26 Dec 2017

Goldschmidt ( Le système stoïcien et l'idée de temps, “Divisibilité du temps”, summary


by Corry Shores


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[The following is summary. Bracketed commentary is my own, as is any boldface. Proofreading is incomplete, which means typos are present, especially in the quotations. So consult the original text. Also, I welcome corrections to my interpretations, because I am not good enough with French or Greek to make accurate translations of the texts.]




Summary of


Victor Goldschmidt


Le système stoïcien et l'idée de temps


Première partie:

La théorie du temps et sa portée


A. La théorie du temps


III. La théorie du temps

Divisibilité du temps




Brief summary:

Chrysippus has a seemingly self-defeating notion of time. He says that no time is completely present, and yet only the present exists. This would seem to suggest that time does not exist. Chrysippus further clarifies that no time exists in the present in the strict sense rather than in the broad sense. The strict sense of the present is not something we actually experience. At best, we can form of concept of it as a limit between the past and future. Under such a conception,  we can think of the present as admitting of no past or future. But time can be said to exist in the broad sense when we think of how we experience the specious present as having some duration. So our senses tell us that there is time in the present, but this is only one sense of the term “present”, namely, the experienceable present. However, the other sense of “present,” the strict sense, is grasped not experientially but only mentally through mathematical procedures. If this sort of present has any reality, we can never actually grasp it as a real component of time. The reason for this has to do with the Stoic ideas regarding the infinite divisibility of continua, including bodies (as spatially extending things) and time (as a temporally extending and perhaps durational thing). When bodies or time are understood mathematically, we can divide them to infinity until arriving upon an infinity of indivisibles. This is already problematic, because suppose we divide a cone into an infinity of stacking circles. We begin by assuming that the cone has a smooth surface. So each ring and its neighbor cannot be of different sizes, because then the cone’s surface would be jagged. But, if they are all the same size, we have a cylinder and not a cone. We encounter a similar problem when we divide bodies and time infinitely. Suppose a body is divided into an infinity of indivisible parts. Those parts would need to lack extension, or else they would be divisible. But parts without extension cannot be parts of extending bodies, because their additive sum would not have extension. (Also, it is not clear how division can arrive upon them, because anything with extension when divided would seem to produce parts with extension, for otherwise the thing being divided would not have extension to begin with.) Similarly for time. Suppose we could infinitely divide time into instants. On the one hand, we cannot obtain indivisibles through division of divisibles. On the other hand, were we to have indivisibles of time and space, their sum could not be said to compose larger structures, because none have any extent or duration. So Chrysippus is saying that such indivisibles produced by a mathematical procedure of infinite division are beings of reason, and if they do have any reality, we can never know them in their actual reality. For in actual practice, we can only continue our divisions endlessly, never arriving upon the limit. Thus, in one sense (the broad sense) time is “real” and in anther sense (the strict sense) time is irreal (it subsists as an incorporeal something without existing corporeally).

[Note: it could instead be that time is simply real (lacking even non-existing susbistance) and the mathematical notion of infinitely divisible time is a misconception that tells us nothing about temporality itself. I will revise this if in the next section that seems to be the case, but I had the impression the next section would propose the Aiôn time which might capture the sense of the mathematical present and also the subsistence of the past and future.]





[The Divisibility of Time]

[The Non-Presence of Time and the Existence of the Present and Subsistence (Non-Existent Somethinghood) of the Past and Future]

[The Present in the Strict and Broad Sense. The Non-Presence of Time (Past and Future)]

[The Continuous Divisibility of Corporeality]

[The Mathematically Infinite Divisibility of Bodies and Time. The Impossibility of Actual Infinite Divisibility]

[Chrysippus and the Reality of Time]





Divisibilité du temps

[The Non-Presence of Time and the Existence of the Present and Subsistence (Non-Existent Somethinghood) of the Past and Future]


(p.37: “C’est ce qu’affirme très clairement sa thèse…”)


[In sum: For the Stoics, no time is completely present, and yet only the present exists, while the past and future subsist. We note the oddity here that it would seem we are to conclude that no time exists, even if the present exists.]


[We are continuing with Chrysippus’ definition of time as given by Stobaeus. See section] “He says most clearly that no time is wholly present” (Long and Sedley 1987: I, 304; II, 301-302. “C’est ce qu’affirme très clairement sa thèse : aucun temps n’est entièrement présent.”) But the text continues to claim that “only the present belongs; the past and the future subsist, but belong in no way (Long and Sedley 1987: I, 304; II, 301-302) / “only the present exists, whereas the past and future subsist but do not at all exist” (Inwood & Gerson 2008: 88. “seul, le présent existe ; le passé et le futur subsistent, mais n’existent pas du tout”). Although this might seem like a contradiction, we should not think of it as such, as Stobaeus claims that Chrysippus says this “most clearly”.

« C’est ce qu’affirme très clairement sa thèse : aucun temps n’est entièrement présent. » Mais la suite du texte soutient que « seul, le présent existe ; le passé et le futur subsistent, mais n’existent pas du tout ». S’il y a là une contradiction, elle ne peut être qu’apparente, puisqu’elle se trouverait dans un même passage où le doxographe, pour sa part, ne voit rien que de « très clair ».



[The Present in the Strict and Broad Sense. The Non-Presence of Time (Past and Future)]


(p.37: “ La thèse : « Aucun temps n’est entièrement présent »…”)


[In sum: the present can be understood in two senses. 1) In the strict sense as the physically real present that is the limit between past and future, admitting of no parts of them. 2) In the broad sense, as the specious present we experience, where it simply appears as if in the present there is also a little bit of the past that is passing away and little bit of the future that is now coming into being. So in reality, the past and the future do not exist. They rather are sayables that can only “exist” by being expressed by thoughts in the mind.]


The claim that no time is entirely present is reformulated in the conclusion: “Consequently no time is present exactly, but it is broadly said to be so” (Long and Sedley 1987: I, 304; II, 301-302) / “Consequently, no time is present in the strictest sense but only in a broad sense” (Inwood and Gerson 2008: 88. In the French: “Aucun temps n’est rigoureusement présent, mais on le dit (présent) selon une certaine étendue.”) So it is in the “strict” or “exact” (“rigoureux”) sense that time is not “wholly present” (“entièrement présent”). We see the sense of this distinction in the parallel text in Stobaeus attributed to Posidonius. Here “strict” or “exact” (“rigoureux”; perhaps ἀπαρτισμὸν in the Stobaeus/Chrysippus text and in the Stobaeus/Posidonius text) means “known” / “understood” (“saisi par la pensée” and possibly “νοεῖσθαι” in the Stobaeus/Posidonius text). And “broadly” or “in a broad sense” (“en étendue” and perhaps “πλάτος” in the Stobaeus/Posidonius text) means known by perception (“perceptible” or perhaps “πρὸς αἴσθησιν” in the Stobaeus/Posidonius text). The present that Chrysippus says exists is thus a “being of reason”; so, it is quite natural in this sensualist philosophy that such a being as this does not really exist. [The English translation for this part is: “Now and the like are thought of broadly and not exactly. (5) But now is also spoken of with reference to the least perceptible time encompassing the division of the future and the past” (Long and Sedley 1987: I, 305; II, 303-304) / “And the ‘now’ and similar expressions are time understood in a broad sense and not with precision. The ‘now’ and the minimal perceptible time are established around the division between future and past” (Inwood and Gerson 2008: 86-87). The idea here seems to be the following, but I am not sure. No time exists in the exact sense means that in reality, there is no past or future that inheres in the present. But we also have a phenomenological notion of the present as having a certain thickness including a little past that is going away and a little future that is coming to be. Goldschmidt might be saying that we are to understand the present taken in the broad sense to mean the specious present of sense experience, and the present taken in the exact sense to mean the real physical present, which admits of no past or future parts (and thus has no duration).] Thus these incorporeals [the past and the present], which are sayables, only exist in thought.

La thèse : « Aucun temps n’est entièrement présent », est précisée dans la conclusion : « Aucun temps n’est rigoureusement présent, mais on le dit (présent) selon une certaine étendue. » C’est donc au sens « rigoureux », qu’aucun temps n’est « entièrement présent ». Le sens de cette distinction nous est donné dans le texte parallèle de Posidonius. « Rigoureux » signifie : « saisi par la pensée » ; « en étendue » veut dire : « saisi par la sensation »2. Le présent dont Chrysippe conteste l’existence, est donc un « être de raison » ; il est très naturel, dans cette philosophie sensualiste, qu’un tel être n’existe pas. C’est le propre de ces incorporels que sont les exprimables, que de n’exister que dans la « pensée »3.

2. Posidonius ap. Ar. Did., 26 (Dox. gr., 461, 19-21) : [See the last two sentences of the following text

Posidonius in Stobaeus 1.105,17-106,4 in Anthologium, vol.1:

Posidonius in Stobaeus 1.105.SPosidonius in Stobaeus 1.106.S

(Stobaeus 1884a: 105-106)]

3. Diog. Laërt., VII, 63 (S.V.F., II, 181) : Φασὶ δὲ [τὸ] λεκτὸν εἶναι τὸ κατὰ φαντασίαν λογικὴν ὑφιστάμενον ; cf. p. 18, n. 4-5. Et le texte de Proclus, au sujet du temps : [See the part below beginning : ἕν γὰρ ἦν τῶν παρ᾽ αὐτοῖς and ending φιλαῖς] (in Plat. Tim., 271 d = S.V.F., II, 521).

SVF 251 Proclus Plat Tim.S

(SVF II, 521, p.166)


[The Continuous Divisibility of Corporeality]


(pp.37-38: “La preuve de l’inexistence de ce…”)


[In sum: The non-existence of the present is based on the Stoic argument against Epicurean atoms. Atoms are small parts of corporeal bodies, and so they are arrived upon by division. As corporeal, they are defined as having extension. And as atomic, they are defined as being indivisible. But anything with extension is divisible, for otherwise it would lack substantiality as a corporeality. Thus atoms are both divisible and indivisible, which is absurd. There are thus no atoms, and corporeal divisibility would have to continue to infinity, never arriving upon an indivisible part.]


The proof of the inexistence of the present is based on the infinite divisibility of continua. This particular theory of division is borrowed from Aristotle, and we note that for the Stoics, it has primarily a polemical intention, namely, to show that, contra Epicurus, the division of bodies can continue infinitely without ever arriving upon indivisible elements that are atoms. Under this presentation of the theory, we find that it involves a reduction to absurdity. For, it shows how the atomistic conception, when applied rigorously, destroys the so-called “indivisibles” [or “unbreakables”] and thereby destroys itself. [I am not certain, but the idea might be the following, and this is a guess. Suppose there are indivisible atoms. They would be arrived upon by dividing composites. We also assume that a body is something that has some extension, for otherwise it would have no substantiality in corporeality. But whatever has extension can be divided. So atoms both have and do not have extension, which is absurd. (They have extension because they are corporeal but they do not have extension because they are indivisible.) Thus there are no atoms.]

La preuve de l’inexistence de ce présent s’appuie sur la divisibilité à l’infini des continus. Cette théorie de la division est empruntée à Aristote, et l’on admet que, chez les Stoïciens, elle procède d’une intention surtout polémique : il s’agit de montrer, contre Epicure, que la division des corps peut se poursuivre à l’infini, sans que l’on puisse jamais rencontrer ces éléments « indivisibles » que seraient les atomes4. Si telle est bien la prétention de cette théorie, elle enveloppe donc une réduction à l’absurde ; elle fait voir que la conception atomiste, appliquée rigoureusement, dé- | truit les soi-disant insécables derniers et, par là, se détruit elle-même.

4. Voir E. Bréhier, Chrysippe, p.120.


[The Mathematically Infinite Divisibility of Bodies and Time. The Impossibility of Actual Infinite Divisibility]


(p.38: “Cette polémique implique deux idées …”)


[In sum: For the Stoics, it is only mentally that we can divide bodies and time infinitely such that we arrive upon indivisibles. But in actuality, such a dividing process can never finish. Thus Chrysippus says that bodies and time are infinitely divisible (mathematically) but in actuality bodies and time are only continuously divisible, never arriving upon an infinity of indivisibles.]


This polemic against atomism implies two ideas: {1} that whose non-existence we wish to demonstrate are not the real elements of things but are only those elements which are indivisible and which, for Epicurus and Democritus, and deprived of any sensible quality, making them, in the eyes of the Stoics, no more than “thought” [or mentally conceived] elements. {2} the division to infinity is made using a “dianoetic” [purely intellectual] method of mathematical analyses, which is completely unable to help us grasp the real elements of things. As such, we can apply them without much difficulty to both incorporeals and corporeals, even though in both cases we are only making the divisions in thought without ever making any divisions in real being. This is what Aetius explains (as it is in Stobaeus): “Chrysippus said that bodies are divided to infinity, and likewise things comparable to bodies, such as surface, line, place, void and time. But although these are divided to infinity, a body does not consist of infinitely many bodies, and the same applies to surface, line and place [<and void and time>]” (Long and Sedley 1987: I, 297; II, 296. Bracketed insertions added in accordance with the Long and Sedley II, p.296 footnote and the French text (see the comments following footnote 4 below): “Chrysippe a dit que les corps se divisent à l’infini, de même que les choses qui ressemblent aux corps, comme la surface, la ligne, le lieu, le vide, le temps ; si ces choses se divisent à l’infini, le corps n’est pas (pour autant) composé2 de corps infinis, pas plus que la surface3, ni la ligne, ni le lieu < ni le vide, ni le temps”.) [I gather that the idea here is the following, but I am still not entirely sure. For the Stoics, we can divide bodies to infinity only using mathematical methods and by considering them as mental entities. But were we to actually divide bodies or time, that is something that would never be completed, at least in a finite amount of time. So insofar as the presumed smallest parts are only obtained by actual division, we can say that bodies and time can be divided to infinity mathematically but only infinitely divided (continuously but never to completion) in actuality.] [I note something that I find odd at this point. We are saying that in actuality bodies and time are not divisible into an infinity of indivisibles. But we are also saying that the present in reality (or at least in the strict sense) has no parts. I am not confident in my interpretation so far. But I would have thought that we would say that in the mind the present admits of parts but in reality it does not. Perhaps the idea is the following. The present really does not have parts. We can at best obtain a mathematical notion of this, because it is not possible for humans to arrive upon in actuality. However, we should be cautious with this mathematical notion, because it is only a mental construction and it does not give us the real indivisible itself.]

Cette polémique implique deux idées, d’ailleurs solidaires : a) ce dont on veut ainsi démontrer l’inexistence, ce ne sont pas les éléments réels des choses, mais uniquement ces éléments insécables, invisibles et dépourvus, pour Epicure comme pour Démocrite, de toute qualité sensible1, donc, aux yeux des Stoïciens, des éléments simplement « pensés » ; b) la division à l’infini se fait selon une méthode d’analyse mathémathique, « dianoétique », qui se révèle radicalement impuissante à nous faire saisir les éléments réels des choses. Aussi peut-on l’appliquer sans inconvénient, non seulement aux incorporels, mais encore aux corps : dans les deux cas, on ne divise qu’ « en pensée », sans entamer l’être réel. C’est ce que nous explique Aëtius : « Chrysippe a dit que les corps se divisent à l’infini, de même que les choses qui ressemblent aux corps, comme la surface, la ligne, le lieu, le vide, le temps ; si ces choses se divisent à l’infini, le corps n’est pas (pour autant) composé2 de corps infinis, pas plus que la surface3, ni la ligne, ni le lieu < ni le vide, ni le temps >4 »5.


Démocrite, in Vors6., 68 A 49 (Galien, de elem. sec. Hipp., I, 2) ; Epicure, Lettre à Hérodote, 54.

2. Cf. Plut., de comm. not., 38, 1079 b-c (S.V.F., II, 483).

3. Il faut comprendre : «  ... n’est composée de surfaces », et de même pour les autres termes de l’énumération.

4. L’addition, due à Heeren, semble s’imposer ; l’hésitation de Diels (« ceterum dubitari potest de vacui et temporis notione corporea ») ne se justifie pas ; la corporéité n’est affirmée, ni du vide, ni du temps, dans ce texte, dont la thèse principale (la divisibilité à l’infini des corps, des lieux et des temps) est corroborée par Sextus, math., X, 142 (S.V.F., II, 491).

5. Aëtius, I, 16, 4 (Dox. gr., 315, 8-15 = S.V.F., II, 482).


[Regarding note 4, I cannot follow the explanations in Latin, but let me provide them for the record:

Stobaeus. Eclogae. Heeren p.345.S


Stobaeus. Eclogae. Heeren p.344.S


Stobaeus. Eclogae. Heeren p.344.fth.S


Stobaeus. Eclogae. Heeren p.345.fth.cont.S

(Stobaeus; Heeren 1792: 344-345)


Stobaeus. Eclogae. In Diels Dox Gr.p315.S

Stobaeus. Eclogae. In Diels Dox Gr.p315.ft15.S

(Stobaeus; Diels, Doxographi graeci. 1879: 315)

In Long and Sedley II, the additions are given only in footnote, and in Long and Sedley I, there is no inclusion or footnote.]


[Chrysippus and the Reality of Time]


(pp.38-39: “On ne saurait donc conclure de notre texte …”)


[In sum: So Chrysippus is not arguing that time, which is continuously divisible, is divisible in actuality to indivisible parts. Were he to argue such a thing, that would mean that time is composed of durationaless points where there is neither past nor future, but only a cut between them. And such an argument would imply that time is not real. But that cannot be Chrysippus’ argument, because then by the same operation of division bodies would be composed of non-extending parts, and surely Chrysippus is not arguing that bodies are irreal. Rather, these indivisibles are attainable only through mental operations and not in reality.]


So even though Chrysippus speaks of the infinite divisibility of time, we should not simply conclude that he was arguing for the irreality of time. For, were he doing so, we would have to conclude that bodies are not real. [I am not certain what is meant here, so I will guess that the idea is the following. Superficially we might note that Chrysippus says that bodies and time are divisible to infinity. That would leave time being composed of parts with no temporality, because these parts at best would be like cuts within the flow from future to past. And an infinity of cuts would not make time. This cannot be right, because if Chrysippus also meant that bodies are divisible into indivisible parts, that would mean that bodies are fundamentally composed of things without extension, which is also absurd.] Later we will ask if the theory of the division of continua for Chrysippus entails a positive counterpart. But for now, we simply note that irreality is affirmed of a time or of a present that we might claim to know by a dianoetic analysis. This is similar to the reasoning Chrysippus used when discussing the parts of a cone, which Democritus criticized. For surely the Stoics did not mean to conclude that the cone does not exist. [See the discussion here. Perhaps the idea is that for Chrysippus, only in the mind would there be an infinity of depthless circles making a cone. In reality, the cone would be made of very many ribbons set at the angle of the cone’s slope.] [The last idea might be: Rather, the present is to a certain extent real, and it is grasped by sensation.] [Note 6 will be important in the next section, so let us take a look at it now:] Recall that we said that irreality is affirmed of a time or of a present that we know by mental, mathematical operations. The time that is divided to infinity is “total time,” which extends infinitely into the past and future. But the division applies also to the present, which is limited, because the division cannot stop at an indivisible instant. [If we take present, understood as having a duration, we can also divide it continuously without arriving upon a durationless instant.] This is implied in our text, and Plutarch says it formally: “this is the result for the Stoics, who do not admit a minimal time or wish the now to be partless but claim that whatever one thinks one has grasped and is considering as present is in part future and in part past.” (Plutarch, On common conceptions 1081C, from Long and Sedley I, p.304. In the French: « Ils ne veulent | pas reconnaitre un instant sans parties ; si l’on croit saisir par la pensée un présent, ils répondent que ce présent est en partie du passé, en partie, du futur ». And in the Greek: “ὅ τι ἄν τις ὡς ἐνεστὼς οἴηται λαβὼν διανοεῖσθαι, τούτου τὸ μὲν μέλλον τὸ δὲ παρῳχημένον εἶναι φάσκουσιν”.)

On ne saurait donc conclure de notre texte que Chrysippe enseignant la divisibilité à l’infini du temps, ait voulu montrer l’irréalité de celui-ci, car une conclusion analogue réduirait également à néant, la réalité des corps. Nous nous demanderons plus loin si la théorie de la division des continus ne comporte pas, chez Chrysippe, une contre-partie positive. Pour l’instant, il suffit de voir que l’irréalité est affirmée d’un temps (ou d’un présent)6 que l’on prétendrait saisir par l’analyse dianoétique, de même que sont irréels les | disques innombrables dans lesquels Démocrite avait décomposé le cône1 ; de quoi, assurément, les Stoïciens n’entendaient pas conclure que le cône même n’existait pas2. Est réel, en revanche, le présent d’une certaine étendue et saisi par la sensation.


6. C’est « le temps total » qui, en tant que continu, comporte la division à l’infini (le temps total, qui est infini du côté du passé et du côté de l’avenir). Mais la division, puisqu’elle ne peut s’arrêter à aucun instant indivisible, s’applique également au présent (qui, lui, est limité) ; c’est ce qu’implique notre texte même, et c’est ce que nous dit formellement Plutarque ; « Ils ne veulent | pas reconnaitre un instant sans parties ; si l’on croit saisir par la pensée un présent, ils répondent que ce présent est en partie du passé, en partie, du futur » (ὅ τι ἄν τις ὡς ἐνεστὼς οἴηται λαβὼν διανοεῖσθαι, τούτου τὸ μὲν μέλλον τὸ δὲ παρῳχημένον εἶναι φάσκουσιν), de comm. not., 41, 1081 c (S.V.F, II, 519).

(38-39. Greek text copied from Perseus)

1. Plut., de comm. not., 39, 1079 e (S.V.F, II, 489).






Goldschmidt, Victor. (1953). Le système stoïcien et l'idée de temps. Paris: Vrin.



Also cited:


Diogenes Laertius. 1925b. Lives of Eminent Philosophers, vol.2. Translated by Robert D. Hicks. London: William Heinemann / New York: G.P. Putnam’s Sons.



Doxographi graeci. 1879. Edited by Hermann Diels. Berlin: Reimer.

PDF available at:



Inwood, Brad, and Gerson, Loyd P. 2008. The Stoics Reader. Selected Writings and Testimonia, edited and translated by Brad Inwood and Loyd P. Gerson. Indianapolis and Cambridge: Hackett.


Long, Anthony A. and David N. Sedley. 1987. The Hellenistic Philosophers, vol.1: Translations of the Principle Sources, with Philosophical Commentary. Cambridge: Cambridge University Press.


Long, Anthony A. and David N. Sedley. 1987. The Hellenistic Philosophers, vol.2: Greek and Latin Texts with Notes and Bibliography. Cambridge: Cambridge University Press.


Plutarch. De communibus notitiis contra Stoicos. Taken from:



Stobaeus. 1792.  Ioannis Stobaei. Eclogarum physicarum et ethicarum. Libri duo. Pars Prima. Physica continens. Edited by Arnold Heeren. Göttingen: Vandenhoeck and Ruprecht.

PDF available at:



Stobaeus. 1884a. Ioannis Stobaei: Anthologium, vol.1. [Ioannis Stobaei, Anthologium Volumen Primum, Anthologii Librum Primum Volumen I: Libri duo Priores qui inscribi solent Eclogae Physicae et Ethicae] Edited by Kurt Wachsmuth. Berlin: Weidmann.

PDF at:



SVF. 1964b. Stoicorum veterum fragmenta, vol.2: Chrysippi Fragmenta Logica et Physica. Ed.  Hans von Arnim. Stuttgart: Teubner.

PDF available at:










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