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General Articles 3


General Articles 3

1.) Gödel &Turing - Chaitin

2.) The knowledge argument and the inadequacy of scientific knowledge

3.) The Myths We Live By - Mary Midgley - Criticises mainstream scientific thinking relative to the mind-body problem



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
Much of Chaitin’s discussion centres on the concept of algorithmic information theory. In contrast to Shannon and Wiener’s information theory this focuses on individual objects rather than ensembles or distributions. He asks what is the size in bits of the smallest programme that it takes to calculate an individual object.

Between to objects X and Y there may be mutual information, which measures the commonality of the two objects. Mutual information is regarded as a fundamental concept. Knowing X helps one calculate Y if there is some commonality between the two and reduces the amount of information needed.

Chaitin goes on to define algorithmic randomness and algorithmic independence. The concepts are seen as related. Randomness in one object implies the lack of any common feature that can relate it to another object. In a set of n bit strings only a few have less than n bits of information and these are the minority that have regular pattern.

Chaitin views a formal axiomatic system as a computer programme for listing the set of theorems and measuring its size in bits. The amount of space needed to specify a proof checking algorithm and how to apply it to all possible proofs.

Given an N bit axiom system, it is possible to toss a coin (or other pseudo-random generation) N+1 times. The resulting sequence will almost certainly be random but it will not be possible to proof this within the formal system. Within an N bit axiomatic system it is not possible to proof that a series has more than n bits of information, although all but a small minority series have this.

Chaitin also defines omega as a real number between zero and one which expresses the halting probability of a programme in which each bit depends on the toss of a coin. The base 2 expression of this is seen to be random, which is taken to correspond to the unsolvability of the halting problem. It takes n bits of axioms to prove what the first n bits of omega are and these bits appear to be random.

Random series cannot be compressed and cannot be compressed into axioms which are shorter than themselves. Given that quantum mechanics represents a whole series of coin tosses or random events this indicates the limitation of axiomatic systems.

In practise using omega is too long in any case. Where assertions are not algorithmically independent, that is they have some commonality they can be tackled jointly with less bits than tackling them separately.

Turing’s halting problem is really a more accessible version of the Godel idea. Godel is equivalent to the assertion that there is no general method for deciding whether a computer programme will halt. Chaitin claims to have demonstrated.

The probability that a completely random programme will halt, which he has named omega is a real number between 0 and 1. Omega = 0, would mean that a random programme would never halt and Omega = 1 would mean that random programmes were certain to halt. Neither of these extremes is possible on a general computer so the probability is somewhere between 0 an1. Omega cannot be compressed into a programme shorter than itself. It contains normality in that digits appear with equal frequency.

Chaitin considers how Gödel, Turing and his own efforts have effected mathematics. He says that for the most part the profession has shrugged off the implications, as not applying to normal mathematics, but Chaitin suggests that it may apply more often than they think. His algorithmic information theory suggests randomness and incompleteness may be more pervasive than thought. He suggests that more attention should be given to forming new axioms, which might make it easier to deal with long standing unsolved mathematical problems. This may be due to the incompleteness of axioms.

What Gödel showed was the following. Suppose a formal axiomatic number system dealing with number theory, 1,2,3 etc and addition and multiplication, that’s consistent. If you assume it’s consistent then it’s incomplete. If you assume that it tells the truth it won’t tell the whole truth. The proof involves self reference. Godel was able to construct an assertion that says of itself that it’s unprovable.

Chaitin feels that the computer was implicit in Gödels work. Turing went on to work out the theoretical basis for computing. The Turing machine could carry out any calculation that a computer could perform. He then asked what this machine couldn’t do. The answer to that was the halting problem, of deciding whether the Turing machine or computer would eventually halt. It turns out to be a problem of not knowing when to give up. This means that no formal axiomatic system can solve the problem. It doesn’t work to run all possible programmes because there is a programme that halts if and only if it doesn’t halt, the same as Russel’s barber than shaves all and only the people who do not shave themselves.

Chaitin has considered that there may be a deeper reason for incompleteness. The set of ideas relative to this is called algorithmic information theory. In fact this just means looking at the size of computing programmes. It’s a measure of complexity, of computational complexity. Programme size and complexity is seen to connect with the fundamental stuff of physics. Physics contains the notion of entropy or the amount of disorder in a system, which is in its turn connected to the arrow of time. The size of a computer programme is very similar to the concept of the disorder or entropy of a system. The idea of programme size is also to the Occam razor idea of preferring the simpler system. A theory is seen as a computer programme for predicting observations and a concise computer programme is seen as the best programme or theory. Maximum entropy like randomness is something that cannot be compressed at all.

The trouble with the idea of the smallest computer programme being the best is that one cannot be sure that one has the smallest computer programme. This is beyond the power of mathematical reasoning. It is not possible to calculate the programme size complexity. With N bits of axiom it is impossible to calculate the programme size complexity of anything that has more than n bits. Mathematical truth is seen as an infinite amount of information, while systems of axioms contain only a finite amount of information.





The knowledge argument & the inadequacy of scientific knowledge

Elizabeth Schier

Dept. of Philosophy, Macquarie University, Australia

Journal of Consciousness Studies, 15, No. 1, 2008, pp. 39+62

This paper returns to the sad thought experiment of Mary, the colour scientist, who knows the full science of colour, but has been confined to a black and white environment, and the resulting question of whether she would discover anything new when she was eventually released into a coloured environment. The long debate on this requires some patience for the lay person to whom the difference between the study of how photons oscillate and the experience of red seems obvious. Other examples that might be more forceful are the medical student that knew everything about amputation with anaesthetic, and later had his leg amputated without an anaesthetic. Would he/she learn anything new. Or another student in the same medical school who knew the whole science of sex but had never had an orgasm. Would he/she learn anything new when they had an orgasm? Given the perversity that marks so much of consciousness studies, it should come as no surprise that the dominant view in conventional consciousness studies seems to be that the unlucky Mary would learn nothing new on being released from her black and white prison.

In practise, the only convincing arguments for this stance are ones that involve a sleight of hand, such as stimulating parts of Mary’s cortex to produce the colour experience, while she remains within the black and white prison. But this looks like a form of cheating that allows her to effectively escape from her black and white world, and does nothing to resolve the question of the difference, or lack of it, between scientific knowledge and direct experience.

It looks to be that the underlying reason for this stance by many in consciousness studies is metaphysical. It seems to be assumed that if Mary does experience something new when she comes out of her prison, it means there is some non-physical thing outside of science that produces this effect. But this assumption itself rests on the dualist assumption that consciousness is non-physical, which nearly everyone in consciousness studies rejects. If consciousness were to be a physical thing or process, then there is no reason in terms of physics why it should not have an influence. Much of consciousness studies plays fast and loose with the dualist concept of the non-physical, rejecting it as impossible, when it wants to refute the possibility of any kind of spiritual influence, yet arguing that consciousness cannot have any causal influence because it is non-physical. Logically it follows that if dualism is false, there is no such thing as the non-physical, and therefore consciousness must be physical, and physical processes all have influences on the surrounding universe. As Antonio Damasio pointed out in his book ‘Descartes Error’ much of the medical and scientific community is still mired in the Cartesian assumptions that it claims to reject.

Schier indicates two main things that need to be shown in order to demonstrate that Mary does learn something new when she comes out of the black and white prison, firstly that there is a difference between scientific knowledge and direct experience, and secondly that the experience is physical even though it can’t be demonstrated by scientific knowledge. Unfortunately, she chooses to concentrate on the first question.

Schier discusses the role of visual imagery in scientific conceptions, for instance the visualisation of the benzene ring as a snake swallowing its tail. She speculates that lack of such imagery, which itself derives from experience, not necessarily visual experience, of the external world may be necessary for the full development of scientific concepts.

However,the main theme of Schier’s paper is that scientific representations are fragmented whereas perceptions are a whole. She takes various examples to show that it would be an enormous task to provide the information that perception provides from the piece by piece scientific examination of a simple collection of objects. Schier speculates that the visual cortex has resources that allow it to describe colour in a way that cannot be achieved by external scientific analysis.





The Myths We Live By

Mary Midgley

Routledge (Taylor & Francis Group) (2003)  ISBN  0-415-30906-9

This book is interesting for its criticism of modern scientific thinking, a criticism which has come more centre stage since the book was published. Early on in her book, Midgley criticises the dualism of treating mind and body as separate entities. The consequence of this is claimed to be a tendency to favour one over the other, with modern scientific thinkers feeling compelled to favour the body and ignore the mind. She claims that our notion of scientific rationality is based on 17th century physics. She notes that while the methods of this type of physics are no longer central to modern physics, which has at its heart randomness or acausality, biologists, sociobiologists and social scientists have often not yet got this message, and continue with mechanism-based thought patterns, in the believe that they are being true to physics. This in turn has led to a concentration on the microbiological at the expense of whole organisms. Midgley, as a philosopher, also takes a side-swipe at scientists who will have nothing to do with philosophy, but are in fact unconsciously enslaved by 17th century philosophy.

Midgley discusses the position of Descartes. She claims that he did not start his search for absolute certainty with a completely open mind, but that he was already looking to the mathematical physics of Galileo as a guide. He thought that this type of logical clarity was the only safe guide, and believed this method should be extended to all subjects. Knowledge was to be infallible and tightly organised into a unity in the manner of mathematics.

Midgley criticises this as an impractical aim. She feels that mind could not be included in such a scheme, and that this underlies the mind-body problem. The modern version of this problem seems to involve a single system called ‘science’ on the body side and a tangle of subjective experience on the mind side. She suggests that in practise the distinction is not as clear cut as modern science and philosophy try to make it. Thus, in practical life, a dentist might bring together the objective material of professional knowledge with the subjective reports of the patient. Midgley goes on to argue that we are in need of a scientific pluralism that recognises many independent forms and sources of knowledge.

She discusses the atomistic approach of much of science, the reductive approach of breaking things down into the smallest possible particles, and argues that there are limitations to this. A botanist who is asked to identify a plant will not be able to provide a satisfactory answer by breaking it down into sub-atomic particles. Even a reductionist orientated scientist such as Francis Crick cautions that what might seem hopelessly complex to physicists may have been the simplest thing for evolution, because it built on something that was already there. Midgley argues that there are many forms of reduction, and that the form chosen by researchers is likely to be determined by their intellectual outlook or scientific world view.
Midgley says that because Descartes thought that physical particles operated in much the same way as machines, he reasons that anything made out of physical particles, including bodies, had to do the same. Descartes famously excluded the mind or soul from being part of a machine, but other philosophers and scientists were quick to do way with this distinction. Thomas Hobbes was one of the earliest in this respect. Physical explanations were deemed to be real, while subjective experience was only an ‘appearance’. Hobbes did not allow that there could be any objective facts about subjectivity, that an ’appearance’ could be a fact in the sense that our emotions and sensations are a fact to us.

Midgley argues that a view that bases everything on the action of particles will have problems in dealing with history or accounts of everyday life. A sentence such as ‘George was allowed home from prison at last on Sunday’ is difficult to interpret. This is a series of social relations that would have to vanish in a completely reductive physical explanation. This physical explanation does not begin to convey the meaning of what is said in the sentence. For instance, the importance and implications of George’s coming home from prison has nothing much to do with the distance or route from the prison, or the type of transport used, and George himself is not recognised by physics, which does not have individuals.

Epiphenomenalism
Midgley goes on to examine the modern idea of epiphenomenalism, which claims that consciousness is a by-product of mental processes, and that it can have no influence on the working of the brain or mind. Midgley highlights the most important problem with this concept, which is the question as to why evolution should have selected for a feature that did not have any function.

The really surprising thing is that while epiphenomenalism is widely touted in consciousness studies, the evolution problem in epiphenomenalism is seldom discussed, even in books that devote a chapter to the relationship between consciousness and evolution. Midgley sees the whole idea as being a contrivance aimed at getting rid of consciousness, because it is impossible to connect consciousness and the body in the Descartes derived world picture.

The Church
Midgley further suggests that there is a social and historical context to the tendency to try and downplay the mind, and this was a wish on the part of some of the earlier thinkers to break away from the power of the Church. This has led to a general attitude that anything that can portray itself as downplaying the mind is classed as more scientific, and this is seen as accounting for the dominance of behaviourism and the virtual absence of consciousness studies during most of the 20th century. Midgley particularly criticises behaviourism for not defining why it was ‘scientific’ to ignore the subjective aspect. She also remarks that its is unfortunate that the dogma of preferring the outside view has outlasted behaviourism as such. Midgley suggests that there is an essential conflict between atheism and the idea of mind/body separation. If there are no external spiritual forces then the subjective experience of something spiritual requires a physical explanation.

Midgley says that because Descartes thought that physical particles operated in much the same way as machines, he reasons that anything made out of physical particles, including bodies, had to do the same. Descartes famously excluded the mind or soul from being part of a machine, but other philosophers and scientists were quick to do way with this distinction. Thomas Hobbes was one of the earliest in this respect. Physical explanations were deemed to be real, while subjective experience was only an ‘appearance’. Hobbes did not allow that there could be any objective facts about subjectivity, that an ’appearance’ could be a fact in the sense that our emotions and sensations are a fact to us.

Midgley argues that a view that bases everything on the action of particles will have problems in dealing with history or accounts of everyday life. A sentence such as ‘George was allowed home from prison at last on Sunday’ is difficult to interpret. This is a series of social relations that would have to vanish in a completely reductive physical explanation. This physical explanation does not begin to convey the meaning of what is said in the sentence. For instance, the importance and implications of George’s coming home from prison has nothing much to do with the distance or route from the prison, or the type of transport used, and George himself is not recognised by physics, which does not have individuals.

Midgley goes on to examine the modern idea of epiphenomenalism, which claims that consciousness is a by-product of mental processes, and that it can have no influence on the working of the brain or mind. Midgley highlights the most important problem with this concept, which is the question as to why evolution should have selected for a feature that did not have any function.

The really surprising thing is that while epiphenomenalism is widely touted in consciousness studies, the evolution problem in epiphenomenalism is seldom discussed, even in books that devote a chapter to the relationship between consciousness and evolution. Midgley sees the whole idea as being a contrivance aimed at getting rid of consciousness, because it is impossible to connect consciousness and the body in the Descartes derived world picture.

Midgley further suggests that there is a social and historical context to the tendency to try and downplay the mind, and this was a wish on the part of some of the earlier thinkers to break away from the power of the Church. This has led to a general attitude that anything that can portray itself as downplaying the mind is classed as more scientific, and this is seen as accounting for the dominance of behaviourism and the virtual absence of consciousness studies during most of the 20th century. Midgley particularly criticises behaviourism for not defining why it was ‘scientific’ to ignore the subjective aspect. She also remarks that its is unfortunate that the dogma of preferring the outside view has outlasted behaviourism as such. Midgley suggests that there is an essential conflict between atheism and the idea of mind/body separation. If there are no external spiritual forces then the subjective experience of something spiritual requires a physical explanation.

Motives
Midgley goes on to point out another contradiction apparent in psychological reductionism. On the one hand, the mind is supposed to be reduced to just the movement of particles, or mechanics. At the same time, another and contradictory idea is prevalent. Subjects’ apparent or claimed motives are reduced to underlying and usually less worthy motives. However, the mechanical theory claims to debunk the whole idea of internally sustained motives, so on the basis of that theory, the unworthy underlying motives are no more plausible as being causal, than the apparent and worthy motives. Despite this apparent contradiction, the two methods are often combined in reductionist approaches. Midgley accepts that underlying motives are often there, but criticises a tendency to apply the idea indiscriminately out of the pleasure of showing up other people, and extending a guiding theory into other areas.

In this book, and in an earlier book, ‘Evolution as a Religion’ (1985), Midgley criticises the deceptive use of language in sociobiology. She is particularly critical of what she sees as the misuse of the word ‘selfish’ as in ‘selfish gene’. She says that this fails to distinguish between the biological sense of the word referring to the deterministic behaviour of genes in the human body, and the lay use of the word, which implies a moral/social fault in the person exhibiting selfish behaviour. There is an implied discovery of underlying motives in humans, while the general reader may not notice that the theory also insists that personal and subjective motives, underlying or otherwise, are not the basis of behaviour in the first place.

Further to this, Midgley looks at the whole question of actions and motives. The common sense or folk psychology approach is that we act in response to conscious thinking and to conscious purposes or motives. However, the reductionist approach is that this is an illusion or mistake, and that we act in response to some physical process that we are unaware of. The core idea is that humans are never active agents, but only passive responders to physical processes.

She examines in particular Dawkin’s notion that we are robots or survival machines programmed to preserve genes. She says that Dawkins’s language implies that while the human is a robot, the gene is a real motivated agent. She argues that if humans are not accepted as being agents, then the concept of agency should vanish from the description, whereas with Dawkins it may have been transferred to the genes. Similarly, she criticises Colin Blackemore for denying human agency, and then transferring the agency to the brain. In his scheme humans are somehow a separate passive entity driven by an agent brain. Blackemore refers to a causal chain running back to the origin of life and to atoms, but Midgley says this should really be a network penetrating every aspect of life. Midgley thinks that the motive behind these ideas is the idea that causal reasoning should not embrace the idea of purpose.

Relative this, she reminds us that Descartes ideas worked because they retained the soul to take care of the mind/consciousness. When the soul was removed from the model, it became difficult to make the model work except by the implausible shifts that permeate much of modern biology and consciousness studies. David Bohm remarks on the same problem in his discussion of consciousness, remarking that few people noticed that the removal of the soul from Descartes model had left a gap that was difficult to fill. Midgley thinks that modern reductionists try to eliminate conscious and purposive action by the human agent, because it implies to them a Cartesian and supernatural soul.

Midgley, by contrast, wants to bring conscious thought into the physical world as a normal causal factor in the behaviour of the human species. Subjective experiences are suggested to be as real as stones. The evolution of conscious thought as a property of humans is explained by its usefulness in producing well judged actions. This does not propose a soul that is cut off from material influences, but a consciousness that is influenced by the rest of the environment.

Scientific & Unscientific
Midgley questions why interest in subjective states is often considered to be unscientific. She thinks there is a confusion in that many think that the study of subjectivity is itself subjective. This is likened to thinking that a study of folly is itself foolish, or that a study of evil is evil, or in Dr. Johnson’s terms that fat oxen have to be driven by fat men.

A more sophisticated problem is the difficulty by definition of knowing anything about other people’s subjective states. However, she suggests that this research is not as hopeless as sometimes suggested, because we are both aware of our own subjective states, and a sufficiently good judge of other people’s states, for this skill to be adaptively useful. At the same time, there is the dangerous notion, as in behaviourism, that a research method that only looks at the objective evidence somehow comes to be more scientific, despite the fact that it is suppressing part of the available evidence.

Midgley criticises much of sociobiology for promoting a false form of Darwinism, derived from Herbert Spencer, who adapted Darwin’s ideas to promote 19th century American capitalism. This stresses the role of competition and suggests that social feeling are an illusion/hypocrisy. Midgley objects that Darwin himself, who regarded ‘social instincts’ as an important factor, and the evidence of the natural world, both give a much bigger part to cooperation or at least interdependence even for non-humans. The biologist, J.B.S. Haldane, demonstrated how self-sacrifice or risk taking for the benefit of others could be adaptive, if it benefited individuals who shared one’s genes, for instance relatives in the same group of hunter-gatherers. At the same time, Midgley feels evolution applies to humans as much as to non-humans whereas many social scientists have erected a species barrier against evolution, because of the unacceptable implications of the Herbert Spencer version of evolution.

Reference:-

Midgley, M.  -  Evolution as a Religion  -  Routledge (1985)  ISBN  0-415-27833-3 (pbk)













 
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