## Jacques Hadamard's failures

 Jacques Hadamard is one of the few mathematicians who tried to write popular books which explained the thought processes which go into mathematical discovery. Of course there is the other side of the coin, the failure of a mathematician to spot something important which later is found by someone else. In the quotation below, taken from the beautiful book J Hadamard, The psychology of invention in the mathematical field (Dover, New York, 1954):

Every scientist can probably record similar failures. In my own case, I have several times happened to overlook results which ought to have struck me blind, as being immediate consequences of other ones which I had obtained. Most of these failures proceed from the cause which we have just mentioned, viz., from attention too narrowly directed.

The first instance I remember in my life had to do with a formula which I obtained at the very beginning of my research work. I decided not to publish it and to wait till I could deduce some significant consequences from it. At that time, all my thoughts, like many other analysts', were concentrated on one question, the proof of the celebrated "Picard's theorem." Now, my formula most obviously gave one of the chief results which I found four years later by a much more complicated way: a thing which I was never aware of until years after, when Jensen published that formula and noted, as an evident consequence, the results which, happily for my self-esteem, I had obtained in the meanwhile. It is clear that, in 1888, I had thought too exclusively of Picard's theorem.

My next work was my thesis. Two theorems, important to the subject, were such obvious and immediate consequences of the ideas contained therein that, years later, other authors imputed them to me, and I was obliged to confess that, evident as they were, I had not perceived them.

Some years later, I was interested in generalizing to hyperspaces the classic notion of curvature of surfaces. I had to deal with Riemann's notion of curvature in hyperspaces, which is the generalization of the more elementary notion of the curvature of a surface in ordinary space. What interested me was to obtain that Riemann curvature is the curvature of a certain surface S, drawn in the considered hyperspace, the shape of S being chosen in order to reduce the curvature to a minimum. I succeeded in showing that the minimum thus obtained was precisely Riemann's expression; only, thinking of that question, I neglected to take into consideration the circumstances under which the minimum is reached, i.e., the proper way of constructing S in order to reach the minimum. Now, investigating that would have led me to the principle of the so-called "Absolute Differential Calculus," the discovery of which belongs to Ricci and Levi Civita.

Absolute differential calculus is closely connected with the theory of relativity; and in this connection, I must confess that, having observed that the equation of propagation of light is invariant under a set of transformations (what is now known as Lorentz's group) by which space and time are combined together, I added that "such transformations are obviously devoid of physical meaning." Now, these transformations, supposedly without any physical meaning, are the base of Einstein's theory.

JOC/EFR December 2008