Calculated Risks, Condorcet, Bernoulli, d’Alembert and Inoculation

Diderot and d’Alembert’s Encyclopédie contains an entry for “Conjecture.” Penned by Diderot, it starts:

Jugement fondé sur des preuves qui n’ont qu’un certain degré de vrais-semblance, c’est-à-dire sur des circonstances dont l’existence n’a pas une liaison assez étroite avec la chose qu’on en conclut, pour qu’on puisse assûrer positivement que les unes étant, l’autre sera ou ne sera pas: mais qu’est-ce qui met en état d’apprétier cette liaison? L’expérience seule.

The article clearly defends the idea that trials should be the basis for scientific research and allow one to go beyond mere conjecture and reach proof. As I would like to show, philosophical, moral and scientific imperatives collided at times, making the transition from conjecture to proof all but impossible.

Smallpox inoculation was an eighteenth-century debate which was to set standards for hypotheses and conclusions. It was a dangerous operation as it involved artificially administering an incurable illness and hoping for the best, in order to guarantee the inoculated patient—assuming he or she survived the operation—immunity from subsequent attacks of the disease. It was the object of numerous debates, in particular in France. In itself it was based on conjecture: nobody was able, at the time, to understand why the same strain of smallpox would be more severe if caught naturally than if inoculated—we now know that subcutaneous administration of the disease allows the body time to create antibodies. In the debate regarding whether or not to practice this dangerous method, lobbies and individuals raised different questions: could one usurp God’s right to decide who should become ill? If one had one’s child inoculated and he or she died as a result, could one be considered to be his or her murderer? Was it not best to let nature take its course? And so on. To make the method seem acceptable, a notion of calculated risk had to be put forward.

As early as 1754, La Condamine had stressed: “ce n’est point ici une question de morale, c’est une affaire de calcul. Ne faisons point un cas de conscience d’un problème d’arithmétique.” If the human body can be treated and strengthened, the social body can too, with striking results—at least this was the idea promoted by many thinkers. It implied that individuals could be treated as statistics and led to the recurring use of images: troops in battle with a wise general who manages to spare most of them as against an unwise one whose men are massacred; lotteries in which there were fewer or more winning tickets. It also had a convenient catchphrase which La Condamine probably coined: “La petite vérole nous décime, l’inoculation nous millésime.” The method was adopted more readily in Britain than in France, according to a remark by Duboueix in the Journal encyclopédique in March 1774 because “les Anglais, nos voisins, [sont] plus sages que nous en ceci, et bien meilleurs calculateurs.”¹ I should like to look at what the presentation of calculated risks in favor of inoculation becomes in texts by Condorcet, Bernoulli and d’Alembert, and to argue that the latter two in particular used inoculation to promote developments in mathematics which were to have implications far beyond the field of medicine.

In order to try and make the questions raised scientific, the defenders of inoculation had to wrest the debate from philosophical, moral or theological ones. In his eulogy of the most famous inoculator to practice in France, Théodore Tronchin, a Protestant from Geneva, Condorcet questions the use of nature as an alternative course to medical intervention:

Ceux surtout qui parlent de médecine font souvent de la nature une espèce d’être moral qui a des volontés, qui supporte impatiemment la contradiction, qui a quelquefois assez de sagacité pour sauver le malade et bien diriger ses efforts, mais qui, malgré les bonnes intentions qu’on lui suppose, est sujette à se tromper presque aussi souvent que...