22 Nov 2008

Spinoza Letter 12: "The Letter on the Infinite" summarized

by Corry Shores
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[The following summarizes Spinoza's 12th Letter, "The Letter on the Infinite." At the end I place the text in full.]

We may correctly conceive the infinite by distinguishing three sets of related notions:

I. Infinity as a consequence either of something's nature or its cause:
I.1) One sort of infinity is infinite by its very nature, which is the same as saying that it is infinite by virtue of its definition, and
I.2) Another sort of infinity is unlimited not by virtue of its essence (nature, or definition), but by virtue of its cause; and also,

II. Unlimited Infinities and Infinities Contained within Limited Bounds
II.1) One sort of infinity is called infinite because it is unlimited, and
II.2) Another sort is called infinite because its parts cannot be equated or explicated by any number, despite our knowing its maximum and minimum (The Letters 101-102); and finally,

III. Strictly intelligible things, and things both intelligible and imaginable:
III.1) We can apprehend certain things only by the intellect and not by the imagination, and
III.2) We can apprehend certain other things both by the intellect and the imagination.

These distinctions will allow us to differentiate kinds of infinities:
A.1) those that can neither be divided into parts nor possess any parts, and
A.2) those that can be divided into parts without contradiction.

B1) those that can be considered greater or lesser than other infinities, and
B2) those that cannot be considered greater or lesser than other infinities.

Spinoza explains that to understand these distinctions, we must review his notions of Substance, Mode, Eternity, and Duration.

1) Substance's existence follows solely from its essence and definition.
2) It is not manifold; there is only one substance.
3) Substance must be conceived as being infinite.

1) Modes are the affections of Substance.
2) When we define a mode, we do not include in that definition that it exists (unlike Substance whose existence is predicated to it in its definition). Thus, even if the mode does exist, we can conceive it hypothetically as not existing. However, we cannot consider substance as not existing, because that involves a contradiction with its definition.
3) Modes are part of the order of Nature as a whole, that is, the chain of causation; and also, modes have essences that can be considered individually and as apart from the greater chain that contextualizes them. When we consider their essences in this abstract manner, we cannot deduce from their present existence alone that they will or will not exist in the future, or that they did or did not exist in the past (102cd). Thus Substance is of an entirely different kind than modes, because Substance must always necessarily exist.

1) This is Substance's "infinite enjoyment of existence" or being (essendi).

1) This is the mode's finite span of existence.

Thus we may consider a mode outside its context of other modes; and, we may thereby conceive its finite duration of existence as being greater or less than another. We may also consider this duration as divisible into smaller parts, smaller units of duration. However, we cannot consider Eternity and Substance as being greater or less or as being divisible, because this would annul their definition, which says that there is only one Eternal infinite Substance. Rather, we can only conceive them as infinite. Hence we cannot think of substance as being composed of parts or bodies that are really distinct from each other; rather, modes are modifications that are immanent expressions of substance, not divisible parts that can exist on their own individually.

For this reason, we cannot side with philosophers who argue that extended Substance is finite, because this presumes that substance is composed of finite parts.

The reason we are normally inclined to divide extended Substance is because we conceive Quantity in two ways,
1) when we consider it abstractly or superficially, as when we imagine it with the help of the senses. This is how we usually conceive quantity, and it is by this means that we think it to be divisible, finite, composed of parts, and manifold; and,
2) when we conceive it as Substance, that is, when we apprehend it solely by means of the intellect and thus apprehend it as it is in itself, which is quite difficult. This is how we find it to be infinite, indivisible, and one alone (103c).

Because we conceive Quantity in abstraction from Substance, and Duration in abstraction from Eternity, we make use of two other notions: Time and Measure.
1) Time delimits Duration.
2) Measure delimits Quantity.

And because we consider Substance's affections in their own classes, we make use of the notion of Number to determine the quantities of affections in each class.

Thus Measure, Time, and Motion are no more than modes of thinking or imagining; and when we use these ideas to understand Nature, we obtain the many confusions we have about infinity, because these ideas are just tools for the imagination, and not actual properties of Nature.

For example, Duration should not be confused with Time. If we did confuse them by considering Duration as divided into parts, we would confront Zeno's paradox; for, "in order that an hour should pass by, a half-hour must first pass by, and then half of the remainder, and the half of what is left; and if you go on thus subtracting half of the remainder to infinity, you can never reach the end of the hour" (104c). When we conceive of number in a similar way, we mistakenly regard it as being made up of noughts added together.

Because Number, Measure, and Time are delimitations invented by the imagination, none of them can be infinite. Thus when we mistakenly apply these notions to reality, we might commit the error of denying the actual existence of the infinite. Mathematicians realize this, because they do not apply the notion of number to every quantity.

Spinoza gives the geometrical example of an off-centered circle set within a larger circle. This example is difficult to understand unless we examine Proposition 9 in his Principles of Cartesian Philosophy. He has us imagine the circulation of water moving through the space between offset circles:

The water circuit moves from A to B to C and so on, and the space at A is four times as wide as it is at B. Because the same amount of water must move through space-A as moves through space-B, which is one-fourth the width, the water must move four times as fast to replace the water that displaces from A. Spinoza demonstrates in the previous proposition this necessity for the water at B to rush-over to replace A. Here he has us imagine a circuit of bodes 1 through 8:

When body 1 moves into the place of body 2, then body 2 must move into its new place, where body 3 is, and so on. And at the same time that body 1 moves to body 2, some body must move into the place where body 1 was, in this case, body 8. Hence the displacement of body 1 necessitates its displaced space be filled. So in the case of the water circuit, just as much water leaving A must be replaced, and in order to move that quantity of water through the narrower channel, it must move at a greater speed.

And, because of the geometry of the displaced circles, every part of the circuit has a different width. Thus at every point along the circuit, the water moves at a different speed.

In the lemma to proposition 9, Spinoza argues a similar point regarding the spaces between offset semi-circles:

Spinoza writes that in the left diagram, both semi-circles share the same center. The space between their circumferences is everywhere the same (because in a circle, all points on the circumference lie at the same distance from the center).

But, Spinoza writes, if the semi-circles do not share the same center, then the "space between their circumferences will be everywhere unequal" (61):

When see, then, that Spinoza's example in Letter 12 is based on these propositions. The diagram he uses now is of an off-set circle set within a larger circle, like the water flow diagram, but with a cross-sectional line, like with the offset semi-circles:

He writes that "all the inequalities of the space lying between the two circles ABCD in the diagram exceed any number, as do all the variations of the speed of matter moving through that area" (The Letters 105). Every line spanning between the circumferences, that is, every space between them, is of a different magnitude, and hence, if matter were moving through these spaces, it would be moving at different speeds at each different place.

So, the number of lines with different lengths (or the number of places where matter moves at a different speed) cannot be represented by any finite number. We know this is so, on account of the property of density: between any two lines there is always another line.

What is interesting is that Spinoza could have used an example where the circles shared the same center, and said that there are an infinity of lines of equal length between the circumferences. I hypothesize that the reason he used off-centered circles was so that each line would be unique in appearance and not just distinguishable by location. So if they were all the same magnitude, then he would not be stressing the individuality of each line. In the off-centered circles, however, it is easier to imagine that there are an infinity of singularities, and not just an infinity of generalized sames. And also, because there is a dense continuum of singularities, there is a continuous variation of magnitudes. The circumference of a circle is a line that continuously changes its "direction," so when one circle's circumference is offset from another one set within it, there is a continuous divergence of space between them.

Spinoza writes that the reason there is an infinity of unequal spaces is not because there is so much space between the circles; for, we can just look at any portion of that space, no matter how small, and within that portion there will still be an infinite number of unequal spaces. Also, there is an infinity of inequalities not because we do not know the limits of the total space, for we know them to be the largest and smallest lines, line AB (the maxima) and line CD (the minima). Rather, we know that there is an infinity of inequalities because there is no finite number which can count them. Thus on account of the continuous divergence of space between offset circles, "if anyone sought to express all those inequalities by a definite number, he would also have to bring it about that a circle should not be a circle" (105d).

Hence also, if one were to determine all the motions of matter that have ever existed, "reducing them and their duration to a definite number and time," then one would deprive Substance of its nature, because Substance expresses itself through a continuum of varying singular affections; which is to say, there is an infinite number of modes even within the smallest location and duration of Substance's modification.

Thus there are three different ways something can be infinite.
1) Substance is infinite by nature, and cannot be conceived as finite, because it is expressed by an infinite but immanent diversity of individual modes.
2) However, the finite modes taken together are infinite on account of their cause, that is, on account of their being brought about through the infinitely diverse modifications of substance.
3) And lastly, there are modes, like the circle diagram, which are finite in extensive magnitude, but infinite in composition. Such modes are infinite because they are indefinite; for, even though one mode is extensively larger than another, neither mode's contents can be given a definite number. Thus, things which cannot be given a number can still be unequal to each other; in other words, there can be different sizes of infinity: neither the quantity of the singularities (simple bodies) composing a large diverse extended mode nor those composing a small diverse extended mode can be given a number, however, the larger one can be said to contain a larger infinity (or indefinity) of content.

Spinoza then addresses the proof for God that is based on the absurdity of an infinite regress of causes, as articulated by "Rab Chasdai." The rabbi argues that if something is caused, then it exists by virtue of what causes it, and not by virtue of itself. If everything is caused to be, then nothing can exist, because each thing's existence is contingent, and none is necessary. Hence because such an infinite regress of causes is absurd, then there must be an uncaused cause, which we consider to be God.

Spinoza says that the Peripatetics have misunderstood this argument, because they interpreted Chasdai to have said: the notion of infinity itself leads to contradiction, and hence an infinite sequence of causes is absurd, and thus there must be a first uncaused cause. Rather, Spinoza says, the "force" of Chasdia's argument comes from the reasoning that if everything is contingent, then there is no ground or substance underlying the existence of everything, hence if anything contingent exists, there must also necessarily exist some underlying substance.

Original text of the public domain Elwes translation of Letter 12, presented with deepest gratitude to sacred-texts.com, and with all due credit, and highest recommendations:



Dearest Friend,—I have received two letters from you, one dated Jan. 11, delivered to me by our friend, N. N., the other dated March 26, sent by some unknown friend to Leyden. They were both most welcome to me, especially as I gathered from them, that all goes well with you, and that you are often mindful of me. I also owe and repay you the warmest thanks for the courtesy and consideration, with which you have always been kind enough to treat me: I hope you will believe, that I am in no less degree devoted to you, as, when occasion offers, I will always endeavour to prove, as far as my poor powers will admit. As a first proof, I will do my best to answer the questions you ask in your letters. You request me to tell you, what I think about the infinite; I will most readily do so.

Everyone regards the question of the infinite as most difficult, if not insoluble, through not making a distinction between that which must be infinite from its very nature,. or in virtue of its definition, and that which has no limits, not in virtue of its essence, but in virtue of its cause; and also through not distinguishing between that which is called infinite, because it has no limits, and that, of which the parts cannot be equalled or expressed by any number, though the greatest and least magnitude of the whole may be known; and, lastly, through not distinguishing between that, which can be understood but not imagined, and that which can also be imagined. If these distinctions, I repeat, had been attended to, inquirers would not have been overwhelmed with such a vast crowd of difficulties. They would then clearly have understood, what kind of infinite is indivisible and possesses no parts; and what kind, on the other hand, may be divided without involving a contradiction in terms. They would further have understood, what kind of infinite may, without solecism, be conceived greater than another infinite, and what kind cannot be so conceived. All this will plainly appear from what I am about to say.

However, I will first briefly explain the terms substance, mode, eternity, and duration.

The points to be noted concerning substance are these: First, that existence appertains to its essence; in other words, that solely from its essence and definition its existence follows. This, if I remember rightly, I have already proved to you by word of mouth, without the aid of any other propositions. Secondly, as a consequence of the above, that substance is not manifold, but single: there cannot be two of the same nature. Thirdly, every substance must be conceived as infinite.

The modifications of substance I call modes. Their definition, in so far as it is not identical with that of substance, cannot involve any existence. Hence, though they exist, we can conceive them as non-existent. From this it follows, that, when we are regarding only the essence of modes, and not the order of the whole of nature, we cannot conclude from their present existence, that they will exist or not exist in the future, or that they have existed or not existed in the past; whence it is abundantly clear, that we conceive the existence of substance as entirely different from the existence of modes. From this difference arises the distinction between eternity and duration.Duration is only applicable to the existence of modes; eternity is applicable to the existence of substance. that is, the infinite faculty of existence or being (infinitum existendi sive (invitâ Latinitate 1) essendi fruitionem). From what has been said it is quite clear that, when, as is most often the case, we are regarding only the essence of modes and not the order of nature, we may freely limit the existence and duration of modes without destroying the conception we have formed of them; we may conceive them as greater or less, or may divide them into parts. Eternity and substance, being only conceivable as infinite, cannot be thus treated without our conception of them being destroyed. Wherefore it is mere foolishness, or even insanity, to say that extended substance is made up of parts or bodies really distinct from one another. It is as though one should attempt by the aggregation and addition of many circles to make up a square, or a triangle, or something of totally different essence. Wherefore the whole heap of arguments, by which philosophers commonly en' to show that extended substance is finite, falls to the ground by its own weight. For all such persons suppose, that corporeal substance is made up of parts. In the same way, others, who have persuaded themselves that a line is made up of points, have been able to discover many arguments to show that a line is not infinitely divisible. If you ask, why we are by nature so prone to attempt to divide extended substance, I answer, that quantity is conceived by us in two ways, namely, by abstraction or superficially, as we imagine it by the aid of the senses, or as substance, which can only be accomplished through the understanding. So that, if we regard quantity as it exists in the imagination (and this is the more frequent and easy method), it will be found to be divisible, finite, composed of parts, and manifold. But, if we regard it as it is in the understanding, and the thing be conceived as it is in itself (which is very difficult), it will then, as I have sufficiently shown you before, be found to be infinite, indivisible, and single.

Again, from the fact that we can limit duration and quantity at our pleasure, when we conceive the latter abstractedly as apart from substance, and separate the former from the manner whereby it flows from things eternal, there arise time and measure; time for the purpose of limiting duration, measure for the purpose of limiting quantity, so that we may, as far as is possible, the more readily imagine them. Further, inasmuch as we separate the modifications of substance from substance itself, and reduce them to classes, so that we may, as far as is possible, the more readily imagine them, there arises number, whereby we limit them. Whence it is clearly to be seen, that measure, time, and number, are merely modes of thinking, or, rather, of imagining. It is not to be wondered at, therefore, that all, who have endeavoured to understand the course of nature by means of such notions, and without fully understanding even them, have entangled themselves so wondrously, that they have at last only been able to extricate themselves by breaking through every rule and admitting absurdities even of the grossest kind. For there are many things which cannot be conceived through the imagination but only through the understanding, for instance, substance, eternity, and the like; thus, if anyone tries to explain such things by means of conceptions which are mere aids to the imagination, he is simply assisting his imagination to run away with him. 1 Nor can even the modes of substance ever be rightly understood, if we confuse them with entities of the kind mentioned, mere aids of the reason or imagination. In so doing we separate them from substance, and the mode of their derivation from eternity, without which they can never be rightly understood. To make the matter yet more clear, take the following example: when a man conceives of duration abstractedly, and, confusing it with time, begins to divide it into parts, he will never be able to understand how an hour, for instance, can elapse. For in order that an hour should elapse, it is necessary that its half should elapse first, and afterwards half of the remainder, and again half of the half of the remainder, and if you go on thus to infinity, subtracting the half of the residue, you will never be able to arrive at the end of the hour. Wherefore many, who are not accustomed to distinguish abstractions from realities, have ventured to assert that duration is made up of instants, and so in wishing to avoid Charybdis have fallen into Scylla. It is the same thing to make up duration out of instants, as it is to make number simply by adding up noughts.

Further, as it is evident from what has been said, that neither number, nor measure, nor time, being mere aids to the imagination, can be infinite (for, otherwise, number would not be number, nor measure measure, nor time time); it is hence abundantly evident, why many who confuse these three abstractions with realities, through being ignorant of the true nature of things, have actually denied the infinite.

The wretchedness of their reasoning may be judged by mathematicians, who have never allowed themselves to be delayed a moment by arguments of this sort, in the case of things which they clearly and distinctly perceive. For not only have they come across many things, which cannot be expressed by number (thus showing the inadequacy of number for determining all things); but also they have found many things, which cannot be equalled by any number, but surpass every possible number. But they infer hence, that such things surpass enumeration, not because of the multitude of their component parts, but because their nature cannot, without manifest contradiction, be expressed in terms of number. As, for instance, in the case of two circles, non-concentric, whereof one encloses the other, no number can express the inequalities of distance which exist between the two circles, nor all the variations which matter in motion in the intervening space may undergo. This conclusion is not based on the excessive size of the intervening space. However small a portion of it we take, the inequalities of this small portion will surpass all numerical expression. Nor, again, is the conclusion based on the fact, as in other cases, that we do not know the maximum and the minimum of the said space. It springs simply from the fact, that the nature of the space between two non-concentric circles cannot be expressed in number. Therefore, he who would assign a numerical equivalent for the inequalities in question, would be bound, at the same time, to bring about that a circle should not be a circle.

The same result would take place—to return to my subject—if one were to wish to determine all the motions undergone by matter up to the present, by reducing them and their duration to a certain number and time. This would be the same as an attempt to deprive corporeal substance, which we cannot conceive except as existent, of its modifications, and to bring about that it should not possess the nature which it does possess. All this I could clearly demonstrate here, together with many other points touched on in this letter, but I deem it superfluous.

From all that has been said, it is abundantly evident that certain things are in their nature infinite, and can by no means be conceived as finite; whereas there are other things, infinite in virtue of the cause from which they are derived, which can, when conceived abstractedly, be divided into parts, and regarded as finite. Lastly, there are some which are called infinite or, if you prefer, indefinite, because they cannot be expressed in number, which may yet be conceived as greater or less. It does not follow that such are equal, because they are alike incapable of numerical expression. This is plain enough, from the example given, and many others.

Lastly, I have put briefly before you the causes of error and confusion, which have arisen concerning the question of the infinite. I have, if I mistake not, so explained them that no question concerning the infinite remains untreated, or cannot readily be solved from what I have said; wherefore, I do not think it worth while to detain you longer on the matter.

But I should like it first to be observed here, that the later Peripatetics have, I think, misunderstood the proof given by the ancients who sought to demonstrate the existence of God. This, as I find it in a certain Jew named Rabbi Ghasdai, runs as follows:—"If there be an infinite series of causes, all things which are, are caused. But nothing which is caused can exist necessarily in virtue of its own nature. Therefore there is nothing in nature, to whose essence existence necessarily belongs. But this is absurd. Therefore the premise is absurd also." Hence the force of the argument lies not in the impossibility of an actual infinite or an infinite series of causes; but only in the absurdity of the assumption that things, which do not necessarily exist by nature, are not conditioned for existence by a thing, which does by its own nature necessarily exist.

I would now pass on, for time presses, to your second letter: but I shall be able more conveniently to reply to its contents, when you are kind enough to pay me a visit. I therefore beg that you will come as soon as possible; the time for travelling is at hand. Enough. Farewell, and keep in remembrance Yours, &c.

Rhijnsburg, 20 April, 1663.


317:1 See Introduction, pp. xv, xx.

319:1 Spinoza apologizes here in the original for the use of the unclassical form "essendi," being. The classical Latin verb of being is, as the ancients themselves admitted, defective in a most inconvenient degree.

320:1 "Nihilo plus agit, quam si det operam ut sua imaginatione insafiat." Mr. Pollock paraphrases, "It is like applying the intellectual tests of sanity and insanity to acts of pure imagination."

Summary based on:

Spinoza. The Letters. Transl Samuel Shirley. Cambridge: Hackett Publishing Company, Inc., 1995.

Text reproduction from R. H. M. Elwes translation, available online at:

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