The ideas discussed in this book look crucial to our understanding of spacetime, energy, matter, the physical law and the relationship of consciousness to all of these. Spacetime and the energy it contains are viewed as fundamental, while quantum particles are suggested to be less fundamental being distortions of the underlying spacetime. This could be seen as related to the Penrose suggestion of objective reduction as a result of the separation of the spacetimes of superposed particles, which is also a distortion of spacetime. Also discussed are the ideas of Gregory Chaitin, which appears close to Penrose in arguing that mathematicians can go beyond what any computer can perform, because they can go beyond the constraint of the Gödel incompleteness theorem. Chaitin also proposes that logical mathematics is the exception and can be seen as islands of logic in a vast sea of random truths with no logical basis.
Gravity and Mass: Possibly the most important part of this book is concerned with gravity and mass. This involves the question of inertia, the built in resistance of objects to being moved if they are stationary, or having their motion changed if they are already moving. This kind of inertial mass is the most familiar form of mass. The associated concept of weight represents the force of gravity acting on the mass, and for this reason weight varies according to the local strength of the gravitational field. This is referred to as gravitational mass, as opposed to the constant of inertial mass.
Mass is also conceived of as a concentrated knot of energy. Einstein identified that there was energy associated with mass. This is related to the fundamental particles out of which matter is built. Ordinary matter and energy, as distinct from dark matter and energy is built from quarks that make up the protons and neutrons of the atomic nucleus, and from leptons of which electrons are a subset. These particles are bound together by the four fundamental forces of nature. The strong and weak nuclear force govern the nucleus of the atom, the electromagnetic binds together mid-sized objects such as organic matter and machines, while the gravitational force governs the movements of stars and planets. All these forces are conveyed by carrier particles, with photons carrying the electromagnetic force and gluons carrying the strong nuclear force that binds together the protons and neutrons of the atomic nucleus and the quarks of which these are composed.
It is generally thought that these four forces are manifestations of a deeper symmetry (meaning acting in the same way in all directions) that prevailed at the beginning of the universe, but has since been broken. The single symmetry that prevailed at the beginning of the universe carries with it the assumption that at that time particles had no mass, although many of them have since acquired mass. This account of the universe indicates that there must be a mechanism by which mass is bestowed on previously massless particles.
One possible mechanism is the proposed Higgs field. The Higgs field is suggested to provide the ‘rest mass’ that is intrinsic to the particle rather than any mass associated with the energy of its movement. The particle may also possess mass by virtue of it being in motion. Fields such as the electromagnetic field and the Higgs field are viewed as being fundamental, with quantum particles being less fundamental, because they are just local excitations of a field. P. However, there is much that the Higgs concept does not explain. It does not explain why different particles have different mass, although it is assumed that they have different coupling constants with the Higgs field. In any case, the Higgs field, if shown to exist, accounts for only a small part of the energy of ordinary matter. The majority is tied up in the field of the strong nuclear force intermediated by the gluons which themselves have no rest mass.
The Quantum Vacuum: It is still not clear whether the Higgs field can explain inertial and gravitational mass. Some researchers, such as Bernard Haisch of the Calphysics Institute think that these forms of mass come from interaction between a quantum particle and the quantum vacuum, as the particle moves through the vacuum. The fundamental particles are seen as localised knots in the quantum fields.
Haisch has considered the possibility that the quantum vacuum has some connection to inertial mass. In this idea quantum behaviour is traced back to the oscillation of photons jumping in and out of existence in the quantum vacuum. Haisch’s idea is developed through a discussion of Hawking radiation. Hawking proposed that the strong gravity near a black hole distorts the quantum vacuum so that virtual photons that normally pop in and out of existence here receive enough energy to become permanent particles. It is suggested that these permanent photons would to an external observer look like the radiation from a hot furnace. Working from the equivalence of gravity and acceleration, researchers Paul Davies and Bill Unruh think that if an observer near a black hole saw heat radiation coming from the black hole, it also means that an observer accelerating through the quantum vacuum would see heat radiation coming from in front of them. From the point of view of an accelerated observer, the quantum vacuum is a real thing capable of having an effect.
Another researcher, Alfonso Rueda proposes that the oscillation of the virtual particles of the vacuum interact with objects so as to produce inertial mass. Photons are seen as being exchanged between the virtual particles of the quantum vacuum and the quarks and electrons that are most fundamental in matter. This accords with the idea that inertial force comes from outside the body, from the quantum vacuum and from the interaction between the particles of matter and the virtual particles of the quantum vacuum. It turns out in this approach that the fundamental thing is not mass, but the quantum vacuum. The Higgs field is relegated to producing rest mass, while inertial mass comes from the vacuum. Photons can be exchanged between the quantum vacuum and the quarks and electrons that make up matter. Although an electron is regarded as a point particle, it behaves as if it had a certain size, and this is viewed as an oscillation that reflects the oscillation of the quantum vacuum around it. It is speculated that the different masses of particles reflect differences in the resonating frequency with the quantum vacuum.
It is further suggested that inertial and gravitational mass share a common origin, which is that they both arise from the interaction of electron charges with the quantum vacuum. Haisch and Rueda believe that the electric charge in matter distorts the quantum vacuum in their vicinity, attracting or repelling virtual particles with the same or opposite charges. This distortion interacts with the charges in other matter creating a force of attraction between the two pieces of matter. One bit of mass only pulls on another via the quantum vacuum. The bending of light that is seen as a proof of the warping of space in general relativity is here explained in terms of a distortion of the quantum vacuum. Acceleration through the quantum vacuum results in resistance from the vacuum and this is seen as explaining inertia. Similarly, with gravitational mass, this is having the quantum vacuum accelerate past you as you fall towards a massive object.
According to the theory of general relativity spacetime is warped by energy, with mass being categorised as a form of energy. In the quantum theory approach to this concept virtual photons that jump in and out of existence in the vacuum warp spacetime around themselves. The source of the energy that warps space in general relativity is the energy density of space or the amount of energy in a unit volume of space. Similarly it is thought that inflation which consensus thinking believes to have driven the expansion of the very early universe, may have been a function of the quantum vacuum.
In quantum theory, the quantum wave has a height or amplitude that can be calculated at any point in space by means of the Schrodinger equation. The square of the amplitude represents the probability that a particle will be located at a particular point in space. The quantum wave spreads out over time according to the Schrodinger equation so that the longer that the wave is isolated from the environment the greater the uncertainty as to the position of the particle. Where quantum waves overlap and interfere with one another they are referred to as coherent. This quantum coherences gets lost or decoheres when a particle interacts with the environment. In the human eye coherent quantum particles of light (photons) decohere as a result of interaction with a large number of molecules in the eye. Because quantum coherence is lost when particles interact with a large number of other particles, quantum coherence is usually seen as a property of isolated particles. Relatively large collections of quantum particles have been demonstrated to remain coherent if they are isolated from the environment. Thus Zeilinger and team at the University of Vienna has succeeded in making a ‘buckyball’, a molecule of 60 carbon atoms remain coherent.
The Omega number: The mathematician, Gregory Chaitin, developed the idea of the Omega number. This number is seen as a demonstration that most mathematics cannot be discovered solely by logic and reasoning. The fact that mathematicians can discover new mathematics may mean that they are employing some form of intuition that no computer can replicate. Although the author does not mention Penrose, possibly because he does not want to involve a popular book in an acrimonious and often ill-informed controversy, Chown nevertheless seems to side with Penrose and against the very vocal ‘group-think’ consensus, in arguing that brains can do things that computers cannot.
Chaitin equates the length of a programme with the complexity of a number. The existence of a pattern in a number is the key factor in how complex a number is. If there is a pattern there is a short cut to writing down a programme for the number. The programme in this case is shorter than the number itself. Such a number contains reducible information. Where information is irreducible, the programme is as long as the number. Omega is defined by Chaitin as an infinitely long number without any pattern.
Set Theory: Set theory is concerned with a group of objects known as ‘sets’. Examples of sets are the set of all countries with names beginning with the letter ‘A’ or the set of all odd numbers or the set of all mammals. Some sets are contained within larger sets, as the set of all mammals is contained within the set of animals. Set theory sounds innocent enough, but research into set theory during the nineteenth century drew attention to the existence of a catastrophic set, the set of all sets that are not a member of themselves. In this case the set is a member of itself only if it is not a member of itself. The example of this is the case of the village barber who shaves every man who doesn’t shave himself. He shaves himself if and only if he doesn’t shave himself.
This contradiction in set theory was a nightmare for nineteenth century mathematicians. Mathematics was founded on logical reasoning, and was regarded as a superior realm of clear-cut truths. But in the case of set theory logical reasoning led to absurdity. The German mathematician, David Hilbert, aimed to eradicate this problem. Maths is based on axioms, self-evident truths on which mathematicians agree. Theorems are a logical consequence of such axioms. Hilbert hoped to identify a small group of axioms as the basis of all mathematics. Following from this he hoped to set out all detailed logical rules for getting from the axioms to all the theorems. This would make it possible to prove any mathematical statement. The important thing was to show that the theorem could be derived from the bedrock axioms. There would be a procedure of algorithm for checking each step in a proof. The list of theorems could be infinite and all contradiction could be removed. What Hilbert had accidentally conceived was what we now understand as computing, a totally mathematical procedure.
Gödel: However in 1931, Gödel showed that the Hilbert programme could never be achieved. Whatever axioms were selected as the basis for mathematics there would always be legitimate theorems that could not be derived from the axioms. It was discovered that the world of mathematics was full of undecidable theorems that are true, but can never be proved by logical reasoning. Gödel proved his result by embedding in mathematics the self-referential statement that “this statement is unprovable”. Mathematics was thus shown to be incomplete. The subsequent idea of getting round Gödel by simply adding more axioms does not work because Gödel’s incompleteness theorem shows that no matter how many axioms are added, there will always be some theorems that cannot be derived from them.
Non-computabilty: A bit later than Gödel, Turing produced the idea of uncomputability or non-computability. Non-computability is viewed as being connected to Chaitin’s Omega concept, where complexity is a function of the length of programme needed to generate a number. The similarity is that just as the Omega number cannot be compressed into a programme, an undecidable Godel theorem cannot be compressed into axioms. Undecidabality is therefore seen as a consequence of non-computability which involves such questions as whether it is possible to know whether a programme looking for a particular number, for instance an even number that is not the sum of two prime numbers (Goldbach conjecture) will ever halt. If it was possible to have axioms of the kind that showed that a programme like this would or would not halt, it would be possible to solve the halting problem, but Turing showed that this was impossible. In this way, he showed that there were theorems that could not be proved by step-by-step logical rules.
In Chaitin’s view, undecidability and non-computability are normal in mathematics, rather than an esoteric state at the margin, which is how they had been treated during the twentieth century. Most of mathematics is seen as being composed of random truths that are true for no reason. Randomness is a statement that events are unpredictable and happen for no reason. Chaitin envisages mathematics as islands of provable truth, such as algebra and calculus, connected by threads of logic in a sea of random truths. Chaitin views the Goldbach conjecture as just such a random truth, not connected by logic to anything else, with no way for it to be deduced from a set of axioms. This means that the Goldbach conjecture should be accepted as an axiom in its own right. Chaitin takes the view that any given set of axioms only captures a tiny part of the complexity of the universe.
Chaitin’s views raise a question as to how mathematicians actually do mathematics and find new theorems. Mathematicians move between the islands of mathematical provability. Reason and logic is insufficient. Chaitin thinks that they use insights that go beyond reason and logic. Mathematics of this kind appears to involve imagination and creativity, and as such is not limited by Godel’s incompleteness theorem, with the brain performing functions that no computer can perform. This is precisely what Penrose had argued in 1989 in respect of the brain and mathematical understanding, although the connection is not mentioned here.
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