Authors: Bill Bryson
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.
None of these latter-day echoes of Leibnizian thinking has generated traceable, exact results in the same way that, for example, Newtonian mechanics was able to predict the orbit of the Moon. If such a thing happens in the future – if, for example, the practitioners of loop quantum gravity use their theory to make predictions that are verified by experiment – then credit will have to go to them and not to Leibniz, who could never have imagined such a science. It’s not the point of this chapter, in other words, to argue that Leibniz was right, much less that Newton was wrong. Leibniz was not even doing science as we now define the term. My conclusions are two. First of all, that the infamous duel between Newton and Leibniz – which was only superficially about who had invented the calculus – came back from the dead a hundred years ago to exert remarkable influence over the course of modern science. Secondly, that Leibniz’s most fundamental assumption, namely that the universe makes sense and that the human has the power to make sense of it and that, consequently, pure metaphysics is no waste of time, remains perhaps the central question of all science. In 1960, Eugene Wigner wrote a paper,
The Unreasonable Effectiveness of Mathematics in the Physical Sciences,
in which he addressed the nearly miraculous way in which pure mathematics – seemingly a product of human cognition, and nothing else – predicts the behaviour of the physical world. The examples cited by Wigner would have made sense to Leibniz. Leibniz, however, would have been baffled by Wigner’s use of the adjective ‘unreasonable’ in the title of his paper. Wigner was a modern: a product of a sceptical age. He was uneasy (or felt obliged to pretend to be uneasy) with the philosophical implications of the way in which the physical world answered to mathematics. This unease could not have been more alien to Leibniz, who, during his long philosophical career, questioned many things that would have been easier to leave alone, but believed, with a kind of medieval serenity, in the reasonableness of Creation.
*
This perennial theological chestnut seems to have occasioned some soul-searching for Newton as well, since he risked serious trouble by semi-openly espousing the Arian heresy, which denies the Trinity.
Rebecca Newberger Goldstein is a philosopher and a novelist. Her work, both literary and scholarly, has received numerous prizes, including a MacArthur ‘genius’ award. Her non-fiction works are
Incompleteness: The Proof and Paradox of Kurt Godel and Betraying Spinoza: The Renegade Jew Who Gave Us Modernity.
Her fiction includes the novels
Properties of Light: A Novel of Love, Betrayal and Quantum Physics, Mazel, The Dark Sister
and
The Mind-Body Problem,
as well as a volume of stories,
Strange Attractors.
Her latest book is
36 Arguments for the Existence of God: A Work of Fiction.
A
MID THE DIVISIONS IN THOUGHT WHICH MARKED THE SCIENTIFIC REVOLUTION, THE FOUNDERS OF THE
R
OYAL
S
OCIETY INSISTED ON BINDING TOGETHER TWO CONTENDERS FOR THE BASIS OF NATURAL EXPLANATION. AS REBECCA
G
OLDSTEIN EXPLAINS, THERE WERE DEEP COMMITMENTS TO THE PRIMACY OF EXPERIMENTAL RESULTS, AND TO RECOGNISING UNDERLYING MATHEMATICAL PATTERNS. BUT THE REALLY POWERFUL TRICK, THEN AS NOW, LAY IN FINDING HOW TO BRING THEM TOGETHER.
After a lecture given by Christopher Wren, then the Gresham College Professor of Astronomy, twelve prominent gentlemen, deciding that they would meet weekly to discuss science and perform experiments, recorded their intention to form a ‘Colledge for the Promoting of Physico-Mathematicall Experimentall Learning’.
It might not have been the most elegant of designations, but it did, in its very wordiness, portend great things. It gave notice to the hope – because it was still, in 1660, only a hope – that two distinct orientations, one mathematical, the other experimental, would be pounded together into one coherent scientific method. The hope paid off, and it was from within the ranks of the Royal Society that the new compound emerged. Two cognitive stances that had seemed to have little to do with one another, except in their opposition to the system of natural philosophy dominant for centuries, were rendered equally necessary in the explanation of physical phenomena.
It was a time of epistemological urgency. A grandly unifying cathedral of thought was crumbling.
1
The all-inclusive view of the cosmos, laid down by Aristotle and buttressed by the medical theories of Galen, the astronomy of Ptolemy, and the theology of Christianity, had offered a way of explaining … absolutely everything. From the falling of objects to the rising of smoke; from generation and decay to the four basic personality types; from the relation between body and soul to the pathways of the planets; the supposed nature and reason for every aspect of the world could be extracted from an interlocking system that employed a homogeneous form of explanation throughout.
The form of explanation had been purpose-driven, or teleological, and its scaffolding was the metaphor of human action. We explain human actions by citing the end state that the agent has in mind in undertaking it. The old system took this familiar model of explanation and expanded it to apply to the world at large. ‘To be ignorant of motion is to be ignorant of nature,’ Aristotle had written, but by motions he meant not just displacements of bodies but such processes as becoming a parent, gaining knowledge, growing older. All were subsumed under the same conception: a striving to actualise an end state that was implicit in the motion and provided the explanation, the final cause, for the course that the motion took. The explanatory logic of human actions – based on intentions – was one with the explanatory logic of the cosmos.
The working hypothesis behind teleology was, of course, that all natural phenomena and processes do in fact have goals, allowing them to be viewed as potentialities on the way to being actualised. But every form of explanation makes use of some working hypothesis or other, ascribing to nature the features that allow such explanation to work. The mode of physical explanation that was to supplant teleology, making essential use of mathematics, also staked its claim on the world’s being a certain way.