Henri Poincaré

Henri Poincaré is invoked for a definition of continuity that exposes, rather than resolves, a central problem in Sartre’s account of temporality: how a continuum can join what it also separates. The reference gives Poincaré a precise philosophical role—as the source of a compact formula that Sartre examines, applies to time, and ultimately finds insufficient as an explanation.

The Definition of Continuity

Poincaré’s formula describes a series in which a equals b, b equals c, yet a does not equal c. Sartre calls the definition excellent because it foreshadows a kind of being that is what it is not and is not what it is. Its terms are linked through b, while the relation between a and c remains different from the relations that connect each of them to b.

That structure makes continuity more than a simple assertion of sameness. By the axiom, a equals c; by continuity itself, a does not equal c. Likewise, b is equal to both a and c, yet differs from itself insofar as a and c are not equal. Poincaré’s formulation thus presents continuity as a relation whose terms are held together without becoming simply identical.

Sartre’s Application to Time

Sartre uses the formula to illuminate the temporal relation between before and after. If instants a and c are distinct, continuity can place an intermediary b between them, however close together they may be. Because b is indistinguishable from a and indistinguishable from c, it can serve as the site at which the relation of before and after is realized: b is before itself in the sense that it is related to both distinct terms.

The example makes continuity’s explanatory appeal vivid, but also sharpens the question it cannot answer. What kind of being could b be, if it is at once related to two distinct instants and not divided into them? Sartre asks how this being can possess an ecstatic nature, why the division outlined within it is not completed, and how it avoids breaking apart into one term that dissolves into a and another that dissolves into c. The intermediary therefore does not eliminate the problem of temporal cohesion; it concentrates that problem in its own unity.

Limits of the Formula

Sartre’s criticism is not that Poincaré’s definition fails to describe a useful pattern. It is that the definition’s ingenious play among equality and inequality does not provide the principles or foundation of the being it describes. Even when the formula offers a way to understand how an intermediary can connect distinct instants, the question of how such an intermediary exists remains open. As Sartre puts it, everything still remains to be done.

The reference consequently marks both the power and the limit of formal definition. Poincaré’s formula gives Sartre a model for a being that is and is not, but does not account for the unity that would make such a being possible. Sartre suggests that a fuller examination might show that only the For-itself can exist in this ecstatic unity of self, while emphasizing that this examination has not been attempted. Continuity, in this passage, names a fact requiring explanation rather than supplying its own solution; the chronological order of the continuous cannot simply be identified with an order of identity.