Gödel’s Theorem

Godel and Einstein

Kurt Gödel and Albert Einstein

Kurt Gödel[1] was a Platonist,[2] logician and mathematician who developed the intention of making a profound and lasting impact on philosophical mathematics. His next task was to think of something! Amazingly, at the age of twenty five, he achieved his goal, publishing his incompleteness theorem.

A good friend of Albert Einstein’s, Einstein once said that late in life when his own work was not amounting to much, the only reason he bothered going to his office at the Institute for Advanced Study at Princeton was for the pleasure of walking home with Gödel.

John von Neumann wrote: “Kurt Gödel’s achievement in modern logic is singular and monumental – indeed it is more than a monument, it is a landmark which will remain visible far in space and time. … The subject of logic has certainly completely changed its nature and possibilities with Gödel’s achievement.”[3] Continue reading

The Halting Problem – there is, definitively, more to thinking than computation

Alan Turing

Alan Turing

Kurt Gödel’s Incompleteness Theorem was inspired by David Hilbert’s question “Are the axioms of a formal system sufficient to derive every statement that is true in all models of the system?” Hilbert played the same role regarding Alan Turing’s proof of the halting problem. Hilbert had asked: “Is there some mechanical procedure [an algorithm] for answering all mathematical problems, belonging to some broad, but well-defined class?”[1] In German this is called Entscheidungsproblem – the decision problem.[2]

Turing found that he could answer this question by framing it in terms of a Turing machine[3] – could there be a program that could determine whether any other arbitrary computer program and input would eventually stop or just loop forever? This was called the halting problem.

“Alan Turing proved in 1936 that a general algorithm to solve the halting problem for all possible program-input pairs cannot exist.”[4]

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Apparitions of the Gods

Gods 01 Muses by Andreas Mantegna (1496)

The Muses (1496) by Andreas Mantegna (1431 – 1503)

“The dove – the rood – the loaf – the wine.”

Men know the gods because they have seen or intuited them, but not all men have seen or intuited the gods, and some men are incapable of seeing or intuiting them.  The gods, moreover, sometimes disguise themselves so as to test men, or they appear in and as omens and auguries, which the dull of mind and the wicked of heart invariably either miss entirely through their mental obtuseness or, through self-serving prejudice, blatantly misread.

I. The gods appear in and as their attributes, which again only those who have vested themselves in the proper lore and the requisite discipline can correctly interpret. Who would see the gods must enjoy a gift of pre-attunement, even before he bows under the discipline and engraves the lore in his heart that will let him see them. Such a man is called a poet.  The ancient Boeotian teller of the gods, Hesiod, whom scholars assign to the late Eighth Century or early Seventh Century BC, bears a name that means simply “The” (he or hos) “Poet” (aiodos), suggesting that the Boeotians, or at least those of them in the vicinity of Mt. Helicon, recognized his special talent and accorded him the status owing thereto.  That status may claim itself paramount because the community must communicate with the gods, just as the gods must communicate with the community, and an efficient go-between nicely serves the requirement both ways.  One misthinking modern school argues through Hesiod’s name that any particular poet is a non-existence, as though no one could write a poem, as though poems constituted themselves, authorless, and as though therefore no one really ever saw Hesiod’s gods or heard them speak.  This thesis of a literary fantasy amounts, however, merely to another kind of noetic obtuseness.  Someone wrote Hesiod’s poems, obviously, and if Hesiod were the invention of that someone then that someone nevertheless would have seen Hesiod’s gods – through his invention, as it were, and taking Hesiod’s name, but equally in a vision such that the seeing must guarantee its own authenticity and such that He remains The Poet.

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Our Dreams of the Implicate Order

On the walk from my office to the train a week ago last Monday, I reflected on the fact that I had all day been curiously alive to moments from my past. In part this was due to the fact that it was my birthday, and people from every era of my life were reaching out to wish my happiness. But other factors were at work, too. I ran into a blog post that linked to a recording of Allegri’s Miserere Mei – one of the most sublime works ever written – and vividly remembered singing it as a boy, and so enacting Heaven. A story I had told my little granddaughter the day before, about the time when I was only four, and went camping with my Dad, and woke up unable to find my way out of the mummy sleeping bag, so that I tried to stand up and get his help, in the process falling down the steps out of the open forest shelter (and almost into the fire he had started), made me chuckle again. So did the memory of her reaction: “Silly Poppy!” I began to remember lots and lots of things from long and not so long ago – some of them tagged (oops!) for later use in the confessional – and suddenly as I walked the moments all crowded in upon me at once. Not in a chaos or a hurry, but as it were quietly, softly.

It was no stampede. Rather, it was a stately pavane.

Suddenly I staggered, thunderstruck by a completely unexpected notion: what if those moments *really were* immediately present to this one? What if I could feel that moment of suffocated terror in the mummy bag as if it were still happening? Clearly, I could: all that I had to do, in order to make that happen, was simply attend to it carefully enough, and without distraction. It might take a few moments of concentration, but if I wanted to I could, I knew, bring back any moment I wanted with as much clarity and intensity as I wished.

Then – this was the strike of the thunder – I thought: “That’s what dreams are like; and it is the way things really are; for, in Eternity, and to Eternity, everything (whether actual or not) is all at once together.”

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Upstate Consolation University Celebrates First Graduating Class in New Degree Program


College Graduate in Educationally Mismatched Job

“Higher education is not about knowledge or skills,” says Upstate Consolation University Executive Deputy Chancellor of the Committee on Investor Communications Marl Flaybiter from behind the large mahogany desk in his office overlooking West Campus’s scenic Green Parking Lot; “no – higher education is about respect.”  A few years ago, on being appointed to his incumbency, Flaybiter began noticing how little respect graduating degree-holders from UCU were receiving when they entered the job market and presented their credentials to prospective employers.  While escorting potential investors around Uppchoock-on-the-Lake, the small, northerly city where his institution is located, Flaybiter observed that many of the service personnel in the local coffee bars and chain restaurants were recent UCU graduates.

Flaybiter counts off the many types of prestigious UCU-granted degrees held by these disrespectfully under-employed new alumni: “At least three of those kids – bright kids – had come out of our Social Justice and Sustainability Programs; five or six had bachelor’s or bachelorette’s degrees in Women’s Studies, and others came from Adventure Education, Puppet Arts, Safe Space Organizing, Slut-March Planning, and Critical White-Privilege Sciences.”  Flaybiter pauses to shake his head sorrowfully.  “I just couldn’t bear to see those kids – I mean, those young people – so shamefully disrespected by having to work as baristas, cashiers, waiters, and waitresses while living in their parents’ basements and going to work in their pajamas.”  As Flaybiter sees it, “The mismatch between the education and the job is, well, a tragedy, not just for the kids, and not just for the pajamas, but for the community.”

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When God is Dead, Rationalists Accept Contradictions to Fill the Void

Goedel’s Theorem is an application to mathematics of Aristotle’s thesis that thinking relies on first principles and that first principles are unprovable assumptions. This means that faith and hope are ineradicable features of human existence even in the exact sciences. The briefest summary of the implications of Goedel’s Theorem and the necessity for first principles is the notion that not everything that is true can be proven to be true.

Goedel’s Theorem states that an axiomatic system can be consistent and incomplete, inconsistent and complete, but never consistent and complete. Eternal verities can only be proven in relation to other eternal verities. Axiomatic systems exist on the rational plane of thought. Their rationally approximate and unprovable nature is due to their ultimate reliance on transcendent truths described in Plato’s realm of Forms. For instance, people contrast earthly justice with perfect justice, though the latter has never been instantiated in the physical realm. This implies some intuition of perfect justice, though no one has ever experienced such a thing.

Positivists and post-modern relativists are likely to regard each other as opposites. More than likely both will be liberals and in most cases share a contempt for religion and any notion of transcendence. As rationalists, they will also most likely reject emotional attachment to and especial preference for family, tradition, community, culture and the local physical landscape. The modern liberal instead is committed to being a citizen of the world and welcoming to all comers, no matter their basic hostility to the ethos of the host culture.

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The Illogicality of Determinism – Further Considerations

This article is now available at the Sydney Traditionalist Forum. Among other things, I argue that if physical determinism were true, then the appearance of intelligent behavior and the fact that car accidents, for instance, are the exception, not the rule, would be a mystery. Determinists typically want to banish God and consciousness – that is, our ordinary subjective experience of freely thinking, evaluating, deciding and having purposes – yet end up imbuing The Big Bang, by logical implication, with many of the properties of both God and consciousness, including omniscience, omnipotence, benevolence, purpose, intention and desire.

The link is The Illogicality of Determinism – Further Considerations.

It is a follow up to The Illogicality of Determinism.

And it is related to The Reflexive Problem in Analytic Philosophy – Illogical Logicians.

The Reflexive Problem in Analytic Philosophy: Illogical Logicians


Analytic philosophers either accept or regard as perfectly reasonable two philosophical contentions that violate logic and common sense: determinism and the denial of consciousness. Arguing for determinism implies free will and in denying the existence of consciousness the philosopher is using the very thing he says does not exist. In this article published by the Sydney Traditionalist Forum, I argue that this is a result of certain interesting psychological and emotional deficits, a commitment to materialism and atheism, the “philosophy as the handmaiden of science” notion and the very methods and approach used by analytic philosophers. These methods include conceptual analysis and arguments considered as words on a page or monitor – looking at internal coherence and validity – but overlooking the reflexive implications for the person doing the analysis.

This results in risible performative contradictions; a notion absent from the logical toolbox of analytic philosophers as far as I know.

The Reflexive Problem In Analytic Philosophy: Illogical Logicians

The Illogicality of Determinism

Physical determinism is the notion that all events, including thoughts and actions, are the result of cause and effect. Each effect is the result of a prior cause. Each effect is also the cause of some new effect, creating an endless causal chain.

C→E/C→E/C …


From this point of view, every event is “necessary.” Given the cause, the effect must occur. Exactly what “must” and “necessary” mean here philosophers have found it difficult to say.

Every event is thought to be unavoidable in some way and a “necessary” consequence of preceding events.

If the Big Bang is taken as the first physical cause, then all subsequent events can be regarded as the result of that first cause, when time began. Thus, according to determinism, since the beginning of time, everyone’s thoughts and actions have been pre-determined and unavoidable. No deviation from this predestination is possible on this view. The “events” referred to would seem to include thoughts, on the assumption that brains generate consciousness.


However, if physical determinism is true then the person arguing for it has no choice as to whether he believes in physical determinism or not, nor whether he argues for determinism or not. He is in the grip of physical forces beyond his control. It is as though someone pushed the cosmic “play” button and the arguer starts arguing for something he never had any choice but to believe and to argue for. He is the victim of circumstance. Why should any attention be paid to such a victim – to such a mindless and compulsive machine – to such an idiot? He has an unfortunately not so rare form of Tourette’s syndrome and should be pitied.

It is a farce. The farce gets worse when the person being blasted with this nonsense is considered. According to determinism, the interlocutor too has no choice whether he listens to the sounds the other madman makes, for he too is mad. He listens or does not listen compulsively. He agrees or does not agree with the determinist’s argument through no free will of his own. While the arguer is a cosmic tape machine playing its predetermined recording, the interlocutor is affected by blind physical forces himself. The outcome of this travesty masquerading as “reasoning” has been predetermined since the beginning of time and the exercise is pointless.

The image of two tape machines alone in a room together playing their scripted comments and responses comes to mind. Nobody and nothing is really asserting anything nor really responding. Determinism is consciousness denying. No meaningful “thinking” is occurring if the determinist is right.

Determinism has reflexive implications – it applies to the person arguing for determinism. All determinists that I have met in practice imagine that they can freely decide when and if they will argue for determinism. They imagine that it is possible to step in and out of determinism like it is a river. But determinism does not leave room for an “inside” and an “outside;” that’s the whole point. If it were possible to freely choose when to do something and when not to do something determinism would be false.

Some determinists argue that computers are deterministic machines that argue and can produce valid arguments and that proves that meaningful argument and physical determinism are compatible. This is supposed to support the notion that there is no problem imagining that arguing human beings are deterministic machines. The notion of deterministic computers is meant to provide evidence that humans might be deterministic.

However, computers are the product of human minds. They are explicitly programmed to do the things programmers want them to do. They argue as the programmer determine. As John Searle’s Chinese Room argument demonstrates, computers understand nothing – neither the input nor their own output. Computers are the physical medium by which human beings communicate with each other or derive answers to computational questions or do the things we wish. They are not the product of blind deterministic physical forces. Humans are governing what they do. If computers seem intelligent, it is because humans are.

If the determinist claims that computers are indeed the product of deterministic forces because human thought is the product of deterministic forces, then the determinist has simply assumed humans are not free in order to prove that humans are not free! Instead of using computers to prove that humans are determined, the determinist assumes humans are determined to prove that computers are determined to prove that humans are determined.

The purpose of a philosophical argument is supposed to be to provide evidence for controversial assertions. It is logically possible that determinism is true, but it is not logically possible to persuade someone that determinism is true because determinism precludes the possibility of logic and genuine persuasion in the context of controversial assertions.

Any argument that expresses skepticism about consciousness or the ability to think rationally is problematic and generates self-refuting paradoxes since the arguer is using the very thing he is arguing is untrustworthy to arrive at the conclusion that this thing is untrustworthy.

Reasoning and mindless physical forces are incompatible. If the phrase “mindless physical forces” seems question-begging, and mindful physical forces are postulated instead, then qualities of mind are being attributed to physical forces. This results in the situation where mind is thought to be affecting matter  affecting mind with matter as a simple intermediary between two aspects of mind – a cosmic mind (a giant thinking nature) and a parochial mind (human minds).

Do brains generate minds?

If brains do not generate minds, then physical determinism does not apply to thoughts.


The brains in the diagram represent the same brain in different states – A, B and C. Under determinism, Brain State “A” causes Brain State “B” which in turn causes Brain State “C.” As a physical mechanism, the brain is following the laws of chemistry and physics. It can be imagined that each brain state is giving rise to a discrete thought. Brain State “A” (BSA) gives rise to the thought that “p → q” (If p, then q). BSB give rise to the thought that “p.” BSC gives rise to the thought “q.”

In the diagram above, the brains and the black arrows represent physical determinism; one brain state giving rise to another. The blue arrows point to thoughts produced by the brain states, namely the modus ponens argument, “p→q, p, ∴ q.”
If p, then q,
Therefore q.
The diagram represents brain states physically leading to each other. Also shown is the level of abstract, rational thought, with each thought related to the next by logic; not physics or chemistry.

From a mental level perspective, it would seem that these thoughts generated by a physically determined brain, are effectively random. Each thought is not the product of rational reflection. It is the inexorable product of physical processes, each state deterministically producing the next state. If the brain generates the mind then we are driven to conclude that the mental events which seem conceptually related to other mental events are really random from a conceptual point of view. If the thoughts seem conceptually coherent, and importantly related to each other in the form of an argument, this is just a coincidence.


An analogy could be Powerball machines. In this form of lottery, a machine blows ping pong balls with numbers on them. After a while, the ping pong balls end up in a structure at the bottom of the machine. If the numbers happened to be the first few digits of pi, this would be by pure chance. Mindless physical processes are acting on the balls; any meaning is a coincidence. It could be imagined that on the balls are written letters instead of numbers and that words might get accidentally spelled. Or variables and logical operators such that p → q, p, ∴ q.  Again, such results would be comparable to faces in clouds and the like.

If the physical processes are not mindless then the balls would not be being blown by forces governed by deterministic laws of nature but would be being selected on the basis of meaning. The apparent autonomy of physical processes with their never-ending chains of cause and effect would be an illusion.

With regard to the Powerball machine, the game would be rigged. Instead of purely physical processes determining what happens to the balls, reason and logic would be guiding the balls to the desired logical outcome. Mind would have proved to be more fundamental than matter. Of interest might be Thomas Nagel’s Mind and Cosmos that asserts that consciousness and material reality are fundamental aspects of nature and always have been.

Philosophers refer to bottom up causation: the brain or body affecting the mind such as the effects of caffeine or sleep deprivation on thoughts and feelings. Top down causation is when the mind affects the brain and body. Someone says something that upsets another person and that person goes red in the face and his blood pressure rises. A mind selecting brain states to produce mental events would be similar to top down causation.

Causes versus reasons

The premises and conclusions of a valid argument are connected, but their connection is not physical. Or if they are physically connected by being written on the same piece of paper, or blackboard, or hard drive, if this can be called “connected,” then this physical connection is irrelevant. It is the way the premises and conclusion are connected logically that matters.

Causally determined physical processes are incompatible with arguments which require true and relevant reasons conceptually and logically related to the conclusion. If he is good he will get a bike for Christmas. His parents agree that he has been good. Therefore, he will get a bike for Christmas. The relationship between the premises and conclusions is conceptual and logical.


Physical things work by causes. Arguments work by reasons. Causes are not reasons. If it is said that driving to Canada is a good idea if someone has a disease that requires expensive medications, and thanks to the Canadian health system, medicines are cheaper in Canada, and that the internet should not be used for these purchases because companies in the Caribbean have been known to pretend to be Canadian, and you go to Canada on the basis of this reasoning, you have rationally been persuaded to go to Canada on the basis of reasons.


If someone is locked in an entirely dark room with no company for five straight days, when the door is opened, that person is likely to be craving stimulation so badly that he will be very excited about anything someone says and to be highly amenable to suggestion. If someone then says “Go to Canada” and that person goes, the first person has caused the second to go. He has been brainwashed to go.


Self-refuting paradoxes

If it is claimed that no, reasons are just fancy jumped up causes and that when someone thinks he is being persuaded, he is really just being caused by physical processes, then skepticism is being expressed about the reality of rational persuasion.

But the person who expresses skepticism about rational persuasion is attempting to rationally persuade. He wants to be an exception to his own rule, a telltale sign of being wrong, because he is contradicting himself. Francis Crick says “‘You,’ your joys and your sorrows, your memories and your ambitions, your sense of personal identity and free will, are in fact no more than the behavior of a vast assembly of nerve cells and their associated molecules.” (Science Set Free, Rupert Sheldrake, p. 110, 2012) The trouble is that if all mental activity is “no more than the behavior of a vast assembly of nerve cells and their associated molecules,” then that thought itself is “no more than the behavior of a vast assembly of nerve cells and their associated molecules” and not to be taken seriously; “no more than” being the key phrase here.

Similarly, a professor may argue that gender is nothing more than a social construct. Yet her ability to challenge this supposed social construct means that the professor gets to occupy a rational space outside the social construct. She has her own personal opinion differing from the usual social construct; an opinion that is thus not merely derived from the social construct. The professor is effectively calling other people sheep, merely following the dictates of culture, while she gets to have her own self-derived opinion. She is a wondrous exception to her own rule.

“Everything is what it is and not another thing,” said Bishop Butler. You, your joys and sorrows, memories and ambitions are just what they are and not something else. Nerve cells and molecules may be involved somewhere, but there is no reason at all for reducing them to these things.

Neither can it even be rationally suspected that humans are really mechanistic robots that are wholly the product of unthinking physical processes, because for this suspicion to be rational, it must itself be the product of true and relevant reasons. If there is no rational ground for the suspicion, then, rationally speaking, suspicion should stop.

If it is possible to wonder if humans are deterministic robots whose thoughts are ruled by causally determined physical processes, and there is any rational basis at all for this wonder, then rationality does indeed exist and people are not robots.

So, free will exists because rationality exists. In order to be rational, a person must be free to consider the merits of an argument based on an evaluation of the truth and relevance of the premises of that argument. He must be free of external interference in his evaluation. If something physical is forcing him to a particular conclusion due to an unbroken chain of physical causal processes stretching back to the beginning of time, then rationality per se is impossible. He is unable to make an evaluation on the basis of conceptual and logical relationships, but instead must think whatever the physical unthinking deterministic processes makes him think.

Moral responsibility and love

In addition to the self-defeating nature of arguing for determinism, there is a practical objection. This objection is that determinists are only classroom determinists. Their behavior outside the classroom indicates that they believe in free will. In order for determinism to be true, moral responsibility must be an illusion and meaningful love must be an illusion. Determinists continue to hold other people and themselves morally responsible for their actions, and hopefully, they manage to love other people.

Courts of law recognize that actions performed under duress, in which someone has no choice, do not make him morally or legally culpable. If someone is compelled to do something by being threatened with something dire and he has a reasonable expectation that the threat will be carried out, then he is not held responsible for his actions. And so it should be.


Regarding love, love is not meaningfully love if it is not freely given. If it is discovered that every time a beloved attempts to leave the lover he or she is tasered, or is taken into an interrogation cell and brainwashed, then this would change the lover’s feelings about his beloved. He would know that she is not freely choosing to love him or be with him but is acting under compulsion.

Love is not an entirely rational process of course but it does involve respect and admiration and these are based on a more or less rational assessment of the other person. If a girlfriend, husband, etc., thinks their beloved is an idiot or morally corrupt, nasty and horrible, boring, humorless and ugly, then they do not love the beloved.

One of the more amazing life experiences is when someone thought to be really cool and beautiful responds in kind. This means going from having a crush, an unreciprocated affection, to the beginnings of love. The beloved does not have to like the lover and it is flattering that someone as impressive as that thinks he is attractive and nice too – someone worth getting to know better. If it were possible to just flip a switch on the back of someone’s head and she would gaze at the lover adoringly, this would make that person’s affection worthless and pathetic. It could be hoped that no one ever gets so desperate for even simulated affection that he would be willing to do this.

Love is a gift. No one can demand to be loved. If a gun is pointed as someone’s head with the command “Love me!” it is not possible to actually comply. The love could only be pretended. In Christian theology, even God cannot compel humans to love Him. Hence, there is the notion of God the lure, or Jesus as making his followers fishers of men. Likewise, the one ability Bruce does not have in Bruce Almighty even though he is God of Buffalo for a week, is the ability to make anyone love him. The writers consulted a Jewish theologian to get their theology right. If God could make you love him, he would be no better than a man with a date rape drug. If a person would not be satisfied with “love” in those circumstances, neither would God.


Love is a chemical? Partly

There is an amusing movie about a man who thinks love is a chemical. Explaining that to a date would likely end the relationship. A version of the first encounter could be – [Robotic voice] “Dopamine levels, satisfactory, serotonin plateaued, oxytocin slightly raised, scheduling second meeting – waiting, waiting, Thursday is free – waiting for reply. You’re a dick! Affirmative. Negative response recorded. Is this decision final? Press 1 for yes, 2 for no.”


There is an element of truth in the love is a chemical idea. Love is not a chemical, but feelings of love and affection may be related to hormones in the “It” quadrant. Oxytocin seems to be associated with bonding and is released when the skin is stroked – but someone has had to have decided they actually like another person before he is going to let you stroke his skin. If someone tries to start with skin stroking prematurely, no oxytocin.

Whether someone think someone else is boring, stupid, humorless, ugly and a jerk, or beautiful, interesting, funny, smart and nice is likely to be influenced by his cultural context and class, the “We” quadrant. Women are likely to look at a man’s job, social status and income, the “Its” quadrant if they are thinking about whether to marry someone or not. Love or not love occurs in a context. It is the “I” quadrant that is being examined here – the idiosyncratic and freely chosen response to another person.

Love and transcendental arguments

So the claim is that there is something wrong with a person if he would be happy with taser or determinism compelled love. If it is thought that love as it is being defined here exists, then free will exists. Kant called this a transcendental argument. This means starting with the phenomenon and then working backwards to the way the universe must be if this phenomenon exists. In other words, one prioritizes evidence/data over theory. Physical determinism is a theory based on a commitment to the metaphysical notion called materialism. If someone is ontologically committed to determinism, and love as a datum and an experience seems to exist, love is just an illusion. Since materialism remains an unproven assumption the determinist has chosen to rule out of existence one of life’s most important experiences on the basis of an unproven hypothesis. Where data and theory conflict, logically it is possible to reject either one. In the transcendental argument, the data is chosen and the theory of materialism is rejected.

Consciousness vs determinism

Lastly, if brains were self-contained physical mechanisms with no input from conscious minds, i.e., top down causation, then the brain would follow its own predetermined chain of causation. But this would mean that the brain would not be subject to adjustment by conscious evaluations of what is going on around someone in the environment. Without consciousness and top down causation, a person would not be aware of his environment and could not adjust his behavior as the demands of the environment changed. These demands are unpredictable. The workings of the brain, if the organism is to survive, must be constantly adjusted to the environmental factors perceived and evaluated by the conscious mind.

From the point of view of the organism, its environment is unpredictable. Even if the universe were a large deterministic machine – ignoring the fact that machines have designers and in-built purposes – the organism still has no idea what events it will encounter. An event that cannot be predicted is effectively random. An appropriate response cannot be preprogrammed to an unforeseen event – and with the complexity of human social interactions there are many such events. There can be no rule for an unanticipated circumstance. The ability to improvise is required and for this improvisation to be successful someone’s reaction must be perfectly suited to this new circumstance and for that, real live consciousness free from deterministic rules must exist.