the true one among the false ones.For Poincaré, the aim of science is prediction rather than, say, explanation. Although every scientific theory has its own language or syntax, which is chosen by convention, it is not a matter of convention whether scientific predictions agree with the facts. For example, it is a matter of convention whether to define gravitation as following Newton’s theory of gravitation, but it is not a matter of convention as to whether gravitation is a force that acts on celestial bodies, or is the only force that does so.So, Poincaré believed that scientific laws are conventions but not arbitrary conventions.Poincaré had an especially interesting view of scientific induction. Laws, he said, are not direct generalizations of experience; they aren’t mere summaries of the points on the graph. Rather, the scientist declares the law to be some interpolated curve that is more or less smooth and so will miss some of those points.Thus a scientific theory is not directly falsifiable by the data of experience; instead, the falsification process is more indirect.


Poincare also proposed the theory of relativity before Einstein but never published it vastly, infact Einstein had to read Poincare about relativity(as Poincare at that time was the only pioneer of the theory of relativity).In fact, Einstein works are not completely original. They were based on previous works as those of Hendrik Lorentz and Henri Poincaré.

Henri Poincare

With 49 proposals between 1901 and 1912, Poincare still remains the most nominated scientist of Noble Prize in Physics. Untimely death of Poincare prevented him to add the Noble prize to his amazing list of awards.


"Every time I see picture of Einstein I think of Poincare because I think Poincare was supreme and better than Einstein.It took almost a century to understand his theories".
Nassim Nicholas Taleb


1. Life
Poincaré was born  on April 29,1854 in Nancy and died on July 17, 1912 in Paris. Poincaré’s  family was influential.   His cousin Raymond was the President and the  Prime Minister of France, and his father Leon was a professor of medicine  at the University of Nancy.His sister Aline married the spiritualist  philosopher Emile Boutroux.Poincaré studied mining engineering, mathematics and physics in Paris.Beginning in 1881,he taught at the University of Paris.There he held the chairs of  Physical and Experimental Mechanics, Mathematical Physics and Theory of Probability, and Celestial Mechanics and Astronomy.At the beginning of his scientific career, in his doctoral dissertation  of 1879, Poincaré devised a new way of studying the properties of  functions defined by differential equations.He not only faced the  question of determining the integral of such equations, but also    was the first person to study the general geometric properties of these  functions. He clearly saw that this method was useful in the solution of  problems such as the stability of the solar system, in which the  question is  about the qualitative properties of planetary orbits (for example, are  orbits regular or chaotic?) and not about the numerical solution of gravitational equations.During his studies on differential equations, Poincaré made use of   Lobachevsky’s non-Euclidean geometry.Later, Poincaré applied to celestial mechanics the methods he had    introduced in his doctoral dissertation.

 His research on the stability of the solar system opened the door to the study of chaotic deterministic systems; and the methods he used gave rise to algebraic topology.Poincaré sketched a preliminary version of the special theory of  relativity and stated that the velocity of light is a limit velocity and that mass depends on speed. He formulated the principle of relativity, according to which no mechanical or electromagnetic experiment can discriminate between a state of uniform motion and a state of rest, and  he derived the Lorentz transformation.His fundamental theorem that every  isolated mechanical system returns after a finite time [the Poincaré  Recurrence Time] to its initial state is the source of many philosophical and scientific analyses on entropy. Finally, he clearly understood how radical is quantum theory’s departure from classical physics.

Poincaré was deeply interested in  the philosophy of science and the foundations of mathematics. He argued  for conventionalism and against both  formalism and  logicism.Cantor’s set theory was also an object of his    criticism. He wrote several articles on the philosophical interpretation of mathematical logic.During his life, he published three books on the  philosophy of science and mathematics.A fourth book was published posthumously in 1913.


2. Chaos and the Solar System
In his research on the three-body problem, Poincaré became the first person to discover a chaotic deterministic  system.Given the law of gravity and the initial positions and velocities of  the only three bodies in all of space, the subsequent positions and  velocities are fixed–so the  three-body system is deterministic.However,Poincaré found that the evolution of such a system is often chaotic in the sense that a small perturbation in the initial state such as a slight change in  one body’s initial position might lead to a radically different later state than would be produced by the unperturbed  system.If the slight change isn’t detectable by our measuring instruments, then we  won’t be able to predict which final state will occur.So, Poincaré’s research proved that the problem of determinism and    the problem of predictability are distinct problems.From a philosophical point of view, Poincaré’s results did  not receive the attention that they deserved.  Also the scientific line of  research  that Poincaré opened was neglected until meteorologist Edward Lorenz, in 1963, rediscovered a chaotic deterministic system while he was studying the evolution of a simple model of the atmosphere.

Earlier, Poincaré had suggested that the difficulties of reliable weather  predicting are due to the intrinsic chaotic behavior of the atmosphere.Another interesting aspect of Poincaré’s study is the real nature of the distribution in phase space of stable and unstable points,which are so mixed that he did not try to make a picture of their    arrangement. Now we know that the shape of such distribution is fractal-like. However, the scientific study of fractals did not begin until  Benoit Mandelbrot’s work in 1975, a century after Poincaré’s first insight.
Why was Poincaré’s research neglected and underestimated? The  problem is interesting because Poincaré was awarded an important  scientific prize for his research; and his research in celestial mechanics was recognized to be of fundamental importance.Probably there were two  causes.Scientists and philosophers were primarily interested in the revolutionary  new physics of relativity and quantum mechanics, but Poincaré worked  with classical mechanics.Also, the behavior of a chaotic deterministic system can be described  only by means of a numerical solution whose complexity is staggering.Without the help of a computer the task is almost hopeless.


3. Arithmetic, Intuition and Logic
Logicists such as Bertrand Russell and Gottlob Frege believed that  mathematics is basically a branch of symbolic logic, because they  supposed that  mathematical terminology can be defined using only the  terminology of logic and because, after this translation of terms, any  mathematical theorem can be shown to be a restatement of a theorem of  logic.Poincaré objected to this logicist program.He was an intuitionist  who stressed the essential role of  human intuition in the foundations of mathematics.According to Poincaré, a definition of a mathematical entity is not the exposition of the essential properties of the entity, but it is the construction of the entity itself; in other words, a legitimate mathematical definition creates and justifies its object.For Poincaré, arithmetic is a synthetic science whose objects are not independent from human thought.Poincaré made this point in his investigation of Peano’s axiomatization  of arithmetic.Italian mathematician Giuseppe Peano (1858-1932) axiomatized the mathematical theory of natural numbers.This is the arithmetic of the nonnegative integers.Apart from some purely  logical principles, Peano employed five mathematical axioms.Informally, these axioms are:
  1. Zero is a natural number.
  2. Zero is not the successor of any natural number.
  3. Every natural number has a successor, which is a natural number.
  4. If the successor of natural number a is equal to the successor of natural number b, then a and b are equal.
  5. Suppose:
    (i) zero has a property P;
    (ii) if every natural number less than a has the property P then a also has the property P.
    Then every natural number has the property P.  (This is the principle of complete induction.)
Bertrand Russell said Peano’s axioms constitute  an implicit definition of  natural numbers, but Poincaré said they do only if they can be  demonstrated to be consistent.They can be shown consistent only by showing there is some object satisfying these axioms.From a general  point of view, an axiom system can be conceived of as an implicit definition only if it is possible to prove the existence of at least one object that  satisfies all the axioms.   Proving this is not an easy task, for the number of  consequences of Peano axioms is infinite and so a direct inspection of each consequence is not possible.Only one way seems  adequate: we must verify that if the premises of an inference in the system  are consistent with the axioms of logic, then so is the conclusion. Therefore, if after n inferences no contradiction is produced, then after  n+1 inferences no contradiction will be either. Poincaré argues that this reasoning is a vicious circle, for it  relies upon the principle of complete induction, whose consistency we  have to prove. (In 1936, Gerhard Gentzen  proved the consistency of Peano axioms, but his proof required the use of a limited form of  transfinite induction whose own consistency is in doubt.) As a  consequence, Poincaré  asserts that if we can’t noncircularly establish the consistency of Peano’s  axioms, then the principle of complete induction is surely not provable by means of general logical laws; thus it is not analytic, but it is a synthetic judgment, and logicism is refuted. It is evident that Poincaré  supports Kant’s epistemological viewpoint on arithmetic. For Poincaré, the principle of complete induction, which is not provable via analytical    inferences, is a genuine synthetic a priori judgment. Hence arithmetic cannot be reduced to logic; the latter is analytic, while arithmetic is synthetic.

The synthetic character of arithmetic is also evident if we consider the  nature of mathematical reasoning. Poincaré suggests a distinction between two different kinds of mathematical inference: verification and proof. Verification or proof-check is a sort of  mechanical reasoning, while proof-creation is a fecund inference. For  example, the statement “2+2 = 4″ is verifiable because it is possible to demonstrate its truth with  the help of logical laws and the definition of sum; it is an analytical statement that admits a straightforward verification. On the contrary, the general statement (the commutative law of addition)
For any natural numbers x and y, x + y = y + x
is not directly verifiable. We can choose an arbitrary pair of natural  numbers a and b, and we can verify that    a+b = b+a; but there is an infinite number of admissible choices of pairs, so the verification is always incomplete. In other words, the verification of the commutative law is an analytical method  by means of which  we can verify every particular instance of a general theorem, but the  proof of the theorem itself is synthetic reasoning which really extends our knowledge, Poincaré believed.Another aspect of mathematical thinking that Poincaré analyzes is  the different roles played by intuition and logic. Methods of formal  logic are elementary and certain, and we can surely rely on them.However, logic does not teach us how to build a proof. It is intuition that helps mathematicians find the correct way of to assemble basic inferences into a useful proof. Poincaré offers the following  example. An unskilled chess player who watches a  game can verify whether a move is legal, but he does not understand why players move certain pieces, for he does not see the plan which guides player's choices.In a similar way, a mathematician who uses only  logical methods can verify every inference in a given proof, but he  cannot find an original proof.In other words, every elementary  inference in a proof is easily verifiable through formal logic, but the  invention of a proof requires the understanding -grasped by intuition  of the general scheme, which directs mathematician’s efforts towards the final goal.

Logic is — according to Poincaré  the study of properties  which are common to all classifications.There are two different kinds of  classifications:  predicative classifications, which are not modified by the  introduction of new elements; and impredicative classifications, which are modified by new elements.Definitions as well as    classifications are divided into predicative and impredicative. A set is  defined by a law according to which every element  is generated. In the case of an infinite set, the process of generating elements is unfinished; thus there are always new elements.If their introduction changes the classification of already generated objects, then the definition is impredicative.For example, look at phrases containing a finite number of words and defining a point of space. These  phrases are arranged in alphabetical order and each of them is associated with a natural number: the first is associated with number 1, the second with 2, etc. Hence every point defined by such phrases is    associated with a natural number. Now suppose that a new point is defined by a new phrase. To determine the corresponding number it is  necessary to insert this phrase in alphabetical order; but such an  operation modifies the number associated with the already classified  points whose defining phrase follows, in alphabetical order, the new  phrase.Thus this new definition is impredicative.

For Poincaré, impredicative definitions are the source of  antinomies in set theory, and the prohibition of impredicative  definitions will remove such antinomies.To this end, Poincaré enunciates the vicious circle principle: a thing cannot be defined with respect to  a collection that presupposes the thing itself. In other words, in a  definition of an object, one cannot use a set to which the object  belongs, because doing so produces an impredicative definition.Poincaré  attributes the vicious circle principle to a French  mathematician J. Richard.In 1905, Richard discovered a new paradox in  set theory, and he offered a  tentative solution based on the vicious circle principle.

Poincaré’s prohibition of impredicative definitions is also connected with his point of view on infinity. According to  Poincaré, there are two different schools of thought about  infinite sets; he called these schools Cantorian and Pragmatist. Cantorians are realists with respect to mathematical  entities;  these entities have a reality that is independent of human conceptions.The  mathematician discovers them but does not create them.Pragmatists    believe that a thing exists only when it is the object of an act of  thinking, and infinity is nothing but the possibility of the mind’s generating  an  endless series of finite object.Practicing mathematicians tend to be  realists, not pragmatists or intuitionists.This dispute is not about the role of  impredicative definitions in producing antinomies, but about the  independence of mathematical entities from human thinking.


Henri Poincare

4. Conventionalism and the Philosophy of Geometry
The discovery of non-Euclidean geometries upset the commonly accepted  Kantian viewpoint that the true structure of space can be known apriori. To understand  Poincaré’s point of view on the foundation of geometry, it helps  to remember that, during his research on functions defined by differential equations, he actually used non-Euclidean  geometry. He found that several geometric properties are easily  provable  by means of Lobachevsky geometry, while their proof is not  straightforward in Euclidean geometry. Also, Poincaré knew    Beltrami’s research on Lobachevsky’s geometry.Beltrami (Italian  mathematician, 1835-1899) proved the consistency of Lobachevsky  geometry  with respect to Euclidean geometry, by means of a translation of every    term of Lobachevsky geometry into a term of Euclidean geometry.The  translation is carefully chosen so that    every axiom of non-Euclidean geometry is translated into a theorem of  Euclidean geometry. Beltrami’s translation and Poincaré’s study of  functions led Poincaré to assert that:
  • Non-Euclidean geometries have the same logical and mathematical  legitimacy as Euclidean geometry.
  • All geometric systems are equivalent and thus no system of axioms may claim that it is the true geometry.
  • Axioms of geometry are neither synthetic a priori judgments nor  analytic ones; they are conventions or ‘disguised’ definitions.
According to Poincaré, all geometric systems deal with the same  properties of space, although each of them employs its own language, whose syntax is defined by the set of axioms. In other words, geometries differ in their language, but they are concerned with the same reality,for a geometry can be translated into another geometry. There is only one  criterion according to which we can select a geometry, namely a criterion of economy and simplicity. This is the very reason why we commonly use Euclidean geometry: it is the simplest. However, with  respect to a specific problem, non-Euclidean geometry may give us the  result with less effort.In 1915, Albert Einstein found it more convenient, the conventionalist would say, to develop his theory of general relativity using  non-Euclidean rather than Euclidean geometry.Poincaré’s realist opponent would disagree and say that Einstein  discovered space to be non-Euclidean.

Poincaré’s treatment of geometry is applicable also to the general  analysis of scientific theories.Every scientific theory has its own language, which is chosen by convention. However, in spite of  this freedom, the agreement or disagreement between predictions and  facts is not conventional but  is substantial and objective. Science has an objective validity. It is not due to chance or to freedom of choice that scientific predictions are often accurate.
These considerations clarify Poincaré’s conventionalism.There is an  objective criterion, independent of the    scientist’s will, according to which it is possible to judge the soundness of the scientific theory, namely the accuracy of its predictions.Thus the principles of science are not set by an arbitrary convention.In so far  as scientific predictions are true, science gives us objective, although incomplete, knowledge. The freedom of a scientist takes place  in the choice of language, axioms, and the facts that deserve attention.However, according to Poincaré, every scientific law can be analyzed into two parts, namely a principle, that is a  conventional truth, and an empirical law.The following example is due to Poincaré. The law:
Celestial bodies obey Newton’s law of gravitation
The law consists of two elements:
  1. Gravitation follows Newton law.
  2. Gravitation is the only force that acts on celestial bodies.
We can regard the first statement as a principle, as a convention; thus it becomes the definition of gravitation.But then the second statement  is an empirical law.
Poincaré’s attitude towards conventionalism is illustrated by the following statement, which concluded his analysis on classical mechanics  in Science and Hypothesis:
Are the laws of acceleration and composition of forces nothing but  arbitrary conventions? Conventions, yes; arbitrary, no; they would seem  arbitrary if we forgot the experiences which guided the founders of  science to their adoption and which are, although imperfect, sufficient    to justify them. Sometimes it is useful to turn our attention to the  experimental origin of these conventions.
 
5. Science and Hypothesis
According to Poincaré, although  scientific theories originate from experience, they are neither verifiable nor falsifiable by means of the experience alone.For example, look at the problem of finding a mathematical law  that describes a given series of observations.In this case,  representative points are plotted in a graph, and then a simple curve  is interpolated.The curve chosen will depend both on the  experience which determines the representative points and on the desired  smoothness of the curve even though the smoother the curve the more that  some points will miss the curve. Therefore, the interpolated curve — and thus the tentative law — is  not a direct generalization of the experience, for it ‘corrects’ the  experience.The discrepancy between observed and calculated values is  thus not regarded as a falsification of the law, but as a correction  that the law imposes on our observations.In this sense, there is always a necessary difference between facts and theories, and therefore a  scientific theory is not directly falsifiable by the experience.

For Poincaré, the aim of the science is to prediction. To accomplish  this task, science makes use of generalizations that go beyond the  experience.In fact, scientific theories are hypotheses. But every  hypothesis has to be continually tested. And when it fails in an  empirical test, it must be given up. According to Poincaré, a  scientific hypothesis which was proved untenable can still be very useful.If a hypothesis does not pass an empirical test, then this fact  means that we have neglected some important and meaningful element;  thus  the hypothesis gives us the opportunity to discover the existence of an  unforeseen aspect of reality. As a consequence of this point of view about the nature of scientific theories, Poincaré suggests that a  scientist must utilize few hypotheses, for it is very difficult to find  the wrong hypothesis in a theory which makes use of many hypotheses.
For Poincaré, there are many kinds of hypotheses:
  • Hypotheses which have the maximum scope, and which are common to  all  scientific theories (for example, the hypothesis according to which the  influence of remote bodies is negligible). Such hypotheses are the last to be changed.
  • Indifferent hypotheses that, in spite of their auxiliary role in scientific theories, have no objective content (for example, the hypothesis  that unseen atoms exist).
  • Generalizations, which are subjected to empirical control; they are  the true scientific hypotheses.
Regarding Poincaré’s point of view about scientific  theories, the following have the most lasting value:
  • Every scientific theory is a hypothesis that had to be tested.
  • Experience suggests scientific theories; but experience does not  justify them.
  • Experience alone is unable to falsify a theory, for the theory often corrects the experience.
  • A central aim of science is prediction.
  • The role of a falsified hypothesis is very important, for it throws light on unforeseen conditions.
  • Experience is judged according to a theory.


6.Randomness. 
 Poincre was first big gun to understand mathematical techniques and limits involved in randomness and hence forecasting limits.Poincare was the first thinker to go against Gaussian or Normal Bell curve and suggested that such curve is useless for big and rare events. Many claim that Poincare was the first one to give idea of relativity but he never made it big to get prominence.Poincare also started the study of fractals and their use in physics,math and randomness etc,Mandlbrolt later continued his work 100 years after Poincre was first big gun to understand mathematical techniques and limits involved in randomness and hence forecasting limits. Poincare's research on solar system got a Prize which was highest at that time. Poincare was the advisor of Louis Bachelier, the pioneer of Financial Maths.Poincaré saw mathematical work in economics and finance as peripheral. In 1900 Poincaré commented on Louis Bachelier's thesis "The Theory of Speculation", saying: "M. Bachelier has evidenced an original and precise mind [but] the subject is somewhat remote from those our other candidates are in the habit of treating." (Bernstein, 1996, pp. 199–200) Bachelier's work explained what was then the French government's pricing options on French Bonds and anticipated many of the pricing theories in financial markets used today .Poincare suggested that as you project in the future you may need an increasing amount of precision ,near precision is not possible . Think of forecasting as in terms of tree branches, this grows in multiple ways and doubling every time so such increasing amount requires a lot of precision.Poincare suddenly died in July, 1912 and left his works for generations.

According to famous French mathematician Jean Leray "Very few men were able to follow Poincare's thoughts , he had no students.After one century of mathematical work and advancement we are able to understand his works and thoughts more easily."



 Bibliography
COLLECTED SCIENTIFIC WORKS (in French).
  • Oeuvres, 11 volumes, Paris : Gauthier-Villars, 1916-1956
PHILOSOPHICAL WORKS.
  • 1902 La science et l’hypothèse, Paris : Flammarion  (Science and hypothesis, 1905)
  • 1905 La valeur de la science, Paris : Flammarion (The value of  science, 1907)
  • 1908 Science and méthode, Paris : Flammarion (Science  and method, 1914)
  • 1913 Dernières pensées, Paris : Flammarion  (Mathematics and science: last essays, 1963)
  • The first three works are translated in The foundations of  science, Washington, D.C. : University Press of America, 1982 (first  edition 1946).

1 comment:

  1. EXCELLENT! Thx, Nassif.

    Swami

    P.S. I found you and this blog entry in my preparation for a math/philosophy meeting on Poincaré. Here is the event notice and details https://www.facebook.com/events/876205565741504. And, this link has details about my "Wing Circle" group http://tinyurl.com/WC-EventHistoryImages.

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