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Synopsis of Marcel Françon, ‘Le langage mathématique de Jean-Jacques Rousseau’

[‘The Mathematical Language of Jean-Jacques Rousseau’]

CpA 8.4:85–93

Born in Lyon in 1900, Marcel Françon initially studied the sciences before a fellowship from the Harvard Club of France allowed him to pursue graduate work in literature in the United States. From 1924 until his retirement in 1966, Françon spent the rest of his career at Harvard, where he specialized in sixteenth-century French literature. In this short article, first published in 1949 in Isis, an American journal devoted to the history of science, he argues that the mathematical language found in Book III, Chapter One of the Social Contract only makes sense if we recognize Rousseau’s equivocal use of the term rapport [relation] as evoking both mathematical and common usage, and if we account for changes in the mathematical meaning of certain concepts between the eighteenth and twentieth centuries.

In this critical chapter of the Social Contract, Rousseau attempts to articulate the relation among three terms, the Sovereign, the Government, and the Citizens, within the total body that is the State. He begins his analysis with a biophysical analogy, arguing that the relation of legislative power to executive power in a political body is similar to that between moral will and physical power in a human body. In order for any act to take place, the two causes must converge, or work together. To take a step forward, I must will it, but the physical capacity for the step must be there as well. Rousseau writes: ‘If a paralytic wills to run and an active man wills not to, they will both stay where they are’. Very quickly, however, Rousseau turns to a set of mathematical examples to describe the relation in question and the mediate term necessary to account for the relation between these two causes.

Françon provides the preliminary mathematical formulations necessary to make sense of the ‘continuous proportion’ of Government’s relation to the Sovereign and to the Citizenry (the people) that Rousseau seeks to articulate. If we are given four numbers, A, B, C, D, they are said to form a proportion when AB = CD. In this formulation A and D are the extreme terms, where B and C are the average terms. A continuous proportion is formed when the consequent term of the first relation (i.e., B) is equal to, or shares the same cause as, the antecedent term of the second relation (i.e., C). In this scenario, if the average of AB and the average of CD are equal, then we have a situation where there are only three terms, since B and C are effectively equal to one another. To formalize it: A : B :: B : C, or A  B = B  C. The key point is that, since the product of the extremes is (or should be) equal to the product of the averages, B serves as the proportional average of A and C.

In Rousseau’s rubric, the Sovereign is A, the Government is B, and the Citizens as subjects are C. What this means is that the site of equality is always in the Government itself, as the site where the relation of A to C is effectively averaged out. Françon suggests that when Rousseau says there is a only one ‘good Government’ per state, what he means is that there is only one possible correct average, only one B that is the proportional average of A and C. Rousseau inputs numbers to make his point more strongly. He asks us to imagine a state with 10,000 citizens. The Sovereign is the composite of all the subjects, hence its ‘value’ is 10,000. As each subject is one, a unity in itself, the relation of the Sovereign to an individual subject is 10,000  1. Rousseau observes that as the Sovereign grows in strength, i.e. number, there is a proportional diminution in the freedom of each subject. Here Rousseau exploits the dual sense of the word rapport [relation]: ‘When I say the relation increases, I mean that it grows more unequal. Thus the greater it is in the geometrical sense, the less relation there is in the ordinary sense of the word. In the former sense, the relation, considered according to quantity, is expressed by the quotient [exposant]; in the latter, considered according to identity, it is reckoned by similarity’. Françon observes that what makes the passage difficult to understand is that, whereas the French word ‘exposant’ currently means ‘exponent’ in French, in Rousseau’s day this was the technical term used to describe what we now call, in both English and French, a ‘quotient’. Citing one of d’Alembert’s entries in the Dictionnaire encyclopedique, Françon writes: ‘The “exponent of a geometrical reasoning” used to be the “quotient of the division of the consequent by the antecedent”’.

Françon then explains Rousseau’s use of the peculiar expression ‘raison doublée’, which G.D. Cole translates as ‘duplicate ratio’. The ‘raison doublée’ occurs when two pre-existing relations are related to one another. Take, for example, two pairs A  B and C  D. The product, or ‘raison doublée’ of these two pairs would be (A × B)  (C × D). But since Rousseau has already established that we are dealing with a ‘continuous proportion’ with one middle term, the formula is more accurately rendered: (A × B)  (B × C). Moreover, since C, standing in each case for an individual citizen, is always equal to 1, the value of B is fully depending on changes in the value of A. With this formulation in mind, Rousseau concludes, ‘From this we see that there is not a single unique and absolute form of government, but as many governments differing in nature as there are States differing in size’. Moreover, since it was already established that B is the proportional average of A and C, that is, A × C = B2, and C is always simply 1, then B = A. Or, the Government is the square root of the Sovereign.

Françon cites Rousseau himself to remind us that such a proposition is illustrative at best. But we can nonetheless appreciate the likely appeal of Françon’s distillation of this dense text for the editorial board of the Cahiers pour l’Analyse. Elsewhere in this same chapter of the Social Contract, Rousseau argues that ‘in order that the body of the government may have a true existence and a real life distinguishing it from the body of the State, and in order that all its members may be able to act in concert and fulfil the end for which it was instituted, it must have a particular personality, or self [il faut un moi particulier - Rousseau’s emphasis], a sensibility common to its members, and a force and will of its own which ends toward preservation’ [Cole translation modified]. In other words, this moi that is the Government in its relation to the Sovereign will, the various Citizens as subjects, and the State as a whole, is represented in Rousseau’s algebra by the variable B. The Government is the mediate term between Sovereign will and the sheer physical force of the voluminous political body, comprised of discrete citizens. In psychoanalytic terms, it is the Ego [moi] mediating the moral Super-ego and the rumbling Id.

Taking his lead from Freud’s ‘The Ego and the Id’, Lacan was keen to emphasize the primordial connection between the Id and the Super-ego, the Real and the Symbolic. Likewise, for Rousseau, the Sovereign will and the composite of subjects are but two sides of the same coin, two expressions of the same entity. Moreover, Lacan himself attempted an algebraic expression of that which was signified by the proper name, ultimately reaching a conclusion similar to Rousseau’s own. In his 1960 essay, ‘The Subversion of the Subject and the Dialectic of Desire’, Lacan suggests that the signifier of the proper name can be represented by a 1. When something is named, or called into being, the lack inscribed in its origin by dint of the fact that it did not exist before remains. Hence Lacan’s option for 1, to communicate this lack that is essential to what is signified. Lacan then tries to determine what this proper name signifies or is ‘equal to’. Lacan defines signification, in this essay, as the relation of the signifier to the signified; the result of this relation, then, is the statement. He ventures that, even though it is the subject’s submission to signification, represented as follows:


that results in the statement s, hence:

 s = s

when we venture that the process of signification is effectively equal to the statement (that is, s = s) we get:

1 = s2

More to the point, however, we are left with the following conclusion:

1 = s

For Lacan, that which is signified in the signifying chain via the proper name is something as unthinkable as the square root of negative one. And yet, this is the supposed site of the moi, that which holds the domains of the signifier and the signified together. Just as Lacan posits a momentary functional equivalence between signification and the statement, commuting the properties of one to the other, so too does Rousseau commute the totality of subjects to the sovereign. A × C = A because C is always equal to one. But A × C must equal the proportional average, which is B2. The government as moi will be the square root of the sovereign. It is different from the sovereign, yet fully dependent upon it, because it only exists through a constant relation to it. In the same way, for Lacan, that which is signified is dependent upon the fluctuating chain of signification, even as it remains distinct from it. However, its very distinction from signification is predicated upon its primordial relation to the signification which brought it into being in the first place.

In 1949, with no recourse to (and presumably no foreknowledge of) Lacanian formulations, Françon concludes with praise for Rousseau’s recognition that distinction is always predicated upon a relation: ‘As contestable as it might seem at first to apply mathematics to questions of politics…we must nonetheless acknowledge that Rousseau succeeded in illuminating one of the most important ideas, that of the relativity of forms of government, and that he was, moreover, able to insist upon a distinction that one must deem fundamental to democratic states: the distinction he made between the government and the sovereign’ (88).

References to this text in other articles in the Cahiers pour l’Analyse:


English translation:


Primary bibliography:

  • Rousseau, Jean-Jacques. Du Contrat social. Maurice Halbwachs, ed., Paris: Aubier, 1943.
  • ---. ‘The Social Contract’ and Other Later Political Writings. Victor Gourevitch, ed., Cambridge: Cambridge University Press, 1997.

Secondary bibliography:

  • Francia: Periodico di Cultura Francese, n. 16, ottobre-dicembre 1975, Napoli, ‘Numero speciale dedicato a Marcel Françon’.
  • Fink, Bruce. ‘The Lacanian Phallus and the Square Root of Negative One’, in Lacan to the Letter: Reading Ecrits Closely. Minneapolis: University of Minnesota Press, 2004, pp. 129-140.
  • Lacan, Jacques. ‘The Subversion of the Subject and the Dialectic of Desire’, in Écrits (1966), trans. Bruce Fink, in collaboration with Héloïse Fink and Russell Grigg. New York: W.W. Norton, 2006.