Henri Poincaré (1854-1912)
-His (partially failed) attempt to solve the three-body problem is one of the starting points of what would be later known as chaos theory. Poincare’s had realised that the Newtonian system worked fine when dealing with two spatial bodies, but that it could not handle more than that. Poincare believed that predicting the motion of the three spatial bodies (such as planets orbiting) would be at times impossible because small changes in their initial conditions could lead to unexpected consequences.
-Critical of logicism
-Critical of Cantor’s set theory
-One of the pioneers in the field of topology
THE FOUNDATIONS OF SCIENCE– TRILOGY
SCIENCE AND HYPOTHESIS (1902):
–” To doubt everything or to believe everything are two equally convenient solutions; both dispense with the necessity of reflection.”  [all the other citations will come from the same source]
-Science and maths allow us to create logical and rigorous frameworks, however these frameworks are our inventions (they do not necessarily reflect nature). Poincare also claims that Lobachevsky’s insights on non-Euclidean geometry show that the principles of geometry are just conventions. Space may be different from what is perceived by our senses.
-Because of the induction problem (we cannot reproduce all circumstances simultaneously), the ‘truths’ of science are only probabilistic not certain.
-At page 52, Poincare’ offers an interesting definition of ‘existence’ in mathematics: “…“existence” has not the same meaning when it refers to a mathematical entity as when it refers to a material object. A mathematical entity exists provided there is no contradiction implied in its definition, either in itself, or with the propositions previously admitted”
–A few pages later, Poincare’ offers a really powerful interpretation on the epistemological debate on the existence of non-Euclidean geometry “we constantly reason as if the geometrical figures behaved like solids. What geometry would borrow from experiment would be therefore the properties of these bodies….If geometry were an experimental science, it would not be an exact science. It would be subjected to continual revision. Nay, it would from that day forth be proved to be erroneous, for we know that no rigorously invariable solid exists. The geometrical axioms are therefore neither synthetic à priori intuitions nor experimental facts. They are conventions. Our choice among all possible conventions is guided by experimental facts; but it remains free and is only limited by the necessity of avoiding every contradiction, and thus it is that postulates may remain rigorously true even when the experimental laws which have determined their adoption are only approximate. In other words, the axioms of geometry (I do not speak of those of arithmetic) are only definitions in disguise. What, then, are we to think of the question: Is Euclidean geometry true? It has no meaning. We might as well ask if the metric system is true, and if the old weights and measures are false; if Cartesian co-ordinates are true and polar co-ordinates false. One geometry cannot be more true than another; it can only be more convenient. Now, Euclidean geometry is, and will remain, the most convenient: 1st, because it is the simplest, and it is not so only because of our mental habits or because of the kind of direct intuition that we have of Euclidean space; it is the simplest in itself, just as a polynomial of the first degree is simpler than a polynomial of the second degree; 2nd, because it sufficiently agrees with the properties of natural solids, those bodies which we can compare and measure by means of our senses.”.
–Poincare philosophises on his belief on the relativity of motion in the space (anticipating some of the key ideas of Einstein’s 1905 annus mirabilis papers). He criticises the Newtonian idea of absolute time and absolute space.
-Poincare’s known phrase on science and facts: “The man of science must work with method. Science is built up of facts, as a house is built of stones; but an accumulation of facts is no more a science than a heap of stones is a house.”
-Poincare’ also explains what he considers to be a good experiment and how we can generalise from it: ” What, then, is a good experiment? It is that which teaches us something more than an isolated fact. It is that which enables us to predict, and to generalise. Without generalisation, prediction is impossible. The circumstances under which one has operated will never again be reproduced simultaneously. The fact observed will never be repeated. All that can be affirmed is that under analogous circumstances an analogous fact will be produced. To predict it, we must therefore invoke the aid of analogy—that is to say, even at this stage, we must generalise”. For Poincare’, however, the problem of generalisation is that if even one experiment turns to be successful, the prediction that other repetitive experiments will be successful is only probabilisitic and never certain. Poincare seems to hint that his prudence is due to the induction problem- indeed he suggests that there is no definite number of experiments that can make our findings 100% certain, however a successful pattern of prediction may increase our degree of probability. This prudent but not self-defeating modus operandi seems to be Poincare’s own compromise to his preamble early in the book that “To doubt everything or to believe everything are two equally convenient solutions; both dispense with the necessity of reflection“.
-On the chapter on probability Poincare’ highlights one paradox of the theory of probability: compared to certainty, probability tries to deal with the unknown- but Poincare’ asks himself “how can we calculate the unknown?”. A lesser degree of ignorance of the unknown depends on our ability to compute all possible events, to have knowledge of a law that explains a certain behaviour and to know the initial state/position of a certain object. The less information we have about such things the greater our degree of ignorance in computing the probability of the unknown.
THE VALUE OF SCIENCE (1905) 
-“If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living. Of course I do not here speak of that. beauty which strikes the senses, the beauty of qualities and of appearances; not that I undervalue such beauty, far from it, but it has nothing to do with science; I mean that profounder beauty which comes from the harmonious order of the parts and which a pure intelligence can grasp. [my emphasis] ”
-“If the Greeks triumphed over the barbarians and if Europe, heir of Greek thought, dominates the world, it is because the savages loved loud colors and the clamorous tones of the drum which occupied only their senses, while the Greeks loved the intellectual beauty which hides beneath sensuous beauty, and that this intellectual beauty it is which makes intelligence sure and strong. ”
-Poincare says that the best expression of the internal harmony of the world is the law- meaning that natural phenomena seem to appear with regularities than by chance. The point of physics is to understand these phenomena through mathematical formulas. However he doubts the possibility of a ‘Platonic’ reality existing outside of the human mind.
-He believes there are two types of mathematicians: logicians and intuitionists. The first are more likely to use analysis, while the latter are more likely to use geometry as they prefer to visualise maths in space.
-Poincare agrees with those claiming logic alone leads us to tautologies. Something more is needed in maths and that is intuition- Poincare seems to describe intuition as a mental process that transcends the evidence of the senses. Intuition is the ‘antidote of logic’. Intuition and logic are both needed: intuition is an instrument of invention, while logic is an instrument of demonstration.
-Mathematical induction is a synthetic a priori process.
CH.2: If we assume all parts of the universe are interchained, then a phenomenon cannot be the effect of a single cause but rather the result of infinitely many causes.
-Poincare criticises the idea of absolute space. If I want to represent myself reaching an object, then the coordinate system will reflect that it is relative to me that the exterior object is referred-“absolute space is nonsense, and it is necessary for us to begin by referring space to a system of axes ,invariably bound to our body ”
As in ‘Science and Hypothesis’ Poincare’ makes the case for the relativity of space. He also claims that we cannot be sure that the space has 3 dimensions- this is just a convention because empirically we find it easier to visualise space in such manner.
Poincare seems to suggest that it is the mind that creates for us the perception of space as a continuum of 3-dimensions. However Poincare is no solipsist: he adds that our mind does not create that perception out of nothing, but out of the ‘materials and models’ of the space in which we are present.
-The role of mathematics in physics to reveal the hidden harmony of things.
-“Today we no longer beg of nature; we command her”
-Poincare describes Aristotle as “the most scientific mind of antiquity”
-Poincare criticises Comte’s idea that science is only useful for practical purposes- for instance Comte believed it was useless to study the composition of the sun.
-The first crisis in mathematical physics has to do with particle physics and thermodynamics contrasting some aspects of Newtonian physics.
-Poincare criticises Leroy’s nominalism. ” Science foresees, and it is because it foresees, that it can be useful and serve as a rule of action”. Poincare recognises science is made of arbitrary conventions, however such conventions allow to tell us whether a fact is true or not after an empirical investigation. Scientists do not create facts, they only create the language to enunciate such facts.
In another digression on the nature of language, Poincare’ adds that “The possibility of translation implies the existence of an invariant…. [for example] to decipher a cryptogram is to seek what in this document remains invariant, when the letters are permuted”. For Poincare’, in science, the ‘invariant’ is the relation between the facts that we witness in nature. We explain such relations with scientific notation/language (and that is a convention rather than an ‘invariant’).
-The laws derived from our empirical observations are only approximations. The laws are not certain, but probable. Science progresses with laws more and more probable.
-On the objectivity of science “What guarantees the objectivity of the world in which we live is that this world is common to us with other thinking beings….Nothing is objective except what is identical for all”
-Science is about classification: it is about bringing together facts that explain the relation among things and phenomena. Science is ‘a system of relations’. These relations should help us to explain the ‘universal harmony’ inherent in all phenomena.
SCIENCE AND METHOD (1908) [some of the content of this book is very similar if not equal to the previous two]
-He criticises the idea that maths can be reduced to the rules of formal logic.
-Science is about having a method: since there are infinite facts, we need to choose the most ‘useful’ ones- this is called (citing Mach) ‘economy of thought’. The most ‘useful’ facts are those that repeat themselves in nature with greater frequency, those that are not isolated but that are related to multiple phenomena. Ideally such facts should lead us to the discovery of a new law. In the words of Poincare’, discovery is selection.
Poincare recalls a period of his life when he was working on Fuchsian functions. He had left Caen (the city where he lived) to arrive at Coutances to join a conference on geology. During the trip he had completely forgotten about his work on Fuchsian functions. Once arrived at Coutances, Poincare had the chance to have a break and visit the city. At the very beginning of his walk around Coutances, Poincare had a sudden realisation that his work on Fuchsian functions was actually related to non-Euclidean geometry. When Poincare’ returned to Caen, he developed this idea further and his initial intuition at Coutances was confirmed.
Amazed by this experience, Poincare’ later decided deliberately to go to the cliff and think about different issues than mathematics and see if his unconscious thinking would assist him again. Indeed, Poincare’ recalls that as we was walking in a cliff he suddenly grasped new important mathematical ideas after a long period where he had not thought about mathematics at all. Poincare’ claims that he conceived many more mathematical ideas in such manner.
He believes that after a period of hard-work and of conscious study into a problem, one should have a distraction for a while and think about other issues to let some mysterious ‘unconscious’ mechanisms to enlighten one at a later point. After this sudden illumination, another period of conscious work is needed to reflect on the new ideas.
“These sudden inspirations are never produced… except after some
days of voluntary efforts which appeared absolutely fruitless, in which one thought one had accomplished nothing, and seemed to be on a totally wrong track. These efforts, however, were not as barren as one thought; they set the unconscious machine in motion, and without them it would not have worked at all, and would not have produced anything”
Poincare claims that the ‘unconscious ego’ that is responsible for our ‘sudden illuminations’ is not as automatic or machine-like as we might think, but actually creative as well. Poincare claims that the outstanding results of the ‘unconscious ego’ may be due to his work in re-arranging the pieces of information that we have acquired during the conscious period of work into various combinations.
Poincare’ claims that the aesthetic beauty of mathematics comes from creation/discovery of well-ordered combinations.
If there exists a mind that knows all the causes and effects since the beginning of time, then for such mind the word ‘chance’ is meaningless (Poincare’ begins a discussion related to Laplace’s demon). Instead, for limited minds like ours, chance is an account of our ‘frailty and ignorance’. Poincare cites the laws of Gay-Lussac suggesting that in a gas the velocities of the molecules vary irregularly (so by chance).
Small causes escape our observations; therefore we attribute certain phenomena to chance. However even if we knew all the laws of the universe and the initial conditions of the universe at a given moment some of our predictions could fail- Poincare seems to suggest that this is due to non-linearity (small changes in the initial conditions leading to big changes in the future). The difficulty encountered in making accurate predictions in meteorology is due to non-linearity.
Poincare introduces concepts related to the relativity of space, the frame of reference and the deformation of space- he rules out the concept of absolute space.
“Absolute space exists no longer; there is only space relative to a certain initial position of the body [the human body]”. The ‘initial position’ that is chosen is arbitrary because I could pick any point my body has occupied in space.
Poincare’ believes that the space, viewed in three dimension, is simply a product of human intelligence. On the choice of Euclidean geometry: “We have chosen the most convenient space, but experience guided our choice.”
There are two types of mathematicians: logicians like Weierstrass or intuitionists like Riemann. For the idea of ‘continuity’ we must give credit to the intuitionists, however logicians are also needed to give clear definitions to mathematical concepts. Hence the problems related to the foundations of calculus have now ‘vanished’. Analysis is now made of numbers (finite or infinite) bound together with equalities and inequalities. Hence, as they say, mathematics has been ‘arithmetized’. However Poincare’ rejects the idea that logicians can represent the whole reality with just elementary propositions.
“it is by logic that we prove, but by intuition that we discover. To know how to criticize is good, but to know how to create is better.”
For Poincare’ mathematics is independent of the existence of material objects.
The method of induction is useful to ‘tame’ infinity since direct verification would be impossible when the number of propositions is infinite.
Poincare’ is sceptical about Russell’s logicism- the idea that once a few propositions has been stated, the rest of mathematics can be constructed without bringing in new elements. Poincare’ accuses logicians like Russell and Hilbert of making ‘petitio principii’ (circular reasoning) arguments in their foundations of logic.
In mathematics “you must be infallible or cease to exist.”.
To exist in mathematics is to be free of contradictions.
Poincare’ wonders whether nature is governed by caprice or harmony. He claims science is worth pursuing because it reveals the harmony of nature.
Scientists need to select information and make a hierarchy of facts. Facts that teach us laws are the ones to select. Unfortunately the “entanglement of these circumstances exceeds the compass of our mind”.
 ‘Science and Hypothesis’ (London and Newcastle-on-Tyne:
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 ‘The value of science’ by Henri Poincare (translation by George Bruce Halsted- New York The Science Press 1907)