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January 7 - January 20, 2019
COMPLEXITY AND COMPUTABILITY IN PHYSICAL THINGS
According to a recent set of ideas introduced by David Deutsch (1985), it is possible in principle to construct a ‘quantum computer’ for which there are (classes of) problems which are not in P, yet which could be solved by that device in polynomial time.
To address such questions we must turn away from matters purely mathematical and ask, in the following chapters, how the physical world actually behaves!
THE STATUS OF PHYSICAL THEORY
WHAT NEED WE know of the workings of Nature in order to appreciate how consciousness may be part of it? Does it really matter what are the laws that govern the constituent elements of bodies and brains?
What science has so far achieved has been dramatic. We have only to look about us to witness the extraordinary power that our understandings of Nature have helped us to obtain. The technology of the modern world has derived, in good measure, from a great wealth of empirical experience.
A good deal of this physical theory is not particularly recent. If one event can be singled out above all others, it is the publication, in 1687, of the Principia of Isaac Newton. This momentous work demonstrated how, from a few basic physical principles, one may comprehend, and often predict with striking accuracy, a great deal of how physical objects actually behave.
Newton’s own work, as he readily admitted, owed much to the achievements of earlier thinkers, the names of Galileo Galilei, René Descartes, and Johannes Kepler being pre-eminent among these. Yet there were important underlying concepts from more ancient thinkers still, such as the geometrical ideas of Plato, Eudoxos, Euclid, Archimedes, and Apollonios.
Einstein’s famous equation ‘E = mc2’ was part of this theory. But the theory’s impact on technology has been very slight so far (except where it impinges on nuclear physics), and its relevance to the workings of our brains would seem to be peripheral at best.
Einstein’s special relativity (anticipated by Poincaré and elegantly reformulated by Minkowski) gives a wonderfully accurate description of phenomena in which the speeds of objects are allowed to come close to that of light – speeds at which Newton’s descriptions at last begin to falter.
Einstein’s supremely beautiful and original theory of general relativity generalizes Newton’s dynamical theory (of gravity) and improves upon its accuracy, inheriting all the remarkable precision of that theory concerning the motions of planets and moons.
Escher’s depiction of Lobachevskian space. (Think of all the black fish as congruent and all the white fish as congruent.)
This real break with Euclidean geometry only came with Einstein’s general relativity, which was put forward many years later. Far from Euclidean geometry being a logical necessity, it is an empirical observational fact that this geometry applies so accurately – though not quite exactly – to the structure of our physical space!
In a sense, this was not so far from the philosophical viewpoint espoused by Plato (c. 360 BC; this was some fifty years before Euclid’s Elements, his famous books on geometry). In Plato’s view, the objects of pure geometry – straight lines, circles, triangles, planes, etc. – were only approximately realized in terms of the world of actual physical things.
Those mathematically precise objects of pure geometry inhabited, instead, a different world – Plato’s ideal world of mathematical concepts. Plato’s world consists not of tangible objects, but of ‘mathematical things’. This world is accessible to us not in the ordinary physical way but, instead, via the intellect.
The fifth book of Euclid’s Elements was basically an exposition of the ‘theory of proportion’, described above, that Eudoxos introduced. This was deeply important to the work as a whole. Indeed, the entire Elements, first published in about 300 BC, must be rated as one of the most profoundly influential works of all time. It set the stage for almost all scientific and mathematical thinking thereafter.
Upon the impressive foundations that Galileo had laid, Newton was able to erect a cathedral of superb grandeur. Newton gave three laws governing the behaviour of material objects. The first and second laws were essentially those given by Galileo: if no force acts on a body, it continues to move uniformly in a straight line; if a force does act on it, then its mass times its acceleration (i.e. the rate of change of its momentum) is equal to that force.
I am trying to raise is whether it is conceivable that a human brain can, by the harnessing of appropriate ‘non-computable’ physical laws, do ‘better’, in some sense, than a Turing machine. It is of no use trying to harness something like: ‘If ball A never meets ball B then the answer to your problem is “No”.’
Unpredictability is surely something that one would wish to minimize rather than ‘harness’ in the construction of a thinking machine!
There is a deep and beautiful mystery in this fact: that these superbly accurate theories are also extraordinarily fruitful simply as mathematics. No doubt this is telling us something profound about the connections between the real world of our physical experiences and the Platonic world of mathematics.
All these equations are derived from just one important quantity: the Hamiltonian function H, which is the expression for the total energy of the system in terms of all the position and momentum variables.
Such unity of form in the structure of dynamical equations, despite all the revolutionary changes that have occurred in physical theories over the past century or so, is truly remarkable!
The form of the Hamiltonian equations allows us to ‘visualize’ the evolution of a classical system in a very powerful and general way.
Try to imagine a ‘space’ of a large number of dimensions, one dimension for each of the coordinates x1, x2, . . ., p1, p2, . . . (Mathematical spaces often have many more than three dimensions.) This space is called phase space (see Fig. 5.10). For n unconstrained particles, this will be a space of 6n dimensions (three position coordinates and three momentum coordinates for each particle).
A vector field on phase space, representing time-evolution according to Hamilton’s equations.
As time evolves, a phase state region R0 is dragged along by the vector field to a new region Rt. This could represent the time-evolution of a particular alternative for our device.
COMPUTABILITY AND THE WAVE EQUATION
In any case, even if fields of this kind were permitted, it would be hard to see how any physical ‘device’ (such as a human brain?) could make use of such ‘non-computability’.
Do we actually have a teleological element in physical laws, where the future somehow influences what is allowed to happen in the past?
THE SPECIAL RELATIVITY OF EINSTEIN AND POINCARÉ
Common sense would seem to tell us that if we were to travel very rapidly in some direction, then the speed of light in that direction ought to appear to us to be reduced to below c (because we are moving towards ‘catching the light up’ in that direction) and the apparent speed of light in the opposite direction ought to be correspondingly increased to above c (because we are moving away from the light) – which is different from the fixed value c of Maxwell’s theory.
It was through worrying about such matters that Einstein was led, in 1905 – as, in effect, was Poincaré before him (in 1898–1905) – to the special theory of relativity. Poincaré and Einstein independently found that Maxwell’s equations also satisfy a relativity principle (cf. Pais 1982); i.e. the equations have a similar property of remaining unchanged if we pass from a stationary to a moving frame of reference, although the rules for this are incompatible with those for Galilean–Newtonian physics!
(Michelson and Morley showed that the apparent speed of light on the earth’s surface is not influenced by the earth’s motion about the sun – very much contrary to expectations.)
EINSTEIN’S GENERAL RELATIVITY
Recall Galileo’s great insight that all bodies fall equally fast in a gravitational field. (It was an insight and not completely a direct observation for, owing to air resistance, feathers and rocks do not fall together! Galileo’s insight was to realize that if air resistance could be reduced to zero, they would fall together.) It took three centuries before the deep significance of this insight was properly realized, and made the cornerstone of a great theory.
What has Galileo’s insight to do with the idea of ‘space-time curvature’? How could it be that such an idea, apparently so different from Newton’s scheme whereby particles accelerate under ordinary gravitational forces, could reproduce, and even improve upon, all the superb precision of that theory? Moreover, can it really be true that Galileo’s ancient insight contained something not subsequently incorporated into Newton’s theory?
The astronaut sees his space vehicle hover before him, seemingly unaffected by gravity.
Travelling in free orbit under gravity is just a sophisticated (and very expensive!) way of ‘falling’.
This is because acceleration and gravitational effects are just like each other, so one’s sensations are unable to distinguish one from the other. This fact – that the local effects of gravity are equivalent to those of an accelerating reference frame – is what Einstein referred to as the principle of equivalence.
Clocks run very slightly slow in a gravitational field, as Einstein maintained, this effect having now been directly measured in many different ways. Light and radio signals are indeed bent by the sun, and are slightly delayed by the encounte–ragain well-tested general relativity effects.
(A gravitational wave is the gravitational analogue of an electromagnetic wave, and it travels at light speed c.)
Recall that, in relativity theory, material bodies cannot travel faster than light – in the sense that their world-lines must always lie within the light cones (cf. Fig. 5.29). (In general relativity, particularly, we need to state things in this local way. The light cones are not arranged uniformly, so it would not be very meaningful to say whether the velocity of a very distant particle exceeds the speed of light here.)
The message that the observer V sends back to W could, by prior arrangement with W, simply repeat the message he received at B. Thus, W receives, at an earlier time on his world-line, the very same message that he is to send out later!
Fig. 5.33. A spacelike surface, for the specification of initial data in general relativity.
COMPUTABILITY IN CLASSICAL PHYSICS: WHERE DO WE STAND?
Throughout this chapter, I have tried to keep one eye on the issue of computability, as distinct from that of determinism, and I have tried to indicate that computability issues may be at least as important as those of determinism when it comes to the questions of ‘free will’ and mental phenomena.
There is yet no ‘home’ for our free wills in such phenomena: in the first case, because the classical Lorentz equation for a point particle (as resolved by Dirac) is not thought to be physically appropriate at the level where these problems arise; and in the second, because the scales at which classical general relativity might lead to such problems (black holes, etc.) are totally different from the scales of our own brains.
Now, where do we stand with regard to computability in classical theory? It is reasonable to guess that, with general relativity, the situation is not significantly different from that of special relativity – over and above the differences in causality and determinism that I have just been presenting. Where the future behaviour of the physical system is determined from initial data, then this future behaviour would seem (by similar reasoning to that which I presented in the case of Newtonian theory) also to be computably determined by that data27 (apart from the ‘unhelpful’ type of
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Indeed, it is hard to see that in any of the physical theories that I have been discussing so far there can be any significant ‘non-computable’ elements. It is certainly to be expected that ‘chaotic’ behaviour can occur in many of these theories, where very slight changes in i...
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Energy, after all, depends upon the speed with which that substance is travelling. The energy of motion in an express train is considerable, but if we happen to be sitting in that train, then according to our own viewpoint, the train possesses no motion at all. The energy of that motion (though not the heat energy of the random motions of individual particles) has been ‘reduced to zero’ by this suitable choice of viewpoint.
