Gueroult's "Spinoza's Letter on the Infinite (Letter XII, to Louis Meyer)," summarized

by Corry Shores
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I.

Infinity and indivisibility are two unique properties of substance which derive immediately from its fundamental property: cuasa sui.

If something exists on its own account, it cannot be deprived of any of its existence, and is thus infinite and indivisible. These two properties go hand in hand, so there is a "radical antinomy" between the infinite and the divisible. Thus because Spinoza affirms the infinite, he denies divisibility. This conflict in substance is resolved when we consider that it necessarily exists of itself. (182a.b)

However, we affirm infinite divisibility on the modal level, and thus we affirm both the infinite and the divisible, despite their antinomy. This problem is not solved in the Ethics but rather in his Letter on the Infinite.

The letter is a correction of common errors made when conceiving the infinite, so it is not a directly expounded doctrine, but rather an obscure text that only indirectly makes positive claims about infinity (183a).

II.

Difficulties with the infinite result from three types of confusions failing to distinguish six separate cases:

A.1 The thing infinite by its essence or by virtue of its definition.

A.2 The thing without limits, not by virtue of its essence, but by virtue of its cause.

B.3 The thing infinite insofar as without limits

B.4 The thing infinite insofar as its parts, although included within a maximum and a minimum known to us, cannot be expressed by any number.

C.5 The things representable by understanding alone and not by imagination.

C.6 The things representable at once by imagination and by understanding.

(above sentences quoted from 183-184)

By confusing these pairs, we fail to distinguish

a) divisible and indivisible infinities

b) infinities divisible without contradiction

c) infinities conceivably larger than others

d) infinities not so conceivable

(184a)

III.

To understand infinity we must always keep in mind that "there exists in Nature nothing other than substance in eternity and modes in duration" (184b).

IV. First Case: Things infinite by their essence

We cannot think substance as finite, for it is defined as being unlimited, nor as divisible, because partitioning implies a finitude. Substance is thus infinite, eternal, and indivisible.

V.

Second Case: Things unlimited only by their cause

Modes' essences do not "envelop" their existences, so they are not infinite by reason of their essence, and they are not conceived under the concept of eternity. Modes can be thought not to exist and to begin and end, and so are divisible and subject to duration. (184d)

However, insofar as modes are produced by God, they are infinite by reason of their cause. God's power affirms, produces, and conserves the modes existences, and in this way they are internally unrestricted.

In this way the infinite is enveloped, since infinity is defined as "the absolute affirmation of the existence of any nature whatever." (185a)

Hence we consider modes as "without limits" or infinite when conceiving them in terms of the internal force of the divine cause affirming them.

But, because modal essences do not necessarily envelop their existences, their absolute affirmation is not necessary; modes could be only partially affirmed, and they do not "invincibly envelop" their actual infinity.

The infinity of the internal force is thus resolved into a simple indefinite tendency to exist and to persevere in being. (185b)

Limits that divide without contradiction the existence or duration of a mode are contingent, foreign accidents to the mode. Without them, the mode would have been infinite and indivisible. Although, this is just a possibility; a mode can be considered as having limits, but nothing says it must have them.

This is why it does not radically exclude time, but only finite time, and must thus be said "to envelop an indefinite time" (185c). The infinity of modes then is not essential infinity, which by definition excludes a priori absolutely all limits (185cd).

Like the Cartesian indefinite, modes do not necessarily imply limitation. But unlike the Cartesian indefinite, whose eminent subjectivity follows from our inability to "decide between the finitude or objective infinity of the thing," modes are objectively indefinite on account of the infinity of their cause and the finitude of their essence (185d).

Thus Spinoza presents duration as bearing two sides:

A. The mode's duration seems infinite when we consider its infinite cause. We grasp this duration when we consider God's creative power that

posits and constitutes it as an indefinite and indivisible tendency (conatus). This is what we experience in lived duration. (186a)

Although the mode's essence envelops "the infinite and indivisible power by which it can be advanced to existence," it "does not at the same time envelop the necessity that this power make it exist." So because it is not necessarily infinite, "its duration appears finite and divisible" (186b). There is an infinite chain of causes that necessarily force the mode's existence to be in some certain place in the universe. But when we consider the mode in abstraction from this infinite chain (which we must do given we do not have access to it anyway), it seems contingent, because it is arbitrary where we conceive it to be, here or there, now or then. The mode's existence is contingent in relation to its essence, because its existence is not determined by its essence. It is for this reason that exterior things necessarily determine it:

it is only the universal context, the common order of Nature, which makes it necessary at such and such a point in the series. (186c)

On account of the modal existence's contingent relation to its essence, we may posit the mode's existence and duration in any way we choose, larger or smaller, and divided into parts. "Duration is thus 'abstractly conceived as a kind of quantity'" (187, Gueroult quoting Spinoza, E., II, xlv, S.)

Insofar as it envelops the divine power that causes it, every singular body carries an internal tendency to persevere in its being. (187a)

Singular bodies, then, do not contain anything that could limit or divide their existence, and hence they remain identical to themselves, and undivided until some exterior cause limits, fragments, or destroys it. But, because the modal essence does not envelop the mode's existence, we may conceive it being able to be or not to be, or as limited, fragmented, altered in size or partitioned.

Here, magnitude is not perceived as substance, but superficially and abstractly grasped as a property common to all modes of extension. It then appears as an infinitely divisible quantity. (187b)

The mode is indivisible and infinite in terms of its interior cause, the infinite power of God that is "the immediate cause in it of the immanent force that makes it exist and continue" (187c). Division is extrinsic to the mode, because it "befalls the mode, as if by accident, from without." But most importantly,

the infinite divisibility of time and space, determinable ad libitum, is only an abstract concept forged by the imagination from the accidental limitation of the internal force, indivisible in itself, which advances and sustains the existence of the mode. (187d)

VI. Third Case: Things infinite insofar as they lack limits

Some things are considered infinite on account of their lacking external factors to limit them. Something whose limits we cannot imagine is more properly indefinite. An example of the unlimited infinite would be infinite magnitude considered abstractly. For the imagination, it is "a given whose limit is never reached, but one for which we see no reason whatever to affirm that such a limit be impossible" (188b).

There is another type of unlimited infinite, which is so because of some extrinsic cause, so for example, the mode which is the whole of finite modes is infinite in this way, because it springs up from the entirety of the infinite substance or God. Such an infinity is "without limits" and immense. But, because it cannot exist on its own, it is interiorly infinitely divisible, although it can "never be rendered finite by an ultimate limit which would restrict its immensity" (188c-189). And as the whole of finite modes, nothing outside it can limit it, hence it is both infinite by reason of its cause (God's power) and infinite in itself, as the infinite effect of its infinite cause. Contrariwise, singular modes are only infinite on account of their cause, because they are merely a finite effect of it (189b).

VII. Fourth Case: Things infinite insofar as their parts are numerically inexpressible, despite bearing minima and maxima.

The previous infinity is such because it is too great for us to count, but if they are potentially countable, then the are not truly infinite. Our imagination causes this error, which can be made evident

by spaces, which, although contained within the boundaries of a maximum and a minimum are able to be as small as one wishes, and still each one an Infinite, without being so by virtue of the immensity of their size nor, consequently, by virtue of the exceedingly great multitude of their parts. (190a)

Such an infinite "escapes number," not "because it contains too many parts, but because, by nature, it is not expressible by it" (190b), which we illustrate later with Spinoza's geometrical example.

VIII.

We have thus answered four questions:

1) Substance is infinite because it cannot be divided or limited.

2) Mode's are infinite because they are divisible without contradiction; by virtue of the infinite cause, no limitation, partition, or division destroys their nature.

3) Infinites which we may conceive as being larger than other infinites do not exclude divisibility and are included in every singular mode.

4) However, we cannot conceive substance's infinity in this way.

IX. The fifth and sixth cases: Things known by the understanding alone, and things known by the understanding and by the imagination.

We neglect the above truths when conceiving the infinite with the imagination rather than the understanding. (191a)

The understanding knows essences; the imagination knows only existences. The understanding knows things as they are in themselves (ut in se sunt); the imagination grasps only the affections that they determine in our body. (191b)

Hence:

1) We cannot know substance by the imagination, which can only know modes, for it is limited to perceiving our bodies' affections. Only the understanding can grasp substance as self-caused, eternal, infinite by essence and thus as well indivisible, because it can conceive substance's inseity (its in-itselfness) and its perseity (its quality or fact of existing independently) (191b).

2) We know modes confusedly by the imagination, and "rightly" through understanding.

The imagination perceives modes, since it perceives the affections of the Body, which are modes. But it does not perceive them as modes of substance, since it does not know substance.

The understanding knows modes and perceives them as modes of substance, since it knows substance. It knows them rightly, for it knows them in and through substance and sees "how they flow from eternal things." (191d; quoting Spinoza Ep., 12)

Thus understanding knows both the modes'

nature of their duration as infinite and indivisible on principle (lived duration), and the nature of bodies and their diverse sizes, which it conceives as the variable and continuous modification of the same extended substance, in itself absolutely infinite and indivisible.

X.

We pervert our knowledge of Nature if we confuse imaginable with unimaginable things, resulting in two errors (191-192):

1) We mistakenly imagine that modes are independent of one another, because we can only know that they are united in and through substance. Hence, the imagination conceives modes as independent really separated substances, and thus it

introduces in them the divisibility of the discontinuous (founded on real distinction), instead of the divisibility of the continuous (founded on modal distinction), which is proper to them. (192ab)

2) Thus the imagination breaks substance into as many finite substances as it perceives modes. This error is threefold:

a) Divisibility belongs only to modes, but is here is ascribed to substance, which absolutely excludes divisibility

b) This divisibility of the discontinuous which we ascribe to modes is not proper to them.

c) This posits every substance as finite, "which is just as absurd as to posit a square circle." (193b.c)

When we reduce substance's infinity to that of finite substances,

that is, of absolutely independent being, we are then forced to conceive it as resulting from the addition of finite things or parts: one must then explain the absolutely indivisible Infinite (that of substance) by the divisible, and what is without parts by parts. In a similar manner, one must also explain the infinitely divisible Infinite (that of modes) by the addition of these modes to infinity. (192c.d)

So either way we explain the infinite in terms of the multitude of its parts. But it is as senseless to conceive the infinite by the finite as to "wish to construct a triangle or a square with circles, or an essence with essences which negate it" (192d).

XI.

The above error underlies the two common mistakes,

the one consisting in denying the indivisibility of substance and in thus affirming that it is finite, the other, in denying the infinite divisibility of its modes and in thus affirming that their multitude is finite. (193a)

these two refutations amount to only one, which consists in proving that no Infinite can be inferred from its parts, that is, that discontinuity is a fiction.

For example, there is no difference between proving substance is "an absolutely indivisible Infinite" and proving that a line is an infinitely divisible Infinity, because if we wanted to formulate the contrary for refutation, we would "postulate the same absurdity, namely, that substance is composed of parts and the line composed of points" (193b).

Proving substance's absolute indivisibility is no different from proving modes' continuity (or infinite divisibility).

the endless divisibility of the continuous, which is that of modes, is conceivable only through the indestructible subsistence in them of an indivisible absolute, which necessitates that no truly separate part can ever be reached and that the division, since it can never be completed, be absolutely infinite. This absolute indivisibililty, which is immanent in the modes, is that of their substance. (193c.d)

[so in other words, we may continually subdivide modes, never reaching a final subdivision, because they are modifications of something that has no subdivisions. The infinity of substance keeps producing more and more space, no matter how much we subdivide, hence Zeno's paradox.]

However, the understanding can properly conceive substance as indivisible.

"This is why, if we consider magnitude as it is for the imagination, which is the most frequent and the easiest case, we find it to be divisible, finite, composed of parts, and multiple. If, however, we consider it as it is in the understanding, and if the thing is perceived as it is in itself, which is very difficult, then, just as I have sufficiently demonstrated to you earlier, we find it to be infinite, indivisible, and unique." (193-194; quoting Spinoza Ep., 12)

We began by noting that substance is infinite and indivisible, and hence there is a tension between infinity and divisibility. But when our understanding considers knowledge of substance, the tension disappears, because its infinity excludes all divisibility. (194b)

However, modes are both infinite and divisible, because here there is not a division of discontinuous separate parts. Rather, the parts are continuous and not really separate, and "can never in themselves constitute a determined multitude" (b.c) [So if they were separate, they would be a determinate multitude, and hence expressible by number, but as continuous and infinitely divisible, they are indeterminate, and thus number cannot express them, similarly to the irrational numbers. So because there are innumerable differences in the diagram, they are infinite; for, no number can express its quantity: there is no number of lines. And yet, this infinity is inscribed within limits, creating an extensive magnitude that may be greater or lesser than some other. So even though both are infinities, one is a larger infinity than the other. It is not larger because it has a larger number, for neither has a number of parts. Rather it is larger because it has a wider extent of internal differences, that is, ofintensive alterations. So in other words, one such infinity is greater than another because it has a greater intensive magnitude.]

So, because we do not consider infinity to come from a multitude of finite parts, but rather an indefinite quantity of a multitude of parts to come from the infinite substance, we no longer explain the indivisible and infinite through the divisible and the finite, but rather the other way around.

Henceforth, finite things being conceived as, each one, interiorly infinite, and all together as constituting an infinite, they are only the second aspect or the immediate expression of the absolute indivisibility of what is by nature the Infinite.

XII. The above conclusions support three others:

1) The absolutely infinite and indivisible substance must produce an infinite number of modes. Thus modes are not discontinuous, in which case they would be a finite multitude of finite parts.

2) Only modes may be infinitely divisible, because infinite divisibility is conceivable solely by means of the modal distinction defining them.

3) Substance is indivisible, since "divisibility is the property of its affections and since it is beyond its own affections." (194-195)

Substance is

a) anterior to modes

b) conceivable in its truth only when disregarding modes

c) the mode's cause, "and there is nothing in common between the cause as cause and the effect as effect."

XIII.

To say that the infinite divisibility of every mode envelops the absolute indivisibility of substance, is to say that this substance, with regard to its nature, is complete in each mode. Moreover, this conclusion is evident in the very concept of indivisibility, for what is indivisible by nature can only be complete wherever it is, that is, "equally in the part and in the whole." (195b; quoting Spinoza E., II, xxxvii, xxxviii, and xlvi)

Thus substance is equally and entirely

in the totality of its modes as it is in each of them, in each of them as it is in each of their parts, and in each of their parts as in each of the parts of these parts, etc., to infinity.(195c)

Moreover, substance is found in modal parts in two ways:

1) through the attribute defining its essence. Extension constitutes corporeal substance's essence, and it is as complete in each body as it is in them all,

inasmuch as it is the common property by which they are identical among themselves and identical to it. Indeed, since the nature of extension remains complete, that is, identically what it is, in the least of its particle, it is necessarily present, with the indivisibility proper to it, in every part of the different bodies. (195cd)

2) Substance is the indivisible cause by which modes exist and therefore

the idea of this substance is enveloped equally in the idea of the whole and in that of the part

Hence, every mode, whether small or large, envelops within itself the indivisibility of infinite substance.

But insofar as modes are finite beings, they are divisible.

This divisibility is infinite, however, since division will never be able to really separateit, either from other modes or from the indivisible substance immanent in it. Thus in each part (or mode), however small it may be, we rediscover in its integrity the same indivisible infinite which allows it an infinite divisibility in act. (196a.b)

So there are as many different infinitely divisible infinites as there are modes, thus in the attribute of extension, there are as many infinites as there are different sizes, "each infinitely divisible in its own fashion" (196bc).

And yet, under each of these different infinities there is also always the same identical Infinite, which could not be smaller or larger than another, namely, the Infinite of substance which is equally complete in each of them. ... The infinite that is larger or smaller is the infinite of substance, invariable in itself, perceived as contained within the limits of a mode, limits which are more or less restricted according to the different modes. (196c)

Likewise for the attribute of Thought. The indivisible substance is found complete in each mode of Thought, thus "the idea of substance and of its modes is equally, that is, in its completeness, in the whole and in the part" (196d). Every soul, then, knows the infinite, and we may adequately know what can be deduced from the infinity of modes, and we may know it to the same extent as God does, because the infinity of substance is no less in the partial modes as in the whole substance. Hence Spinoza's doctrine of substance's indivisibility and mode's infinite divisibility lies at the heart of his theory of knowledge (196-197).

XIV.

Some object that it is absurd to claim there are different sizes of infinities, because no matter its extensive size, if both contain an infinity of parts, then both contain the same amount of parts: what surpasses all number is the largest of all, and hence could be said to be either larger or smaller than something else. Furthermore, some critics object that a number is only infinite because it is too great for our understanding, so really it is a definite number that is indefinite to us, for "every multitude of parts, however great it may be, always constitutes a number, and every number, being determinate, is finite" (197b).

XV.

The above objections are based on the following two principles:

1) the infinite is inferred from the multitude of its parts

2) number is in itself competent to express every magnitude. (197c)

Both these presuppositions are summarized in the common postulate: number sovereignly governs Nature and our understanding. Number forces the understanding to deny what seems obvious to it, "that substance is absolutely indivisible and the mode infinitely divisible" (197c). To critique this postulate we will seek out its origin and genesis.

XVI.

Most minds have been compelled to consider number as being the highest ideas of understanding and the fundamental laws of Nature (197d).

But the concepts number, measure, and time have nothing to do with the understanding.

they offer, rather, this threefold quality of being:

a) Products of the imagination that is, Beings of reason or rather of imagination.

b) Nothings of knowledge

c) Aids of the imagination (auxilia imaginationis), merely capable of facilitating the conception of imagined things. (198a.b)

a) Products of the imagination:

The imagination brings about number, measure, and time through analogous processes. Time determines duration and measure determines quantity, and only their object distinguishes the two. Time is measured duration considered abstractly as a quantity, and number confers to measure "the exactness which distinguishes it from simple evaluation." These three concepts then are inseparable and result from a similar processes of confusion, abstraction, and limitation. (198b.c)

But unlike Aristotle who thought that time arises from number as being the number of movement, or Bergson who traced abstract space onto concrete duration, we should not consider number, measure, and time to have an absolutely identical origin (198c).

"Measure comes from the abstract knowledge of what constitutes the essence of extended substance." We separate magnitude from substance and apply it strictly to its modes as their common property, and we then regard modes as divisible, composite, and multiple, hence we may delimit magnitude in terms of measure.

"Time comes from the abstract knowledge of the existence (or duration) of modes." When we grasp modal duration independently of the "the eternal things from which it flows and from the order of Nature which determines it;" and we conceive this modal duration as contingent, variable, and divisible at will in such a way that we may delimit it in terms of time. (198-199)

"Number comes from the confused knowledge of multitude and of the differences among singular things." When things affect our Bodies, the imagination succeeds in retaining only their general characteristics. Then the imagination "divides them into classes where they subsist only as unities without intrinsic difference, capable of being counted." (199a)

All these processes presuppose "a foundation of discontinuity, the principle of discrete unities" (199b).

b) Nothings of knowledge

Number, measure, and time are nothings of knowledge, because

b)a) they are not ideas, for they do not represent real objects outside us

b)b) Insofar as we define truth as the conformity of idea to its object, these notions are without this sort of truth

b)c) They are neither true nor false, thus we cannot say that they are false. (199c)

Hence we see that these concepts, number, measure, and time, are modes of imagining rather than modes of thinking. (199c)

c) Aids of the imagination

These notions could not be instruments of the understanding, instead only instruments of the imagination.

Introducing into "imagined things," that is, into the qualitative and heterogeneous perception of corporeal affections, the homogeneity of similar parts and the discrete character of identical unities, they permit us to "retain" them better, and by establishing "comparisons" and "relations" among them, to "explain" them better. (199-200)

These "pragmatic instruments" which allows us to better orient ourselves in the world of sensibility need not be true, but merely just efficacious (200b).

XVII.

Affirming the sovereignty of number and of related notions, in effect, shatters Nature, for it establishes everywhere the discrete. (200c)

1) The affirmation of number's sovereignty pulverizes substance, "reducing it to a collection of finite substances, whereas finite substance is a chimera as absurd as the square circle" (200cd).

2) It means "implicating substance in the contradiction of the infinite and the divisible, to which it is foreign, since it is absolutely indivisible."

3) It means "disuniting the modes into a multitude of really separate parts, whose number, however great it may be, could not be infinite, since the infinite and number are mutually exclusive."

4) It means "emptying each mode of its internal infinity, because if the infinite is not in the whole it cannot be in the part." (200d)

5) It means "refusing to admit that modes of different magnitudes can envelop unequal infinities, because if the infinite must in each be inferred from the multitude of its parts, the number of these parts must, as infinite, be the greatest of all in each -- an other words, the same" (200-201). But this is absurd, because they are different according to their magnitudes.

6) Lastly, "it is to cut into pieces duration, which, at the base of all things, is in itself indivisible and infinite, and to claim to reconstruct it by their aggregate" (200a).

This is why we cannot understand why duration elapses; for, if we may infinitely divide time, we will never endure a duration, similarly to Zeno's paradox. Gueroult asks, "in order to avoid dividing it indefinitely, will we reduce it to a multitude of indivisible instants? But this would be claiming to construct it of nothings of duration. We could just as well 'hope to form a number by adding zeros'" (201b; quoting Spinoza EP., 12).

We encounter these problems because number, measure and time are not infinite, but are merely aids to the imagination. (200b) But those who make this mistake come to deny the reality of the infinite, and consequently to deny the existence of God (200c).

XVIII.

Spinoza's devaluation of number separates his mathematical philosophy from that of his contemporary mathematicians (200d).

XIX.

Mathematicians are correct insofar as they have clear and distinct ideas and reject these four propositions:

1) Everything can be expressed by a number. This is false, because there are irrational magnitudes, which no number can express.

2) Every Infinite is such that its magnitude is so excessive (nimia magnitudine) that we cannot perceive its limits, or that its variations are not contained between any boundaries. This is wrong, because we know that there are infinite magnitudes contained between precisely determined maxima andminima. These magnitudes are infinite in that no matter how small, they "always include an infinity of variations."

3) Every Infinite is such that the multitude of its parts is such that we cannot succeed in assigning a number to it. This is an error, because the real reason that they are infinite is because "it is contradictory to their nature that number be applied to them," not because "the multitude of their parts surpasses every assignable number." (203-203)

4) There cannot be unequal infinities. This is false because it presupposes that the infinite can be expressed only by the greatest of all numbers, but we know that it is absurd to think that number even applies to the infinite (203b).

XX.

Spinoza offers his geometrical example to illustrate the fourth case (The thing infinite insofar as its parts, although included within a maximum and a minimum, cannot be expressed by any number) and refute the false interpretation of the third case (The thing infinite insofar as without limits), that false interpretation being the second error ((Every Infinite is such that its magnitude is so excessive (nimia magnitudine) that we cannot perceive its limits, or that its variations are not contained between any boundaries)). And thus it concerns the second pair (The thing infinite insofar as without limits; the thing infinite insofar as its parts, although included within a maximum and a minimum known to us, cannot be expressed by any number).

Consider the diagram below, with two nonconcentric circles BC inscribed within AD.

We can see that

"the sum of the inequalities of the distances included between them (that is, the sum of the variations of these distances) is an infinite. This infinite does not result from the excessive magnitude (nimia magnitudine) of the space interposed between them, since if we consider only a portion of it, as small as we wish, the sum of the inequalities of the distances always surpasses every number." The infinite also does not result from the fact that the variations in distance are not within a maximum and minimum, because line AB is the maximum, and CD is the minimum. (203b.c)

"It results from the fact that the nature of the space interposed between the two nonconcentric circles does not allow a finite, determinate number of inequalities of distance" (203d).

"If, then, these things can be called indefinite, it is because number cannot equal them, that is, define them, but not because they are in themselves deprived of true infinity. 'Indefinite' refers here, then, not to the nature of the thing, but only to the 'impotence of the imagination'" (203-204).

XXI

But Tschirnhaus observes that we still have not demonstrated that the Infinite is not inferred from the multitude of its parts (204c). But, Gueroult says, the contrary is so; for, inferring the infinite from the multitude of its parts is to say that its multitude is greater than any given multitude, that is, greater than any assignable number, and hence it would be impossible to conceive any greater magnitude.

"Now, this consequence is false, since in the total space included between the two circles we conceive a multitude of parts two times greater than in half this space, --although, hypothetically, the multitude of the parts of half the space as well as of the total space is greater than any assignable number." (204c.d)

The contradiction arises because on the one hand, we claim that there are such a great number of parts that it would be impossible to conceive any greater, while on the other hand we conceive that the larger extensive space does have a greater multitude. Gueroult offers an explanation that Spinoza does not provide: it is absurd to think that both half the space and the whole space have a number of parts greater than any assignable number, because then the whole would no longer be greater than the half.

XXII.

Because of such absurdities, Spinoza thinks that the understanding and not the imagination is capable of conceiving the infinite. Descartes also thinks that infinities come in different sizes, but he considers number as an idea of the understanding, and not a product of the imagination as Spinoza conceives it. (205b)

Spinoza is unlike Leibniz who does not reduce number to a finite being of the imagination. Spinoza's notion of the true idea is as "adaequatio, that is, as the intuitive grasp of the totality of reasons or 'requisites' of the thing;" unlike Leibniz, for whom the notion of the true idea is an "oblique expression, necessarily symbolic, of a reality which, remaining in itself always inaccessible to intuition, can be grasped only by blind though by means of an algorithm." (206b)

XIII.

The geometrical example has led to misunderstandings partly on account of errors in translation.

1). "quantumvis parvam ejus portionem capiamus" has been incorrectly translated as "however small we may conceive it" or "suppose it," with the it "designating the space interposed between the two circles." This space decreases as "these two circles themselves are smaller or as the inscribed circle is larger." This passage should have been translated, "However small be the part that we may consider of the interposed space," in other words, whether we consider half the space, a fourth, a thousandth, or whatever. (206c)

2) For the passage "Omnes inaequalitates spatii duobus circulis AB et CD interpositi," Hegel along with the commentators who followed his translation render "inaequalitates spatii," as "unequal distances" rather than as "inequalities of distance." Gueroult proceeds to explain the example.

We consider circumference O and circumference O '

and lines CD, EF, and AB between them. We find that CD is less than all the lines EF that span the distance between CD and AB.

"The sum of the segments EF is then an end-to-end placement of segments of a transfinite number (according to current terminology)" [larger than all finite numbers but not necessarily absolutely infinite] (207a).

But Gueroult does not accept this interpretation, because Spinoza could have offered a less complex example making the same point; he could have said to add the infinity of lines between two concentric circles with equal lines between them. Still the sum of their lengths would equal infinity (207a.b).

But because Spinoza sums the inequalities, the circles cannot be concentric. If all the lines were equal, we could not sum the variations of segments EF.

"We see then that the example can illustrate the thesis that any portion of space envelops an infinite divisibility, inexpressible by number" (207c).

The distance D'B

is found by subtracting the minimum from the maximum. If we wanted, we could arbitrarily determine a set of lines between them, lines EF, whose subtracted lengths equal the difference between the maximumand minimum.

But this requires we only consider a finite number of differences, rather than the infinite that Spinoza speaks-of.

However, when EF varies continuously from AB to CD, the sum of the variations, that is, the integral of the differentials of EF (when E is on the arc of the circle)

(207c.d) [Firstly we note that the diagram lettering does not place A and C along the same arc, so we will take Gueroult here to mean arc of circle AD. So if we were to split up each of the small divisions given in the list above, we would at the same time split the differences, so we would never gain anything by doing such an infinite splitting, because the number would always converge to the value of D'B].

Spinoza here takes up what Descartes says about constricted vortical rings, "where the necessity imposed upon matter in order to overcome this constriction of dividing itself endlessly compensates for the narrowness of place with the greater speed of its smaller parts." Spinoza makes use of the non-concentric circles of Descartes' vortex in his Principia philosophiae cartesianae. Here Spinoza takes up Descartes' distinction between inaequalitates spatii and spatia inaequalia. The water moving through the vortical rings that constricts its movement, hence the quantity of this matter must move across unequal spaces (inaequalia spatia) included between a maximum and a minimum.

And on account of the infinite divisibility of space, the multitude of these spaces surpasses every number, so in order for the same quantity of matter to pass through the decreasing spaces, "each time it must lose an infinitely small part of itself corresponding to the progressive narrowing of the canal, and to compensate for this its speed must increase each time to a correspondingly infintely small degree." (208a.c) So because there is a continuum of change, "the sum of the inaequalitates, that is, of the inequalities or differences of variation of the volume of matter as well as of its speed, in short, the sum of the small parts of diminished extension, just like the sum of the small degrees of accumulated speed, is an infinite, inexpressible in terms of number" (208-209).

For this reason, it is more true to the original text, "omnes inaequalitates spatii," to translate it as "the sum of the inequalities of distance," rather than, "sum of unequal distances" (209b).

Gueroult now has us consider two segments AB and CD as located on a "secant line that pivots around the center O.

Then we add their successive differences and obtain in absolute values:

[To better grasp what Gueroult might be saying, we will draw from the integral section of Edwards & Penney's Calculus. We might think of dividing the area between the circles into a number of sections:

Each segment has an area, but we may then divide it into smaller sections:

and still smaller

and smaller

Each time, as the number of spaces increases, their areas decrease. When those areas approach zero, and are infinitesimally small,

We may take the sum of their areas by means of integration. We find that although there are an infinite number of different spaces we are summing, there is a finite amount of space].

So we see that Spinoza has us sum the inequalities, and not the distances, and we see that within a finite circumscribed space, there are an infinity of such inequalities (209b.c).

We also see that the finite distance between the maximum and minimum "includes an actual infinity of infinitely small distances, and consequently is indivisible into discontinuous parts," because instead it is infinitely divisible into continuous parts. As well, the distance CD is a "minimal quantity obtained by continuously diminishing the infinitely small parts of AB, the sum of these diminishments being a definite integral, that is, a finite quantity D'B resulting from an infinite summation of differentials" (209c.d).

XXIV.

Spinoza's solution to the problem of the Infinite resolves all our antinomies (209d). Our imagination confers upon the infinite properties of the finite, and where our imaginations fail to conceive the infinite, our understandings are able. (210a-c)

Also, when our understandings do grasp substance, they realize that its infinity and indivisibility generate all that exists (2510c).

And because this absolute indivisibility rests at the basis of all things, finite modes are thus infinitely divisible and continuous (210d).

Regarding the antinomy of the world's beginning, we know that we cannot assign to substance determinate time and duration. "The universe, that is, the whole of finite modes, is thus an infinite and has never begun." (210b)

Our duration is our existence, and it is "posited by the immanent and eternal act of substance" (211bc). Substance's act of positing our durations constitutes our existence, because it "constitutes it from within." The act is indivisible, because substance is indivisible, and the act is "complete in my existence and entire in each of its moments," again, on account of substance being indivisible. "Each of the moments of my real duration is thus an actual infinite" (211c). Two things result.

1) This fact reverses the Cartesian problem: instead of needing to explain what keeps beings in their perdurance, we need instead account for what causes their end.

Leibniz thought that each instant "encloses itself in the infinite," but the way that Spinoza means this is that each moment envelops the infinity of its cause, and not "the infinity of all the predicates, past and future, of my existence" (211cd). Our essence, then, "includes only the reason of what defines it sub specie aeternitatis"(212a).

Hence, every instant of my duration envelops, not the infinity of past and future moments of this existence, that is, the infinity of its predicates, but only the identity of the indivisible duration of my existence, directly expressing the infinity of its cause, whose eternity, although having no common measure with the succeeding instants, is nevertheless immanent to them (212b).

2) Because the changes to our existence "do not ever really divide it and are only changes of modes, my duration can only be continuous" (212bc). We wrongly see our durations as discontinuous when we confuse them with time, "the imaginative aid which serves to measure it."

Finite, like any measure, time, in order to measure it [duration], endeavors to reconstruct it by joining finite fragments end to end, fragments which are really separate one from another, inert, because detached from the act which makes them exist, in short, discontinuous.(212c)

Thus when we see substance for what it is, we see Nature unveiled, "the unity of an infinite without break or fault" (212d).

from:

Gueroult, Martial. "Spinoza's Letter on the Infinite." inSpinoza: A Collection of Critical Essays. Ed. Marjorie Grene. Transl. Kathleen McLaughlin. Indiana: University of Notre Dame Press, 1979.