The Emperor's New Mind

Gödel’s Theorem 
The centre piece of the early part of the book is the discussion on Gödel's theorem.  Gödel demonstrated that in any significant system of axioms there would be statements that were undecidable in terms of the axioms, but which were obviously true. The Gödel theorem as such is not controversial in relation to modern logic or mathematics, but the argument that Penrose derived from it has proved to be highly controversial.
 
He claims that the fact that human mathematicians can see the truth of a statement that is not demonstrated by the axioms shows that the human mind contains some function that is not based on algorithms, and therefore could not be replicated by any computer, since the functioning of computers is based solely on algorithms.

From Gödel, Penrose moves on to a straightforward account of physics, starting with classical physics, notably Galileo, and going onto relativity and quantum theory. This can be read as a useful popularisation of physics in its own right, but the real purpose of these chapters appears to be to establish the ground for Penrose’s claims about the relationship between physics and consciousness.

Quantum Theory
Penrose's discussion of quantum theory is particularly important in this respect. He implies criticism of the tendency in some quarters to try and marginalise quantum theory. He emphasises that quantum features underly many macroscopic phenomena such as solid objects, the physical properties of materials, colour, chemistry and DNA. But at the same time, he views present day quantum theory as a stop gap that does not provide a complete picture of the world. 

In particular, he disagrees with the traditional Copenhagen interpretation, which says that the theory is just an anstract  calculational procedure and that the quanta only achieve objective reality when a measurement has been made. Reality somehow arises from the unreal or from abstraction, giving a dualist quality to the theory. The discussion of quantum theory repeatedly comes back to the theme that Penrose regards the quantum world, and the uncollapsed wave function as having objective existence. In Penrose's view, the objective reality of the quantum world allows the quantum world to play a role in consciousness.

Penrose emphasises that the evolution of the wave function portrayed by this Schrödinger equation is both deterministic and linear. This aspect of quantum theory is not random. Randomness only emerges when the wave function collapses, and gives the choice of a particular position or other properties for a particle. Penrose discusses the various takes made on wave function collapse by physicists. Some would like everything to depend on the Schrödinger equation, but Penrose rejects this idea, because it is impossible to see how the mechanism of this equation could produce the transformation from the superposition of alternatives, as found in the quantum wave, to the random choice of a single alternative. He discusses the suggestion that the probabilities of the quantum wave that emerges into macroscopic existence arise from uncertainties in the initial conditions and that the system is analogous to chaos in macroscopic physics. This does not satisfy Penrose, who points out that chaos is based on non-linear developments, whereas the Schrödinger equation is linear. He also disagrees with Eugene Wigner’s suggestion that it is consciousness that collapses the wave function, on the basis that consciousness is only manifest in special corners of space and time. It is important to note this, as some commentators mix up Wigner’s idea with Penrose’s propositions on quantum consciousness, and then advance a refutation of Wigner as a supposed refutation of Penrose. He is also dismissive of the ‘many worlds’ version of quantum theory, which would have an endless splitting into different universes with, for instance, Schrödinger’s cat alive in one universe and dead in another universe. Penrose objects to the lack of economy and the multitude of problems that might arise from attempting such a solution, and in addition argues that the theory does not explain why the splitting has to take place, and why it is not possible to be conscious of superpositions.

Penrose instead argues for an objective interpretation of the wave function collapse. If the superpositions described by the quantum wave extended into the macroscopic world, we would in fact see superpositions of large scale objects. As this does not happen, it is argued that something that is part of objective reality must take place to produce the reality that we actually see. Penrose takes the view here that what he is proposing requires some new physics. This is often criticised as an unjustified demand for a revolution in physics. However, these criticisms tend to ignore the fact that while quantum theory provides many accurate predictions there has never been satisfactory agreement about its interpretation, and some of its founders regarded the theory as only provisional.

Penrose sees consciousness as not only related to the quantum level but also to spacetime, which means that it is related to relativity as well as quantum theory. This involves him in an extensive discussion of the origin of the universe. It is important to stress this, as the discussion of some of these areas does not seem immediately relevant to consciousness.

Entropy and the second law of thermodynamics
Penrose starts this stage of his argument by discussing the increase in entropy or disorder in the universe. He takes the example of a water glass falling from a table, shattering and the water pouring out onto the floor. The debris and spillage is more disordered than the previous structure of the water glass, and in addition some of the energy of the water and the glass has dispersed as heat. This is an example of increased entropy.

Because the laws of physics are time symmetric, they in theory allow for the water and glass to reassemble themselves and lift themselves back onto the table. However, we never see this happen in real life, the reason being that it would require coordinated action amongst the molecules in the debris to happen by chance, and the odds against this are vanishingly small.

The importance of this example is that in practise it represents an irreversible process in which entropy can only increase. This principle is encapsulated in the second law of thermodynamics, and is at least related to the arrow or flow of time. Another way of looking at the entropy question is that many fewer possible arrangements of particles are compatible with the organised state of a glass containing water than are compatible with the debris on the floor.

What is interesting to Penrose however is what this seems to tell us about the early state of the universe. If entropy is continually increasing it must have been much lower at the beginning of the universe. Penrose poses the question as to why it was that entropy was so low in the early universe. Normally if a system has experienced low entropy that has subsequently increased, it means that the entropy was somehow constrained to be low in the past, and then later released from that constraint.

Penrose looks for the source of the low entropy of the early universe. In respect of this, he considers life on Earth and its dependence on energy drawn from the sun. Penrose concludes that gravitational contraction in the sun and other stars has created high entropy. The stars formed from diffuse gas that had little gravitational clumping, and therefore constituted a reserve of low entropy. This diffuse gas derived from the Big Bang itself. The fact that the gas was distributed uniformly through space indicates a low level of entropy in the early universe, which in turn gives us the second law of thermodynamics. Penrose notes that there was low entropy despite the fact that the evidence of the background microwave radiation is that the early universe was near to thermal equilibrium, which would normally indicate high entropy. However, in this case, the entropy of the thermal equilibrium was more than offset by the low entropy related to the lack of gravitational clumping.

Penrose argues that the singularity of the Big Bang is quite different from the singularities of black holes, which represent very high entropy because of gravity, and still more different from the mass of congealing black holes that would arise if, as in some cosmological scenarios, the whole universe were to collapse back to a future singularity. So we have low entropy at the beginning of the universe but high entropy at one version of the end of the universe and at intermediate black holes.

Spacetime Curvature
Penrose moves on to the question of the spacetime curvature described in general relativity. He looks at the effect of singularities relative to two spacetime curvature tensors, Weyl and Ricci. Weyl represents the tidal effect of gravity by which the part of a body nearest to the gravitational source falls fastest creating a tidal distortion in the body. Ricci represents the inward pull on a sphere surrounding the gravitational force. In a black hole singularity, the tidal distortion of Weyl would predominate over Ricci, and Weyl goes to infinity at the singularity.

However, in an early universe expanding from the Big Bang, the inward tidal distortion is absent, so Weyl=0, while it is the inward pressure of Ricci that predominates. So the early universe is seen to have had low entropy and Weyl close to zero. Weyl is related to gravitational distortions, and Weyl close to zero indicates a lack of gravitational clumping, just as Weyl at infinity indicated the gravitational collapse into a black hole. Weyl close to zero and low gravitational clumping therefore indicate low entropy at the beginning of the universe. The fact the Weyl is constrained to zero is seen by Penrose as a function of quantum gravity. The whole theory is referred to as the Weyl curvature hypothesis or (WCH).

The question that Penrose now asks is as to why initial spacetime singularities have this structure. Penrose points out that at the beginning of the 20^th century quantum theory provided a solution for infinities in electro-magnetic theory. He thinks that quantum theory also has to help the problem of the infinity of singularities. This would be a quantum theory of the structure of spacetime, or in other words a theory of quantum gravity. However, although Penrose looks for the solution to his problem in quantum theory, he also says that quantum theory itself may have to change to take account of relativity.

Penrose regards the problems of quantum theory in respect of the disjuncture between the Schrödinger equations deterministic evolution and the randomness in wave function collapse as fundamental. He thinks in terms of a time-asymmetrical quantum gravity, because the universe is time asymmetric from low to high entropy. He argues that the normal process of collapse of the wave function is time-asymmetric. He describes an experiment where light is emitted from a source and strikes a half-silvered mirror with a resulting 50% probability that the light reaches a detector and 50% that it hits a darkened wall. This experiment cannot be time reversed, because if the original emitter now detects an incoming photon, there is not a 50% probability that it was emitted by the wall, but instead 100% probability that it was emitted by the other detecting/emitting device.

Penrose has a section which seeks to relate the loss of information that occurs in black holes to the quantum mechanical effects of the black hole radiation described by Stephen Hawking. This relates the Weyl curvature that is seen to apply in black holes and the quantum wave collapse. As Weyl curvature is related to the second law of thermodynamics, this is taken to show that the quantum wave reduction is related to the second law and to gravity.

Towards objective reduction
Penrose does not develop his objective reduction (OR) idea until his second book, but he does feel his way towards the general concept in this book. He points out that there have always been technical difficulties in relating the discrete concepts of quantum mechanics to the curvature of spacetime as described in relativity. He suggests that the quantum linear superposition of the Schrödinger wave can be expected to fail as soon as spacetime curvature becomes significant. He thinks it is at this point that the probability-weighted alternative is chosen from amongst complex amplitude superpositions. The amount of spacetime curvature involved is equivalent to one graviton, the quantisation of the gravitational field, the smallest unit of curvature allowed in a quantised theory. It is suggested that once this level of curvature is reached there is some form of time-asymmetric instability that allows one of the alternative probabilities to win out over the others. These suggestions are reformulated in the more specific and testable objective reduction theory in Penrose’s second consciousness related book, ‘Shadows in the Mind.’

Penrose’s ideas as to how quantum processes might be instantiated in the brain are not really developed until ‘Shadows in the Mind’, which demonstrated the influence of his cooperation with Stuart Hameroff. In the last part of the book, he merely reiterates that non-algorithmic processes might have a role in the brain, and adds to this that some as yet undescribed version of the quantum wave reduction could contain such a non-algorithmic element, and possibly allow the mind to form a bridge between physical reality and the abstract world of mathematics.

As a whole, this book represents an early and tentative stage of the development of the Penrose/Hameroff model. The book does not contain the idea of objective reduction (OR), which later on became central to the model. OR could be seen as a read out from the fundamental space time geometry of the universe, but this was to come later in Penrose's second book, 'Shadows of the Mind'. Even more crucially, when this book was written, Penrose, who lacked a neuroscientic background, had no model for how the brain could support quantum coherent activity or if this did exist, how it would function as a basis of consciousness.

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