Jacques Hadamard was a great mathematician who studied his thought processes in solving mathematical problems. He interviewed many other leading mathematicians and scientists, including Henri Poincaré, many of whom shared Hadamard's experience that solutions to problems often came suddenly and completely, generally after long reflections on the problem.

In his 1945 book Psychology of Invention in the Mathematical Field, Hadamard described the Synthèse conference in Paris in 1936 to study creativity. In Chapter III, Hadamard described how the combination of random ideas could lead to a choice of the best combination. Chance alone is not enough.

Combination of Ideas. What we just observed concerning the unconscious in general will be seen again from another angle, when speaking of its relations with discovery. We shall see a little later that the possibility of imputing discovery to pure chance is already excluded by Poincaré's observations, when more attentively considered.

On the contrary, that there is an intervention of chance but also a necessary work of unconsciousness, the latter implying and not contradicting the former, appears, as Poincaré shows, when we take account not merely of the results of introspection, but of the very nature of the question.

Indeed, it is obvious that invention or discovery, be it in mathematics or anywhere else, takes place by combining ideas.¹ Now, there is an extremely great number of such combinations, most of which are devoid of interest, while, on the contrary, very few of them can be fruitful. Which ones does our mind — I mean our conscious mind — perceive? Only the fruitful ones, or exceptionally, some which could be fruitful.

1. Max Müller observes that the Latin verb "cogito," for "to think," etymologically means "to shake together." St. Augustine had already noticed that and also observed that "intelligo" means to "select among," In a curious connection with what we say in the text.

However, to find these, it has been necessary to construct the very numerous possible combinations, among which the useful ones are to be found.

It cannot be avoided that this first operation take place, to a certain extent, at random, so that the role of chance is hardly doubtful in this first step of the mental process. But we see that that intervention of chance occurs inside the unconscious: for most of these combinations — more exactly, all those which are useless — remain unknown to us.

Moreover, this shows us again the manifold character of the unconscious, which is necessary to construct those numerous combinations and to compare them with each other.

The Following Step. It is obvious that this first process, this building up of numerous combinations, is only the beginning of creation, even, as we should say, preliminary to it. As we just saw, and as Poincaré observes, to create consists precisely in not making useless combinations and in examining only those which are useful and which are only a small minority. Invention is discernment, choice.

To Invent Is to Choose. This very remarkable conclusion appears the more striking if we compare it with what Paul Valéry writes in the Nouvelle Revue Française: "It takes two to invent anything. The one makes up combinations; the other one chooses, recognizes what he wishes and what is important to him in the mass of the things which the former has imparted to him.

"What we call genius is much less the work of the first one than the readiness of the second one to grasp the value of what has been laid before him and to choose it."

We see how beautifully the mathematician and the poet agree in that fundamental view of invention consisting of a choice.

Esthetics in Invention. How can such a choice be made? The rules which must guide it "are extremely fine and delicate. It is almost impossible to state them precisely; they are felt rather than formulated. Under these conditions, how can we imagine a sieve capable of applying them mechanically?"

Though we do not directly see this sieve at work, we can answer the question, because we are aware of the results it affords, i.e., the combinations of ideas which are perceived by our conscious mind. This result is not doubtful. "The privileged unconscious phenomena, those susceptible of becoming conscious, are those which, directly or indirectly, affect most profoundly our emotional sensibility.

"It may be surprising to see emotional sensibility invoked à propos of mathematical demonstrations which, it would seem, can interest only the intellect. This would be to forget the feeling of mathematical beauty, of the harmony of numbers and forms, of geometric elegance. This is a true esthetic feeling that all real mathematicians know, and surely it belongs to emotional sensibility."

That an affective element is an essential part in every discovery or invention is only too evident, and has been insisted upon by several thinkers; indeed, it is clear that no significant discovery or invention can take place without the will of finding. But with Poincaré, we see something else, the intervention of the sense of beauty playing its part as an indispensable means of finding. We have reached the double conclusion:

that invention is choice
that this choice is imperatively governed by the sense of scientific beauty.

Hadamard described Poincaré as the source of the basic idea, but he credited Paul Valéry with the idea that there are two stages in creativity, perhaps even two entities - one to generate random alternative_possibilities, and the other to select or choose the best alternative.

These suggestions of Hadamard's were a major influence on Daniel Dennett's 1978 two-stages model of decision making which were later dubbed "Valerian." Dennett quotes the Valéry "It takes two to invent anything...," and then imagines the two stages in one mind.

Hadamard quoted Mozart to show that the first stage involves ideas that just "come to us" freely.

When I feel well and in a good humour, or when I am taking a drive or walking after a good meal, or in the night when I cannot sleep, thoughts crowd into my mind as easily as you could wish. Whence and how do they come? I do not know and I have nothing to do with it. Those which please me I keep in my head and hum them; at least others have told me that I do so....Then my soul is on fire with inspiration.

Hadamard and Poincaré both describe ideas that "present themselves" as William James described it.

In 1906 Hadamard wrote a review of Josiah Willard Gibbs' Elementary Principles of Statistical Mechanics. (Bulletin of the American Mathematical Society, 12, p.194-210) He called it pure mathematics, applying the calculus of probabilities (of Laplace and others) to mechanics.

He wrote:

It remains to address the most important and most delicate that raises the study of the distribution phase. What happens to this distribution in the course of the movement, when part of any state: it tends, for example, to move closer to the canonical distribution or any distribution with similar properties?

This is, in short, the vital issue for kinetic theories. The paradox related and which seems at first to determine in advance any theory of this kind is as follows. How can the equations of dynamics, which are all reversible, lead to irreversible laws, to the growth of entropy? (p.201)

[Il reste à traiter la question la plus importante et la plus délicate que soulève cette étude de la distribution en phase. Que devient cette distribution au cours du mouvement, lorsqu'on part d'un état quelconque: tend-elle, par exemple, à se rapprocher de la distribution canonique ou d'une distribution présentant des propriétés analogues?

C'est, en somme, la question vitale pour les théories cinétiques. Le paradoxe qui s'y rattache et qui semble, au premier abord, miner par avance toute théorie de cette nature est eu effet le suivant. Comment, eu partant d'équations de la dynamique, toutes réversibles, parviendra-t-on à des lois irréversibles, à la croissance de l'entropie?]

We develop an answer in our treatment of micro-irreversibility.