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 spaceand 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 aboutphysical 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
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 spaceand 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 aboutphysical 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
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