Where Newton’s work is grounded in Euclidean geometry, Leibniz begins with certain precepts that he takes to be axiomatic, such as the Principle of Sufficient Reason (nothing exists without a reason; there is no effect without a cause) and the Identity of Indiscernibles (two individual things cannot differ in number alone; it must be possible to explain why they are distinct based on some intrinsic difference). Newton developed calculus because it enabled him to solve problems in his theory of gravitation; Leibniz developed it as an outgrowth of his fascination with the problem of the Continuum, which asks how a line can be made up out of points, a span of time from instants, or a thought from the minute perceptions and endeavours of a mind. Just as Newton would not bother developing a physics that could not explain the fact that planets move in elliptical orbits, Leibniz had no time for any metaphysics that was incompatible with the transubstantiation of the Eucharist (both the Protestant and the Catholic versions!) and the incarnation of God in Christ. Much of the pick-and-shovel work of his Monadology came from a 1671 tract about the Incarnation of God. *
The modern reader, following the development of Leibniz’s ideas over the years between 1661 and his death in 1716, veers between finding it all quite reasonable and feeling as though it must have come from an alien planet. Just when one is about to judge Leibniz as having the strangest mind of anyone who ever lived, one remembers Newton and his lifelong obsession with alchemy and his strenuous efforts to predict the exact date of the End Times by ransacking the Book of Revelation for encrypted clues.
It takes an entire book such as Mercer’s to explain Leibniz’s full chain of reasoning, so there is not room here to attempt any such thing. The end point – Leibniz’s mature system, as described in Monadology – may be summarised as follows:
Matter, assumed by most to be the primary stuff of the universe, extended in space and time, is, in fact, unreal. Atomism in its conventional form – the idea that physical objects can be divided and subdivided up to a certain point, but (for some, usually unspecified, reason) no further, and that the result is a collection of tiny indivisible matter-bits moving around in empty space and banging into one another – is all wrong. The true atoms – the fundamental, indivisible units that make up the universe – are not spatiotemporal and so are not bound by spatial and temporal constraints; rather, space and time are epiphenomena of their activities, which are mental (today we might say computational) rather than physical. Leibniz calls these mind-atoms by the name of monads.
Use of ‘mind’ and ‘mental’ is apt to give modern readers the wrong idea. Many translators of Leibniz (including Russell) choose the word ‘soul’ instead of ‘mind’, which is even more confusing. A word about those words is, therefore, in order. Extension (occupying physical space) and duration (persisting through time) are obvious properties of matter that had long been of interest to natural philosophers. Beginning around 1671, Leibniz added a third element, namely cognitio, which can be translated as ‘thought’ or ‘knowledge’. In his metaphysics, cognitio is a property that things can possess and that makes them different from inert matter. Early in his career, it is as fundamental as extension. Later, it becomes more so. Previously, he had admitted God and the human mind as the only two incorporeal principles in his system; the key move he now made was to admit the possibility of cogitating entities (‘minds’ or ‘souls’) that were neither divine nor human, and to make them and ‘endeavour’ – the smallest possible unit of cogitation, which is to cognitio as a point is to a line or an instant is to time – as fundamental as space and time. Later, he goes on to deny the primary reality of space and time altogether and to assert that the created world consists entirely of these unextended monads and that the universe is created from moment to moment as a result of their cognition. In this he breaks from the metaphysics assumed by Newton (and almost anyone else who has thought in a commonsensical way about space, time and atoms) in which space and time have an absolute reality, and form a sort of lattice on which the laws of physics are enacted, and, indeed, without which they cannot even be written down.
Because the monads do not exist in space and time, they are free to take on certain powers and properties that would otherwise be implausible: (1) each monad perceives the state of every other monad in the universe, and (2) each exists in a certain state, and is capable of changing that state. This process of continual internal state-change is the cogitation that is the raison d’être of the monad and the fundamental process of the universe.
Internal and intrinsic to each monad is a rule (dubbed by Mercer the Production Rule) that governs how it changes its state in response to its current state and the perceived state of all of the other monads. And just as the constraints of space and time are inapplicable to monads, so cause and effect work differently, for each monad is causally independent of all other monads. It makes its own decisions by its own lights, obeying its intrinsic rule.
This raises the obvious objection that if the states of the other monads serve as inputs to the production rule, then there would seem to be a cause-and-effect relationship at work, but Leibniz doggedly maintains that no such relationship exists and that coordination among monads comes about, not through causal linkages, but as the result of a divinely ordained pre-established harmony that brings all of the monads into a kind of synchronisation without encroaching on their independence. For minds and cogitation are, to Leibniz, the ultimate reality, and unless the minds have free will, they are not minds at all but physical mechanisms numbly obeying deterministic rules.
This is the one feature of the Monadology that might (I speculate) have aroused some competitive anxiety in Newton’s mind. The Leibniz–Clarke correspondence probably would not have drawn the attention of so many important people were it not that traditional (spatiotemporal) atomism, combined with the then-new science of mathematical physics, seems to lead ineluctably to what was later called Laplacian determinism. If the behaviour of all objects can be explained in terms of spatiotemporal atoms, and if the atoms’ behaviour, in turn, is subject to Newton’s deterministic mathematical laws, then there is no room for free will. Humans are robots and religion is a fraud.
Newton was aware of this problem. He had no intention of promulgating a philosophy that stripped humans of free will. He seems to have got around it by positing supernatural intervention, i.e., by recourse to entities and powers that lay outside the system described by his science. Leibniz’s approach, bizarre as it might be in many respects, was, in a sense, more scientific; free will was no longer a problem that needed to be explained away, but an intrinsic feature of every monad.
Monadology spent the next two centuries on the ash-heap of intellectual history. After Leibniz’s death, a faulty version was published by one of his disciples, and its errors laid at Leibniz’s feet. Then it swam into the gunsights of Immanuel Kant. In his Critique of Pure Reason, Kant begins by saying a few complimentary things about Leibniz. Three hundred pages later, having carefully set his pieces out on the board, he annihilates Leibniz’s metaphysics in a few sentences. According to Kant’s philosophy, Leibniz is correct in thinking that space and time, cause and effect, are not ultimate realities, but rather constructs of mental activity. But by the same token, Kant says, the human mind is powerless to think in any useful or productive way about anything that is outside space and time, cause and effect, and so Leibniz’s entire Monadology – or any thinking that attempts to transcend spatiotemporality – is rubbish.
In the day of Newton and Leibniz, metaphysics had been as respectable as mathematics, but the hard-headed empiricists of the scientific world began to kick dirt on it during the nineteenth century and, in the first half of the twentieth, the logical positivists buried it. And indeed, Leibniz’s work seems unsound at best, ludicrous at worst, by the scientific standards of the era before relativity, quantum mechanics and Gödel’s proof.
Today, metaphysics in general has regained much of its former respectability among philosophers. For almost everyone else, though, it retains the connotations of woolliness that