Terence Blake’s ‘BLOODSHOT EYES AND VISION CONCEPTS: from indifference to disapproval of thinking’

 

by Corry Shores

 

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Terence Blake

 

BLOODSHOT EYES AND VISION CONCEPTS: from indifference to disapproval of thinking

 

In this post Blake discusses the notions of horizon and the infinite in Deleuze and Guattari’s What Is Philosophy? Here he offers a useful translation and explanation of a particular passage:

The thinkers are unreasonable and head for the horizon. We know that in science the horizon is only relative: “What is primary in science is relative light or the relative horizon” (WHAT IS PHILOSOPHY?, 42, translation modified by me to bring out the idea that the light of science is relative too, and not just the horizon). The philosopher “heads for the horizon”, that is to say “plunges into the infinite”., his or her horizon is absolute, as is the light. This movement is both physical and mental (“in this respect chaos has as much a mental as a physical existence”, 42). If we “return with bloodshot eyes” it is not only because of an excess of alcohol or of light (this is the physical side of the “unreasonable”, pathological or esoteric measures) but also because of our vision of a power that is almost too strong for us. I say “almost too strong”, because in this text the thinker comes back, only changed, with “bloodshot eyes” and with new vision and new concepts. The eyes of the mind have been opened and strained to their limits.

Both these movements (heading out to, and coming back from ,the infinite) are necessary to thinking. Heading out unreasonably, dangerously and coming back bearing the mark (bloodshot eyes, or in some cases worse) of the voyage towards (which is “inner” only in the sense of being noetic or intensive), or of the encounter with, the horizon, but bearing also the vision, the percepts and the concepts. This double movement is what gives consistency to our philosophical territory: a territory is constituted by the movement of leaving it, which also means exposing oneself to risk, and also by the movement of returning back with a new song or a new colour, a new posture, or a new scent.

Already by heading off outside we risk indifference turning into “disapproval” (42), because the danger becomes obvious. Academic philosophy is not usually very perilous , but there is the danger to one’s career and to that of one’s friends or allies. This is what Deleuze and Guattari call “obvious” danger, easily recognizable. The disapproval is redoubled when one brings back “outlandish” concepts (according to Deleuze in the ABC PRIMER “outlandish” is a good synonym for deterritorialised).

(Terence Blake)

 

Note: I still have not found Melville’s use of this term, at least in a way that would seem to correspond to deterritorialization. But I also do not know where to look.

 

 

 

 

 

 

 

 

 

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Goldschmidt (1.1.3.14) 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

 

1.1.3.14

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.]

 

 

 

Contents

 

1.1.3.14

[The Divisibility of Time]

 

1.1.3.14.1

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

 

1.1.3.14.2

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

 

1.1.3.14.3

[The Continuous Divisibility of Corporeality]

 

1.1.3.14.4

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

 

1.1.3.14.5

[Chrysippus and the Reality of Time]

 

 

 

Summary

 

1.1.3.14

Divisibilité du temps

 

1.1.3.14.1

[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 1.1.3.10.] “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 ».

(37)

 

1.1.3.14.2

 

[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:

(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 II, 521, p.166)

 

 

1.1.3.14.3

 

[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.

 

 

1.1.3.14.4

 

[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.

(38)

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).

(38)

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

(345)

(344)

(344)

(Stobaeus; Heeren 1792: 344-345)

 

(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.]

 

 

1.1.3.14.5

 

[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.

(38-39)

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).

 

 

 

 

From:

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.

http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0258%3Abook%3D7%3Achapter%3D1

 

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

PDF available at:

https://archive.org/details/doxographigraec00dielgoog

 

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:

http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A2008.01.0392%3Astephpage%3D1058f

 

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:

https://books.google.com.tr/books?id=61wUAAAAQAAJ&dq=stobaeus+Heeren&source=gbs_navlinks_s

 

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:

https://archive.org/details/adw8682.0001.001.umich.edu

 

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

PDF available at:

https://archive.org/details/stoicorumveterum02arniuoft

 

 

 

 

 

 

 

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Goldschmidt (1.1.3.13) Le système stoïcien et l'idée de temps, “Temps infini et temps limité”, 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

 

1.1.3.13

Temps infini et temps limité

 

 

 

Brief summary:

For the Stoics, time can be understood as a line stretching infinitely into the past in one direction and infinitely into the future in another. Although time is infinite, it has parts which are either infinite or finite: time is “bound” on the far extremities by the limitless limits of the infinite past and infinite future; but past and future are limited on the inside by the finite present, being a limited limit to both past and future.

 

 

 

 

 

Contents

 

1.1.3.13

[Infinite and Finite Time]

 

1.1.3.13.1

[Stoic Time as “Interval” of Motion, and the Opposition of Past and Future to the Present]

 

1.1.3.13.2

[Time in Terms of Part-Whole Structures and the Totality Encompassing the Corporeal and Incorporeal]

 

1.1.3.13.3

[Time’s Unlimited Ends and Limited Present]

 

 

 

 

Summary

 

1.1.3.13

Temps infini et temps limité

 

 

1.1.3.13.1

[Stoic Time as “Interval” of Motion, and the Opposition of Past and Future to the Present]

 

(pp.35-36: “Par delà ses résonances aristotéliciennes …”)

 

[In sum: What makes the Stoic theory of time different from Aristotle’s is that the Stoics speak of time as being the “interval” of motion, while for Aristotle it is the “number”. Also, the Stoics oppose the past and future to the present.]

 

[Previously in section 1.1.3.10, we examined Chrysippus’ definition of time, and in section 1.1.3.11 we examined Aristotle’s. Then in section 1.1.3.12 we discussed how Chrysippus’ definition of time as the interval of movement echoes Aristotle’s definition of time as the number of movement. See especially section 1.1.3.12.3.] Despite the resonances between Chrysippus’ and Aristotle’s definitions of time, we see already at the beginning of the definition attributed to Chrysippus by Stobaeus that the Stoic theory has an originality to it. Now we will examine the important term “interval” in the Stoic theory, which takes the place of “number” in Aristotle. We will also see how for the Stoics the past and future stand in opposition to the present.

Par delà ses résonances aristotéliciennes, le début de notre texte fait déjà entrevoir l’originalité de la théorie stoïcienne. La suite immédiate va expliquer le terme par où cette théorie s’oppose directement à Aristote : « intervalle », substitué à « nombre »; et le commentaire mettra | d’emblée l’accent sur la thèse fondamentale : passé et futur, opposés en bloc au présent.

(35-36)

 

 

1.1.3.13.2

 

[Time in Terms of Part-Whole Structures and the Totality Encompassing the Corporeal and Incorporeal]

 

(p.36: “Le temps se prend dans deux acceptions, ainsi que la terre …”)

 

[In sum: The Stoic notion of time is bound up with certain Stoic part-whole structures, like the part-whole structure involved in their cosmology, where there is a totality that encompasses both the corporeal and incorporeal.]

 

“[T]ime is spoken of in two senses, as are earth and sea and void and the universe and its parts” (Stobaeus, in Inwood and Gerson 2008: 88. In the French as: “Le temps se prend dans deux acceptions, ainsi que la terre, la mer et le vide : (on peut en considérer) le tout ou les parties.”) We see this part-whole distinction made in another context by Diogenes Laërtius: “Both substance and matter are terms used in a twofold sense according as they signify (1) universal or (2) particular substance or matter. The former neither increases nor diminishes, while the matter of particular things both increases and diminishes” (Diogenes Laertius 1925b. Book 7, Ch.1, section 150. Copied from Perseus. In the French as “La substance, c’est-à-dire la matière, se dit dans deux sens : celle de toutes choses et celle des êtres particuliers ; la première ne s’accroît ni ne diminue ; l’autre s’accroît et diminue.”) The part-whole distinction is also made by the first two examples [either just earth and sea, or perhaps alternatively, earth and sea on the one hand and void on the other. I am not exactly sure yet how in either case there is a part-whole relation. Earth and sea would not seem to be parts related to the void as a whole. I wonder if the idea is simply that earth and sea are wholes that are divisible.] There is a third example for this part-whole structure, namely time: “Just as the void in its totality is infinite in every respect, so time in its totality is infinite on either side. For both the past and the future are infinite” (Long & Sedley 1987: I, 304; II, 301-302. In the French: “De même que le vide total est infini de toutes parts, de même le temps total est infini à ses deux extrémités ; en effet, le passé et le futur sont infinis.”) The phrase “in its totality” (“total”), which is used to designate the whole of void and time, is also what in another division incorporates the world and the void. So in Stoic cosmology, the totality is the most general term, including both being and the incorporeal condition of being. In other words, “totality” is a term that applies to both the corporeal and the incorporeal, to the “whole” (“tout”) as well as to the “parts” (“parties”) that are virtually contained in it.

« Le temps se prend dans deux acceptions, ainsi que la terre, la mer et le vide : (on peut en considérer) le tout ou les parties. » La distinction : tout-parties nous est rapportée, dans un autre contexte, par Diogène Laërce : « La substance, c’est-à-dire la matière, se dit dans deux sens : celle de toutes choses et celle des êtres particuliers ; la première ne s’accroît ni ne diminue ; l’autre s’accroît et diminue¹. » Cette distinction : tout-parties, suggérée par les deux premiers exemples², va être précisée à l’aide du troisième : « De même que le vide total est infini de toutes parts, de même le temps total est infini à ses deux extrémités ; en effet, le passé et le futur sont infinis. » – L’expression de « total », par où est désigné le tout du vide et du temps, est celle-là même qui, dans une autre division, englobait le monde et le vide³, c’est-à-dire, en cosmologie, le terme le plus général, qui comprend l’être et la condition incorporelle de l’être ; un terme, autrement dit, qui peut s’appliquer aussi bien à l’incorporel qu’à ce qui est corporel, au « tout », aussi bien qu’aux « parties » qui y sont virtuellement contenues.

(36)

1. Diog. Laërt., VII, 150 (S.V.F., II, 316) ; cf.Arius Did., 20 (Dox. Gr., 1457 sq.).

2. C’est tout ce que prétendent ces deux exemples, mais 1’analogie ne va pas plus loin ; la terre et la mer sont des ὅλα, c’est-à-dire des touts limités, comme le monde lui-même, alors que le vide est un πᾶν, c’est-à-dire un tout infini.

3. Cf. p. 27 sq.

(36)

 

 

1.1.3.13.3

 

[Time’s Unlimited Ends and Limited Present]

 

(pp.36-37: “ Quelles sont les parties du vide …”)

 

[In sum: For the Stoics, time can be understood as a line extending backwards to the past and forwards to the future. There is no limit to how far back and forward it goes. So you might say that time is “bound” by unlimited limits on either side. But internally it has a finite present that acts as a limited limit cutting of the past at its least past and the future at its least future.]

 

[Note, in the prior section 1.1.3.13.2, we used the Long & Sedley translation: “Just as the void in its totality is infinite in every respect, so time in its totality is infinite on either side”. This does not directly suggest that the void has parts, but perhaps the French version does: “De même que le vide total est infini de toutes parts, de même le temps total est infini à ses deux extrémités”. Yet, I am not sure if the idea here is about compositional parts or rather about extremities or sides. So where Long & Sedley say “in every respect” for what is “de toutes parts”, maybe we are to understand also “on all sides”. Here is the Greek for this passage: “ Ὥσπερ δὲ τὸ κενόν πᾶν ἄπειρον εἶναι πάντῃ καί τὸν χρόνον πάντα ἄπειρον εἶναι ἔφ᾽ ἑκάτερα;”. I am not certain, but I suspect that the term in question is πάντῃ/πάντα, which apparently can mean either “every way” and “on every side”. I do not know ancient Greek, but I wonder if  the sense of “both sides” lies in “ἔφ᾽ ἑκάτερα”. Yet I would note that the clause also has πάντα, so I am not sure if that should be understood as “on every side” if also we are later specifying that there are just two sides.] The Stobaeus/Chrysippus text does not explain what the parts/sides of the void are. But it is clear that the parts [/sides] are not parts [/sides] of void but are places. It would seem then, that it is no longer a matter of the parts [/sides] of time. Since the void is said to be infinite “in every respect” [/“on every side”] and time is infinite “on either side,” we could gather that the infinity in both cases is equally complete. (And we note that here time is represented by a line.) [Judging from what follows, I am supposing that the idea here is that we might at this point think that every part of time is infinite, just like every part of the void is.] But in fact, this is not the case, as is shown in the following quotation by Diogenes: “And time past and time future are infinite, but time present is finite” (“Le passé et le futur sont infinis, mais le présent est limité”). [What we have here is the following structure. Time understood as a line has two extremities, past and future, which are like its unlimited limits on the far ends. But there is a third part, the present, which is a limited limit on the inner side between past and future.]

Quelles sont les parties du vide, notre texte ne le dit pas ; mais il est clair que ces parties ne sont plus « du vide » à proprement parler, mais déjà des « lieux ». A première vue, il ne semble pas non plus être question des parties du temps. Si le vide est dit infini « de toutes parts » ; le temps, « à ses deux extrémités », on pourrait croire que l’infinitude, dans les deux cas, est également complète (le temps étant figuré par une droite). Mais il n’en est rien, ainsi qu’il ressort de l’explication suivante : « En effet, le passé et le futur sont infinis. » Ces « deux extrémités » ne sont donc pas les seules limites illimitées de la ligne temporelle ; comme le vide total peut avoir des parties : les lieux ; de même, semble-t-il, le temps, infini en passé et en avenir, pourrait, en partie, se limiter. Et c’est ce que nous dit Diogène : « Le | passé et le futur sont infinis, mais le présent est limité »¹.

(36-37)

1. Diog. Laërt., VII, 141.

(37)

 

 

 

From:

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.

http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0258%3Abook%3D7%3Achapter%3D1

 

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.

 

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:

https://archive.org/details/adw8682.0001.001.umich.edu

 

 

 

 

.

Russell, Ch.39 of Principles of Mathematics, ‘The Infinitesimal Calculus’, summary notes

 

by Corry Shores
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Bertrand Russell

Principles of Mathematics

Part 5: Infinity and Continuity

Ch.39: The Infinitesimal Calculus

Brief Summary

Leibniz’ infinitesimal calculus used a concept of the infinitely small (the infinitesimal quantity). Russell explains how differential and integral calculus now function with the concept of limit rather than the concept of infinitesimal.

 

Summary

 

§303

‘Infinitesimal calculus’ refers to differential and integral calculus, however “there is no allusion to, or implication of, the infinitesimal in any part of this branch of mathematics.” [330]

Leibniz was its inventor, but he considered it to be more practically applicably than metaphysically truthful. “He appears to have held that, if metaphysical subtleties are left aside, the Calculus is only approximate, but is justified practically by the fact that the errors to which it gives rise are less than those of observation”. [330]

But because Leibniz believed in the actual infinitesimal, he was unable to see that calculus rests on the doctrine of limits. Newton’s fluxions are closer to this truer foundation.

When he was thinking of Dynamics, his belief in the actual infinitesimal hindered him from discovering that the Calculus rests on the doctrine of limits, and made him regard his dx and dy as neither zero, nor finite, nor mathematical fictions, but as really representing the units to which, in his philosophy, infinite division was supposed to lead. And in his mathematical expositions of the subject, he avoided giving careful proofs, contenting himself with the enumeration of rules. At other times, it is true, he definitely rejects infinitesimals as philosophically valid; but he failed to show how, without the use of infinitesimals, the results obtained by means of the Calculus could yet be exact, and | not approximate. In this respect, Newton is preferable to Leibniz: his Lemmas give the true foundation of the Calculus in the doctrine of limits, and, assuming the continuity of space and time in Cantor’s sense, they give valid proofs of its rules so far as spatio-temporal magnitudes are concerned. [330-331]

Leibniz’ error has misled philosophers and mathematicians from his time to Weierstrass.

it is at any rate certain that, in his first published account of the Calculus, he defined the differential coefficient by means of the tangent to a curve. And by his emphasis on the infinitesimal, he gave a wrong direction to speculation as to the Calculus, which misled all mathematicians before Weierstrass (with the exception, perhaps, of De Morgan), and all philosophers down to the present day. It is only in the last thirty or forty years that mathematicians have provided the requisite mathematical foundations for a philosophy of the Calculus. [331]

§304

“The differential coefficient depends essentially upon the notion of a continuous function of a continuous variable”. [331]

In §254 we noted that a function relates the elements of one set to those of another, usually in an order-preserving way. [Also see Edwards and Penney’s account of functions here.]
And in §277 we defined a continuum as  a dense series of terms whose values are all definable by means of limits contained with in it. [Russell says we examined continuous variables in this chapter. There is no use of this term, just continuum and continuous series, which might therefore be equivalent.] If the function is one-valued and ordered correlatively with a continuous variable, then the function is continuous. [332d]. But if the function has an order indepent of correlation, it could possible be that the series obtained in its correlation is not continuous. When the correlation does produce a continuous series in some interval, then the function is continuous in that interval. [From the informal and formal definitions, it seems that a function is continuous at a point when the limits to either side of that point have equal value. If we were thinking in infinitesimal terms, if we were to move infinitesimally to the right or left of the point, it would be the same value (except for the infinitesimal, inassignable difference). But it is discontinuous when the side-limits are different. As  Mr. Flatcher writes “The graph of a continuous function has no holes, jumps, or gaps. Think of a continuous function as one that you can graph without ever lifting your pencil.” (Mr. Flatcher / Flatchermatics) Consider for example this function.

At x = 1, the limits on either of its sides should be 0. However, as we can see, the y values are much different from zero.

(Image and function from ‘Classification of Discontinuities’, wikipedia)

As you can see, at the limit right before x = 1, the y value is greater than zero, and at the limit right after, the y value is even greater than that. Below we have an animation showing a transition from continuity to discontinuity. The caption reads: “A sequence of continuous functions f_n(x) whose (pointwise) limit function f(x) is discontinuous. The convergence is not uniform.”

(Animated diagram and above caption from ‘Continuous function’, wikipedia)

]

If the function is one-valued, and is only ordered by correlation with the variable, then, when the variable is continuous, there is no sense in asking whether the function is continuous; for such a series by correlation is always ordinally similar to its | prototype. But when, as where the variable and the field of the function are both classes of numbers, the function has an order independent of correlation, it may or may not happen that the values of the function, in the order obtained by correlation, form a continuous series in the independent order. When they do so in any interval, the function is said to be continuous in that interval. The precise definitions of continuous and discontinuous functions, where both x and f(x) are numerical, are given by Dini as follows. The independent variable x is considered to consist of the real numbers, or of all the real numbers in a certain interval; f(x), in the interval considered, is to be one-valued, even at the end-points of the interval, and is to be also composed of real numbers. We then have the following definitions, the function being defined for the interval between α and β, and ɑ being some real number in this interval.

“We call f(x) continuous for x = ɑ, or in the point ɑ, in which it has the value f(ɑ), if for every positive number σ, different from 0, but as small as we please, there exists a positive number ε, different from 0, such that, for all values of δ which are numerically less than ε, the difference f(ɑ + δ) − f(ɑ) is numerically less than σ. In other words, f(x) is continuous in the point x = ɑ, where it has the value f(ɑ), if the limit of its values to the right and left of a is the same, and equal to f(ɑ).”

“Again, f(x) is discontinuous for x = ɑ, if, for any positive value of σ, there is no corresponding positive value of ε such that, for all values of δ which are numerically less than ε, f(ɑ + δ) − f(ɑ) is always less than σ; in other words, f(x) is discontinuous for x = ɑ, when the values f(a + h) of f(x) to the right of a, and the values f(ɑ − h) of f(x) to the left of ɑ, the one and the other, have no determinate limits, or, if they have such, these are different on the two sides of ɑ; or, if they are the same, they differ from the value f(ɑ), which the function has in the point ɑ.” [331-332]

But the limit of a function is slightly different than the limit in general (of series) that we have discussed so far. Russell defines the limit in this way. [Put in simple terms, it seems that the limit is the value most immediate to the point, were we using infinitesimal terms.]

A function of a perfectly general kind will have no limit as it approaches any given point. In order that it should have a limit as x approaches a from the left, it is necessary and sufficient that, if any number ε be mentioned, any two values of f(x), when x is sufficiently near to a, but | less than a, will differ by less than ε; in popular language, the value of the function does not make any sudden jumps as x approaches a from the left. Under similar circumstances, f(x) will have a limit as it approaches a from the right. But these two limits, even when both exist, need not be equal either to each other or to f(ɑ), the value of the function when x = ɑ. The precise condition for a determinate finite limit may be thus stated:

“In order that the values of y to the right or left of a finite number a (for instance to the right) should have a determinate and finite limit, it is necessary and sufficient that, for every arbitrarily small positive number σ, there should be a positive number ε, such that the difference y_{ɑ + ε} − y_{ɑ+ δ} between the value y_{ɑ + ε} of y for x = a + ε, and the value yɑ + δ, which corresponds to the value a + δ of x, should be numerically less than σ, for every δ which is greater than 0 and less than ε.” It is possible, instead of thus defining the limit of a function, and then discussing whether it exists, to define generally a whole class of limits. In this method, a number z belongs to the class of limits of y for x = ɑ, if, within any interval containing ɑ, however small, y will approach nearer to z than by any given difference. Thus, for example, sin 1/x, as x approaches zero, will take every value from −1 to +1 (both inclusive) in every finite interval containing zero, however small. Thus the interval from −1 to +1 forms, in this case, the class of limits for x = 0. This method has the advantage that the class of limits always exists. It is then easy to define the limit as the only member of the class of limits, in case this class should happen to have only one member. This method seems at once simpler and more general.
[332-333]

§305

Russell will now discuss the derivative or differential coefficient of the function. [To better grasp Russell’s example, we will draw from our summary of one of David Jerrison’s class lectures on the differential. We will find an equivalent formulation to Russell’s so that we can make what he is saying more concrete. So first let’s understand the formulation. Consider a curve with point P.

The line has a different slope (tendency of variation) at each point. We will ask, what is its slope at x₀, or point P? We determine the y value on the basis of the function f(x). And since P = (x,y), then P = (x0,(fx0)). Our calculation will involve looking at the change in x to the change in y (or change in f we might say).

The slope is found at the limit as Δx goes to zero.

We see the coordinates given here:

Slope is rise-over-run, or Δy / Δx. So

m = (y₂ – y₁) / (x₂ – x₁)

or in our case,

m = (f₂ – f₁) / (x₂ – x₁)

Notice in the above diagram that the y values in P and Q are: f(x₀) and (f(x₀ + Δx). For (x₂ – x₁) we only need Δx. So if we substitute these values into the slope formula, we have:

(f(x0 + Δx) – f(x0)) / Δx

The derivative we will denote as f’(x₀). Thus

We call that formulation “the difference quotient”.

Now let’s use a specific function.

f(x) = 1 / x

We want to find the derivative at x₀, and the dotted line is the tangent whose slope we seek. So we need to find Δf / Δx. The formula for Δf was (f(x₀ + Δx) – f(x₀)). When we plug in our function, we (multiplicatively) invert the x values, that is, put a one over them, hence we obtain [18]:

When we remove the embedded fractions (by dropping the top couched-denominators to the entire bottom denominator), we get:

As you can see, 1 / Δx is common to both parts. When we factor it out, we get:

The subtracted parts need a common denominator for us to simplify them. To give them both a common denominator, we multiply each by the other’s denominate set over itself (thus equaling 1).

Let’s combine these figures to get:

We can now subtract the terms:

We then distribute the negative in the right side of the numerator to make x₀ – x₀ –Δx, thereby leaving –Δx; hence:

Since both sides share Δx inversely, we can cancel them, leaving us with 1/1 on the left side, which can as well be eliminated, and remaining on the left is:

The last step is to take the limit, as delta tends to zero, and substitute zero for Δx. We can do this now, because before the numerator and denominator gave us number divided by zero, which is undefined. But through algebraic operations, we were able to make the Δx negate-out of the equation without leaving a zero in the denomenator. Thus we now substitute-in the limit, that is, make Δx equal zero, leaving us with:

Let’s put all of this together into one large formulation:

We then compare with our chart. This is negative, and likewise the slope is negative. Also, as x_o goes to infinity, so as x moves to the right, it becomes less steep:

As we will see, Russell uses a very similar formulation, except with δ instead of our Δx]

 

If f(x) be a function which is finite and continuous at the point x, then it may happen that the fraction

{f(x + δ) − f(x)}/δ

has a definite limit as δ approaches to zero. If this does happen, the limit is denoted by f '(x), and is called the derivative or differential of f(x) in the point x. If, that is to say, there be some number z such that, given any number ε however small, if δ be any number less than some number η, but positive, then {f(x ± δ) − f(x)}/ ± δ differs from z by less than ε, then z is the derivative of f(x) in the point x. If the limit in question does not exist, then f(x) has no derivative at the point x. If f(x) be not continuous at this point, the limit does not exist; if f(x) be continuous, the limit may or may not exist.
[334]

§306

[Russell will say that the notion of the infinitesimal was not used in this definition. This probably results from the notion of the limit as standing outside the series, and the values approaching can only ever get closer and closer with no final value. I challenge this view, because it implies the value of the interval between the limit and the series that approaches it is finite. Let’s take this idea that between any two values is a middle value, infinitely. So a series of diminishing values perhaps could be something like 1/1, 1/2, 1/4, 1/8. What is important regarding Cantor’s infinity is that the cardinal value for infinity  α₀ is not among the natural numbers. It is the limit to which their law of genesis implicitly strives toward but does not precisely attain. But so long as the interval is finite, which Russell insists it must be, then does it fulfill the definition for “given any number ε however small”? It seems here the idea is not “the interval is so small it is infinitely small and thus continuous with zero” but rather “the interval is very small but still finite, yet it is close enough to zero that we can substitute one for the other.” Perhaps this is where the term “arbitrarily” small comes from. Is it strange that this ‘fudging’ sort of operation where we exchange a finite value for zero is considered more precise than when we think of this value as infinitely small, especially since in both cases we assume that there are an infinity of subdivisions? If intervals really are infinitely sudividable, why is it so hard to conceive of the intervals between them as being infinitely small? If there were not infinitely small, then they would be finitely small, and an infinity of them would compose an infinitely large interval. But an infinity of infinitely small values could conceivably compose a finite interval (for we would multiply infinity times one over infinity, equaling a finite unit after the infinities cancel.)] Russell emphasizes that the fact he has defined the derivative using limits and not infinitesimals is philosophically the most important part of his treatment on calculus. [The philosophical implication of this might be that the law of continuity does not hold, and thus change is not a matter paradoxically co-given contrary states. Also, change or motion would be ‘at-at’; infinitesimal intervals suggest ‘between-between’ or ‘at-at plus at-at’. I think Russell insists on this philosophical point for logical reasons. The problem with the infinitesimal calculus and its law of continuity is that it is a dialetheia, a true contradiction, and Russell will not allow exceptions to his rigid logic of perfect self-consistency. Because of the advances in dialetheic logic, it is no longer illogical to say that change is inherently paradoxical. And on account of the invention of non-standard analysis, it is no longer conceptually sloppy to use the notion of the infinitesimal in calculus. So here too is equally my greatest point of emphasis and the purpose for all these mathematical technicalities: despite Russell’s insistence, we can have a dialetheic between-between theory of motion, and it will not have such oddities like we find in Russell’s account, such as an infinity of finite intervals composing a finite interval and not an infinite one, and a moving object always being in no state other than rest, just rest in different places at different times.]

The only point which it is important to notice at present is, that there is no implication of the infinitesimal in this definition. The number δ is always finite, and in the definition of the limit there is nothing to imply the contrary. In fact, {f(x + δ) − f(x)}/δ, regarded as a function of δ, is wholly indeterminate when δ = 0. The limit of a function for a given value of the independent variable is, as we have seen, an entirely different notion from its value for the said value of the independent variable, and the two may or may not be the same number. In the present case, the limit may be definite, but the value for δ = 0 can have no meaning. Thus it is the doctrine of limits that underlies the Calculus, and not any pretended use of the infinitesimal. This is the only point of philosophic importance in the present subject, and it is only to elicit this point that I have dragged the reader through so much mathematics. [383]

[In the above, it seems Russell is absolutely clear that the small interval does not equal 0, but it also does not equal an infinitesimally small value. Rather, it equals a finite value that is as small as you want it to be.]

§307

[In Russell’s description of the definite integral, we divide the interval up into n portions. We find their ‘areas’ or products. Then we want the sum of all such interval areas or products. As we increase n, the sum tends toward a definite limit, which gives us the integral sum.  For more on this operation, see David Jerison’s definite integral class or Edwards & Penney’s section on Riemann sums.]

Just as the derivative of a function is the limit of a fraction, so the definite integral is the limit of a sum. The definite integral may be defined as follows: Let f(x) be a function which is one-valued and finite in the interval α to β (both inclusive). Divide this interval into any n portions by means of the | (n − 1) points x₁, x₂, . . . x_n − 1, and denote by δ₁, δ₂, . . . δ_n the n intervals x₁ − α, x₁ − x₂, . . . β − x_{n − 1} . In each of these intervals, δ_s, take any one of the values, say f(ζ_s), which f(x) assumes in this interval, and multiply this value by the interval δ_s. Now form the sum

This sum will always be finite. If now, as n increases, this sum tends to one definite limit, however f(ζs) may be chosen in its interval, and however the intervals be chosen (provided only that all are less than any assigned number for sufficiently great values of n)—then this one limit is called the definite integral of f(x) from α to β . If there is no such limit, f(x) is not integrable from α to β.
[384-385]

§308

 

Russell now explains that neither the concept of the infinitesimal nor of infinity were used in this account of integral calculus. In fact, it is not even a sum [which might be impossible were the terms infinite in number]. Rather, this value is the limit of a sum [to boundary value to which its summing is tending]. But it never reaches that value. [If it did, then that could only be by means of an infinitesimally small increment between the limit and the next value near it. But so long as it never gets there but instead continually gets nearer without arriving, then it is made of many finite units. Imagine a segment that is built by adding 1/2 + 1/4 + 1/8 + etc., it is tending toward the total 1. See the divided square diagram on this page for an illustration (found near the end, under “geometric series that halves each time.”) We don’t need infinitely many to know it is tending to that limit. So we do not need the concepts of infinitesimal and infinity. But if we think that it does actually reach that limit, then it could only do so with an infinity of divisions with the smallest being infinitesimal.]

As in the case of the derivative, there is only one important remark to make about this definition. The definite integral involves neither the infinite nor the infinitesimal, and is itself not a sum, but only and strictly the limit of a sum. All the terms which occur in the sum whose limit is the definite integral are finite, and the sum itself is finite. If we were to suppose the limit actually attained, it is true, the number of intervals would be infinite, and the magnitude of each would be infinitesimal; but in this case, the sum becomes meaningless. Thus the sum must not be regarded as actually attaining its limit. But this is a respect in which series in general agree. Any series which always ascends or always descends and has no last term cannot reach its limit; other infinite series may have a term equal to their limit, but if so, this is a mere accident. The general rule is, that the limit does not belong to the series which it limits; and in the definition of the derivative and the definite integral we have merely another instance of this fact. The so-called infinitesimal calculus, therefore, has nothing to do with the infinitesimal, and has only indirectly to do with the infinite—its connection with the infinite being, that it involves limits, and only infinite series have limits.
[386]

 

 

Sources [unless otherwise notes, all bracket page citations are from]:

Bertrand Russell. Principles of Mathematics. London/New York: Routledge, 2010 [1st published 1903].

 

Otherwise:

Mr. Flatcher. Continuity and Differentiability.
http://fletchmatics.weebly.com/continuity-and-differentiability.html

Wikipedia. ‘Classification of Discontinuities’.
http://en.wikipedia.org/wiki/Classification_of_discontinuities

Wikipedia. ‘Continuous function’.
http://en.wikipedia.org/wiki/Continuous_function