Bertrand Russell.

      Monadology. In his History of Western Philosophy (1945) he ends his chapter on Leibniz as follows: ‘What I…think best in his theory of monads is his two kinds of space, one subjective, in the perceptions of each monad, and one objective, consisting of the assemblage of points of view of the various monads. This, I believe, is still useful in relating perception to physics.’

      Leibniz then came to the attention of a wide range of thinkers. To tell the story in chronological order, including all of the requisite details about those who have knowingly or unknowingly echoed Leibniz’s views, would require a substantial book in itself, of which the following might serve as a brief sketch or outline.

      1. The debate on free will vs. determinism is no more settled today than it was at the time of the Leibniz–Clarke correspondence, and so in that sense (at least) Monadology is still interesting as a gambit, which different observers might see as heroic, ingenious, or desperate, to cut that Gordian knot by making free minds or souls into the fundamental components of the universe.

      2. Leibniz’s interpreters made use of the vocabulary at their disposal to translate his terminology into words such as ‘mind’, ‘soul’, ‘cognition’, ‘endeavour’, etc. This, however, was before the era of information theory, Turing machines and digital computers, which have supplied us with a new set of concepts, a lexicon, and a rigorous science pertaining to things that, like monads, perform a sort of cogitation but are neither divine nor human. A translator of Leibniz’s work, beginning in AD 2010 from a blank sheet of paper, would, I submit, be more likely to use words like ‘computer’ and ‘computation’ than ‘soul’ and ‘cognition’. During Leibniz’s era, the only person who had thought seriously about such machines was Leibniz himself; building on earlier work by Blaise Pascal, he designed, and caused to be built, a mechanical computer, and envisioned coupling it to a formal logical system called the Characteristica Universalis. He invented binary arithmetic, and, according to no less an authority than Norbert Wiener, pioneered the idea of feedback.

      3. In particular, the monads’ production rule scheme clearly presages the modern concept of cellular automata. Quoting from Mercer’s work:

      The Production Rule of F is a rule for the continuous production of the discrete states of F so that it instructs F about exactly what to think at every moment of F’s existence. Following Leibniz’s suggestion, if F exists from t1 to t_n and has a different thought at each moment of its existence, then at every moment, there will be an instruction about what to think next. The present thought occurring at t1, together with the Production Rule, will determine what F will think at t₂.

      Combined with the monadic property of being able to perceive the states of all other monads, this comes close to being a mathematically formal definition of cellular automata, a branch of mathematics generally agreed to have been invented by Stanislaw Ulam and John von Neumann during the 1940s as an outgrowth of work at Los Alamos. The impressive capabilities of such systems have, in subsequent decades, drawn the attention of many luminaries from the worlds of mathematics and physics, some of whom have proposed that the physical universe might, in fact, consist of cellular automata carrying out a calculation – a hypothesis known as Digital Physics, or It from Bit.

      4. Leibniz insisted that each monad perceived the states of all of the others, a premise that runs counter to intuition, given that this would seem to require that an infinite amount of information be transmitted to and stored in each monad. Of all the claims of Monadology, this must have seemed the easiest to refute a hundred years ago. Since then, however, it has been given a new lease on life by quantum mechanics. Consider, for example, the Pauli exclusion principle, which states (for example) that in a helium atom with two electrons in the same orbital, the two must have opposite spins. It is not possible for both of them to possess exactly the same state. Each of the two electrons somehow ‘knows’ the direction of the other’s spin and ‘obeys’ the rule that its spin must be different. The Pauli exclusion principle is Leibniz’s identity of indiscernibles principle translated directly into physics. Moreover, the ability of an electron to ‘know’ the state of another electron, without any physical explanation as to how this information is transmitted and stored, is strongly reminiscent of Monadology. Elementary descriptions of quantum mechanics tend to limit themselves to extremely simple systems, such as individual particles or atoms, since beyond there the mathematics becomes intractable. But the same principles apply, albeit in vastly more complex form, in larger systems: the quantum state of each particle is dependent upon the states of all the other particles in the system.

      5. Leibniz’s notion that the ultimate entities in the universe were nonspatiotemporal received a kind of weak boost from general relativity, which called into question the idea of absolute space and time as a fixed lattice on which the laws of physics were enacted. More recently, absolute space and time have come under more concerted attack as some physicists have sought to develop so-called background-independent theories. The idea of background independence is explained in more detail in Lee Smolin’s The Trouble with Physics, and the history of the concept of absolute space and time, from the Babylonians forwards, is told by Julian Barbour in his magisterial The Discovery of Dynamics. That space and time have an absolute reality, and that the laws of physics must be hung on a fixed spatiotemporal lattice, are metaphysical assumptions. Very reasonable, empirically grounded assumptions to be sure, but assumptions nonetheless. Resulting theories are called background-dependent. Various efforts have been made to derive background-independent theories that make no assumptions as to the fundamental reality of space and time. Barbour in particular has done seminal work along these lines, showing that general relativity is a realisation of a relational, i.e. Leibnizian, view of space and time. More recently, other researchers, notably Smolin, have sought to unify Barbour’s formulation of general relativity with quantum mechanics, the aim being to develop a background-independent theory of quantum gravity according to which space and time are emergent properties resulting