General Articles 3

General Articles 3

General Articles review and summarise articles of background relevance to quantum consciousness.

General Articles 3 covers further background articles related to neuroscience.

1.) Descartes Error - Antonio Damasio

2.) The Astonishing Hypothesis - Francis Crick

Descartes Error

Antonio Damasio

Although Damasio remains within the conventional paradigm, this classic book, published in 1994, kicks away some of the props of the mechanistic ‘brain as classical computer’ orthodoxy, by arguing for the involvement of the emotions and the body in neural processing. In this respect, the book can be considered to have made more of a contribution than Damasio’s later volume ‘The Feeling of What Happens’, also reviewed in our general section, which left some readers slightly disappointed, partly because of a refusal to distinguish between self consciousness and consciousness as such.

Near the beginning of his book, Damasio describes a patient, whose condition contradicts the idea that absence of emotion would be a good attribute for a Star Fleet officer. The patient had suffered a lesion in the prefrontal cortex. The patient appears normally intelligent, logical, possessed of adequate memory and attention span, and yet in daily life he continually makes decisions that make it impossible for him to hold down a job, or appear socially acceptable. But the only apparently abnormal aspect to the patient’s behaviour was a lack of ability to experience feelings.

Damasio argues that the famous 19^th century case of Phineas Gage is supportive of his case. Gage was working on the construction of a new railway in Vermont, when he was injured by an explosion, in which a steel pole passed through the front of his brain. Remarkably, for the time, he recovered both physically and mentally, but his personality was changed, so that he could not hold down a job or otherwise lead a stable life, exactly the same conundrum as Damasio faced with his modern patient. In fact, the author argues that the particular orientation of 19^th century neuroscience meant that it missed the main point of the Gage case, which was the connection between personality/social reasoning and frontal lobe damage.

The Gage case suggested that there were systems in the human brain dedicated to the social dimensions of reasoning. Social reasoning could therefore be lost as a result of brain damage, even when other aspects of reason and language remained intact. This involved the loss of some system that allowed future planning within a complex social environment. Examination of the preserved skull of Gage, suggests that his brain was damaged in an area known as the ventromedial prefrontal, which is involved in decision making. In modern times, the Gage problem of intact mental processing but dysfunctional social behaviour following a brain lesion has emerged in many patients.

Damasio moves on to discuss a patient of his own, referred to as Elliot. Elliot had unimpaired movement , language, memory, and normal intellect in most respects. His brain damage was confined to the ventromedial prefrontal area. Elliot had already been through a battery of intelligence tests that showed he was normal. Damasio did not disagree with these tests as such, but discovered that Elliot lacked the ability to make appropriate decisions in his social and working life.

For the lay observer, these testing problems may tend to confirm an unease about the extent to which psychological and intelligence tests look for the sort of things that comprise intelligence in real life. However, Damasio is making a much more limited point about social decision making. Elliot faired badly in a sorting test in which the criteria for correct sorting are changed half way through. Patients with Elloit’s type of frontal lobe damage tend to persist with the old criteria. In general, the ability to make estimates on the basis of incomplete knowledge is often impaired with this sort of patient.

Damasio began to realise that he had been paying too much attention to Elliot’s intellect and too little to his emotions. Elliot came over as a very dispassionate observer of events. Damasio began to surmise that reduced emotion and feeling were connected to Elliot’s problem. It was discovered that Elliot had a theoretical grasp of appropriate social behaviour, but that he could not apply it in his own case. Elliot couldn’t actually make the appropriate decision, although in theory, he knew what it was. The problem was therefore right at the end of the chain of reasoning, just where the actual decision had to be made.

The findings with Elliot were confirmed by numbers of patients with similar lesions. Their ability to choose advantageous courses of action was lost, despite other mental capacities being intact. At the same time emotions and feeling were dampened. The patients were inflexible in their approach to life, unable to organise future activity and were less able to experience pleasure. This is associated with damage to the orbital and medial regions of the frontal lobe. Damasio sees damage to the ventromedial sector of the prefrontal as being associated with impairment of decision taking and dampened emotions.

Emotional dampening is also reported with some other forms of brain damage. There are patients with certain types of anosognosia, where they are paralysed on the left side of the body, but deny any problem, and show a lack of emotion of relative to the subject. This is related to damage to parts of the right somatosensory cortex. In other cases, patients with damage to both the left and right amygdale, a brain structure related to fear and anger, show a similar pattern of inappropriate social decisions and reduced emotions.
Damasio takes the view that reason and emotion effectively intersect in the ventromedial prefrontal and that there is a similar area in the right somatosensory cortex. He thinks that the regions connected to decision taking are also involved in emotion and feeling.

He also discusses the role of a brain structure that is part of the limbic system called the anterior cingulate cortex, on the basis of observing patients with damage to this area, the nearby supplementary motor area (SMA) and the third motor area. This condition may also involve the prefrontal areas related to movement, emotion and attention and the motor cortex. These patients are very impaired both in reasoning and emotion.

Damasio notes that there is a high concentration of serotonin receptors in the ventromedial, neighbouring medial temporal cortex and the amygdala. He suggests a connection between the ventromedial prefrontal and the amygdala, an important component of the limbic system, associated with fear and anger. Like the other neurotransmitters, serotonin is delivered from small nuclei of neurons in the brain stem or the basal forebrain. Axons from these terminate in the cortex and the limbic system. One role of serotonin is the inhibition of aggressive behaviour, and it is seen as favourable to social behaviour.

In pursuing his hypothesis about the influence of the body on reasoning, Damasio points out that the central nervous system is connected to the rest of the body by the peripheral nervous system. In addition, it is also connected chemically by hormones and peptides.

He goes on to discuss the process of reaching decisions. He regards the body and brain as indissociably integrated by biochemical and neural circuits. The nervous system carries signals from every part of the body to the brain and vice versa. Incoming signals go to the somatosensory cortex. The bloodstream provides an alternative route carrying chemical signals such as hormones. Chemical substances arising from the bodies activities can reach the brain via the bloodstream. The brain can act on the body via the autonomic nervous system and the voluntary nervous system. The signals for the autonomic systems arise in the amygdala, the cingulate and the hypothalamus, while the voluntary nervous system acts from the motor cortex. The brain also acts by controlling the release of hormones. Parts of the brain stand between the sensory input systems and the output systems. These parts are the association areas of the cortex, the basal ganglia, the brain stem, the thalamus and the limbic system. Damasio suggest that the modulator neurons in the base of the brain that distribute neurotransmitters, such as dopamine, noradrenaline, serotonin and acetylcholine, to both the cortex and sub cortical areas, respond to feedback from conditions in the body.

Such a system suggests that the traditional separation between mind and brain is a myth, as is the separation between mind and body. Innate neural patterns are seen as being located in the brain stem and the hypothalamus. The hypothalamus is crucial in regulating the endocrine system and the immune system, the latter also depending on chemicals released into the bloodstream. The controls of the brain stem and the hypothalamus are complemented by the limbic system, which is the brain centre for emotions, and also by signals from the cortex.

Damasio suspects that the limbic is less dependent on innate or genetic factors than the brain stem or the hypothalamus. The hypothalamus and related structures also react to chemical signals from the body. The system is seen as interactive. Hormones act on body cells but also on the glands that produce them and on the hypothalamus. Neural signals give rise to chemical signals in the bloodstream, which give rise to other chemical signals, which can alter cells, and ultimately alter cells in the brain that started the original biological cycle. The hypothalamus can also act on the limbic or the cortex as well as being acted on by them.

As examples, anxiety is known to alter the regulation of sexual hormones, while bereavement can depress the immune system. The brain is similarly influenced from the body, an external substance such as alcohol being a good example. Other chemicals act either directly on neurons, or by affecting the neurons that distribute transmitters, so as to produce states such as elation or depression. For its part the brain can release a substance such as oxytocin relative to child birth or sex.

Damasio’s main point is that systems involved in biological regulation are also involved in cognitive processes. The brain stem, hypothalamus and limbic are all involved in emotion, learning and perception. Rationality therefore in this argument does not depend just on activity in the cortex.

Primary emotions such as fear, anger, disgust, happiness, sadness are seen to depend on the limbic system, with the amygdala and the cingulate as the most prominent components. Secondary emotions, such as guilt or embarassement, requiring cognitive content, involve the prefrontal and the somatosensory, and may be more acquired than innate. Activity in these areas is suggested to be referred back via the limbic to the autonomic nervous system and the endocrine system, while the neurotransmitter modulators may also release in response to this.
These activities are suggested to cause an emotional body state that is again referred back to the limbic and somatosensory. Cognitive processes are suggested to be influenced by all of this. Thus a representation of the body state is distributed over a large number of brain structures.

Damasio criticised conventional accounts of cognition for excluding emotions and feelings, partly because they are seen as sub cortical. Damasio, by contrast, argues that body feelings and emotions make themselves felt at the cortical level. An example, now accepted, of how the brain influences feelings is the production of endorphins in the brain to change both feelings and mood. The distribution of neurotransmitters within the brain acts in a similar way.

Damasio further proposes the somatic marker hypothesis. Reasoning by the brain is aimed at reaching a decision, and the act of deciding involves the selection of a particular option. Conventionally, emotion was excluded from this process. The decision process was normally thought of in terms of deriving logical consequences from premises. The trouble with this approach is that it can require the brain to survey an impractically large range of scenarios. In a complex decision such as choosing a career, there would be an out-burgeoning of possible scenarios. Furthermore it may involve comparisons that do not use the same currency, such as comparing job interest to financial reward. Like the non-polynomial problems found in computing, a decision might take an unrealistically long time, or there might be no decision at all.

This is where Damasio brings in his somatic marker hypothesis. The somatic marker is a body feeling relative to the possible decision, and this allows the brain to cut through the complexity of purely rational pondering. The somatic marker is seen as being a feeling generated by the secondary emotions. Somatic markers are viewed as assisting or prompting rational deliberations. They are an internal preference system aimed at achieving future goals.

The prefrontal is seen as critical for secondary emotions. This area also receives sensory input from most areas of the brain, and the bioregulatory system including neurotransmitters from within the brain, and also from the limbic system. This area may be responsible for providing scenarios of future outcomes, based on inputs from the rest of the brain. The problem with patients such as Elliot is that they become cut off from somatic markets, and can therefore only respond to the impulses of the moment.
Somatic markers may also covertly assist reasoning by creating preferences for attention to particular aspects of the question. This system may also act covertly, and constitute what we regard as intuition

Damasio relates these ideas to creativity, suggesting that creativity arises from a covert ability to juxtapose facts or concepts that appear diverse, but have an unexpected kinship. Most juxtapositions of unrelated concepts are irrelevant, but the prefrontal may be able to screen out a good proportion of these. Therefore, reason as such does not have to be applied to the whole field of possible options, but only to an unconsciously selected group, from which the really implausible candidates have already been excluded. Logical thinking by itself is not seen as enough for creativity.

Damasio supports the hypothesis that reason depends on the prefrontal cortex, the brain stem and the hypothalamus working in concert. The brain stem and hypothalamus maintain connections with most of the bodily organs, placing the body in the same chain as the processes that support reasoning and social behaviour. Thus emotion, feeling and biological regulation are suggested to all influence reasoning. He does not see feelings as at all intangible. The feelings are seen to derive from a combination of the prefrontal cortex, the limbic, which is seen as seat of emotions in the brain and the state of the body. Feeling as opposed to emotion is viewed as a continually updated window onto the state of the body. Damasio suggests that these bodily feelings provide a frame of reference for our neural processing. He sees the brain and body as being interactive. He thinks that the workings of neurons are not something separate from the rest of the organism.

The implications of Damasio’s ideas may never have been fully worked out either by himself or by those in the mainstream who have been influenced by him. It seems likely that his ideas have been distorted to preserve the ‘meat computer’ notion of consciousness, with the initial idea that everything was concentrated in the neurons being merely extended to included the idea of consciousness being embodied or embedded in the rest of the body.

What does not seem to be tackled in any of this, is that while rational processing on the part of neurons can be accomplished unconsciously, the essential feature of at least a good part of emotions and bodily feelings is that they are experienced subjectively, and it is the subjective experience that gives them their power to influence, and in Damasio’s hypothesis to form the final stage in decision making. It is also worth considering that Damasio’s idea that emotion and feeling cuts through a potentially unresolvably complex problem for reason, bears a resemblance to Penrose’s suggestion that the human brain has some feature that can go beyond the axioms of a formal mathematical system.

To highlight the points made above, I provide an excerpt from a recent popular science fiction novel, ‘Persephone Wakes’ by Jack Junius.

The story so far: Persephone, the first conscious android has escaped from her inventors. By a serious of mischances, she arrives at the luxurious villa of Freya, the head of a rival and innovative android design company. Freya wonders whether or not she is really being presented with a wonderful new piece of technology ......

     First, she (Freya) had to investigate more carefully. She knew now that this girl for all her warmth and vitality was just a machine, but was this machine conscious or just a very clever simulation? Well, her own prior efforts to build conscious androids had at least taught her what she had to look for. She would need to detect apparently subjective responses. She needed to see preferences or choices of one course of action over another that were sufficiently unlikely to have been included in Persephone’s original programming or training.
     Freya knew that robots were just thinking machines. But the human brain had a two-way connection between the rational and emotional centres. If something damaged that connection, it became difficult for a human to make even trivial decisions. There was nothing mystic about the emotions. They were based on the same type of brain cells as the rational area of the brain. However, it was not just a matter of a brain signal from the emotional area. It was the conscious experience of the emotion that gave it its special punch in making decisions or forming preferences. It could help to decide between things that were narrowly balanced, or insist on something that was not wise from the rational point of view, such as more wine or sex with the wrong person.
     Freya reached for the wine bottle, meaning to fill one of the bathroom tumblers. ‘My God, you’ve drunk nearly the whole bottle!’ she exclaimed.
    ‘Yes, I like that wine.’
    ‘Like! Androids can’t like or dislike. Although, I suppose if you’re really conscious … ’ that was a lucky break, Freya suddenly realised. There was absolutely no point in programming an android to drink wine in the absence of its owners. Doing that in itself suggested some form of will and preference.
     She had realised that one could never be totally certain that another entity was conscious. It had to be inferred. You knew you were conscious yourself. You inferred that other people were conscious, to the extent that their behaviour resembled your own. Over a very long period of time, most people had inferred that androids and other computers and computer-driven machines were not conscious.
     It seemed to her that the important difference leading people to that conclusion had been that androids never expressed their own preference for particular things or particular courses of action, never indicated that they felt that they liked something, or that they wanted to perform a particular action. You could argue that in humans these preferences were themselves just the product of genes or nurture, but we all knew that at the point of action, they derived from a subjective experience, and that was what gave them their special punch.
    Of course, it was child’s play to programme the expression or simulation of a preference into an android, but in many cases there would be no sensible reason for doing such a thing. Persephone’s behaviour with the wine had been human rather than robotic. There was no apparent reason for her to drink the wine, except for a small amount when humans were present in order to give a companionable and humanoid impression.
    Freya decided that she would try some more-or-less formal tests. First she would make some further use of Persephone’s wine-drinking propensities. She had Snodgrove (Freya’s non-conscious android butler) bring in several glasses of different types of wine. Persephone mustn’t see the bottle, because she might have been programmed to admire particular wines.
    When Persephone had sampled a rather ample amount of each wine, Freya asked her to rank them in order of preference. A normal android could only grade wines according to its knowledge of the quality as signified by the label. Confronted with this sort of blind tasting, the android would be stumped. It would remain silent, or apologise for its inability to answer the question. Android operating manuals warned owners against this type of open question, which could have harmful effects on the quantum brain.
   Persephone, however, ranked her preferences with hardly a moment’s hesitation, and added for good measure that she preferred the original Gewürztraminer to any of the subsequent wines.
   Freya was shocked when she heard that answer. She knew that Persephone was a machine, but she had for some time believed that no one would ever build a machine that could do what Persephone had just done.
   For her second test, Freya had Snodgrove wheel in several racks of clothing. ‘Your stuffs all ruined,’ Freya remarked. ‘You’d better choose something you like from these.’ This should be revealing, Freya thought. It was once again the sort of open choice that the manuals warned against. The android could become confused, and might simply put on the first garments that came to hand, with a possibly incongruous effect. It was better for the user to give clear instructions.
   Persephone spent a considerable time flicking through the rails, but the actual process of choosing did not appear to bother her. She tried on lots of things and generally took an uneconomic amount of time. She did ask Freya’s opinion a few times, but then tended to disagree with her, and didn’t seem to have any notion that Freya’s view should prevail. Once again, Persephone was showing very specific preferences that could not realistically have been programmed in.
   Freya searched around for another test. Then she had it. Since the invention of the quantum brain, robots had always been stumped by art and artistic preferences. The villa had been built by Freya’s great-uncle. It had been decorated with frescoes by an artist, who had been fashionable at the time, but had subsequently become something of a joke in artistic circles. At least, that was until the last few years when he had started to come back into fashion, and Freya had felt obliged to open parts of the villa once a year for members of the society formed by the artist’s new admirers.
   Freya asked Persephone for her opinion of the bathroom frescoes, and to say which of the various panels she preferred. At this point, an android would normally fall back on a simple factual catalogue of the subject matter. The android mind had never gone that further step to give a subjective response. But Persephone admired the light and airy feel of the landscapes, and showed little hesitation in picking out the panels she liked best.

The Astonishing Hypothesis

Francis Crick

It is interesting to revisit a book that was seen as seminal, when it was published in 1994, but is less frequently mentioned now. Crick makes frequent references to his cooperation with Christof Koch, although in contrast to subsequent joint work, Koch is not actually credited as co-author.

There is an element of hubris to the title itself, in that in seeking to uphold the mainstream paradigm, to the effect that consciousness can be derived from neuroscience as currently understood, this book is the reverse of astonishing. Rather, it is exactly the point of view that a mainstream theorist would be expected to take.

What would have been truly astonishing would have been if Crick had sought an explanation outside of established neuroscience. For astonishment, and in deed fury and ridicule, we need look no further than the response to Penrose’s rather more surprising hypothesis, produced in the same period as Crick’s book. The only hint of the unconventional in this book appears in the first page of the preface, where Crick appears to take a sideswipe at Dennett’s even more hubristically titled ‘Consciousness Explained’ published three years earlier in 1991. Crick remarks that:

‘some philosophers are under the delusion they have already solved the mystery (of consciousness), but to me their explanations do not have the ring of scientific truth.’

The opening lines of the book proper are memorable for their tone of high confidence stating that:

‘The Astonishing hypothesis is that “You”, your joys and your sorrows, your memories and ambitions, your sense of personal identity and freewill, are in fact no more than the behaviour of a vast assembly or nerve cells and their associated molecules. As Lewis Carroll’s Alice might have phrased it: “You’re nothing but a pack of neurons.”

It is interesting to contrast the rather domineering tone of these opening lines in 1994, with the cautious approach of a joint article by Crick and Koch published in 2006^(1.), which states that:

‘Our strategy is to leave the core of the problem (qualia) on one side for the time being and instead try to discover the minimal neural mechanisms.’

Beyond this, the main surprise of Crick’s 1994 book is that the first 200 or so pages contain very little discussion of consciousness. There is a detailed description of the brain and in particular the visual system, but little attempt to explain how these mechanisms generate consciousness. On pages 7-8 it is rather arbitrarily decided that explanations relative to the brain and neurons do not need to descend below the level of chemical interactions, but there is little discussion as to why this should be so. Crick does not bother with even one of the routine dismissals of quantum consciousness. The proposition does not exist so far as this book is concerned.

He does, however, caution against a too simplistic comparison between brains and computers. Computers depend on very fast serial calculations, while brains use relatively slow parallel calculations. This makes brains more resilient than computers, because the loss of a few components from a parallel system is not vital to its function. Further to this neurons have a much less predictable response than computer switches, and can be subject to signals that modulate their behaviour. Crick also take the view that the widely used computer simile of hardware and software is not valid when applied to the brain. He thinks that there is no clear distinction between hardware and software in the brain. He also points out the difference between the number crunching processes, where computers far outstrip humans, and the task of recognising the significance of objects and processes, where computers have tended to struggle. As is the case in much 1990s consciousness literature, a considerable amount of space is devoted to neural net computing. In this period, there were hopes that these could mimic the parallel processing of the brain, but much less has been heard about this type of technology in recent years.

It is not until p. 207 that Crick starts to look in earnest for the neural correlate of consciousness. He reasons that consciousness is likely to involve some form of attention, and with it some very short-term memory, otherwise there would be no memory of the thing that is being attended to. A problem arises when it is considered that more than object or experience can be attended to. This is the ‘binding problem’ or the problem of the unity of consciousness. Crick points out that in comprehending less familiar objects, the brain must deal with an almost infinite possible combination of features.

Possibly the most interesting part of the book is the discussion as to whether binding could be achieved by the correlated firing of neurons involved in attentional activity. The most striking brain oscillations are the so-called 40 Hz (actually 35-75 Hz) or gamma oscillations, as originally studied by Wolf Singer, Charles Gray and others, shortly before this book was written. It was shown that synchrony could occur between different cortical areas and even between different hemispheres. Singer and Gray suggested that this could be a solution to the binding problem. Crick and Koch were more explicit, suggesting that the attentional mechanism would select an object and synchronise a coalition of neurons relevant to the object of attention.

Crick also tried to locate the seat of consciousness in the brain. He suggests that the lower layers (5&6) of the cortex that receive the results of computations in the other layers are a possible site. Pyramidal neurons in layer 5 are seen as prime candidates, because they project outside the cortical layers. Short-term memory input is required for this system to work, and this is suggested to depend on a reverbatory circuit from layer 6 of the cortex to the thalamus and back to layers 4 and 6 of the cortex. The thalamus may also be a location of consciousness in this scheme. Synchronised firing in this system is suggested to be the neural correlate of consciousness.
What is lacking in this scheme is any suggestion of what physically different characteristic in the pyramidal neurons or connected neurons in the thalamus gives rise to the distinct qualia and subjective property of consciousness. Crick admits rather in passing that his scheme has not explained the subjective nature of consciousness or the qualia.

However, the really disappointing aspect of all this is that the most interesting aspect of this book, the gamma synchrony, was not properly followed up. For a time Crick’s prestige promoted the work started by Singer and Gray. However, the discovery that the synchrony was with the dendritic activity of neurons rather than their axonal spike activity seems to have led to the topic being dropped by the mainstream, leaving Hameroff, who connects gamma synchrony to possible qauntum activity, as its main proponent.

A book of this kind would not be complete without an attempt to get rid of freewill. In this case, Crick appears to hide behind the screen of unconscious computation. He does assume that he is conscious of his future plans, which to some extent conflict with the Libet readiness potential method of disposing of freewill. However, Crick comments that we are not conscious of the computation involved in deciding our plans. In a sense, this is obviously true in that it is known that a large part of the brain’s activities are unconscious. However, if we consider the complex thinking that can be involved in arriving at a decision, it is apparent that this is very different, from Crick’s implication that a ready made and unconsciously manufactured decision pops out of some kind of slot in the mind.

More importantly, although Crick mentions Antonio Damasio, he fails to mention important aspects of his early 1990s work, which strongly argued that the ability to come to decisions, as opposed to merely pondering pros and cons, depends on links between the prefrontal areas of the brain, involved in reasoning, and our experience of emotions and bodily feelings. If this link is broken, as in some forms of brain damage, decision making becomes impossible or refers only to immediate gratification rather than future planning. The important characteristic of emotion and bodily feeling is that they involve qualia or subjectivity, which therefore need to be brought into the processing, computational or otherwise, by which the brain produces decision.
In the end, the main significance of this book was the Crick’s prestige made consciousness studies respectable after having been a taboo area during much of the 20^th century.

References:-

1.) Francis Crick & Christof Koch (2006) - What are the neuronal correlates of consciousness - In: 23 Problems in Systems Neuroscience Eds; van Hemmen, L. & Sejnowski, T. - Oxford University Press ISBN-13: 978-0-19-514822-0

Gödel & Turing

Chaitin

Gödel showed that the connection between proof and truth was shaky. In mathematics and in other formal systems statements can be true but unprovable. Some mathematical propositions might be undecidable and this demolishes the idea of a closed consistent body of rules, and replaces it with incompleteness.

Chaitin discusses randomness. Something is random if it has no pattern or abbreviated description. Then there is no algorithm shorter than the thing itself. On this basis mathematics can be shown to be shot through with randomness.

This has implications beyond mathematics because the laws of physics are mathematical. The laws of physics are seen as algorithms that map input data or initial conditions into output data or the final state.

The possibility of the universe as a computer is examined. Quantum mechanics imposes a lower limit on the time taken for each step in processing and the universe has a finite age, so only a finite amount of information can have been processed in the life of the universe. This is suggested to mean that there is a cosmological bound on the fidelity of mathematical laws.

Random or patternless sequences of a given length require the longest programmes. A random sequence of length k requires a programme of length k. These random or patternless sequences comprise the majority of sequences. Such programmes where th number of bits equal the number of observations are random and useless. The minority of non-random sequences require a programme that is shorter than k. The shorter the programme, the less random the sequence. The shorter the programme the greater the pattern present in the sequence.

This links to information theory. Messages are coded or compressed to eliminate redundant information. Any information source that is not random can be compressed. The definition of randomness is the converse of information theory.

Chaitin also relates this to the Occam’s razor concept in which the simplest theory, or in other words the shortest sequence is seen as being the best theory. Simplicity here means not ease of calculation, but the number of arbitrary assumptions that have to be made. A scientific theory is valuable if it allows one to compress many observations into a few hypotheses.

The minimum quantity of information needed to define a string is equated to the complexity of the string. Most strings of length n have complexity n and these are random. Only the non-random minority have less complexity. Where a string is random, the programme describing it has to expand in direct relation to the length of the string.

Although randomness can be measured and defined a given number cannot be proved to be random. This is seen as being related to the Gödel incompleteness theorem.

The fundamental unit of information is the ‘bit’ and this is defined as the smallest item of information indicating a choice between two things. In binary notation one ‘bit’ is represented by either a ‘0’ or a ‘1’.

In the 1960s the Russian mathematician, Solomonoff, represented a scientist’s theory as an algorithm predicting future observations, and in the case of competing theories about the observations, the model will select the smallest algorithm that is the algorithm comprising the fewest bits. This is the same idea as Occam’s razor.

Any specified series of digits such as 123 can be generated by an infinite number of algorithms. However, the best programme is the smallest one. The smallest programmes are called minimal programmes, and there may be one or many minimal programmes for a given series of digits.

The concept of the minimal programme is closely related to the concept of complexity. The complexity of a series of digits is the number of bits required to get the series of digits as output. There the complexity = the minimal programme for the series.

It is emphasised that non-random distributions are exceptional.

It is easy to show that a series of digits is non-random by finding a programme that is shorter than the series but will generate the series. This need not be the minimal programme for the series but just a programme that is shorter than the series.

To demonstrate that a particular digit is random it is necessary to prove that there is no small programme for generating it. Chaitin’s version of Gödel predicts that such proof of randomness cannot be found.

Godel had shown that it was not possible in mathematics to have a mechanical system of proofs without need for human judgement or insight.

Hilbert looked for a formal system with a finite list of axioms or initial assumptions and rules of inference. This is the definition of a formal system. The formal system has an algorithm for testing proofs.

Hilbert’s requirement for a proof checking algorithm allows for checking allows for checking one by one all the theorems in a particular system.

The complexity of a formal system is a measure of the amount of information that a system contains.

The formal system rests on axioms, fundamental irreducible statements which are the same as minimal programmes. If it turned out that an axiom could be expressed more compactly, that expression would become an axiom, and the first axiom would become a theorem of the new axiom. The randomness of not of numbers that are larger than the formal system cannot be proved and any series of numbers can be arbitrarily large. This is taken to show the Gödel’s incompleteness is not an isolated paradox but a widespread feature of information theory.

Godel’s theorem was based on the liar or Cretan liar paradox. Chaitin however comes to the same place via information theory. Gödel’s original proof constructed an assertion that is true but not provable within the formalisations of number theory.

Chaitin approaches from the angle of information theory and also thermodynamics and statistical mechanics. Chaitin also takes the view that modern mathematics should be approached more like physics, with mathematical truths sought in the same way as physical truths.

Gödel’s original truth is based on the paradox of the liar ‘This statement is false’ altered by Gödel to ‘This statement is unprovable.’ If the assertion is unprovable it is true and therefore the formalisation of number theory is incomplete. If it is provable it is false and this would make number theory inconsistent. Gödel’s first paper dealt with number theory, and a later paper with a wider range of formal axiomatic systems. The modern approach applies to all axiomatic systems. The more general modern fashion derives from Turing’s formalisation of the workings of a computer.

Russel’s paradox involved a barber in a small town who shaved all those and only those who did not shave themselves. The operative words being all and only. This left the barber in a paradoxical position. He could not be in either the set of those who shaved themselves or those who were shaved by him. Mathematically this means this is taken to mean that a programme would never output a specific natural number.

Algorithmic information theory
var _gaq = _gaq || []; _gaq.push(['_setAccount', 'UA-17430256-2']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })();