Spinoza and Leibniz: Strong Affinity on the Nature of Infinity
by Jim Macdonald
Presented to Washington, DC, Spinoza Club, Tuesday, September 3, 2002

        Let’s begin not with either Spinoza or Leibniz, but with Zeno and also the moment of death.  I want to know what you all think of Zeno’s Paradox.1   We all know that an hour passes in an hour.  Yet, how does it pass?  Is it because 60 minutes or 3600 seconds pass?  Well, if so, how does a minute or a second pass?  Is the second the smallest unit of passage, a kind of temporal atom?  Might I go half a second, and half a half-second, and half of that ad infinitum?  We know that the hour passes and that a second passes, but how do we explain the passage of each?  We can express the problem of Zeno this way:

                                    ½ + ¼ + … 2/n + 1/n = 1

Yet, for any number that we actually assign n, we will always have a sum less than 1.  What is it that gives us the “leap” we need to reach 1?  We give it the name 1/n, but what is it?  How do we account for it?  I’d like to hear your thoughts on that.

         But, before we do, consider this similar problem.  It was raised most famously by the skeptic Sextus Empiricus.  How do we explain the “moment of death”?  At one moment, you are alive.  At the next moment, you’ve joined Colonel Sanders.  But, what explains the movement from one state to the other?  If we say that there is no movement—just two contiguous states that touch—then how we do we explain the movement of one living state to a different living state?  That is, if I’m now giving a talk on Leibniz and Spinoza, but later will be home with my wife, what explains my movement from one state to the other, though both are living states?  Like our division of the second into halves, we can just as easily divide each of our states into halves.  What explains the “leap”?  Is a “leap” an explanation?

        Now, tell me, how would you solve these related problems?

        Now, let me ask you this.  We’ve pondered how we might resolve the problem, but now I am asking the following: How do you think Spinoza would resolve the following problems?

        Let us now put these questions aside for a moment.

        I came tonight to talk about Leibniz and Spinoza, and I have to assume that while most of you have a fair amount of knowledge about Spinoza many of you may know very little about Leibniz or his relationship with Spinoza.  However, I am not going to give you much of a historical exegesis of the two men’s lives.  For the most part, we are going to approach the similarities and differences of the two men from inside out.  That is, I want to explore how each proposes to handle a specific philosophical problem and then draw out the implications on how each’s handling of that problem shows convergences and divergences in their overall philosophical schemes.  In fact, I tend to think that what we take to be of interest historically in each of these two thinkers’ lives arises because of an interpretation of his thought and less the other way around.   Regardless of whether you like that approach, that is the one I will be taking.

        Nevertheless, a few historical details are of some interest and will help set up a sort of minimalist context from which to work.  So, let me say just a few words about Leibniz and his historical relation to Spinoza.  I will then relate each to a letter by Spinoza that shall be the focus of my discussion.

        Gottfried Wilhelm von Leibniz is most commonly thought of as the third of the three great rationalists; the others being Descartes and Spinoza.  Being the third means nothing more here than to say that Leibniz was the youngest.  He was born in 1646, making him too young for Descartes, but just old enough to have known Spinoza.  And, in fact, the two met on Leibniz’s trip to The Hague just a few months before Spinoza died in 1677.2   And, that is what is really rare.  Leibniz is by far the most famous thinker to have known Spinoza and certainly the only one of note to take his thinking seriously, at least in the years up to and immediately following Spinoza’s death.  Leibniz knew of Spinoza’s work through a younger contemporary, Tschirnhaus, who supplied him with drafts of Spinoza’s writings, including drafts of the Ethics and numerous other letters.  Spinoza himself, in response to a letter by Leibniz on lenses, gave Leibniz a copy of his Theologico-Political Treatise.3

        Leibniz took extensive notes on Spinoza’s writings, including the letter we are considering tonight, which is known as Spinoza’s Letter XII, written to Meyer in 1663.4   That letter, more helpfully, is often called, “On the nature of the infinite.”  In 1676, months before Leibniz met Spinoza and left rather unimpressed by his metaphysics,5  but was, however, rather impressed with most of what Spinoza wrote on the infinite.  And, it is why Leibniz was impressed and the degree of that impression that interests us here this evening.

        Now, why do I consider this letter by Spinoza to be a means for drawing many similarities and some differences between the thinkers?  It goes pretty much without saying that I think the question of the infinite has a lot to do with Leibniz’s development as a thinker.  In fact, the scholar Richard Arthur notes that Leibniz considered the tangles of the infinite, particularly how to resolve paradoxes like Zeno’s, one of the two great labyrinths in philosophy (the other being the question of necessity and fate).6   However, there is more to my consideration than Leibniz’s development.  I also am considering this letter because I think how one answers this question about the infinite has consequences that can make all the difference in how one interprets basic questions that all of us reflective people face.

        Let us return a minute or so to Zeno’s Paradox.  What if I said that there is a smallest unit of time?  That is, what if I make time atomic?  Then, I believe at the core of explanation, things are best explained by their smallest parts and how those parts succeed each other.7   Might I be more likely to be the kind of person that thinks all problems reduce to neat little solutions, just the facts ma’am?  However, if I think the question is hopeless, a mysterious leap, might I not think much of the facts?  I might rather think that mysticism is a better way of living and resent people who have tidy solutions for everything.  You might be the sort of person who enjoys mysteries and likes to marvel at the incapacity of reason.  Perhaps, when you have a problem you throw your arms up in the air and leave it to the gods or curse your fate because you cannot even so much as tell whether this life has a final meaning.

        Yet, surely there are more alternatives to this puzzle than atomism or skepticism, right? Some of us are atomists about some things and mystics about other things, but we are unaware of the problem and are often incapable of seeing how even our most insignificant leap in life in some way is an extension of Zeno’s Paradox.  But, does the world reduce to these two choices or a mixture of these two choices?  Perhaps, there are ways that are neither atomic nor mystical.

        Leibniz and Spinoza both represent different, though less widely considered, alternatives to the riddles of the infinite, but their alternatives can make a great deal of difference in grasping the ways that we divide between each other and within ourselves.  We are going to look at this letter because I believe that Zeno is consequential to our lives, and just as importantly, our lives are consequential to Zeno’s problem (but such a bizarre claim is for another discussion)8,  and both these thinkers met Zeno—and let’s not forget about the “moment of death”—head on in an altogether different way.

        So, let’s now look at Spinoza’s Letter XII.

        Spinoza believes that all the problems that people have with the infinite would be solved if they stopped and noticed that there are different kinds of infinity, each appropriate only to its subject matter.  There are three kinds of questions that a rational thinker of the infinite should keep straight:

                            1. Between that which is infinite from its very nature (in virtue of its definition)
                               Between that which is infinite not by nature but rather in virtue of its cause

                            2. Between that which is infinite because it has no limits
                                Between that which is infinite because the parts cannot be expressed by any  number

                            3. Between that which can be understood and NOT imagined
                                Between that which can be understood and ALSO imagined.

Spinoza writes, “If these distinctions, I repeat, had been attended to, inquirers would not have been overwhelmed with such a vast crowd of difficulties.”

        Let us notice something here about these 3 kinds of questions.  It does not follow from them that there are six types of infinite, merely that there are 3 pairs of considerations when considering the infinite.  We shall see that it turns out that Spinoza, like Leibniz, holds that there are 3 sorts of ontological categories to which the word “infinity” properly applies.  In any event, notice Spinoza’s strategy.  If we notice that infinity permits of different meanings, we might also notice that those different meanings sometimes get conflated improperly and create all sorts of absurdities in our reasoning.  It’s as if he’s pleading with the skeptic, “Hold on before you give up hope in an answer.  Have you considered that the paradox might present you with a false dilemma?”

        Spinoza moves on in the letter to explain what he means by a substance and a mode.  The way he draws out the distinction here is crucial.  He defines substance simply as that whose existence follows solely from its essence.9   In other words, a substance’s very definition involves existence.  It cannot not exist without being a contradiction of terms.  Mode, on the other hand, even if it happens to exist, does not have existence as part of its essence, as part of its definition.  “Hence, though they exist, we can conceive them as non-existent.”  Then, Spinoza says something curious.  He says that it “is abundantly clear, that we conceive the existence of substance as entirely different from the existence of modes.”  The word “entirely” is a rather harsh word.  All that that can mean in this context is that in respect to essence, substance has existence and mode does not.

        Now, concerning substance, because it cannot not exist, it cannot be limited.  That is, it must be in every way.  It cannot be “limited” in respect to existence.  Well, Leibniz jumps in at this point, and says that Spinoza has yet to prove that substance is infinite in this way, but it is not crucial.10   The point is is that substance, in virtue of what it is, is not limited.  While a mode, let us say like extension, is only infinite, not because it has existence as its essence, but because it is caused by what is infinite.

        Before we go into the problems of confusing the sense of infinite that belongs to the mode and the sense of infinite that belongs to substance, we should talk about the third, and for Spinoza, final way of considering the infinite.  Do you remember our infinite geometric series that never quite adds up to 1 for whatever number n that we assign?  That quasi-number n can be greater or smaller depending on how far out we extend the series.  However, the only sense that n is infinite is that we cannot assign a finite number to it.  But, this n does have limits.  In the series we are considering, it cannot be such that the sum is greater than 1.  It will always be less than the number preceding it.  So, it is not like the first infinite.  Yet, neither is it like the second infinite.  It does not follow as a consequence of substance, but rather as a consequence of considering a finite sum divided a certain way.  In some ways, it is the least infinite of all the infinities because its ontological scope is so poor.11   It is neither everything nor even a mode but simply a consideration of a mode taken a certain way.

        So, we have three infinities in Spinoza.  We have substance itself, or God.  We have the modifications of substance, which Spinoza calls modes, and we have an infinite, sometimes called the “indefinite”, because “it cannot be expressed in number, but can be conceived of as greater or less.”

        Already there is something a little odd in this talk about greater or lesser infinities.  How can an infinity be greater or less?12   If it is infinite, it is unbounded.  But, to fall into that confusion is to miss Spinoza’s point here.  Yes, one kind of infinity is unbounded absolutely, and that we call substance.  Another type is unbounded as long as we see it as a consequence of the first type.  The final type is unbounded in that we will never be able bind it to a particular number.  So, it may sound odd, but it is only odd because we aren’t accustomed to thinking of infinity as being dependent upon the question, “To what does this infinity pertain?”

        Leibniz liked Spinoza’s characterization of the three infinities, although he said he was accustomed to call them by different names.  Nevertheless, he saw Spinoza’s division as pretty much the same as his.  Leibniz writes in his notes:

                                I usually say that there are three degrees of infinity.  The lowest is, for the sake of an example, like that of the asymptote of
                                          a hyperbola; and this I usually call the mere infinite.  It is greater than any assignable, as can also be said of all the other
                                          degrees.  The second is that which is greatest in its own kind, as for example the greatest of all extended things is the whole of
                                          space, the greatest of all successives is eternity.  The third degree of the infinity, and this is the highest degree, is
                                        everything, and this kind of infinite is in God, since he is all one; for in him are contained the requisites for existing of all
                                          the others.13

So, Leibniz and Spinoza start down the very same road together.  They both assign three kinds of infinity that also reflect in some sense three degrees of infinity that rise the ontological ladder.

        Now, let us move back to Zeno.  We’ve been traveling toward Zeno and are not yet there, but now is the time to meet our evening’s maker, so to speak.

        We want to know how it is that Spinoza resolves the problems entangling infinity, like those raised when we are considering how we are to understand the composition of a single hour, or how we explain the moment of death.  We know that Spinoza believes that it has entirely to do with the conceptual confusions involving the infinite.  Now, we need to see how Spinoza accounts for the confusions.

        Spinoza takes the example of duration, and he notes that duration is to the modes as eternity is to substance.  Since a mode is something that can be limited, and considered as existing, a mode can be treated as finite and divided.  Substance cannot be because to consider substance as being one sort of thing one time and another sort of thing another time is to limit substance.  He says repeatedly that substance is single.

        Is that clear why that’s so?  You cannot say of something that is infinite in every way of being that it can exist as one particular way at the exclusion of its other ways of being. That would involve a contradiction because substance has existence in its essence.  Now, of course, I find this somewhat disingenuous on Spinoza’s count because substance still is limited by the law of noncontradiction, by its rationality, by being other than what it is not.  This consideration of “being otherwise” as it relates to substance is Hegel’s launching pad for moving beyond Spinoza, and I would argue that it is also Leibniz’s, but we cannot consider any of that here.14   Still, it is worth noting.

        While substance cannot be limited in any way, is therefore incapable of being finite, and is therefore incapable of division, mode is capable of being limited, being considered as finite, and therefore of being divided.  Spinoza says, for instance, in respect to duration, when we treat it as finite and divisible, that’s where the concept of time arises.  Time is the limiting of a duration.  It is expressed in numbers.  But, since time arises from a consideration of a mode taken by itself, Spinoza says that it clear that “time” is something imaginary.  That is, we “image” it.  We take it by itself, separate it, divide it, and count it up.  That helps us see why Spinoza, in his third consideration of the infinite, says there is a difference between that which is only understood (like substance) and that which is both understood and imagined (like time and number).

        So, what’s the problem?  The problem, strangely enough, is that people often confuse duration with time.  Remember, duration is a mode of substance, but time is the consideration of the mode taken as finite and separate.  When people take time to be duration, they take what is a measure of what is finite and separate as a complete unit in itself; they take it as single.  But, nothing is truly single except substance, which is indivisible.  So, when people try to explain time, which is finite, restricted, and divisible, as what is only proper to duration in respect to the mode, and really and truly only proper to substance itself, by means of number and division, all sorts of absurdities arise.

        What absurdities arise?  Zeno’s paradox is one example.  Spinoza writes:

                                    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.

When you try to explain what is in essence divisible as though it were single, it fails to add up.  Spinoza’s claim is that this kind of absurdity would never happen if one realized that nothing more is required to understand things than understanding the definition of substance itself.  To try to compose it from finites, from what has limits, is futile.

        So, Zeno’s Paradox is resolved for Spinoza by denying the framework of the problem.  Since the process of reasoning invoked by Zeno yields us no reason for why an hour passes, the infinite process of finite additions is dependent upon something else.  That something else is substance itself, which alone is fully unlimited.

        But, then, we are tempted to ask whether an hour does indeed pass?  I read Spinoza as saying that the question is mute and is a species of considering the nature of things superficially.  Thus, its ontological status at best is imaginary, and all that it is really proper for us to say is that duration follows as a consequence of substance, just as thought and extension do and an infinity of other modifications.

        Now, what of Leibniz?  How far does Leibniz follow Spinoza?  He follows him only so far.  Leibniz agrees with Spinoza that there is a distinction between duration and time, a distinction he would hold until his death.15   Otherwise, as he says that time would be in need of time. Leibniz’s own writings on the nature of infinite number closely parallel Spinoza in agreeing that absurdities follow whenever anyone tries to explain a continuous whole (like duration is to time, for example) by means of divisions.  Let me share with you an example, from Galileo, known as “Galileo’s paradox.”16   If we consider whole numbers, it is true that every number has a square but not every number is a square.  So, for instance, the square of 3 is 9, but 3 is not the square of any other number.  In fact, as we increase the numbers under our scope, the number of numbers that are also squares becomes vanishingly small.  Yet, each natural number has a square.  In that sense, there is a one-to-one correspondence of natural numbers to squares.  So, if we increase the set of natural numbers to infinity, the number of squares is equal to the number of natural numbers.  That seems absurd because the set of squares is smaller than the set of natural numbers.  If we can in fact increase the set of natural numbers to infinity, the part would be equal to the whole, i.e. the set of squares would be equal to the set of natural numbers.  Leibniz takes this conclusion to be absurd.  So, he argues that there is no last number, no infinite number by which we could explain things.  Rather, for any number we assign, there are infinitely more.

        The point here with Galileo’s paradox that Leibniz draws is the parallel to Spinoza’s.  They both accept that any attempt to overcome Zeno’s paradox by simply considering infinite divisions is nonsense.  Leibniz goes further than Spinoza by arguing why there cannot be a smallest part of time or matter either, that is an atom of sorts.  For one thing, there is no reason why an atom should be one size rather than another.  For Leibniz, that’s an affront to divine wisdom, but that involves us in a whole array of issues that we cannot touch in this discussion.  But, more to the point, even if there are atomic units of time, it does not explain how one unit of that time passes into the next unit.  That is, we run up against the moment of death again.  Let us say that we have two distinct seconds, how does the one second pass into the next?  To say simply that it is instantaneous is not to give an explanation at all.  That is arbitrary.  To say that one second occupies the same space as the following second is just absurd.  To say that they meet at a point says nothing about how that point is traversed.  There is a reason Zeno took it that if, magnitude was divisible even once, it is divisible infinitely.  Denying that conclusion creates a whole nexus of new problems.17

        So, the short of it is that Leibniz, just like Spinoza, believes that conceiving of duration as a composition of moments is to confuse two different kinds of infinity, if one wants to explain how the passage of time can account for a given duration, or how life can pass into death.  Yet, we saw that for Spinoza, this was in some ways proof that the parts of time, the divisions into hours, minutes and seconds, made time and parts rather imaginary.  This is Spinoza, after all, who denied the contingent.18   He held onto some semblance of the contingent in modes, but he seems to be denying any meaningful reality for seconds, minutes, hours, slices of pie, and many things we take for granted.  Of course, Spinoza, in some way holds onto all those things so long as they are seen as “species of eternity,” but this is to say that they are in some way more like adjectives than like nouns.

        Leibniz, however, says something very different, yet in its own way, just as bizarre.  What he says, given his obvious affinity to Spinoza, is in fact rather astonishing.  Instead of saying that our temporal divisions are imaginary, he says quite the opposite.  He says of matter, for instance, that it is actually divided infinitely.19   The divisions are real, and actually divided.  However, he adds that this infinite division never resolves into points or into smallest pieces.  For every smallest piece assigned, there exists a world infinitely smaller.  Worlds exist in worlds that exist in worlds.  Zeno’s division neither ends nor reaches an infinitely small piece.  The reasoning is simple.  Since the process of division is finite, there is no point in the process where the division suddenly becomes infinite.  If I have divided something a quadrillion times, I will still be able to divide further and attain a magnitude greater than zero that can be divided again.

        However, why say that the divisions are actual?  Why not simply say that divisions can go to infinity but are really just exercises of the mind?  After all, Spinoza was content simply to say that division was a product of imagination, of the mind itself.  There are a couple reasons for this that I can only briefly mention.  The first reason relates to Leibniz’s plenist physics.  Leibniz did not believe that nature admitted of any vacuum.  However, for motion to exist in a plenist physics, Descartes had noticed that everything had to give way.20   It could not simply give way in a single direction.  In essence, every movement had to reverberate throughout the entire universe.  That could only happen if the parts of the universe fractured infinitely.  So, for motion to exist, there needs to be parts.  But, then why not simply deny motion also as a species of imagination?  In substance for Spinoza, there is neither motion nor rest, or else both would be limits on the substance.  Why add those categories to the nature of substance and reality?  Leibniz’s retort, I suspect, would echo the one I complained about earlier, namely that it is absurd to conceive of substance without some sense of otherness, of what it is not, of what it means to be necessary as distinct from what it means to be otherwise.  The “otherwise” is essential to the being of substance.  While substance itself may not be otherwise, it must grasp it and incorporate it into its being in some other sense.  For Leibniz, substance is “what acts.”  So, the motive force is essential to the very being of substance.  You cannot have motion unless there are also parts moved, and so the reality of every division imaginable and infinitely more besides becomes a key part of Leibniz’s metaphysics.

        That takes Leibniz down a radically different road from Spinoza.  Leibniz’s parts and divisions are incapable of explaining wholes, but they are nevertheless there and quite actual.  Spinoza’s divisions are nothing more than exercises in abstraction.  Leibniz’s understanding leads him to a Christian metaphysics, friendly to the idea of the trinity, but it also leads him to be friendly to pluralism both metaphysically and ethically.  Very few other philosophers in Leibniz’s day had anything as nice to say as him about Native Americans, the Chinese, or the reasoning ability of animals.  The multiplicity of being and “points of view” that are whole worlds unto themselves become central to everything in Leibniz.  Spinoza, on the other hand, is extremely different for better or worse, and I am sure there are many people here who would like to emphasize the better.

        But, let me start the parade on Spinoza’s behalf.  For all the differences that develop between Spinoza and Leibniz, diverging on what to do with the ontological status of differences, you can see that they have an enormous amount of similarities.  Both accepted the same account of the infinite.  Both noted the futility of trying to understand the infinite as “the sum of its parts.”  Both took us down roads that veered away from the path that atomists and mystics took to deal with Zeno’s problem.  By carefully noting the distinctions of the infinite as those categories applied to different ways of being, it was possible to overcome the false confusion of the problem.  And, that’s what’s really the most impressive similarity.  Where philosophy today tends to shy away from any talk of “ways of being,” Spinoza and Leibniz faced those distinctions head on and in a very similar manner.  And, while they subsequently diverge widely from each other, it is easy to see, I hope, that there is a vanishingly small area of difference when we see it in the context of their basic starting points.  I would argue that most discussion contrasting Leibniz and Spinoza roots in that vanishingly small region, but I would also argue that such a world is a world of interesting worlds unto itself.  And, if so, there is no moment of death in our discussion, no moment where there is nothing of interest to say about the distinctions between Leibniz and Spinoza.

        I am sure you all have infinitely many questions; at this moment, let’s consider a few.

1  We are considering a temporal version of Zeno’s Paradox.  Strictly speaking, Zeno’s Paradox, concerned space.  Go back to text
2  Peter Remnant and Jonathan Bennett eds., New Essays on Human Understanding, by G. W. Leibniz, (Cambridge: Cambridge University Press, 1997), p. xciv. Go back to text
3  Ibid.  Also, by J J O'Connor and E F Robertson, “Ehrenfried Walter von Tschirnhaus,” (http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Tschirnhaus.html), gives a nice short biography of Tschirnhaus, 1651-1708.  Richard Arthur, in his translation of Leibniz, The Labyrinth of the Continuum, (New Haven, CT: Yale University Press, 2001), hypothesizes on p. 129, that Tschirnhaus may have been the important character named “Charinus” in Leibniz’s important 1876 dialogue on the infinite, Pacidius to Philalethes: A First Philosophy of Motion.  If you want to read Spinoza’s Letter XLIV to Leibniz on lenses, you can find it, dated November 9, 1671, in Correspondence, translated by R. H. M. Elwes, (http://bdsweb.tripod.com/en/let/l46.htm, 1677).  Go back to text
4  All quotes from this letter are from Correspondence, translated by R. H. M. Elwes, (http://bdsweb.tripod.com/en/let/l12.htm, 1677).  The letter is dated April 20, 1663.  Go back to text
5  Remanant and Bennett note on p. xciv: “I saw him on my way through Holland and I talked to him several times at great length.  He has a strange metaphysics, full of paradoxes.  Among other things he believes that the world and God are one and the same thing in substance, that God is the substance of all things and that creatures are only modes or accidents.  But I could see that some of the alleged demonstrations that he showed me were unsound.  It is not as easy as people think to give genuine demonstrations in metaphysics.”  Go back to text
 6  Richard T. W. Arthur ed., The Yale Leibniz: The Labyrinth and the Continuum: Writings on the Continuum Problem 1672-1686, by G. W. Leibniz, (New Haven, CT: Yale University Press, 2001), p. xxiii.  Go back to text
Over Labor Day weekend, I had an argument with my uncle over the nature of time and space.  He argued that there was a smallest unit, Planck’s Constant (h).  Its being “smallest” was fixed by the nature of observation, and observation was what allows us to fix because “reality is perception.”  Leibniz’s own views on the nature of perception are complicated.  He argued at length that perception should not be equated with consciousness.  So, he would not so much challenge the idea that reality is perception so much as the idea that perception is nothing more or less than “observable reality.”  Anyhow, the point in this paper is not to give the case against atomism so much as it is to show how Leibniz and Spinoza propose alternate views to the problem of infinite division than some to which we may be accustomed.  Go back to text
8  My claim is similar to that of Plato and Aristotle, who each argue for the intrinsic value of philosophy, although for somewhat different reasons.  Life and the dialectical process of philosophy are, I believe, reciprocal ideas.  However, that is clearly a tangential discussion that I cannot pursue here.  Go back to text
9  Spinoza writes: “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.”  Go back to text
10  Leibniz, The Labyrinth of the Continuum, “Spinoza’s Letter on the Problem of the Infinite,” p. 107.  Go back to text
11  In fact, Leibniz himself calls this the “lowest” infinite in The Labyrinth and the Continuum, “Notes on Spinoza’s Ethics and ‘On the Infinite,’” p. 43.  Go back to text
12  It might not seem that odd to a contemporary student of mathematics, or anyone who has studied Georg Cantor’s theory of transfinite numbers. Go back to text
13  Leibniz, “Notes on Spinoza’s Ethics and ‘On the Infinite,’” p. 43.  He makes a similar point in “Spinoza’s Letter on the Problem of the Infinite,” p. 115.  Go back to text
14  Hegel is famous for saying that philosophy begins with Spinoza, but it does not end there.  The compliment is, therefore, also something of a critique.  Likewise, as we shall see later, Leibniz notes that change must be integral to our understanding of substance.  See Monadology, for example.  Go back to text
15  Consider what he says to Samuel Clarke, in Correspondence, edited by Roger Ariew, (Indianapolis: Hackett Publishing, 2000), p. 51, sec. 106.   Here the distinction is not between duration and time as it is eternity and duration.  However, the idea behind the distinction is the same. Go back to text
16   I take my discussion here from Pacidius to Philalethes, pp.177-179, but I am also endebted to the exposition by Samuel Levey, "Leibniz on Mathematics,” in The Philosophical Review 107 (Jan 1998): 61-62 and also Richard Arthur, p. lxxvii.  Go back to text
17  We could consider even more; however, our focus here is not to disprove atomism.  Go back to text
18  This is not as easily said as this.  Leibniz himself seems to think that Spinoza’s “modes” are similar to his “contingent.”  However, the idea that there is some kind of substantial existence to modes is clearly denied in Ethics as a contradiction of terms.  As he says in Letter XII, they have an “entirely” different nature.  Go back to text
19  He says this in numerous places.  Most famously, he says it to Foucher in Journal de Sçavans, March 16, 1693, found in C. I. Gerhardt’s translation, Die Philosophischen Schriften von Gottfried Wilhelm Leibniz, (Berlin: Weidmann, 1875-1890), v. 1, p. 416.  Go back to text
20  An interesting account on Descartes’ reasoning can be found in Levey, “Leibniz on Mathematics,” pp. 51-53.  Go back to text

Short Bibliography

Arthur, Richard T. W. “Leibniz on Infinite Number, Infinite Wholes, and the Whole World: A Reply to Gregory Brown.”  In The Leibniz Review 11, (December 2001): 103-116.

Brown, Gregory.  “Leibniz on Wholes, Unities, and Infinite Number.”  In The Leibniz Review 10, (December 2000): 21-51.

Cantor, Georg.  Contributions to the Founding of the Theory of Transfinite Numbers.  Translated by Philip E. B. Jourdain.  New York: Dover Publications, 1955.

Clarke, Samuel and Leibniz, G. W.  Correspondence.  Edited by Roger Ariew.  Indianapolis: Hackett Publishing, 2000.

Leibniz, G. W.  Monadology.  Translated by George Montgomery.  La Salle, IL: Open Court Press, 1993.

…….. Die Philosophischen Schriften von Gottfried Wilhelm Leibniz.  Seven volumes.  Translated by C. I. Gerhardt. Berlin: Weidmann, 1875-1890.

……..  The Yale Leibniz: The Labyrinth of the Continuum: Writings on the Continuum Problem, 1672-1686.  Edited, introduced, and translated by Richard T. W. Arthur.  New Haven, CT: Yale University Press, 2001.

Levey, Samuel.  “Leibniz on Mathematics.”  In The Philosophical Review 107, (January 1998): 49-96.

……..  “Leribniz’s Constructivism and Infinitely Folded Matter.”  In New Essays on The Rationalists.  Edited by Rocco J. Gennaro and Charles Huenemann.  New York and Oxford: Oxford University Press, 1999.

Spinoza, Benedict.  Correspondence.  Translated by R. H. M. Elwes.  At http://bdsweb.tripod.com/en/let/leti.htm, 1677.

……..  Ethics.  Edited and translated by G. H. R. Parkinson.  Oxford: Oxford University Press, 2000.

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