22 Dec 2008

Spinoza, Ethics, Part 1, Proposition 15: Divine Infinite Quantity, with Deleuze's Commentary

by Corry Shores
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The following is quotation; my summary and commentary is bracketed in red. Deleuze's commentary is found near the end. The Latin text comes last.]

Baruch Spinoza
Part I "Concerning God"
Proposition XV

Prop. XV. Whatsoever is, is in God, and without God nothing can be, or be conceived.

[Anything that exists must exist in God and be conceived through him.]

Proof.-Besides God, no substance is granted or can be conceived (by Prop. xiv.), that is (by Def. iii.) nothing which is in itself and is conceived through itself. But modes (by Def. v.) can neither be, nor be conceived without substance ; wherefore they can only be in the divine nature, and can only through it be conceived. But substances and modes form the sum total of existence (by Ax. i.), therefore, without God nothing can be, or be conceived. Q.E.D.

[We know that there is only one substance: God. And because substances are conceived in and through themselves, there is nothing other than God which can be conceived in and through itself.

Modes, however, must be conceived by means of the substance they modify (and they must exist in that substance). Thus modes must be in the divine nature (the one infinite substance) and be conceived through it.

But because everything that exists does so in itself or through something else, and because substance and modes are the only things that exist in themselves or in something else, we know that there is nothing other than substance and modes. And because both in one way or another are conceived through God, all things exist in him and are conceived through him.]

Note.-Some assert that God, like a man, consists of body and mind, and is susceptible of passions. How far such persons have strayed from the truth is sufficiently evident from what has been said. But these I pass over. For all who have in anywise reflected on the divine nature deny that God has a body. Of this they find excellent proof in the fact that we understand by body a definite quantity, so long, so broad, so deep, bounded by a certain shape, and it is the height of absurdity to predicate such a thing of God, a being absolutely infinite. But meanwhile by other reasons with which they try to prove their point, they show that they think corporeal or extended substance wholly apart from the divine nature, and say it was created by God. Wherefrom the divine nature can have been created, they are wholly ignorant ; thus they clearly show, that they do not know the meaning of their own words.

[Some people mistakenly think God to be like man, that is, to have body, mind, and passions; but this would contradict his definition of being infinite. Some also think that extended substance is wholly apart from God and created by him. But such people are then unable to explain how God came about.]

I myself have proved sufficiently clearly, at any rate in my own judgment (Coroll. Prop. vi, and note 2, Prop. viii.), that no substance can be produced or created by anything other than itself.

[Spinoza has given at least three explanations for why substance is produced only by itself:

1) Substances could only be distinguished either by having different attributes or different modes. If we distinguish them by their attributes, then we are saying that one substance essentially lacks some quality that another one has. But this is to define one substance in terms of another, and substances need only themselves to be conceived. Hence we cannot distinguish substances in terms different attributes.

Modifications are not essential to substance, and are not a part of the truth of their essence. So if we distinguish substances in terms of their modifications, we are not distinguishing them as they truly are. Hence we cannot distinguish substances in terms of different modifications.

But if we cannot distinguish substances by the only two means possible, that is, in terms of different attributes or modes, then substances are indistinguishable, hence whether we presuppose one or many substances, in both cases we conclude there are not more than one substance.

And if there is only one substance, there is nothing outside it that might produce it, hence it must produce itself.

2) We conceive something in terms of its cause. If substance's cause were outside it, we would be conceiving substance in terms of something else, which contradicts its definition as self-conceivable. Hence substances cannot be produced by something else, so they must be produced by themselves.

3) From a definition, we cannot infer how many individuals fall under that definition. So for the definition of substance, we only know that it follows from its definition that at least one substance necessarily exists, but if more exist we cannot know that from the definition. Also, all existing things must have a cause, so we know that there is at least one cause for at least one substance, which is that very substance itself.

Now if we suppose there are more than one substance that exists, each one by definition cannot be conceived through another; thus they cannot have anything in common with each other. But we saw in Proposition 3 that things with nothing in common cannot be causally related. So if there were more than one substance, they all would have to have their causes in themselves, and thus would all have to produce themselves.

But if all these substances are self-caused, then they share the same essential nature, which means to conceive one is to conceive the essential characteristic of the others. But substances cannot share essential characteristics, because they must be exclusively self-conceivable. Hence there can only be one self-produced substance.]

Further, I showed (in Prop. xiv.), that besides God no substance can be granted or conceived.

[God has infinite attributes, which means all of them. If there were another substance, it would have to have at least one of those attributes, for there would be no others. But if they shared attributes, then they would not be self-conceivable, which contradicts the definition of substance. Hence God is the only conceivable substance.]

Hence we drew the conclusion that extended substance is one of the infinite attributes of God.

[We know there is extended substance, and we know that it must be an attribute of God, because all attributes belong to him.]

However, in order to explain more fully, I will refute the arguments of my adversaries, which all start from the following points :— Extended substance, in so far as it is substance, consists, as they think, in parts, wherefore they deny that it can be infinite, or consequently, that it can appertain to God.

[Those who think that extended substance does not belong to God also presuppose that extended substance consist of parts, and they think that these parts are not infinite and thus do not belong to God.]

This they illustrate with many examples, of which I will take one or two. If extended substance, they say, is infinite, let it be conceived to be divided into two parts ; each part will then be either finite or infinite. If the former, then infinite substance is composed of two finite parts, which is absurd. If the latter, then one infinite will be twice as large as another infinite, which is also absurd.

[One way that they argue that extended substance is not infinite is by saying that if it were infinite, and we divided in two parts, then each part is either finite or infinite. If the parts are finite, then something that is infinite is made up of two finite parts, which is absurd. But if we say that the two parts are infinite, then that would also mean that the whole infinite is twice as large as either of its halves, which is also absurd.]

Further, if an infinite line be measured out in foot lengths, it will consist of an infinite number of such parts ; it would equally consist of an infinite number of parts, if each part measured only an inch : therefore, one infinity would be twelve times as great as the other.

[Along the same lines, they might argue against the infinity of Extension by saying if we measured an infinite line's length in feet, it will have an infinity of foot-length parts. And if we measured another infinite line in inches, we would likewise have an infinity of inch-length parts. But feet are twelve times the length of inches, so we would have one infinity that is twelve times the length of the other infinity, but that is also absurd.]

Lastly, if from a single point there be conceived to be drawn two diverging lines which at first are at a definite distance apart, but are produced to infinity, it is certain that the distance between the two lines will be continually increased, until at length it changes from definite to indefinable.

[If we take two lines diverging from a single point and extending infinitely, we can see from a limited scope that there is a definite distance between the lines. Yet supposedly as the lines continue to infinity, the distance between them will no longer be definite but rather will be indefinite. But then somehow in some inconceivable way the definite turns into the indefinite, which is absurd.]

As these absurdities follow, it is said, from considering quantity as infinite, the conclusion is drawn, that extended substance must necessarily be finite, and, consequently, cannot appertain to the nature of God.

[On account of the above absurdities which some use to argue against infinite quantity, they conclude that extended substance must necessarily be finite. And because God is infinite, finite substance could not belong to him.]

The second argument is also drawn from God’s supreme perfection. God, it is said, inasmuch as he is a supremely perfect being, cannot be passive ; but extended substance, insofar as it is divisible, is passive. It follows, therefore, that extended substance does not appertain to the essence of God.

[Another line of argument begins from the notion of God's supreme perfection. If he is supremely perfect, he could not be acted upon and hence be passive. But extended substance is divisible, hence it can be acted-upon. Thus passive extended substance could not belong to a non-passive God.]

Such are the arguments I find on the subject in writers, who by them try to prove that extended substance is unworthy of the divine nature, and cannot possibly appertain thereto. However, I think an attentive reader will see that I have already answered their propositions ; for all their arguments are founded on the hypothesis that extended substance is composed of parts, and such a hypothesis I have shown (Prop. xii., and Coroll. Prop. xiii.) to be absurd.

[If we divide substance, its parts either retain its nature or they do not. It cannot be that they do retain its nature, because substances cannot share attributes. It cannot be that the divided parts do not retain its nature, because then we would be destroying substance's existence by divided it, which is impossible because substance's existence is necessary. So for that reason substance cannot be composed of parts.]

Moreover, anyone who reflects will see that all these absurdities (if absurdities they be, which I am not now discussing), from which it is sought to extract the conclusion that extended substance is finite, do not at all follow from the notion of an infinite quantity, but merely from the notion that an infinite quantity is measurable, and composed of finite parts : therefore, the only fair conclusion to be drawn is that infinite quantity is not measurable, and cannot be composed of finite parts. This is exactly what we have already proved (in Prop. xii.). Wherefore the weapon which they aimed at us has in reality recoiled upon themselves.

[The counterarguments, however, do not argue against the infinite as it actually is, because they conceive an infinite quantity as being measurable and composed of parts. They then deduce the absurdities that follow from these suppositions and try to conclude that the infinite does not exist. But for Spinoza, however, these absurdities show that the actual infinite cannot be measurable and composed of parts.]

If, from this absurdity of theirs, they persist in drawing the conclusion that extended substance must be finite, they will in good sooth be acting like a man who asserts that circles have the properties of squares, and, finding himself thereby landed in absurdities, proceeds to deny that circles have any center, from which all lines drawn to the circumference are equal. For, taking extended substance, which can only be conceived as infinite, one, and indivisible (Props. viii., v., xii.) they assert, in order to prove that it is finite, that it is composed of finite parts, and that it can be multiplied and divided.

[So some conclude that extended substance is finite, which contradicts its own definition. They think that finite substance can be multiplied and divided, which for them proves it is finite.]

So, also, others, after asserting that a line is composed of points, can produce many arguments to prove that a line cannot be infinitely divided. Assuredly it is not less absurd to assert that extended substance is made up of bodies or parts, than it would be to assert that a solid is made up of surfaces, a surface of lines, and a line of points. This must be admitted by all who know clear reason to be infallible, and most of all by those who deny the possibility of a vacuum. For if extended substance could be so divided that its parts were really separate, why should not one part admit of being destroyed, the others remaining joined together as before? And why should all be so fitted into one another as to leave no vacuum? Surely in the case of things, which are really distinct one from the other, one can exist without the other, and can remain in its original condition. As, then, there does not exist a vacuum in nature (of which anon), but all parts are bound to come together to prevent it, it follows from this that the parts cannot really be distinguished, and that extended substance in so far as it is substance cannot be divided.

[Spinoza then addresses counterarguments based on geometrical analogies. Some try to show that a line is made-up of points, and hence is made-up of finite things. But Spinoza says that lines and points are of two different sorts of things, so one cannot make-up the other, just as a solid cannot be made-up of planes, or a surface made-up of lines. Likewise, extended substance cannot be made-up of finite bodies or parts (see Deleuze's Cours Vincennes: 10/03/1981 for how Spinoza's simple bodies are pure relations and do not contain terms).

Those who deny the possibility of a vacuum should also see that extension cannot be made up of finite bodes; for, if it were, and these bodies were tiny discrete units, nothing would stop there from being a vacuum between them so to maintain their integrity. If there were no such vacuum, then nothing would separate the units, hence they would be an infinitely divisible continuum. And if they were discrete, then if one were destroyed, it would have no effect on the others, but we know that the integrity of whole bodies depends not on the independence of its parts but on their reciprocal relations.

So parts must really be connected in a continuum that does not permit division, so substance cannot be divided.]

If anyone asks me the further question, Why are we naturally so prone to divide quantity? I answer, that quantity is conceived by us in two ways ; in the abstract and superficially, as we imagine it ; or as substance, as we conceive it solely by the intellect. If, then, we regard quantity as it is represented in our imagination, which we often and more easily do, we shall find that it is finite, divisible, and compounded of parts ; but if we regard it as it is represented in our intellect, and conceive it as substance, which it is very difficult to do, we shall then, as I have sufficiently proved, find that it is infinite, one, and indivisible. This will be plain enough to all who make a distinction between the intellect and the imagination, especially if it be remembered, that matter is everywhere the same, that its parts are not distinguishable, except in so far as we conceive matter as diversely modified, whence its parts are distinguished, not really, but modally. For instance, water, in so far as it is water, we conceive to be divided, and its parts to be separated one from the other ; but not in so far as it is extended substance ; from this point of view it is neither separated nor divisible. Further, water, in so far as it is water, is produced and corrupted ; but, in so far as it is substance, it is neither produced nor corrupted.

[Spinoza then gives the reason why we are so apt to consider quantity as divisible. He explains that we have two ways of conceiving quantity:

1) abstractly and superficially by means of our imagination,
2) as substance conceived solely by means of the intellect.

When we imagine the infinite, we regard it as finite, divisible, and composed of parts; but when we conceive it in the way our intellect represents it, namely, as substance, then we correctly regard it as infinite, one, and indivisible. Matter is everywhere substantially the same: its parts are not distinguishable. However, we may conceive matter as being diversely modified, and we may designate regions of similarity and regard them as distinct modes. In this way matter's parts can be modally distinguished, even though in reality there are no such distinctions.]

I think I have now answered the second argument ; it is, in fact, founded on the same assumption as the first-namely, that matter, in so far as it is substance, is divisible, and composed of parts. Even if it were so, I do not know why it should be considered unworthy of the divine nature, inasmuch as besides God (by Prop. xiv.) no substance can be granted, wherefrom it could receive its modifications. All things, I repeat, are in God, and all things which come to pass, come to pass solely through the laws of the infinite nature of God, and follow (as I will shortly show) from the necessity of his essence. Wherefore it can in nowise be said, that God is passive in respect to anything other than himself, or that extended substance is unworthy of the Divine nature, even if it be supposed divisible, so long as it is granted to be infinite and eternal. But enough of this for the present.

[This also answers the second argument which is that extended substance cannot be God, because extended substance is divisible and hence passive, but God is not passive. This argument incorrectly regards extended substance as divisible. Moreover, it should not be a problem that God be modified, because anything that would modify him would come from within him. So God can be passive, but only to himself. So even if extended substance were divisible, so long as it were also infinite and eternal, it should also not be a problem for it to apply to God.]

Deleuze's Commentary:

When we make of numerical distinction a real or substantial distinction, we carry it to infinity, if only to ensure the convertibility that then becomes necessary between the attribute as such and the infinity of finite parts which we distinct in it. Great absurdities then follow: “If an infinite quantity is measured by parts equal to a foot, it will consist of an infinitely many such parts, as it will also, if is measured by parts equal to an inch. And therefore, in one infinite number will be twelve times greater than another.” [ft.13, citation] (Deleuze, 33c)

Quand nous faisons de la distinction numérique une distinction réelle ou substantielle, nous la portons dans l’infini, ne serait-ce que pour assurer la conversion devenue nécessaire entre l’attribut comme tel et l’infinité de parties finies que nous y distinguons. En sortent de grandes absurdités : « Si une quantité infinie est mesurée en parties égales à un pied, elle devra consister en une infinité de telles parties ; et de même si elle est mesurée en parties égales à un doigt ; et par suite un nombre infini [ft.13, citation]. » (Deleuze, 26-27)

So that modes are also the affections of God, but God never suffers the activity of his modes; his only affections are active. [ft.28, citation] (Deleuze, 94bc)

Aussi bien les modes sont-ils les affections de Dieu; mais jamais Dieu ne pâtit de ses modes, il n'a d'affections qu'actives [ft.28]. (Deleuze 83d)

From the Latin text:


Quicquid est, in Deo est, & nihil sine Deo esse, neque concipi potest.


Præter Deum nulla datur, neque concipi potest substantia (per Prop. 14), hoc est (per Defin. 3) res, quæ in se est, & per se concipitur. Modi autem (per Defin. 5) sine substantia nec esse, nec concipi possunt; quare hi in sola divina natura esse, & per ipsam solam concipi possunt. Atqui præter substantias, & modos nil datur (per Axiom. 1). Ergo nihil sine Deo esse, neque concipi potest. Q.E.D.


Sunt, qui Deum instar hominis corpore, & mente constantem, atque passionibus obnoxium fingunt; sed, quam longe hi a vera Dei cognitione aberrent, satis ex jam demonstratis constat. Sed hos mitto: nam omnes, qui naturam divinam aliquo modo contemplati sunt, Deum esse corporeum, negant. Quod etiam optime probant ex eo, quod per corpus intelligimus quamcunque quantitatem, longam, latam, & profundam, certa aliqua figura terminatam, quo nihil absurdius de Deo, ente scilicet absolute infinito, dici potest. Attamen interim aliis rationibus, quibus hoc idem demonstrare conantur, clare ostendunt, se substantiam ipsam corpoream, sive extensam a natura divina omnino removere, atque ipsam a Deo creatam statuunt. Ex qua autem divina potentia creari potuerit, prorsus ignorant; quod clare ostendit, illos id, quod ipsimet dicunt, non intelligere. Ego saltem satis clare, meo quidem judicio, demonstravi (vide Coroll. Prop. 6 & Schol. 2 Prop. 8) nullam substantiam ab alio posse produci, vel creari. Porro Prop. 14 ostendimus, præter Deum nullam dari, neque concipi posse substantiam; atque hinc conclusimus, substantiam extensam unum ex infinitis Dei attributis esse. Verum, ad pleniorem explicationem, adversariorum argumenta refutabo, quæ omnia huc redeunt. Primo, quod substantia corporea, quatenus substantia, constat, ut putant, partibus; & ideo eandem infinitam posse esse, & consequenter, ad Deum pertinere posse, negant. Atque hoc multis exemplis explicant, ex quibus unum, aut alterum afferam. Si substantia corporea, ajunt, est infinita, concipiatur in duas partes dividi; erit unaquæque pars, vel finita, vel infinita. Si illud, componitur ergo infinitum ex duabus partibus finitis, quod est absurdum. Si hoc, datur ergo infinitum duplo majus alio infinito, quod etiam est absurdum. Porro, si quantitas infinita mensuratur partibus pedes æquantibus, infinitis talibus partibus constare debebit, ut &, si partibus mensuretur digitos æquantibus; ac propterea unus numerus infinitus erit duodecies major alio infinito. Denique, si ex uno

puncto infinitæ cujusdam quantitatis concipiatur, duas lineas, ut AB, AC, certa, ac determinata in initio distantia in infinitum protendi; certum est, distantiam inter B & C continuo augeri, & tandem ex determinata indeterminabilem fore. Cum igitur hæc absurda sequantur, ut putant, ex eo, quod quantitas infinita supponitur: inde concludunt, substantiam corpoream debere esse finitam, & consequenter ad Dei essentiam non pertinere. Secundum argumentum petitur etiam a summa Dei perfectione. Deus enim, inquiunt, cum sit ens summe perfectum, pati non potest: atqui substantia corporea, quandoquidem divisibilis est, pati potest; sequitur ergo, ipsam ad Dei essentiam non pertinere. Hæc sunt, quæ apud scriptores invenio argumenta, quibus ostendere conantur, substantiam corpoream divina natura indignam esse, nec ad eandem posse pertinere. Verumenimvero, si quis recte attendat, me ad hæc jam respondisse comperiet; quandoquidem hæc argumenta in eo tantum fundantur, quod substantiam corpoream ex partibus componi supponunt, quod jam (Prop. 12 cum Coroll. Prop. 13) absurdum esse ostendi. Deinde si quis rem recte perpendere velit, videbit, omnia illa absurda (siquidem omnia absurda sunt, de quo jam non dispuro), ex quibus concludere volunt, substantiam extensam finitam esse, minime ex eo sequi, quod quantitas infinita supponatur: sed quod quantitatem infinitam mensurabilem, & ex partibus finitis conflari supponunt; quare ex absurdis, quæ inde sequuntur, nihil aliud concludere possunt, quam quod quantitas infinita non sit mensurabilis, & quod ex partibus finitis conflari non possit. Atque hoc idem est, quod nos supra (Prop. 12 &c.) jam demonstravimus. Quare telum, quod in nos intendunt, in se ipsos revera conjiciunt. Si igitur ipsi ex suo hoc absurdo concludere tamen volunt, substantiam extensam debere esse finitam, nihil aliud hercle faciunt, quam si quis ex eo, quod finxit circulum quadrati proprietates habere, concludit, circulum non habere centrum, ex quo omnes ad circumferentiam ductæ lineæ sunt æquales. Nam substantiam corpoream, quæ non nisi infinita, non nisi unica, & non nisi indivisibilis potest concipi (vide Prop. 8, 5 & 12), eam ipsi ad concludendum, eandem esse finitam, ex partibus finitis conflari, & multiplicem esse, & divisibilem, concipiunt. Sic etiam alii, postquam fingunt, lineam ex punctis componi, multa sciunt invenire argumenta, quibus ostendant, lineam non posse in infinitum dividi. Et profecto, non minus absurdum est ponere, quod substantia corporea ex corporibus, sive partibus componatur, quam quod corpus ex superficiebus, superficies ex lineis, lineæ denique ex punctis componantur. Atque hoc omnes, qui claram rationem infallibilem esse sciunt, fateri debent, & imprimis ii, qui negant, dari vacuum. Nam si substantia corporea ita posset dividi, ut ejus partes realiter distinctæ essent, cur ergo una pars non posset annihilari, manentibus reliquis, ut ante, inter se connexis? & cur omnes ita aptari debent, ne detur vacuum? Sane rerum, quæ realiter ab invicem distinctæ sunt, una sine alia esse, & in suo statu manere potest. Cum igitur vacuum in natura non detur (de quo alias), sed omnes partes ita concurrere debent, ne detur vacuum, sequitur hinc etiam, easdem non posse realiter distingui, hoc est, substantiam corpoream, quatenus substantia est, non posse dividi. Si quis tamen jam quærat, cur nos ex natura ita propensi simus ad dividendam quantitatem? ei respondeo, quod quantitas duobus modis a nobis concipitur, abstracte scilicet, sive superficialiter, prout nempe ipsam imaginamur, vel ut substantia, quod a solo intellectu fit. Si itaque ad quantitatem attendimus, prout in imaginatione est, quod sæpe, & facilius a nobis fit, reperietur finita, divisibilis, & ex partibus conflata; si autem ad ipsam, prout in intellectu est, attendimus, & eam, quatenus substantia est, concipimus, quod difficillime fit, tum, ut jam satis demonstravimus, infinita, unica, & indivisibilis reperietur. Quod omnibus, qui inter imaginationem, & intellectum distinguere sciverint, satis manifestum erit: Præcipue si ad hoc etiam attendatur, quod materia ubique eadem est, nec partes in eadem distinguuntur, nisi quatenus materiam diversimode affectam esse concipimus, unde ejus partes modaliter tantum distinguuntur, non autem realiter. Ex. gr. aquam, quatenus aqua est, dividi concipimus, ejusque partes ab invicem separari; at non, quatenus substantia est corporea; eatenus enim neque separatur, neque dividitur. Porro aqua, quatenus aqua, generatur, & corrumpitur; at, quatenus substantia, nec generatur, nec corrumpitur. Atque his me ad secundum argumentum etiam respondisse puto: quandoquidem id in eo etiam fundatur, quod materia, quatenus substantia, divisibilis sit, & ex partibus confletur. Et quamvis hoc non esset, nescio, cur divina natura indigna esset: quandoquidem (per Prop. 14) extra Deum nulla substantia dari potest, a qua ipsa pateretur. Omnia, inquam, in Deo sunt, & omnia, quæ fiunt, per solas leges infinitæ Dei naturæ fiunt, & ex necessitate ejus essentiæ (ut mox ostendam) sequuntur; quare nulla ratione dici potest, Deum ab alio pati, aut substantiam extensam divina natura indignam esse; tametsi divisibilis supponatur, dummodo æterna, & infinita concedatur. Sed de his impræsentiarum satis.


Deleuze, Gilles. Spinoza et le problème de l'expression. Paris: Les Éditions de Minuit, 1968.

Deleuze, Gilles. Expressionism in Philosophy: Spinoza. Trans. Martin Joughin. New York: Zone Books, 1990.

Spinoza. Ethics. Transl. Elwes. available online at:

Spinoza. Ethica. available online at:

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