Seeing Further

by Bill Bryson

  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 it picked up during that century or so of rough treatment at the hands of empiricists and positivists. Many hard scientists still use ‘metaphysics’ as a byword for undisciplined, conjectural thinking. Nevertheless, metaphysics is still being practised today: by philosophers openly, by physicists under other names.

  A straightforward way of defining metaphysics is as the set of assumptions and practices present in the scientist’s mind before he or she begins to do science. There is nothing wrong with making such assumptions, as it is not possible to do science without them. The lepidopterist who records in her notebook that a butterfly is blue may not stop to consider that this is true only because the giant ball of nuclear fuel ninety-three million miles away happens to maintain a surface temperature just right for shedding certain wavelengths of electromagnetic radiation on the Earth; that the eyes of humans have evolved to be sensitive to those wavelengths; that the eye can discriminate slightly different wavelengths as colours; that one of those colours has, by cultural consensus, been defined as ‘blue’, and so on. Nevertheless, science benefits from the lepidopterist’s note that the butterfly is blue.

  Even the hardest of hard sciences is replete with assumptions that may fairly be classified as metaphysical. Almost all mathematicians, for example, presume that they are discovering, rather than creating, mathematical truths. Ask a roomful of mathematicians whether three was a prime number a billion years ago (i.e. before there were humans to define it as such) and every hand will go up. And yet to say so is to espouse the metaphysical position that primeness and all the other subject matter of mathematics have a reality independent of the human mind. This assumption goes under various names, one of which is Mathematical Platonism. Likewise, physicists can hardly go about their work without assuming that the physical world answers to laws that may be expressed and proved mathematically – an assumption for which there is plenty of empirical evidence, dating back (at least) to Galileo, but no proof as such.

  The revival of Leibniz’s fortunes may be dated to approximately 1900, when Bertrand Russell began to publish his studies of Leibniz’s unpublished work. While unsparing in his criticisms of Leibniz’s character and of his more popular writings, Russell had a high opinion of Leibniz’s work on mathematical logic and was fascinated by some of the ramifications of the 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 tn 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 pos
sess 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 non-spatiotemporal 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 from interactions of more fundamental entities joined together in a graph of connections. This theory, which is called loop quantum gravity, is proposed as an alternative to string theory, which is background-dependent.

  6. The Leibnizian concept of pre-established harmony was viciously mocked by Voltaire in Candide, and has become no easier for sophisticated people to accept since then. Stripped of its theological overtones and saccharine connotations, though, the concept has a reasonably clear analogue in modern physics.

  a) Newtonian mechanics exactly describes the behaviour of individual bodies (provided, as Einstein later discovered, that they are reasonably large and slow-moving). Its laws are expressed in terms of individual particles: a particle moves in a straight line unless acted upon by a force. The force acting on a particle is equal to the product of its mass and acceleration (F = ma). As any first-year physics student learns the hard way, naïvely using the F = ma approach to describe systems comprising many independent parts soon becomes mathematically intractable.

  b) Leibniz is credited with having written down the law now known as conservation of energy (which he denoted vis viva). In any system of particles, the product of the mass and the square of the velocity of each particle, summed over all of the particles in the system, remains constant. When this, and the law of conservation of momentum, are imposed as constraints on a system, the mathematics frequently gets easier, to the point where it becomes possible to produce results not obtainable otherwise. Conservation of energy does not contradict Newton’s laws, and, in fact, is derivable from them, and so from a strictly mathematical point of view it adds nothing to Newtonian physics. It does, however, introduce a different way of thinking about physical systems. The naïve reductionist strategy of the first-year physics student gives way to a global approach in which the system as a whole must obey certain rules, to which the detailed movements and interactions of its components are seen as subordinate.

  c) The physicists of the late eighteenth and early nineteenth century developed new tools based on the notion of state or configuration spaces framed not of spatial dimensions but of all the generalised coordinates and momenta needed to specify the state of the system. Any possible state can be represented as a point in that space, and its evolution over time as a trajectory. The behaviour of such trajectories is governed by an ‘action principle’ that encodes all of the applicable physical laws, such as conservation of momentum and of energy. Action principles in classical state space are a mathematical reformulation of Newton’s laws, not an alternative to them. The change in point of view from physical trajectories in Cartesian space to action in state space is nonetheless significant. It is a further step away from the reductionist and toward the global approach. It seems to inject a teleological aspect that is not present in the older formulation, and so has occasioned some introspection among philosophically inclined scientists. In his Lectures on Physics, Richard Feynman interpolated a single, anomalous chapter on the topic, simply because of his abiding fascination with it. It allows the physicist to predict the behaviour of a complex system without having to work out the detailed interactions among its physical atoms. It leads to important results from thermodynamics and it is directly applicable to quantum mechanics. It is a way of thinking, systematically and rigorously, about compossibility, a concept important to Leibniz. Many possible states of affairs might exist or, to put it another way, there are an infinite number of possible worlds. But not all states of affairs are compossible; some are mutually contradictory, and while it is possible to imagine a universe in which contradictory states of affairs coexist, it is not possible for such a universe to come into practical being. The configuration space that describes the universe contains an infinity of points, each of which represents a different state of affairs, but most of these are incoherent. Only certain points – certain universes – make sense internally, and those points lie on trajectories that describe the logical evolution, according to physical law, of those universes over time. If one adopts this frame of reference for considering Leibniz’s concept of the pre-established harmony, and excludes (or at least adopts an agnostic stance toward) the notion that it was all set up at the beginning by God, it is easier to come to grips with Leibniz’s idea that the monads act in a coherent way somehow transcending detailed cause-and-effect interactions.

  d) That much is true of classical (i.e. pre-quantum) state space theory, even though it adds nothing beyond Newton’s original laws. The quantum version of the theory, on the other hand, requires that actions over all possible worlds be brought together in a calculation yielding the probability that any one state of affairs will eventuate. As Feynman puts it, ‘It isn’t that a particle takes the path of least action but that it smells all the paths in the neighbourhood and chooses the one that has the least action … ’ The picture is reminiscent of Leibniz’s ‘best of all possible worlds’.

  7. Possible-world theory has come in for serious study in recent decades both by philosophers and physicists. For impressively technical reasons that are likely to leave lay readers nonplussed, David Lewis (Plurality of Worlds) posited that all possible worlds really exist and are no less real than the one we live in. Such notions are the subject of current philosophical research, under the rubrics of modal realism and actualist realism. Among physicists, Hugh Everett launched the many-worlds interpretation of quantum mechanics in the late 1950s, since which time it has slowly but steadily garnered support. A particularly eloquent latter-day treatment can be found in David Deutsch’s The Fabric of Reality.

  8. Kurt Gödel (1906–1978) who early in his life became known as ‘the greatest logician since Aristotle’ because of his astonishingly original work on the foundations of mathematics, devoted much of the second half of his life to the development
of a rigorous metaphysical system that was to be based upon the work of Leibniz, with whom he had a fascination that became notorious. Gödel was a strong mathematical Platonist who thought in a serious way about the notion that the entities that are the subject matter of mathematics really exist, though not in our physical universe, and that when we do mathematics we in some sense perceive those entities. An almost painfully meticulous scholar, he was well aware of Kant’s objections to Leibniz’s metaphysics, and understood that those objections would have to be dealt with in order for him to make any progress. According to his friend and biographer Hao Wang, Gödel discovered the works of Edmund Husserl (1859–1938) in the late 1950s and devoted much of the remainder of his life to studying them. He felt that Husserl had solved many, if not all, of the metaphysical problems that Gödel had set for himself, including doing away with Kant’s objections to Leibniz’s work. Husserl is prolix, prolific and infamously difficult to read (even Gödel complained of this) and so a reader of sub-Gödel IQ, eyeing a heap of Husserl translations on a table, might despair of ever putting his finger on the passages that Gödel is thinking of. Fortunately, Hao Wang did us the favour of listing the specific Husserl books that Gödel most admired. One of them is Cartesian Meditations, based on a series of lectures that Husserl delivered late in his career. In the fifth and last of these, Husserl gets around to mentioning, in an approving way, Leibniz and monads. Husserl has come round to Leibniz’s way of thinking, but he has got there by taking a different route, pioneered by Husserl, through phenomenology – the premises and development of which I’ll spare the reader. Since Gödel’s death, mathematical Platonism has come in for serious study both by philosophers such as Edward N. Zalta, a metaphysician at Stanford University, and scientists such as Max Tegmark, an MIT cosmologist. Zalta and Tegmark (like Deutsch) have been influenced by David Lewis’ work on modal realism. Beginning from different premises, they have arrived at markedly similar approaches.