The French Connection: Borda, Condorcet and the Mathematics of Voting Theory – The Technical Side of Borda’s and Condorcet’s Works

As noted in the introduction, the two works on the mathematics of voting theory written by Borda and Condorcet, respectively, were essentially forgotten until the twentieth century. In this section, we consider the purely mathematical aspects of their works. We first briefly comment on their relative publication dates.

At first glance, Borda’s work had priority, by more than a decade, over that of Condorcet. Borda’s “Mémoire sur les élections au scrutin” (“Memoir on elections by ballot”) appeared in the Annual Proceedings of the French Academy of Sciences for the year 1781, together with an “Analysis” of that memoir that was likely written by Condorcet (in an editorial capacity derived from his position as Permanent Secretary). In a footnote to the title of his paper, Borda further asserted that he had already presented the ideas it contained to the Academy some 14 years earlier, on June 16, 1770—a claim repeated by the author of the “Analysis” of Borda’s “Mémoire” (again, likely Condorcet himself), but for which no archival evidence has yet emerged.^[14] In fact, based on the available archival documentation, Brian [2008, p. 2] has shown that:

The date of 1781 is a fake. The story really begins the 14th of July of 1784, exactly five years before a date of some importance in the history of democracy.

It was on this “Bastille Day” date in 1784 that Condorcet, then Permanent Secretary of the Academy, announced his intention to publish a book on the application of probability to the analysis of electoral decisions. To do so under the auspices of the Imprimerie Royale, both a positive report by two referees and a vote of approval by the members of the Academy was required. The two referees were chosen by Condorcet, and the required report was read to a meeting of the Academy just three days later—such an astonishingly short turnaround time given the length and complexity of Condorcet’s tome that historians generally believe the report was written by Condorcet himself.^[15] Regardless of who actually wrote that report, it sufficed to secure the vote of approval necessary to go forward with publication. A possible delay then arose when Borda read his own paper on the arithmetic of voting to the Academy on July 21, 1784, apparently as a means to establish priority. In response to Borda’s paper, Condorcet took quick action to avoid a potential stumbling block to his own publication plans: using his authority as Permanent Secretary, he arranged for Borda’s paper to be included in the Academy’s proceedings that were then in preparation, as those would appear in print prior to the intended publication date for his own book. To quote Brian [2008, p. 3] once more:

This is why a paper, supposedly read in 1770, actually presented in 1784, was published in an official academic volume for the year 1781—the one under press in July 1784.

Because Condorcet’s Essai—actually written, supposedly reviewed by referees and already approved for publication by July 1784—was not officially published until 1785, Borda thus technically published on voting theory a year before Condorcet. To see why Condorcet’s work is nevertheless the more celebrated of the two as a pioneering contribution to voting theory, we now turn to a brief overview of the technical contents of both works, beginning with that of Borda.