Gottfried Wilhelm Leibniz (1646-1716)


Gottfried Wilhelm Leibniz was born in Leipzig, Germany, on July 1, 1646.  He was the son of a professor moral philosophy, and after university study in Leipzig and elsewhere, it would have been natural for him to go into academia.  Instead, he began a life of professional service to noblemen, primarily the dukes of Hanover (Georg Ludwig became George I of England in 1714, two years before Leibniz's death).  His professional duties were various, such as official historian and legal advisor.  Above all, he was required to (or allowed to) travel widely, meeting many of the foremost intellectuals in Europe - of particularly formative importance were the astronomer, mathematician and physicist Huygens, and the philosopher Spinoza.

Leibniz was one of the great polymaths of the modern world.  Moreover, a list of his significant contributions is almost as long as the list of his activities.  As an engineer he worked on calculating machines, clocks and even mining machinery.  As a librarian he more or less invented the modern idea of cataloguing.  As a mathematician he not only produced ground-breaking work in what is now called topology, but came up with the calculus independently of (though a few years later) than Newton, and his notation has become the standard.  In logic, he worked on binary systems among numerous other areas.  As a physicist he made advances in mechanics, specifically the theory of momentum.  He also made contributions to linguistics, history, aesthetics and political theory.

Leibniz's curiosity and genius ranged widely, but one of the most constant of his concerns was to bring about reconciliation by emphasizing the truths that lay in both of even the most contradictory positions.  Throughout his life, he hoped that his work on philosophy (as well as his work as a diplomat) would form the basis of a theology capable of reuniting the Church, divided since the Reformation in the 16th Century.  Similarly, he was willing to engage with, and borrow ideas from, the materialists as well as the Cartesians, the Aristotelians as well as the most modern scientists.  It is quite ironic, then, that he was partial cause of a dispute between British and Continental mathematicians concerning who was first to the calculus (and who might have plagiarised who).  This dispute slowed down the advance of mathematics in Europe for over a century.
However, the great variety of his work meant that, sadly, he completed few of his ambitious projects.  For our purposes here, this means above all that Leibniz's rich and complex philosophy has to be gathered primarily from a large set of quite short manuscripts, many fragmentary and unpublished, as well has his vast correspondence. The last section of this entry gives bibliographical details of several editions of Leibniz's work.

Partly because of the above fact, a major controversy in Leibniz scholarship is the question of where to begin.  Insofar as Leibniz is a logician, it is tempting to begin with his conception of truth (and, indeed, this will be our starting point).  But insofar as Leibniz is a metaphysician, it is equally tempting to begin with his account of the nature of reality and in particular substance.  Less common, but perhaps equally likely starting points might lie in Leibniz the mathematician, the theologian or the physicist.  These controversies, however, already contain a lesson: to an important degree it doesn't matter.  So integrated were his various philosophical interests - so tightly laced together into a system - that one ought to be able to begin anywhere and reconstruct the whole.  Or at least Leibniz evidently thought so, since very often we find him using an idea from one part of his philosophy to concisely prove something in an apparently quite distant philosophical region. Likewise, just this systematic nature often makes 'getting into' Leibniz the most difficult step, because every idea seems to rely upon the others.

(Note: this entry will not be dealing with Leibniz's work on, for example, aesthetics, political philosophy, or [except incidentally] physics.  Also, Leibniz 'mature' metaphysical career spanned over thirty years.  During this period, it would be surprising if some of his basic ideas did not change.  Remarkably, however, the broad outline of his philosophy does remain constant.  Therefore, in this entry we will predominately taking the broad view.)

The Idea of Truth

Leibniz the logician would have us ask a seductively simple question: what is truth?  It will turn out to be the case that a conception of what truth is has important consequences for a conception of what reality in general is, and how it is to be understood at its most profound level.  Common-sensically, we say that a proposition is true when its content is adequate to the situation in the world to which it refers.  So: 'the sky is grey' is true if and only if the thing out there in the world we call the sky is actually the colour we call grey at the time the proposition is stated.  This, however, gets us into problems about the relationship of language to the world, and what this 'adequacy' consists in.  (Problems both sceptics and pragmatists are only too fond of drawing our attention to.)

Leibniz says that we can bypass all that, at least for the moment.  Truth is simply a proposition in which the predicate is contained in the subject.  The predicate is what is asserted; the subject is what is asserted about.  So, formally speaking, Leibniz says, all true propositions can be expressed: 'subject is predicate'.  This is not an idea unique to Leibniz by any means - what is unique is the single-mindedness with which he pursues the consequences of such an idea of truth. (See e.g. 'Correspondence with Arnauld', letter of July 14, 1686.)

This idea of truth seems straight-forward enough for what we now commonly call analytic propositions, such as 'Blue is a colour', which has nothing to do with the world, but is simply part of a definition of blue.  The notion of colour is part of the notion of blue.  Similarly, 'A = A', which is a basic logical truth.  The predicate is not just contained in the subject, it is the subject. But Leibniz says that this 'being contained' is 'implicitly' or 'virtually' the case with other truths.  ('Primary Truths', 'The Nature of Truth'.)  Take the statement 'Peter is ill'.  Usually, we take this proposition to be true only because it refers to a real world in which Peter is, in fact, ill.  But Leibniz will analyse this as follows: if we knew everything there was to know about Peter, that is, if we had a complete concept of Peter, we would also know (among many other things) that he is ill at the moment.  Therefore, the statement 'Peter is ill' is true not primarily because of some reference to the world, but in the first instance because we or someone has the concept of Peter, which is the subject of the proposition, and that concept contains (as a predicate) his being ill.  Of course, it may be the case that we happen to know that Peter was ill because we refer to the world (perhaps we see him cough repeatedly).  But the fact that we find out about Peter in this way does not make the statement that 'Peter is ill' true and thus a piece of knowledge because of that reference.  We must distinguish the concept of truth from pragmatic or methodological issues of how we happen to find out about that truth, or what we can do with the truth.  The latter are completely irrelevant to the question of what is truth (or knowledge) in itself.

Leibniz also wants to claim that a statement is true for all time - that is, whenever the statement is made.  So, the statement 'Peter is ill (on January 1st, 1999)' was true in the year 1998 (although neither we, nor Peter, knows it yet) as well as in the year 2000 (although we and Peter may have forgotten about the illness by then). It was also true a million years ago, and will be true a million years from now, although it is very unlikely that anyone will actually know this truth at those times.
Leibniz's own example was Julius Caesar (Discourse on Metaphysics, 13).  He writes,

But there are several further ideas Leibniz introduces in this passage which we need to explore.  What is meant by 'completing the whole demonstration'?  Or by something having a 'foundation', or by 'a reason can be found'?

Sufficient Reason

As we have just seen, for any proposition truth is defined in the same way: the predicate
contained in the subject. It only takes a little thought to realise that for any one subject (like Peter, or Caesar), the number of predicates which are true of it will be infinite (or at least very large!), for they must include every last thing Peter or Caesar did or will do, and also everything that did or will ever happen to them.  Why do all these predicates come together in the one subject?  It could be that the predicates are a quite arbitrary or random collection.  Leibniz does not believe this - and it is certainly not how we normally think.  Rather, we normally think that one predicate or set of predicates explains another.  For example, Peter's coming into contact with a virus explains his illness.  Or, Caesar's ambition and boldness explains why he decided to cross the Rubicon.  So, many (at least) of the predicates that are true of a subject 'hang together' as a network of explanations.

Leibniz goes further still: for every predicate that is true of a subject, there will be a set of other true predicates which constitute a sufficient reason for its being true.  This he calls 'the principle of sufficient reason'.  This is why he uses words like 'foundation' and 'reason' in the quotation above.  Unless this were true, Leibniz feels, the universe would not make any sense, and science and philosophy both would be impossible (see, e.g. New Essays on Human Understanding, preface, p. 66).   Moreover, it would impossible to account for a basic notion like identity unless there were a sufficient reason why I (with my particular properties now) am identical with that 'me' who existed a week ago (and had such different properties).  ('Remarks on Arnauld's Letter...', May 1686)

This idea of sufficient reason accounts for why Leibniz talks about about 'completing the whole demonstration'.  If the complete concept of the subject (all of its true predicates) together constitutes a complete network of explanation, then these explanations can be followed forwards and backwards, at least in principle.  That is, someone could deduce Caesar's crossing of the Rubicon from a full picture of his previous predicates; or, working backwards, deduce from a full picture of those predicates true of Caesar at his death the reasons why he won the battle of Pharsalus.  The 'whole demonstration', then, would be a revelation of the logical structure of the network of explanations that make Caesar who he is.

At least in principle! Clearly, this is not something that you or I can do.  Human minds are not subtle and capacious enough for a task which may be infinite.  Still, in our more limited way, we happily talk about 'personalities', 'characters', and causes or reasons for things.  The quotation from Leibniz given above continues:

These qualifications are very important for Leibniz.  It was often suggested by Leibniz's contemporaries (and is still being suggested!) that his idea of the sufficient reason of all the predicates of a subject meant that everything true of a subject is necessarily true.  This might entail that Caesar did not 'choose' to cross the Rubicon, but that he was acting in a determined manner, like a machine.  In other words, Leibniz seems to be denying anything like free will.  We will return to the free will problem below, but for the moment, a few observations.

First, Leibniz claims that Caesar's crossing of the Rubicon is not necessary in the sense that 'A is A' is necessary.  Because while 'A is not A' is a contradiction, Caesar's deciding not to cross the Rubicon does not imply a contradiction.  To be sure, history would have been different - even Caesar would have been different - but there is no contradiction in that strong sense.  Caesar's properties are not logically necessary.

Second, any truth about Caesar - indeed, the whole complete concept of Caesar - is not 'necessary in itself'.  Caesar is Caesar, but nothing about Caesar in himself proves that Caesar has to be.  By contrast, 'A is A' doesn't need any other explanation for its truth.  So, while every property of Caesar is explained by some other property of Caesar,  no property explains why it is true that 'Caesar existed'.  Caesar is not necessary being.

It remains a strenuously debated issue in Leibniz scholarship what the exact nature of these distinctions is, whether he is justified in making them, and - even if justified - do they yield the results he claims in the area of free will.  We will add more detail to this account, but the existence of this debate should be kept in mind throughout.

Substance, Briefly

At this point, we must turn from a conception of truth to a conception of substance - a subject which has been just out of sight in the above.  We will not deal with Leibniz's full philosophy of substance until section 8.  For the moment, we have only to observe that, for humans (though not for God), complete concepts are always concepts 'of' existing substances - that is, 'of' really existing things.  Leibniz writes:

To be the individual substance Caesar, then, is to be such as to have a notion which includes everything that can truthfully be predicated of the subject 'Caesar'.  We might say that substance means a complete concept made real; and a complete concept means a real substance 'expressed' or 'perceived' in thought.  Moreover, just as for any one predicate, the complete concept contains other predicates which explain that predicate - so, for any given property of a substance, the complete individual substance will itself be the explanation for that property.  Caesar chose to cross the Rubicon for many complex reasons, but they all boil down to this: that was the kind of man Caesar was.

As we shall see below, Leibniz has much more to say about substance - but he claims that it all follows from this insight.  However, the exact relationship Leibniz  intended between the logical idea of a complete concept, and the metaphysical idea of a substance is still hotly debated in Leibniz scholarship.

Necessary Being

The complete concept of Caesar cannot explain itself in its entirety; expressed ontologically, this means that Caesar himself provides no explanation of why Caesar should have existed at all - Caesar is contingent being.  By 'contingent' is meant something that could have been otherwise; that is, here, something that could have not existed at all.  The principle of sufficient reason, if we accept it, must not only apply to each predicate in the complete concept of a subject.  But also it must apply to the concept itself in its entirety as the concept of an existing thing.  Why should this substance exist, rather than some other substance, or nothing at all?  What explains Caesar?  Possibly other complete substances, such as his parents; and they in turn are explained by still others.

But the entire course of the universe, the total aggregate of substances across space and time are one and all contingent.  There are other possible things, to be sure; but there are also other possible universes that could have existed but did not.  The totality of contingent things themselves do not explain themselves.  Here again, the principle of sufficient reason applies.  There must be, Leibniz insists, something outside the totality of contingent things which explains them, something which is itself necessary and therefore requires no explanation other than itself.  Note that we are not assuming an origin or beginning in any sense.  Even if time stretched infinitely into the past, there would still be no explanation for the total course of things.

This forms a proof for the existence of God.  (Monadology 37-9, 'A Specimen of Discoveries')  In fact, it is a version of the third of the cosmological arguments given by St Thomas Aquinas - and subject to many of the same difficulties.  One might, for example, object in a Kantian vein that the concept of explanation, rightly demanded of all individual contingent beings, is applied beyond its proper sphere in demanding an explanation of the totality of contingent beings.  But Leibniz might well counter that this object assumes a whole theory of the 'proper spheres' of concepts.

God, then, is the necessary being which constitutes the explanation of contingent being, why the universe is this way rather than any other.  For the moment, God's necessity is the only thing we know about such being.  (We, with Leibniz, say 'God', although there is not much religious or theological about this bare metaphysical concept.)  God as a being may be necessary, but if the contingent universe were simply a random or arbitrary act of God, then God would not constitute the required explanation of all things. In other words, God must not only be necessary, but also the source of the intelligibility of all things.  It must be possible,  therefore, to inquire into the reasons God had for 'authorising' or allowing this, rather than any other, universe to be the one that actually exists.  And if God is to be the explanation of the intelligibility of the universe, then God must have 'access' to that intelligibility, such that God could be said to know what it is that is being allowed to exist - that is, God must have the ability to grasp complete concepts, and to see at once the 'whole demonstration' we talked about above. God so far is therefore (i) necessary being, (ii) the explanation of the universe, and (iii) the infinite intelligence.

Here Leibniz famously brings in the notion of perfection.  (See e.g. 'A Specimen of Discoveries')  We have to try to imagine God, outside of time, contemplating the infinite universe that 'he' is going to - not create - but allow to be actual and sustain in existence.  In the mind of God are an infinite number of infinitely complex complete concepts, all considered as possibly existent substances, with none having any particular 'right' to exist.  There is just one constraint on this decision: it must not violate the other basic principle of Leibniz's: non-contradiction.  In other words, each substance may 'individually' be possible, but they must all be possible together - the universe forming a vast, integrated system.  For example, God could not create a universe in which there were both more sheep than cows and more cows than sheep.  God could choose a universe in which there is the greatest possible quantity of pizza; or in which everything is purple; or whatever.  God, however, chooses the universe that is the most perfect, and this principle of perfection is not surprising since it is most consummate with the idea of God as an infinite being.  To choose any other less perfect universe would be to choose a lesser universe. Thus, the existing world is the best of all possible worlds. (This claim, and its apparent implications were very effectively and famously satirised by Voltaire in his Candide. Note also that Leibniz is often taken as an ancestor of modern possible worlds semantics; however, it is undeniable that at least the context and purpose of Leibniz's notion of a possible universe was very different.) The theological consequences of this Leibniz explores at, for example, the end of the 'Discourse on Metaphysics'.  (There may be a difficult theological implication: must God be thought of as constrained, first by the concept of perfection, and then by the systemic nature of his creation?  Leibniz attempts, for example in the 'Correspondence with Arnauld' for example, to escape this conclusion.)

To try to understand further this notion of perfection, Leibniz tries out several concepts in various writings: notions of the best, the beautiful, the simply compossible, greatest variety or the greatest quantity of essence. The last of these is the explanation he keeps coming back to: perfection simply means the greatest quantity of essence, which is to say the greatest richness and variety in each substance, compatible with the least number of basic laws, so as to exhibit an intelligible order that is 'distinctly thinkable' in the variety ('A Resume of Metaphysics'; there is a relationship to the Medieval, and particularly Augustine, notion of 'plenitude').  Leibniz seems to understand this principle as just self-evident.  It certainly seems to be a big jump to this aesthetic/moral/wise God from the ontological conception of God deduced above. Although Leibniz may have a point in arguing that it would be absurd in some sense for an infinite being to choose anything other than an infinitely rich and thus perfect universe.  And he finds this aesthetic observed also throughout nature: that nature forms tend towards a maximum of variety compatible with orderliness.  Never the less, contemporary philosophers generally find Leibniz's thought to be rather confused and even 'unphilosophical' at this point.

Problems of Freedom, Sin and Evil

But all this may cause more problems than it solves.  If the complete concept of any being - such as a human being - is known for all time, and was chosen by God for existence. then is such a human being free in any sense.  And if not, then what nonsense is made of the idea of morality or of sin?  Further, it seems possible that what we mean by 'freedom' is that the outcome is not predictable, in the way that the operation of a washing machine, or of the addition of two numbers, is predictable.  Why for example should God punish Adam and Eve for sinning when they clearly had no free choice - since God knew in advance (predicted - indeed made it happen) that they were going to sin?

To clear this up, Leibniz needs to distinguish between several ways in which things might be determined in advance.  Whatever is determined is clearly true.  Truth, however, come in several varieties. (Much of the following is taken from the set of distinctions Leibniz makes in 'Necessary and Contingent Truths'; Leibniz makes similar but rarely identical sets of distinctions in a variety of texts.)

Only truths of essence can be 'necessary' absolutely and strictly speaking.  All other truths, such as the actions of Caesar are only 'hypothetically' necessary - that is, only on the hypothesis that a universe exists as it is, with beings such as these in it.  (See 'Discourse on Metaphysics' 13 and 'Correspondence with Arnauld', April 12th, 1686.)  My actions are, therefore, not necessary by definition (regardless at this point of which type of 'truth of existence' they fall under).  Thus the concept of me 'inclines without necessitating' (e.g. 'Discourse on Metaphysics' 30).  Leibniz further writes: By this 'indifference', Leibniz means a 'physical' indifference - that is to say, there is no universal-physical truth, as defined above, which governs human action. For Leibniz, that means that human action is further freed: the will has the power to suspend its action with respect to the physical sequence of efficient causes - but also even with respect to what would otherwise be seen as a decisive final cause.  'For they [free or intelligent substances] are not bound by any certain subordinate laws of the universe, but act as it were by a private miracle' ('Necessary and Contingent Truths').  Minds are different from mechanical causes.  (As we shall see below, Leibniz goes against the trend of 17th and 18th century thought by reintroducing the Aristotelian and Scholastic notion of a final cause and, indeed, substantial forms.) Although Leibniz occasionally uses the analogy of a machine to describe the soul, the kinds of forces and causes operative in the former are simply inapplicable to the latter.  Thus, if by individual free choice we mean: individual action that cannot be known in advance by even an infinitely subtle application of the laws of physics, chemistry, or biology -  then we have free choice in that sense too.

Leibniz also here and there offers the following additional arguments for his particular conception of human free will:

1. Freedom as 'unpredictability' might be taken to mean freedom as an act uncaused.  But this makes no sense, for free choice is not randomness.  My free act has a cause - namely, me. Why should we complain when the individual concept of 'me' intrinsically determines what I do?  Is this not what is meant by freedom?  That I am the source of my action, and not anyone or anything else?

2. A necessary ignorance of future is practically, perhaps even logically, equivalent to freedom.  As we know, grasping the full explanation of any predicate that lies in the complete concept is an infinite task.  To help illustrate the distinction between contingent and necessary truths, Leibniz makes a famous analogy with the incommensurability of any whole number or fraction with a 'surd' (for example, the square root of two, the value of which cannot be represented numerically by any finite series of numbers.)  For finite human minds, that incommensurability is a positive fact, just like contingency - no matter that for God neither 'calculation' is impossible, or even more difficult.  Thus contingent truths can in principle be known from all time, but necessarily not by a human being.  (See e.g. 'On Freedom'.)  Leibniz writes: 'Instead of wondering about what you cannot know and what can tell you nothing, act according to your duty, which you do know.'  ('Discourse on Metaphysics', 30)   (It should be pointed out that this is somewhat more than an analogy, since it is closely related to the kinds of problems infinitesimal calculus was designed to deal with - and Leibniz takes the possibility of a calculus as having real metaphysical implications.)
3. A famous scholastic debate concerned the so-called 'Sloth Syllogism.'  If everything is fated, the argument goes, then whatever action I 'do' will or will not happen whether or not I will it, therefore I need will nothing at all.  I can just be a sloth, and let the universe happen.  Leibniz thinks this is absurd - indeed, immoral.  Individual will (what I will) matters. If I am the kind of person who is a sloth, then (everything else being the same) the course of my life will indeed be quite different than if I am the kind of person (like Caesar) who takes events by the scruff of the neck.

4.  What many philosophers mean by 'contingent' is that an individual predicate 'could have been' different, and everything else the same.  For Leibniz, this is impossible.  To change one predicate means to alter the whole complete concept/substance, and with it the whole universe.  Leibniz thus claims that philosophers of a more radical sense of freedom do not take seriously the extent to which the universe is an integrated network of explanations, and that this in turn has implications for the idea of contingency. (See the discussion of Adam in Leibniz's letter to Landgraf Ernst von Hessen-Rheinfels, April 12, 1686.) Thus contingent events, even my free acts, must be part of the perfection of the universe - but that does not mean that all contingent events are so in the same way.

Any remaining objections to this idea are because we have a metaphysically incoherent idea of what freedom means, Leibniz claims.  There is no question that Leibniz introduced a spirited and powerful position into the age-old philosophical debate concerning free will.  Whose position is 'metaphysically incoherent', however, remains under debate, as we noted already above.

Leibniz's approach to the classic problem of evil is similar.  If God is good, and the creator (or 'author') of the best possible universe, then why is the world full of pain and sin?  Leibniz wants to claim that this apparent paradox is no real problem.  His replies are to be found spread over many texts.  Here, very briefly, are three: (i) We only see a small fraction of the universe.  To judge it full of misery on this small fraction is presumptuous.  Just as the true design - or indeed, any design - of a painting is not visible from viewing a small corner of it, so the proper order of the universe exceeds our ability to judge it.  (ii) The best possible universe does not mean no evil, but that less overall evil is impossible.  (iii) Similarly to the previous argument, and in the best Neo-Platonist tradition, Leibniz claims that evil and sin are negations of positive reality.  All created beings are limitations and imperfect; therefore evil and sin are necessary for created beings. ('Discourse on Metaphysics', 30)

Space, Time and Indiscernibles

Between 1715 and 1716, at the request of Caroline, Princess of Wales, a series of long letters passed between Leibniz and the English physicist, theologian, and friend of Newton, Samuel Clarke.  It is generally assumed that Newton had a hand in Clarke's end of the correspondence.  They were published in Germany and in England soon after the correspondence ceased and became one of the most widely read philosophical books of the 18th Century.  Leibniz and Clarke had several topics of debate: the nature of God's interaction with the created world, the nature of miracles, vacua, gravity and the nature of space and time.  Although Leibniz had written about space and time previously, this correspondence is unique for its sustained and detailed account of this aspect of his philosophy.  It is also worth pointing out that Leibniz (and after him Kant) continues a long tradition of philosophising about space and time from the point of view of space - as if the two were always in a strict analogy.  It is only rarely that Leibniz deals in any interesting way with time on its own - we shall return to this in section 10.

Newton, and after him Clarke, argued that space and time must be absolute (that is, fixed 'background' constants) and in some sense really existence substances in their own right (at least, this was Leibniz's reading of Newton).  The key argument is often called the 'bucket' argument.  When an object moves, there must be some way of deciding upon a frame of reference for that motion.  With linear motion, the frame does not matter (as far as the mathematics are concerned, it does not matter if the boat is moving away from the shore, or the shore is moving away from the boat); even linear acceleration (changing velocity but not direction) can be accounted for from various frames of reference.  However, acceleration in a curve (to take Newton's example, water forced by the sides of a bucket to swirl in a circle, and thus to rise up the sides of the bucket), could only have one frame of reference.  For the water rising against the sides of the bucket can be understood if the water is moving within a stationary universe, but makes no sense if the water is stationary and the universe is spinning.  Such curved acceleration requires the postulation of absolute space which makes possible fixed and unique frames of reference.  (Similar problems made Einstein's General Theory of Relativity so much more mathematically complicated than the Special Theory.)
Leibniz has a completely different understanding of space and time.

Leibniz first of all finds the idea that space and time might be substances or substance-like absurd.  (e.g. 'Correspondence with Clarke', Leibniz's Fourth Paper, 8ff)  An empty space would be a substance with no properties; it will be a substance that even God cannot modify or destroy. But Leibniz's most famous arguments lay in a different direction.

Let us return to sufficient reason.   This law claims that every thing which happens has, at least in principle, an explanation of why it happened, and why this way rather than that.  Every question 'why' has in principle an answer.  From this principle, together with non-contradiction, Leibniz believes, follows a third: the principle of the identity of indiscernibles.  Leibniz is fond of talking about leaves as an example.  Two leaves often look absolutely identical.  But, he argues, if 'two' things are alike in every respect, then they are the same object, and not two things at all.  So, it must be the case that no two leaves are ever exactly alike.

But why should this be the case?  For if they were in every way the same, but actually different, then there would be no sufficient reason (i.e. no possible explanation) why the first is where (and when) it is, and the second is where (and when) it is, and not the other way around.  If, then, we posit the possible existence of two identical things (things that differ in number only.  That is, we can count them, but that is all), then we also posit the existence of an absurd universe, one in which the principle of sufficient reason is not universally true. Leibniz often expresses this in terms of God.  That is, if two things were identical, there would be for God no sufficient reason for choosing to put one in the first place, and the second in the second place.  (Leibniz's argument relates to a scholastic debate centred around the colourful concept of 'Buridan's Ass'.)

The same, however, can be said about empty space.  Two portions of empty space are indiscernible, and therefore, according to Leibniz, they must be identical.  But if space is to be real - or even an absolute framework for motion - then this clearly cannot be the case.  Again, on the Newton/ Clarke account, Leibniz argues, it would make sense to ask why the whole universe was created here instead of two meters that-a-way.  But since these two universes are indiscernible, the question 'why' cannot make sense.  This should be understood as a reductio ad absurdum of the Newtonian position.
That is the negative portion of Leibniz's argument.  But what does all this say about space?  For Leibniz, the location of an object is not a property of an independent space, but a property of the located object itself - and also of every other object relative to it.  This means that an object here can indeed be different from an object located elsewhere simply by virtue of its different location - because that location is a real property of it.  That is, space and time are internal features of the complete concepts of things, and not extrinsic.  Let us go back to the two identical leaves.  All of their properties are the same, except that they are in different locations.  But that fact alone makes them completely different substances.  To swap them while the rest of the universe wasn't looking would not be just to move things in an indifferent space, but would be to change the things themselves.  If the leaf were located elsewhere, it would be a different leaf.  A change of location is a change in the object itself.  Similarly, with location in time.

This has two implications.  First, that there is no absolute location in either space or time, location is always the situation of an object or event relative to other objects and events. Secondly, space and time are not in themselves real, are not substances.  Space and time are ideal.  Space and time are just ways (metaphysically illegitimate ways) of perceiving certain virtual relations between substances.  They are 'phenomena'; that is, in an important sense illusions - although they are illusions that are well-founded upon the internal properties of substances.  Thus 'illusion' and 'science' are fully compatible.  For God, who can grasp all at once the complete concept, there is not only no space but also no temptation of an illusion of space. Leibniz uses the analogy of the experience of a building as opposed to its blueprint, its overall design. (E.g. 'Correspondence with Arnauld', April 12th, 1686, Monadology 57)  It is sometimes convenient to think of space and time as something 'out there', but this convenience must not be confused with reality.  Space is nothing but the order of co-existent objects; time nothing but the order of successive events.  This is usually called a relational theory of space and time.

Space and time, then, are the hypostatisations of ideal relations - which are real insofar as they symbolize real differences in substances; but illusions to the extent that (i) we take space or time as a thing in itself, or (ii) we believe spatial/temporal relations to be irreducibly exterior to substances, or (iii) we take extension or duration to be a real or even fundamental property of substances.

This raises a serious logical problem for Leibniz.  Above, we talked about truth as the containedness of a predicate in a subject.  This seemed acceptable, perhaps, for propositions such as 'Caesar crossed the Rubicon' or 'Peter is ill'.  But what about 'This leaf is to the left of that leaf'.  That proposition involves not one subject, but three (the two leaves, and whatever is occupying the point-of-view from which the one is 'to the left').  Leibniz has to argue that all relational predicates are in fact reducible to 'internal' properties of each of the three substances.  This includes time, as well as relations such as 'the sister of' or 'is angry at'.  But can all relations be so reduced, at least without radically deforming their sense?  Modern logicians often see this as the major flaw in Leibniz's logic and, by extension, in his metaphysics.
Take the analogy of a virtual reality computer.  What we see on the screen (or in the specially designed VR headset) is the illusion of space and time.  Within the computer's memory are just numbers (and ultimately mere binary information) linked together.  These numbers describe in an essentially non-spatial and temporal way a virtual space and time, within which things can 'be', 'move' and 'do things'.  For example, in the computer's memory might be stored the number seven, corresponding to a bird.  This in turn is linked to four further numbers representing three dimensions of space and one of time - that is, the bird's position.  Suppose further the computer contains also the number one, corresponding to me, the viewer - and again linked to four further numbers for my position, plus another three giving the direction in which my virtual eyes are looking.  The bird appears in my headset, then, when the fourth number associated with the bird is the same as my fourth number (we are together in time), and when the first three numbers of the bird (its position in virtual space) are in a certain algebraic relation to the number representing my position and point of view.  Space and time are reduced to non-spatial and non-temporal numbers.  For Leibniz, God in this analogy apprehends these numbers as numbers, rather than through their 'translation' into space and time. Leibniz is the first philosopher of virtual reality.
So how does Leibniz respond to the Newtonian 'bucket' argument?  Leibniz thinks this is no problem, although philosophers certainly still debate the issue.  He believes that we have simply to provide a rule for the reduction of relations.  For linear motion the virtual relation is reducible to either or both the object and the universe around it.  For non-linear motion, we must posit a rule such that the relation is not symmetrically reducible to either of the subjects (bucket, or universe around it).  Rather, non-linear motion is assigned only when, and precisely to the extent that, the one subject shows the effects of the motion.  That is, the motion is a property of the water, if the water shows the effects.  ('Correspondence with Clarke', Leibniz's Fifth Paper, 53)   Perhaps it seems strange that the laws of nature should be different for linear as opposed to non-linear motion.  It sounds like an arbitrary new law of nature, but Leibniz might respond that it is no more arbitrary that any other law of nature - just that we have become so used to the illusion of space that we are not used to thinking in these terms.

Substance as Monad

We are now, finally, ready to get a picture of what Leibniz thinks the universe is really like.  It is a strange, and strangely compelling, place.  Around the end of the Seventeenth Century, Leibniz famously began to use the word 'monad' as his name for substance.  'Monad' means that which is one, has no parts and is therefore indivisible.  These are the fundamental existing things, Leibniz thinks - his theory of monads is meant to be a superior alternative to the theory of atoms that was becoming very popular in natural philosophy.  Leibniz has many reasons for distinguishing monads from atoms - the easiest to understand is perhaps that while atoms are meant to be the smallest unit of extension out of which all larger extended things are built, monads are unrelated to extension (remember, space is an illusion).

We must begin to understand what a monad is by beginning from the idea of a complete concept.  As we said above, a substance/ monad is that reality which the complete concept represents. A complete concept contains within itself all the predicates that are true of the subject of which it is the concept, and these predicates are related by sufficient reasons into a vast single network of explanation.  So, relatedly, the monad must not only exhibit properties, but contain within itself 'virtually' or 'potentially' all the properties it will exhibit in the future, and also contain the 'trace' of all the properties it did exhibit in the past. In Leibniz's extraordinary phrase, found frequently in his later work, the monad is 'pregnant' with the future and 'laden' with the past.  (e.g. Monadology 22)  All these properties are 'folded' up within the monad, and they unfold when and as they have sufficient reason to do so.  (e.g. Monadology 61)  The network of explanation is indivisible - to divide it would either leave some predicates without a sufficient reason, or merely separate two substances that never belonged together in the first place.  Correspondingly, the monad is one, 'simple' and indivisible.

Just as in the analysis of space and time we discovered that all relational predicates are actually interior predicates of some complete concept, so the monad's properties will include all of its 'relations' to every other monad in the universe.  A monad, then, is self-sufficient.  Having all these properties within itself, it doesn't 'need' to be actually related to or influenced by another other monad.  Leibniz writes:

The monad that is me, for example, 'expresses' (but as an internal property of it) the fact that I now see various words on a computer screen.  Normally, we would think of this as a cause and effect relation - the screen produces light which causes nerve impulses in my eyes, etc.  But all this would happen irrespective of the actual existence of the screen!  Therefore, just like space and time, cause and effect is a 'well-founded' illusion.  We say that one thing (A) causes another (B) when the virtual relation between them is more clearly and simply expressed in A than in B.  But metaphysically, it makes no difference which way around we understand the relation, because the relation itself is not real.  Leibniz writes: Leibniz goes on to insist that the first direction of explanation is much simpler, since the second would involve leaping directly to the action of God to explain the extraordinary action of so many individual bits of water.  But that simplicity is hardly the same as truth.

So, instead of cause and effect being the basic agency of change, Leibniz is offering a theory of pre-established harmony to understand the apparently inter-related behaviour of things.  Consider the common analogy of two clocks.  The two clocks are on different sides of a room and both keep good time.   Now, someone who didn't know how clocks work might suspect that one was the master clock and it caused the other clock to always follow it.  When two things behave in corresponding ways, then we often assume (without any real evidence) that there is causation happening. But another person who knew about clocks would explain that the two clocks have no influence one on the other, but rather that they have a common cause (for example, in the last person to set and wind them).  Since then, they have been independently running, not causing each other.  On Leibniz's view, every monad is like a clock, behaving spontaneously in the way that it does, independently of other monads, but never the less tied into the others through the common reason: God and his vast conception of the perfect universe.  (We must be careful, however, not to take this mechanical image of a clock too literally.  Not all monads are explicable in terms of physical, efficient causes.)  Leibniz has another extraordinary set of phrases for this: every monad 'expresses' every other, as if it were a 'mirror' of the universe, but no monad has a 'window' through which it could actually receive or supply causal influences.  Relatedly, since a monad cannot be influenced, there is no way for a monad to be born or destroyed (except by God through a miracle - defined as something outside the natural course of events).  All monads are eternal.  (It is fair to say that Leibniz's attempt to account for what happens to 'souls' before the birth of body, and after its death, lead him to some colourful but rather strained speculations.)

Everything we perceive around us which is a unified being must be a single monad.  Everything else is a composite of many monads.  My coffee cup, for example, is made of many monads (an infinite number, actually).  In everyday life, we tend to call it a single thing only because the monads all act together.  My soul, however, and the soul of every other living thing, is a single monad which 'controls' a composite body.   Leibniz thus says that at least for living things we must posit substantial forms, as the principle of the unity of certain living composites.  (The term is derived from Aristotle: that which structures and governs the changes of mere matter in order to make a thing what it is.  See e.g. 'A New System of Nature'.)  My soul, a monad otherwise like any other monad, thus becomes the substantial form of my otherwise merely aggregate body.

Implications of Conceiving Substances as Monads

We will examine briefly four important implications of Leibniz's account of substance.  First, the distinction between metaphysical truth and phenomenal description.  Second, the idea of 'little perceptions'.  Third, the infinitely composite nature of all body.  Fourth, innate ideas.

I.    Leibniz has to posit a distinction between levels or 'spheres' ('Discourse on Metaphysics', 10).  The 'metaphysical' level is what is actually happening with monads (no causality, no space, no time at least as ordinarily understood, each monad spontaneously unfolding according to the kind of thing that it is). The 'phenomenal' or descriptive level is what appears to be happening because of our finite, imperfect minds (things cause one another, in space and time).  Science's object is the latter, which is an illusion, but in which nothing happens that is not based upon what really happens in the metaphysical level (the illusion is 'well-founded').  Therefore, the laws of physics are perfectly correct, as a description.  (Berkeley will borrow this idea (see especially his 'De Motu'); Kant will produce a highly original version of it.) Indeed, Leibniz believes, following Descartes and many other materialists, that all such laws are mechanical in nature, exclusively involving the interaction of momenta and masses.  Thus his accusation that Newton's idea of gravity is merely 'occult'.  Whereas, at the metaphysical level, no account of reality could be less mechanical!  Not surprisingly, then, Leibniz's own contributions to physical science were in the fields of the theory of momentum, and engineering.

A serious error would arise only if we took the 'objects' of our science (matter, motion, space, time, etc.) as if they were real in themselves.  Consider the following analogy: in monitoring a nation's economy, it is sometimes convenient to speak of a 'retail price index', which is a way of keeping track of the average change in the prices of millions of items.  But there is nothing for sale anywhere which costs just that amount. As a measure it works well - provided we don't take it literally! Science, in order to be possible for finite minds, involves that kind of simplification or 'abbreviation' ('Letter to Arnauld', 30th April, 1687).

II.    That the monad is the 'mirror' of the whole universe entails that my soul will actually have an infinite number and complexity of perceptions.   Obviously, however, I do not apperceive - am not conscious of - all these 'little perceptions'.  Perception then does not mean apperception - Leibniz argues that this is a major error on Descartes' part.  Leibniz is one of the first philosophers to have analysed the importance of that which is 'unconscious' in our mental lives.  Further, where I am conscious of some perception, it will be of a blurred composite perception.  Leibniz's analogy is of the roar of the waves of the beach - the sound is in fact made up of a vast number of individual sounds of droplets of water smacking into something else.  For Leibniz, little perceptions are an important philosophical insight:  First and foremost, this relates to one of Leibniz's main general principles, the principle of continuity.  Nature, Leibniz claims, 'never makes leaps' (New Essays On Human Understanding, 56).  This follows, Leibniz believes, from the principle of sufficient reason together with the idea of the perfection of the universe consisting of something like plenitude.  But the idea of little perceptions allows Leibniz to account for how such continuity actually happens even in everyday circumstances.  The principle of continuity is very important for Leibniz's physics (see 'Specimen Dynamicum') - and as we shall see turns up in Leibniz's account of change in the monad.

Second, little perceptions explain the acquisition of innumerable minor habits and customs, which make up a huge part of our distinctiveness as individual personalities.  Such habits accumulate continuously and gradually, rather than all at once like decisions, and thus completely bypass the conscious will.  Further, these little perceptions account for our pre-conscious connection with the world.  For Leibniz, our relation with the world is not one just of knowledge, or of apperceived sensation.  Our relation with the world is richer than either of these, a kind of background feeling of being-a-part-of - thus a thorough-going scepticism, however plausible at a logical level, is ultimately absurd.  Finally, for Leibniz, his idea of little perceptions gives a phenomenal (rather than metaphysical) account for the impossibility of real indiscernibles: there will always be differences in the petite perceptions of otherwise very similar monads.  The differences may not be observable at the moment, but will 'unfold in the fulness of time' into a discernible difference (New Essays, 245-6).
III.    As we saw above, what we perceive at the phenomenal level as bodies (my body, my coffee cup) are actually composites of monads.  Actually, such bodies must be made of an infinite number of other inanimate as well as animated monads.  This follows from the universe being the most perfect possible, which as we saw seems to mean the richest in controlled complexity, in 'plenitude'.  Leibniz argues that it would be a great waste of possible perfection to only allow living beings to have bodies at that particular level of aggregation with which we are phenomenally familiar.  Leibniz was understandably impressed by the different levels of magnitude being revealed by relatively recently invented instruments like the microscope and telescope. Leibniz writes:

(Not for a moment denying the extraordinary sublimity of such an image, but Leibniz is often accused of making rather too much of an inadequate and half-baked conception of the infinite.)

Further, the particular monads making up my body are constantly changing as I breath in and out, shed skin, etc. - although not all at one.  The substantial form is thus a unified explanation of bodily form and function.  A mere chunk of stuff has, of course, an explanation, but not a unified one - not in one monad, the soul.  Leibniz thus posits a four-way division of the types of monads: humans, animals, plants, matter.  All have perceptions, in the sense that they have internal properties that 'express' external relations; the first three have substantial forms, and thus appetition; the first two have memory; only the first has reason.  (See Monadology 18-19,29)
IV.    An innate idea is any idea which is intrinsic to the mind rather than arriving in some way from outside it.  During this period in philosophy, innate ideas tended to be opposed to the thorough-going empiricism of Locke.  Like Descartes before him - and for many of the same reasons - Leibniz found it necessary to posit the existence of innate ideas.  Now, at the metaphysical level, since monads have no 'windows', it must be the case that all ideas are innate.  That is to say, an idea in my monad/ soul is just another property of that monad, which happens according to an entirely internal explanation represented by the complete concept.  But at the phenomenal level, it is certainly the case that many ideas are represented as arriving through my senses. In general, at least any relation in space or time will appear in this way.

Thus, one could imagine Leibniz being a thorough-going empiricist at the phenomenal level of description.  This would amount to the claim that the metaphysically true innateness of all ideas is epistemologically useless information.  Leibniz finds it necessary, therefore, to advance the following arguments in favour of phenomenally innate ideas:

(i) Some ideas are characterised by universal necessity.  Such as ideas in geometry, logic, metaphysics, morality, and theology.  It is impossible to derive universal necessity from experience.   (This argument is hardly new to Leibniz!)  (ii)  An innate idea need not be an idea consciously possessed (because of 'little perceptions' for example).  An innate idea can be potential, as an inclination of reason, as a rigid distortion in Locke's tabula rasa.  (Here, Leibniz provides the famous analogy of the veins in the marble prior to the sculptor's work.)  It requires 'attention' (especially in the form of philosophical thinking) to bring to explicit consciousness the operation, and to clarify the content, of these innate ideas.  (iii)  The possibility of foreseeing an event that is not similar to (and thus merely an associated repetition of) a past event.  By using rational principles of physics, for example, we can analyse a situation and predict the outcome of all the masses and forces - even though we have never experienced a similar situation or outcome.  This, Leibniz says, is the privilege of humans over animals ('brutes'), who only have the 'shadow' of reason, because they can only move from one idea to another by association of similars.  (See Leibniz's joke about empiricists at Monadology 28)  Leibniz's most extensive discussion of innate ideas, not surprisingly, is in the New Essays Concerning Human Understanding.

Thus, at the phenomenal level, Leibniz can distinguish between innate and empirical ideas.  An empirical idea would be a property of my monad which itself expresses a relation to some other substance, or which arises from another internal property which was the expression of an external substance. Although the difference empirical/innate is in fact an illusion, it does make a difference, for example to the methodology of the sciences.  This is similar to the distinction made above between the idea of truth (as the containedness of the predicate in the subject), and the pragmatic/methodological issue of how we come to know that truth.  The latter is not irrelevant, except to the foundation and definition of truth.

Activity and Time

Correlate to the inter-connectedness of predicates in the complete concept is an active power in the monad, which thus always acts out its predicates spontaneously. Predicates are, to use a fascinating metaphor of Leibniz's, 'folded up' within the monad. In later writings such as the 'Monadology', Leibniz describes this using the Aristotelian/Medieval idea of entelechy: the becoming actual or achievement of a potential.  This word is derived from the idea of perfections. What becomes actual strives to finish or perfect the potential, to realise the complete concept, to unfold itself perfectly as what it is in its entirety.  This active power is the essence of the monad.  (Leibniz has several different names for this property (or closely related properties) of monads: entelechy, active power, conatus or nisus (effort/striving, or urge/desire), primary force, internal principle of change, and even light (in 'On the Principle of Indiscernibles').)

This activity is not just a property of human souls, but of all monads.  This inner activity must mean not only being the source of action, but also being affected (passivity), and of resisting (inertia).  As we saw above, what we call 'passivity' is just a more complex and subtle form of activity.  Both my activity and my resistance, of course, follow from my complete concept, and are expressed in phenomena as causes and as effects.  Change in a monad is the intelligible, constantly and continuously (recalling here the principle of continuity discussed above) unfolding being of a thing, from itself, to itself.  'Intelligible' means: (i) according to sufficient reason, not random or chaotic; and (ii) acting as if designed or purposed, as if alive - thus Leibniz's contribution to the philosophical tradition of 'vitalism'.

It is important to understand that this is not just a power to act, conceived as separable from the action and its result.   Rather, Leibniz insists that we must understand that power together with (i) the sufficient reason of that power; (ii) the determination of the action at a certain time and in a certain way; (iii) together with all the results of the action, first as the merely potential and then as the actual. (See 'On the Principle of Indiscernibles', and Monadology 11-15)  We are not, therefore, to understand a sequence of states, the individual bits of which are even ideally separable (except as a object of mere description for science), nor a sequence of causes and effects, again understood to be ideally separable (as if you could have had the cause without the effect).  All this follows from the complete concept, the predicates of which are connected in one concept.  Each 'state' therefore contains the definite trace of all the past, and is (in Leibniz's famous phrase) 'pregnant' with all the future.

But time too is an illusion, just like space.  How are we to understand change without time?  The important question is: what conception of time are we talking about?  Just like space, Leibniz is objecting to any conception of time which is exterior to the objects that are normally said to be 'in' time (time as an exterior framework, a dimension).  Also, he objects to time as mere chronology, to a conception of time as a sequence of 'now' points that are ideally separable from one another (not essentially continuous), and are countable and orderable separately from any thing being 'in' them (abstract).
However, in discussing relational properties above, and in particular Leibniz's response to the Newton/ Clarke argument about non-linear motion, we found that 'space' was in a sense preserved as a set of rules about the representative properties of monads.  Here, too, but in a more profound way, 'time' is preserved immanently to the monad.  The active principle of change we have been discussing above is immanent to monads, and no one state can be separated from all the others - without completely altering the thing in question into a thing that never changes, that has only the one state for all eternity.  For Leibniz, the past and future are no more disconnected - in fact less - from the present than 'here' is from 'there'.  Both distinctions are illusions - but temporal relations in a substance form an explanatory, intelligible sequence of a self-same thing.  The principle of change becomes an original, internal and active power of the thing constantly becoming the thing that it is, as the spontaneous happening and internal principle of the particular order of things which make up that substance.  Substances unfold, become the things God always knew them to be, in a time that is nothing other than precisely that becoming.

Time, then, has three levels: (i) the atemporality or eternality of God; (ii) the continuous immanent becoming-itself of the monad as entelechy; (iii) time as the external framework of a chronology of 'nows'.  The difference between  (ii) and (iii) was opened up by our account of the internal principle of change.  The real difference between the necessary being of God, and the contingent, created finitude of human being, is the difference between a (i) and (ii).


Leibniz mathematics - in parallel to Newton's - made a real difference to European science in the 18th century.  Other than that, however, his contributions as for example engineer or logician were quickly forgotten, and had to be re-invented elsewhere later.

However, Leibniz's metaphysics was highly influential, renewing the Cartesian project of rational metaphysics, and bequeathing a set of problems and approaches that had a huge impact on much of 18th century philosophy.  Kant above all would have been unthinkable without Leibniz's philosophy - especially the accounts of space and time, of sufficient reason, of the distinction between phenomenal and metaphysical reality, and his approach to the problem of freedom, not to mention Leibniz's largely welcoming attitude toward British empiricism.  Rarely did Kant agree with his great predecessor - indeed, rendering the whole Cartesian/ Leibnizian approach conceptually impossible - but the influence was never the less necessary.  After Kant, Leibniz was more often than not a mine of individual fascinating ideas, rather than a systematic philosopher, ideas appearing (in greatly modified forms) in for example Hegelian idealism, romanticism and Bergson.

In the 20th century, Leibniz has been widely studied by Anglo-American 'analytic' philosophy as a great logician who made significant contributions to, for example, the theory of identity and modal logic.  In Continental European philosophy, Leibniz has perhaps been less commonly treated as a great predecessor, although fascinating texts by Heidegger and, much later, by Deleuze, show the continuing fertility of his philosophical ideas.

Editions of Leibniz

As noted above, Leibniz did not publish much in his lifetime which fits our familiar description of a philosophy book.  Much was published, however, shortly after his death.  But there remained for the dedication of future editors a huge estate of short papers, letters and drafts of letters, and notes.  The standard edition of the works of Leibniz is the Akademie-Verlag of Berlin. The most comprehensive collection of these in English - together with some published material - is to be found in Leibniz.  Philosophical Papers and Letters.  Translated and Edited by Loemker.  2 Volumes.  University of Chicago Press, 1956.  The text has gone through subsequent editions and is now published (or should be) by Kluwer.

There are several good, inexpensive and readily available shorter anthologies of key texts:

Finally, there are editions in English of more specialised selections, the longer texts, and correspondences of Leibniz.  These include: In the above text I have given references as precisely as possible compatible with the reader finding the passages in which soever edition they may be printed.  For similar reasons I have tried always to quote and reference to texts available in more than one form in English.

 Author Information:

Dr. Douglas Burnham
Staffordshire University