Mathematics and Reality

Let us imagine that we have been travelling on a great journey to some far-off world. We shall call this world Tor’Bled-Nam. Our remote sensing device has picked up a signal which is now displayed on a screen in front of us. The image comes into focus and we see (Fig. 3.1): What can it be? Is it some strange-looking insect? Perhaps, instead, it is a dark-coloured lake, with many mountain streams entering it. Or could it be some vast and oddly shaped alien city, with roads going off in various directions to small towns and villages nearby? Maybe it is an island - and then let us try to find whether there is a nearby continent with which it is associated. This we can do by ‘backing away’, reducing the magnification of our sensing device by a linear factor of about fifteen. Lo and behold, the entire world springs into view (Fig. 3.2): Our ‘island’ is seen as a small dot indicated below ‘Fig. 3.1’ in Fig. 3.2. The filaments (streams, roads, bridges?), from the original island all come to an end, with the exception of the one attached at the inside of its right-hand crevice, which finally joins on to the very much larger object that we see depicted in Fig. 3.2. This larger object is clearly similar to the island that we saw first - though it is not precisely the same. If we focus more closely on what appears to be this object’s coastline we see innumerable protuberances - roundish, but themselves possessing similar protuberances of their own. Each small protuberance seems to be attached to a larger one at some minute place, producing many warts upon warts. As the picture becomes clearer, we see myriads of tiny filaments emanating from the structure. The filaments themselves are forked at various places and often meander wildly. At certain spots on the filaments we seem to see little knots of complication which our sensing device, with its present magnification, cannot resolve. Clearly the object is no actual island or continent, nor a landscape of any kind.

Cosmology and the Arrow of Time

Central to our feelings of awareness is the sensation of the progression of time. We seem to be moving ever forward, from a definite past into an uncertain future. The past is over, we feel, and there is nothing to be done with it. It is unchangeable, and in a certain sense, it is ‘out there’ still. Our present knowledge of it may come from our records, our memory traces, and from our deductions from them, but we do not tend to doubt the actuality of the past. The past was one thing and can (now) be only one thing. What has happened has happened, and there is now nothing whatever that we, nor anyone else can do about it! The future, on the other hand, seems yet undetermined. It could turn out to be one thing or it could turn out to be another. Perhaps this ‘choice’ is fixed completely by physical laws, or perhaps partly by our own decisions (or by God); but this ‘choice’ seems still there to be made. There appear to be merely potentialities for whatever the ‘reality’ of the future may actually resolve itself to be. As we consciously perceive time to pass, the most immediate part of that vast and seemingly undetermined future continuously becomes realized as actuality, and thus makes its entry into the fixed past. Sometimes we may have the feeling that we even have been personally ‘responsible’ for somewhat influencing that choice of particular potential future which in fact becomes realized, and made permanent in the actuality of the past. More often, we feel ourselves to be helpless spectators - perhaps thankfully relieved of responsibility - as, inexorably, the scope of the determined past edges its way into an uncertain future. Yet physics, as we know it, tells a different story. All the successful equations of physics are symmetrical in time. They can be used equally well in one direction in time as in the other. The future and the past seem physically to be on a completely equal footing. Newton’s laws, Hamilton’s equations, Maxwell’s equations, Einstein’s general relativity, Dirac’s equation, the Schrödinger equation - all remain effectively unaltered if we reverse the direction of time.

Where Lies The Physics of Mind?

In discussions of the mind-body problem, there are two separate issues on which attention is commonly focused: ‘How is it that a material object (a brain) can actually evoke consciousness?’; and, conversely; ‘How is it that a consciousness, by the action of its will, can actually influence the (apparently physically determined) motion of material objects?’ These are the passive and active aspects of the mind-body problem. It appears that we have, in ‘mind’ (or, rather, in ‘consciousness’), a non-material ‘thing’ that is, on the one hand, evoked by the material world and, on the other, can influence it. However, I shall prefer, in my preliminary discussions in this last chapter, to consider a somewhat different and perhaps more scientific question - which has relevance to both the active and passive problems - in the hope that our attempts at an answer may move us a little way towards an improved understanding of these age-old fundamental conundrums of philosophy. My question is: ‘What selective advantage does a consciousness confer on those who actually possess it?’ There are several implicit assumptions involved in phrasing the question in this way. First, there is the belief that consciousness is actually a scientifically describable ‘thing’. There is the assumption that this ‘thing’ actually ‘does something’ - and, moreover, that what it does is helpful to the creature possessing it, so that an otherwise equivalent creature, but without consciousness, would behave in some less effective way. On the other hand, one might believe that consciousness is merely a passive concomitant of the possession of a sufficiently elaborate control system and does not, in itself, actually ‘do’ anything. (This last would presumably be the view of the strong-AI supporters, for example.) Alternatively, perhaps there is some divine or mysterious purpose for the phenomenon of consciousness - possibly a teleological one not yet revealed to us - and any discussion of this phenomenon in terms merely of the ideas of natural selection would miss this ‘purpose’ completely.

Real Brains and Model Brains

Inside our heads is a magnificent structure that controls our actions and somehow evokes an awareness of the world around. Yet, as Alan Turing once put it, it resembles nothing so much as a bowl of cold porridge! It is hard to see how an object of such unpromising appearance can achieve the miracles that we know it to be capable of. Closer examination, however, begins to reveal the brain as having a much more intricate structure and sophisticated organization. The large convoluted (and most porridge-like) portion on top is referred to as the cerebrum. It is divided cleanly down the middle into left and right cerebral hemispheres, and considerably less cleanly front and back into the frontal lobe and three other lobes: the parietal, temporal and occipital. Further down, and at the back lies a rather smaller, somewhat spherical portion of the brain - perhaps resembling two balls of wool - the cerebellum. Deep inside, and somewhat hidden under the cerebrum, lie a number of curious and complicated-looking different structures: the pons and medulla (including the reticular formation, a region that will concern us later) which constitute the brain-stem, the thalamus, hypothalamus, hippocampus, corpus callosum, and many other strange and oddly named constructions. The part that human beings feel that they should be proudest of is the cerebrum - for that is not only the largest part of the human brain, but it is also larger, in its proportion of the brain as a whole, in man than in other animals. (The cerebellum is also larger in man than in most other animals.) The cerebrum and cerebellum have comparatively thin outer surface layers of grey matter and larger inner regions of white matter. These regions of grey matter are referred to as, respectively, the cerebral cortex and the cerebellar cortex. The grey matter is where various kinds of computational task appear to be performed, while the white matter consists of long nerve fibres carrying signals from one part of the brain to another. Various parts of the cerebral cortex are associated with very specific functions.

Quantum Magic and Quantum Mystery

In classical physics there is, in accordance with common sense, an objective world ‘out there’. That world evolves in a clear and deterministic way, being governed by precisely formulated mathematical equations. This is as true for the theories of Maxwell and Einstein as it is for the original Newtonian scheme. Physical reality is taken to exist independently of ourselves; and exactly how the classical world ‘is’ is not affected by how we might choose to look at it. Moreover, our bodies and our brains are themselves to be part of that world. They, also, are viewed as evolving according to the same precise and deterministic classical equations. All our actions are to be fixed by these equations - no matter how we might feel that our conscious wills may be influencing how we behave. Such a picture appears to lie at the background of most serious 1 philosophical arguments concerned with the nature of reality, of our conscious perceptions, and of our apparent free will. Some people might have an uncomfortable feeling that there should also be a role for quantum theory - that fundamental but disturbing scheme of things which, in the first quarter of this century, arose out of observations of subtle discrepancies between the actual behaviour of the world and the descriptions of classical physics. To many, the term ‘quantum theory’ evokes merely some vague concept of an ‘uncertainty principle’, which, at the level of particles, atoms or molecules, forbids precision in our descriptions and yields merely probabilistic behaviour. Actually, quantum descriptions are very precise, as we shall see, although radically different from the familiar classical ones. Moreover, we shall find, despite a common view to the contrary, that probabilities do not arise at the minute quantum level of particles, atoms, or molecules - those evolve deterministically - but, seemingly, via some mysterious larger-scale action connected with the emergence of a classical world that we can consciously perceive. We must try to understand this, and how quantum theory forces us to change our view of physical reality.

The Classical World

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? If our conscious perceptions are merely the enacting of algorithms, as many AI supporters would have us believe, then it would not be of much relevance what these laws actually are. Any device which is capable of acting out an algorithm would be as good as any other. Perhaps, on the other hand, there is more to our feelings of awareness than mere algorithms. Perhaps the detailed way in which we are constituted is indeed of relevance, as are the precise physical laws that actually govern the substance of which we are composed. Perhaps we shall need to understand whatever profound quality it is that underlies the very nature of matter, and decrees the way in which all matter must behave. Physics is not yet at such a point. There are many mysteries to be unravelled and many deep insights yet to be gained. Yet, most physicists and physiologists would judge that we already know enough about those physical laws that are relevant to the workings of such an ordinary-sized object as a human brain. While it is undoubtedly the case that the brain is exceptionally complicated as a physical system, and a vast amount about its detailed structure and relevant operation is not yet known, few would claim that it is in the physical principles underlying its behaviour that there is any significant lack of understanding. I shall later argue an unconventional case that, on the contrary, we do not yet understand physics sufficiently well that the functioning of our brains can be adequately described in terms of it, even in principle. To make this case, it will be necessary for me first to provide some overview of the status of present physical theory. This chapter is concerned with what is called ‘classical physics’, which includes both Newton’s mechanics and Einstein’s relativity.

Algorithms and Turing Machines

What Precisely is an algorithm, or a Turing machine, or a universal Turing machine? Why should these concepts be so central to the modern view of what could constitute a ‘thinking device’? Are there any absolute limitations to what an algorithm could in principle achieve? In order to address these questions adequately, we shall need to examine the idea of an algorithm and of Turing machines in some detail. In the various discussions which follow, I shall sometimes need to refer to mathematical expressions. I appreciate that some readers may be put off by such things, or perhaps find them intimidating. If you are such a reader, I ask your indulgence, and recommend that you follow the advice I have given in my ‘Note to the reader’ on p. viii! The arguments given here do not require mathematical knowledge beyond that of elementary school, but to follow them in detail, some serious thought would be required. In fact, most of the descriptions are quite explicit, and a good understanding can be obtained by following the details. But much can also be gained even if one simply skims over the arguments in order to obtain merely their flavour. If, on the other hand, you are an expert, I again ask your indulgence. I suspect that it may still be worth your while to look through what I have to say, and there may indeed be a thing or two to catch your interest. The word ‘algorithm’ comes from the name of the ninth century Persian mathematician Abu Ja’far Mohammed ibn Mûsâ alKhowârizm who wrote an influential mathematical textbook, in about 825 AD, entitled ‘Kitab al-jabr wa’l-muqabala’. The way that the name ‘algorithm’ has now come to be spelt, rather than the earlier and more accurate ‘algorism’, seems to have been due to an association with the word ‘arithmetic’. (It is noteworthy, also, that the word ‘algebra’ comes from the Arabic ‘al-jabr’ appearing in the title of his book.) Instances of algorithms were, however, known very much earlier than al-Khowârizm’s book.

Can A Computer Have A Mind?

Over the past few decades, electronic computer technology has made enormous strides. Moreover, there can be little doubt that in the decades to follow, there will be further great advances in speed, capacity and logical design. The computers of today may be made to seem as sluggish and primitive as the mechanical calculators of yesteryear now appear to us. There is something almost frightening about the pace of development. Already computers are able to perform numerous tasks that had previously been the exclusive province of human thinking, with a speed and accuracy which far outstrip anything that a human being can achieve. We have long been accustomed to machinery which easily out-performs us in physical ways. That causes us no distress. On the contrary, we are only too pleased to have devices which regularly propel us at great speeds across the ground - a good five times as fast as the swiftest human athlete - or that can dig holes or demolish unwanted structures at rates which would put teams of dozens of men to shame. We are even more delighted to have machines that can enable us physically to do things we have never been able to do before: they can lift us into the sky and deposit us at the other side of an ocean in a matter of hours. These achievements do not worry our pride. But to be able to think - that has been a very human prerogative. It has, after all, been that ability to think which, when translated to physicaJ terms, has enabled us to transcend our physical iimitations and which has seemed to set us above our fellow creatures in achievement. If machines can one day excel us in that one important quality in which we have believed ourselves to be superior, shall we not then have surrendered that unique superiority to our creations? The question of whether a mechanical device could ever be said to think- perhaps even to experience feelings, or to have a mindis not really a new one. But it has been given a new impetus, even an urgency, by the advent of modern computer technology.

In Search of Quantum Gravity

What is there that is new to be learnt, concerning brains or minds, from what we have seen in the last chapter? Though we may have glimpsed some of the all-embracing physical principles underlying the directionality of our perceived ‘flow of time’, we seem, so far, to have gained no insights into the question of why we perceive time to flow or, indeed, why we perceive at all. In my opinion, much more radical ideas are needed. My presentation so far has not been particularly radical, though I have sometimes provided a different emphasis from what is usual. We have made our acquaintance with the second law of thermodynamics, and I have attempted to persuade the reader that the origin of this lawpresented to us by Nature in the particular form that she has indeed chosen - can be traced to an enormous geometrical constraint on the big bang origin of the universe: the Weyl curvature hypothesis. Some cosmologists might prefer to characterize this initial constraint somewhat differently, but such a restriction on the initial singularity is indeed necessary. The deductions that I am about to draw from this hypothesis will be considerably less conventional than is the hypothesis itself. I claim that we shall need a change in the very framework of the quantum theory! This change is to play its role when quantum mechanics becomes appropriately united with general relativity, i.e. in the sought-for theory of quantum gravity. Most physicists do not believe that quantum theory needs to change when it is united with general relativity. Moreover, they would argue that on a scale relevant to our brains the physical effects of any quantum gravity must be totally insignificant! They would say (very reasonably) that although such physical effects might indeed be important at the absurdly tiny distance scale known as the Planck length - which is 10-35 m, some 100000000000000000000 times smaller than the size of the tiniest subatomic particle - these effects should have no direct relevance whatever to phenomena at the far far larger ‘ordinary’ scales of, say, down only to 10-12m, where the chemical or electrical processes that are important to brain activity hold sway.

Truth, Proof, and Insight

What is truth? How do we form our judgements as to what is true and what is untrue about the world? Are we simply following some algorithm - no doubt favoured over other less effective possible algorithms by the powerful process of natural selection? Or might there be some other, possibly non-algorithmic, route - perhaps intuition, instinct, or insight - to the divining of truth? This seems a difficult question. Our judgements depend upon complicated interconnected combinations of sense-data, reasoning, and guesswork. Moreover, in many worldly situations there may not be general agreement about what is actually true and what is false. To simplify the question, let us consider only mathematical truth. How do we form our judgements - perhaps even our ‘certain’ knowledge - concerning mathematical questions? Here, at least, things should be more clear-cut. There should be no question as to what actually is true and what actually is false - or should there? What, indeed, is mathematical truth? The question of mathematical truth is a very old one, dating back to the times of the early Greek philosophers and mathematicians - and, no doubt, earlier. However, some very great clarifications and startling new insights have been obtained just over the past hundred years, or so. It is these new developments that we shall try to understand. The issues are quite fundamental, and they touch upon the very question of whether our thinking processes can indeed be entirely algorithmic in nature. It is important for us that we come to terms with them. In the late nineteenth century, mathematics had made great strides, partly because of the development of more and more powerful methods of mathematical proof. (David Hilbert and Georg Cantor, whom we have encountered before, and the great French mathematician Henri Poincaré, whom we shall encounter later, were three who were in the forefront of these developments.) Accordingly, mathematicians had been gaining confidence in the use of such powerful methods. Many of these methods involved the consideration of sets with infinite numbers of members, and proofs were often successful for the very reason that it was possible to consider such sets as actual ‘things’ - completed existing wholes, with more than a mere potential existence.