‘Monadology’ was not a title chosen by Leibniz, it was added by a later editor. It is nevertheless appropriate, for the paper explains Leibniz’s metaphysics in which the monads are central.
The argument for monads proceeds from admitting the existence of composites. If these exist, they are made up of either further smaller composites or of simple parts. One of Leibniz’s great elements of methodological bedrock, as it were, was the principle of sufficient reason, by which he meant that nothing was the case without there being a reason for why it exactly was the case and not something somewhat different. He employed this in the argument for monads by noting that there would be no sufficient reason for divisibility to cease at any particular level: thus composites are infinitely divisible. The simple substances that found such infinite divisibility are the non-extended monads.
If we were to say that Leibniz had ‘design aims’ underpinning his metaphysics, they would include the concepts that the state of the world should adequately reflect the greatness of the perfect being that created it and also that the perfect being should have acted in such a way as to realise the best possible world.
We know that the former was an aim of Leibniz, because he states that “only this theory […] shows up the greatness of God in an appropriate way” . And the latter of these two aims also illustrates another fundamental principle for Leibniz, that of contradiction. This holds that the opposite of a contradiction is a necessary truth. Here, the implication is that a perfect being by definition could not be restricted in terms of will or power, and so it would be a contradiction for this being not to have created the best possible world.
Voltaire and others mocked Leibniz for taking this line, and it would indeed be a natural point to bring in the traditional problem of evil. This points to the inconsistency of the perfect being having unlimited benevolence and power with the existence in the world of acts accorded the epithet ‘evil’ by people. While this argument carries force in general it is a misplaced criticism in the context of Leibniz, because it rests on a misapprehension of what he meant by the ‘best possible world’. This is a world that maximises the amount of variety, complexity and diversity extant, which satisfies the first of the design aims.
Monads “have no windows” ; they cannot interact causally or otherwise with each other. Because they have no parts (they are simple by definition), there is nothing internal that can alter to reflect such a putative interaction. They appear to interact, but this is because of the doctrine of pre-established harmony, whereby each one of the infinite number of monads has an internal organising principle controlling its evolution in such ways that the developments fit together. So a spiritual monad or human mind perceives developments in bodily monads that match with its expectations. Thus when I decide to raise my arm, the decision represents a change in my spiritual monad and then when the arm moves, a separate change uncaused by the first change takes place in the infinite bodily monads that are part of my arm.
It can already be seen that this is a picture of significant complexity, thus again admirable fulfilling the first design principle. Yet it is not however complete at this stage. Each of the infinite monads in each of an infinite number of items has an infinite number of relationships with an infinite number of other monads. This addition clearly significantly extends the ‘amount of complexity’ in the world.
This can perhaps be illustrated by a metaphor: each monad is “a perpetual living mirror of the universe” . If we imagine standing outside at night and holding a mirror, we could perhaps see the reflections of many stars. If we tilted the mirror, the reflections would all change. Thus the ‘perceptions’, by which Leibniz means the relations of a monad to other monads would all change, even though there would be no causal interaction between the bodily monads in the mirror and the bodily monads in the stars. There is theoretically no limit to the distance from which light could reach the mirror; confused perceptions could come from an infinite number of stars.
There are also eastern elements to Leibniz’s thought, hinted at in his statements in the context of what can be termed reincarnation that “only a small number of the elect who move up onto a larger stage” .
Leibniz believed that there were worlds within worlds (“there is a world of creatures […] in the smallest part of matter”) and was delighted with the discoveries of the microscope. He in fact visited Leeuwenhoek in Amsterdam . He would have been equally happy with modern progress in particle physics, whereby the process of revealing new levels of complexity has continued apace without any evidence of such continuation coming to an end. His interest in and contribution to physics was of course immense, and also enlightening on his philosophy.
His most noted contribution to physics was the development of differential calculus, at the same time as Newton; tellingly, while Newton more often is credited with the discovery, it is the elegant intuitive notational system of Leibniz that is used today. The essence of the calculus is the infinite divisibility of mathematical quantities by imagining them to ‘tend’ to an arbitrary or infinitesimal difference to a set limit, without actually reaching it. This enormously valuable idea, where we consider mathematical items approaching zero ‘extension’ that nevertheless are as real as anything else in mathematics and also produce extremely powerful and useful consequences, has clear parallels with the equally valuable and important monad in Leibniz’s metaphysics.
Leibniz was able to use this approach to falsify Descartes’ laws of motion; Descartes erroneously believed that “there is always the same quantity of force in matter” . This led him to argue that if two bodies A and B collide, then if A were ‘stronger’, the two bodies would move off in the direction A had been travelling, whereas if the two bodies were of equal strength, they would rebound from each other. Leibniz argued correctly that this could not be correct, because it would allow a discontinuity. One could imagine making the difference between A and B arbitrarily small; there would nevertheless be a dramatically different consequence of there being an infinitesimal difference in strength (moving off together in the direction of B) and zero difference (rebound).
Leibniz replaced Descartes’ approach with a correct conservation law using the square of velocity (v) as a vector quantity multiplied by mass (m) . This means that in a mechanical system of moving bodies, while individual bodies may change velocity, the total quantity mv2 in the system remains the same; nowadays this would be referred to as conservation of kinetic energy. This must have been highly suggestive for Leibniz; indeed he states that if Descartes had seen this, “he would have ended up with my system of pre-arranged harmony” .
There is a parallel between the way that mechanical bodies appear to interact, meaning the causation we believe we see when one billiard ball hits another, and the monads. For Leibniz, extension and causal interaction perceived by us are merely phenomenal, and yet the overall system is governed by a conservation rule. Likewise, for the monads, causal interaction is phenomenal, and yet the monads are so constructed that the overall system is harmonised.
Criticism has been made of Leibniz on the epistemic status of both his frequent invocation that the will inclines without necessitating, and that the best of all possible worlds has been created. Leibniz claims that these two theses are known a priori, and yet they are not the contraries of necessarily false contradictions. He would have to be using a different derivation for such truths than the principle of contradiction. In fact, there is textual evidence that he also uses an analytic notion of truth: “when a truth is necessary, the reason for it can be found by analysis, by resolving it into simpler ideas and truths until we arrive at the basic ones” . It is also worth noting that Kripke has shown that the notions of necessary truth and a priori truth are not coextensive, giving as an example currently unsolved theorems in mathematics. These are true or false necessarily, and yet will be known a posteriori if solved.
This means that some a priori and necessary analytic truths may be so for Leibniz by virtue solely of the meaning of the terms employed. This could take the form in the case of the former assertion about the will of insisting that we are morally responsible, and if this is so, then the definition of ‘will’ specifies that it be contingently realisable. Also, part of the definition of a perfect being for Leibniz could include the assertion that we have been created by it as morally responsible, thus grounding the first claim. Likewise, Leibniz may be using a definition of the perfect being that specifies that it would only create the best possible world.
G W Leibniz, Philosophical Texts, Tr. and Ed. R S Woolhouse and R Francks, Oxford University Press, 1998 (hereafter “PT”), p. 275
Cambridge Companion to Leibniz, Ed. N Jolley, Cambridge University Press (hereafter “CC”), p. 27