Alan G. Carter

9th May 1999

This paper presents a deep logical error in most people's thinking, that is introduced right at the beginning of thought, and then stays in the thinking until the conclusions are reached. The conclusions are then wrong, but when they are examined there doesn't seem to be anything wrong with them. Nor does there seem to be anything wrong with the reasoning - as people check back, each step seems to be correct. This is because the error was already in place before reasoning began. Even worse, the conclusions people reach in one area seem to match the conclusions they reach in other areas. This is because the error has a particular character wherever it appears, so one conclusion that contains the error has a relationship with another that also contains the error, which makes them match up.

The problem distorts deductive reasoning. It does not distort inductive reasoning, but it makes inductive reasoning appear inadmissible to many people who have done some deductive reasoning with the Ghost Not and turned their minds effectively inside out.

The effect of this has been to create the "Two Half Cultures" (not "Two Cultures" as C. P. Snow claimed - he was seeing double), as well as the single Culture we should have. The three groups are usually called:

Artists. They use intense inductive reasoning (intuition) to spot implicit patterns and express them. They shun deductive reasoning and claim that it is in itself flawed. For some this leads to a paradox of trying to achieve clarity without being clear, which the best avoid by achieving expressions that are deeper than words. Since their social interactions are Ghost Not polluted, they learn to maintain a buffer zone around their art, which they simply refuse to allow any deductive discussion of. This impoverishes society, since there are a great many truths that can be understood deductively but cannot be reached deductively, as Kurt Godel proved in mathematics.

Scientists. They use deductive reasoning to analyse explicit patterns and express them. They shun inductive reasoning because they have the Ghost Not inspired delusion that phenomena do not exist unless they themselves know of a viable causal mechanism at that moment. The best of them avoid this problem by achieving paradigm shifts - recastings of the perceived universe that are deeper than words. Since most scientists believe that is it their primary duty to enforce the orthodox paradigm rather than discover the truth, achieving a paradigm shift usually involves waiting for a generation of scientists to die, as Thomas Kuhn explained.

Alchemists. They use deductive reasoning to illuminate the harmonies they perceive inductively. They use their logic to track what is there rather than what they have admitted to their inner simulations. By seeing what is there they have no conflict between deductive and inductive perceptions, and the correct single mode of perception emerges. The results of their contemplations are immediately encoded as modifications to their own perceptions, in a way that is deeper than words. They come to see on all the levels and into all the worlds. They never forget. They freely mix the tools and language of artists and scientists in a way that the half cultures insist is impossible. As they do so, they realise that the single way of seeing is a reality that "art" and "science" (as these words are usually understood) are mere shadows of. Isaac Newton was an alchemist in the true meaning of the word, and all the great mystical teachers prize rational thinking.

This is an alchemical paper. It starts by borrowing from van Gogh to describe the core idea of seeing like an artist that it is going to explore. Scientists should not be worried about this - where do axioms come from? Then van Gogh, Magritte and Freud help to compare seeing like an artist to the way most people currently do seeing. Then it makes an important philosophical point about the property of symmetry that is found in our universe, and just where we can and cannot reasonably expect to find it, with a little help from Allen Ginsberg. All of these ingredients are then put into a single context, using the Primary Algebra of George Spencer-Brown.

As ideas from art criticism and mathematics are introduced, they are explained. It is very important that both artists and scientists reading this paper realise this, since it is only by using both aspects of the One Culture together that what this paper discusses can be fully understood. It is not the case that those parts that use mathematics are inherently beyond the comprehension of artists, or the other way around. Both are nothing without the other.

At this point the Ghost Not is wholly exposed. The paper then builds up a series of examples, from the simplest logical OR operation through a few levels of mathematical examples to complex human emotional, social, philosophical and scientific contexts. At each level it shows how the Ghost Not appears to distort conclusions in its characteristic way. In the final sections, it makes some comments on a culture that has evolved with all of its thinking distorted by the Ghost Not. These are the areas where understanding the problem will allow us to make improvements.

We shall begin by asking what art is for. What does art do? One answer to this question is to say that it is pretty - decorous. This though, is a very shallow answer. Great art does more than cause nice feelings for us by being pretty. Great art captures in one (convenient) form, some deep and powerful thing usually found in another form. A few dyes daubed on canvas or a piece of hewn stone can capture something of a natural movement or a human emotion that excites, or challenges, or puzzles us. What art does is to convey impressions. Here are the awe inspiring forces of a storm, captured in daubings by Vincent van Gogh:

When he pulls the trick off, van Gogh has to do two quite distinct things. Firstly, he has to spot something worth conveying. This might just be something direct, such a particularly spectacular view. More usually, an artist will try to convey deeper impressions than this though. Here, what van Gogh has done is notice that the wheat in the field has been scraped into the same texture as the stormclouds. Although invisible, the same force (the wind) is behind both patterns. His painting thus celebrates the power of the wind, which cannot itself be painted because it is invisible. He then goes beyond this though, and shows that the track itself seems to have been pulled in the same way the wheat and the clouds have been. Can it be that there is a deeper principle behind both the wind and the historic forces that define the path of a track? Perhaps a modern physicist might think about fluid dynamics here - certainly the philosopher Plato would be sympathetic to the idea that the equations of fluid dynamics are "deeper" than their resultants - wind and track paths. Then look at the crows in the sky. They alone are not distorted by the action of this principle - the wind and perhaps what lies behind it. Perhaps the crows both individually and in their flock, are in some way creatures of - their behaviour is governed by - the same deep principle. Then look once again at the wheat. The raw energy that has scraped it is not the only thing that causes it to resemble the clouds, which have been shaped solely by the wind. It is quite clear that the scraped appearance of the wheat ears derives in part from the regular planting pattern that we see at the front of the picture. Is there another creature about whose nature derives from the deep principle behind wind and crows, that has done this?

Quite a lot of ideas for one eyeful. The sensibilities and editorial skills of van Gogh spotted an impression worth conveying, that caused him to experience a great deal of the world, in its own context, in one go. To do the job, van Gogh must allow sensory data to enter his eyes uncritically, and wait for the relationships that are implicit in the data to emerge as he rolls the raw data around in his head. When he had completed this task, the job of finding something to paint, his second task began. The second task was now to un-notice what he had noticed. Rather than paint a cloud that looked like a field, he must use his technical skill with his materials and a kind of honesty to simply record the visual image that had brought the impressions to his mind. He must leave it to the viewer of the painting to either agree with him or not, that the data he has recorded suggest a relationship between say, planted fields of wheat, windtorn clouds, and farm tracks.

In both of these tasks, van Gogh must perceive uncritically. He must allow sensory data to enter his awareness without asking what it "is", and only later in either his mind, or the mind of the viewer, should the pattern he will (or has) becomes aware of, take form. This is seeing like an artist. It is exactly the same as the honesty that the physicist Richard Feynman insisted is necessary to make any progress in understanding nature. See what is there, not what you expect.

Most people do not usually see like artists. Instead they do something more complicated that they have subconsciously learned to do. First, the sensory data enter their heads. That bit is unavoidable. At that point they categorise what they have seen. If it is a field of wheat before them, they categorise what they have seen as a general issue field of wheat. They then put a general issue field of wheat on a kind of internal whiteboard that they keep in their minds for this very purpose. Then they make with the exaggerated Marcel Marceau impersonation, step back from their whiteboard in amazement and cry, "Gosh! A field of wheat!" as if it were the first time they have seen it.

Why would anyone do such a thing? The reason is that they do it so as to be able to edit what they see as they make the transcription from reality to the internal whiteboard. A brutal dictator can see seen as the beloved father of the nation. Winston Smith in Orwell's 1984 can become able to see exactly as many fingers being held up by O'Brien as O'Brien wishes. Not see one number and say he sees another - Smith in all sincerity "perceives" what O'Brien wants him to. The same thing happens to the mystified people documented by Laing. They perceive the behaviour of their relatives as kindness, not cruelty, and then cannot understand why they are unhappy.

There are several reasons why conventional perception is bad for people who take it up - tagged here "dyadics" - people who live in a universe consisting of the real thing and a poor image, as opposed to "monadics" - people who live in a single universe. Self delusion is a bad idea at the best of times. Never mind that it might seem to keep the brutal dictator off your back - he'll be around sometime soon enough, and if you've convinced yourself he's the nice one, you'll never stand a chance. This is the paradox of cowardice - only those who are afraid to move need fear being squashed. Also, conventional perception is lossy. It does not perceive correctly, in the same way that a standard telephone line is lossy and cannot reproduce music faithfully. This is because the standard issue wheat fields, clouds, tracks etc. used to populate the internal whiteboard have none of the richness (and interconnected richness) of the real thing. The real field, clouds and tracks are all shaped by fluid dynamics (amongst other things), and so have properties in common that van Gogh could discern, with or without a physical theory saying that the relationship existed. The standard issue fields, clouds and tracks might as well have been cut out of an infant's colouring book. They do not contain any of this interconnected richness. A person who insists on "seeing" a field or cloud symbol in this way, will never spot the fluid dynamics in play, because they edited it out long before they compared the two phenomena in their minds. For an example of such lossiness, consider another famous painting by van Gogh, of his own bedroom:

There is a commonly held view that the great painter van Gogh was er... so poor a painter he couldn't get perspective right, and so painted a wonky looking bedroom. Not so. When a TV documentary crew visited the house, they discovered that the room does not have right angle corners, and the wall with the pictures on it is so short, that for an adult to lie in bed, the bed would have to protrude across the door. Again, van Gogh painted exactly what he saw. People who transcribe into standard issue beds, corners and walls however, "know" that the angles recorded by van Gogh "cannot" be correct, and then glibly assert that the great painter was a terrible painter!

For a final example from van Gogh, consider the famous "Starry Night" painting. Conventional perception would suggest that van Gogh was simply a nutter. The painting does not contain "proper" stars, sky or anything else. If however we keep faith with van Gogh, and ourselves see simply what he recorded, we have the opportunity to notice something hidden amongst several little clues. Firstly, stars really are different colours although it is surprising how few people realise this (since every time they look up they "see" little white dots as in Classic Star Trek SFX). But these stars have the red end colours all in the middle, and the blue end colours strongly scattered - by atmospheric effects. And each star has a huge umbra. Next, look at the air flows against the hills that van Gogh has somehow discerned and recorded. Anyone who has lived on a hillside knows the wind blows up a hillside - not down. Something is pushing the wind downwards. Look also at the clear sky, and the band of fog hugging the hills. Finally look at the tree - that is no blustery wind - it is being pushed left by a sustained force. The conclusion is obvious. If we can forebear from "correcting" van Gogh, we learn that before the Wright brothers made such issues important enough to study, there was a meteorological temperature inversion just above the ground effect on the night the painting was made. Making the big sky turn over like a pancake and turning it into a chromatically distorted lens. Which van Gogh got right in detail, because he painted what he really saw. Which was how he captured the majesty of it too. For as Feynman said, perception only adds, never subtracts.

The surprise is - these problems with conventional perception are just the secondary ones. The remarkable thing about the Ghost Not is that no self delusion need occur, and the transcription can be completely faithful to the reality, and the worst effect will still hit home. The very act of setting up the internal whiteboard is enough to do it.

To see why the subconscious habit of "narratising" - transcribing perception onto an internal whiteboard before admitting it is there - is so problematic, we need to take a little time to consider a subtlety in how we represent existence in our minds. This part may seem rarified but please bear with me - the consequences become practical very quickly.

Think of this little windscreen as the eye through which an artist is seeing the world. We'll make it a male artist since we started with van Gogh. At the moment he is looking out on emptiness - perhaps a snowstorm:

Now, the winter can turn to summer, and he can see a tree in the garden:

The important thing is, as the seasons changed, the artist just sat. He did nothing to change his windscreen. He looked at the universe, and if the universe saw fit to have a tree where he was looking, he saw a tree. Otherwise, he saw no tree. The important point is, he uses no marker or symbol to denote existence. He provides a mark-free windscreen, and whatever appears before him is what exists. Now he might decide to propose for the purposes of discussion that that which exists does not exist. Perhaps he is discussing re-arranging the garden. Now, for the first time, he must use a mark of some kind, which he slaps on top of that which exists to denote that it should be imagined as not existing. So the featureless snowstorm gets a mark put in it - something where there was nothing before:

While in the view with the tree in it, the tree gets cancelled out:

So for the artist, existence is denoted with nothing, but non-existence is denoted by a mark. What happens in conventional perception is exactly the other way around. This is because in conventional perception we cannot use a void to denote existence - the real universe won't take care of populating the whiteboard correctly, as it does reality. So when a conventional perceiver sees a tree he must do the work of noting a tree on his whiteboard:

And when he wishes to propose that that which exists does not exist, he simply doesn't make a mark!

So the artist uses a blank to indicate existence in his mind (whatever the universe chooses to put there), and a mark to indicate that what the universe is doing, it should be imagined to be not doing. The dyadic uses a mark to indicate that he has acknowledged the existence of something, and a void to indicate that he has not acknowledged it. Of course the dyadic has "forgotten" that he is counting acknowledgement, and proceeds as if he is counting existence. Thus an artist uses a mark for non-existence and a void for existence, while the dyadic does it the other way around.

The effect is to produce the situation in this painting called "la condition humaine", by Rene Magritte. It is usually taken that the human condition that Magritte was describing here was primarily one of alienation, as described by Sigmund Freud when he founded psychoanalysis. The Ghost Not concept however, suggests that it was Magritte who had gotten to the essence of the issue, while Freud's alienation is an effect that derives from the real isolation. The real isolation comes from the need to transcribe every external perception into a shallow internal one before acknowledging it. To replace the infinite space behind the clouds in the reality with the blankness on top of canvas in the inner painting. To cross a system boundary, to invert sign, to artificially isolate one's self from the rest of the universe. The inner painting in la condition humaine can be taken as a literal internal whiteboard onto which transcription is done. Like van Gogh, Magritte somehow managed to paint exactly what he saw people doing.

Why does this sign flipping, this inverted use of the mark by conventional perceivers matter? Clearly sign flipping is unavoidable by conventional perceivers, but why does it matter? Are not the marked and unmarked states symmetrical? Might we as well say Green/Blue, True/False or Up/Down as readily as Marked/Unmarked? The answer is no, for a very remarkable reason. First, Spencer-Brown showed in "Laws of Form" that Boolean logic can be done with one symbol to denote operands and operations. True/False and Blue/Green are two valued logics, while Marked/Unmarked is one valued. Next, we must ask why we can do Boolean logic with one symbol and the answer is most remarkable - our universe is logically asymmetrical!

We are used to symmetry. Every action has an equal and opposite reaction. If I give you some sweets my count of sweets goes down as yours goes up. It is easy to con ourselves that symmetry is universal, but it isn't. The reason is very elegant. By definition, everything we can ever have knowledge of is contained within our universe. If we can never have knowledge of a thing, it is not a part of our universe. Now if I dig a hole, I will end up with a hole and a matching pile of spoil. The fact of the spoil's existence "balances" the fact of the hole's existence. But when it comes to universes, any other fact that balances the fact of our universe's existence is true outside our universe, not within it. Therefore we can never have any knowledge of this fact. Therefore, from our point of view within it, there is no other fact that balances the fact of this universe's existence. It simply exists, and that is all there is to it. As Allen Ginsberg put it,

Our universe is logically asymmetrical at the deepest ontological layer, and the symmetries we see within it are emergent from the essential asymmetry.

As we shall see, it is the fact of our universe's logical asymmetry that makes the choice of marked or unmarked state to denote existence very important indeed, because the answers really do come out differently.

We can recap the ideas discussed above in a very stark and elegant way that brings out the key point, and then allows us to see the consequences of the error in how we see the most primitive logical operations, which usually we take as being solid. The Primary Algebra was introduced by George Spencer-Brown in "The Laws of Form". It is a very, very simple way of doing logic. He starts with a blank space, which he cuts in two. He now has something to talk about, because the two halves are different. Then he says that if there is another boundary found inside the existing one, you should think of the contents as being out of one half of the original distinction, and so they must be in the other half.

Every time you cross a boundary, you change from one state to the other. He defines some rules for making sense of which state a person who took into account all of the boundaries in a cleaved space like this would end up with. If there are a pair of boundaries inside each other with nothing else inside them, they disappear:

If there are two empty boundaries next to each other, they merge:

The rules can be used over and over again to make sense of anything, by canceling out different arrangements of negations:

Next, we can simplify the notation, by putting a little mark instead of drawing the whole circle, so that the example above would be:

He doesn't need to have any symbol at all on the other end of the last equals sign, because he indicates FALSE, and NOT with the mark, and TRUE with a complete blank. Whether he needs to represent a FALSE in the inputs to such a sum, or a NOT operation in the sum, he uses exactly the same symbol. And because he can construct any logical operation he likes out of correctly arranged NOTs, he can now do arithmetic below the level of Boolean algebra.

Next, he shows how to specify a relationship between inputs without specifying whether the inputs are TRUE or FALSE by using variables, as in ordinary algebra:

You just replace the letters with a mark if they are FALSE, and nothing at all if they are TRUE, and work out the answer by arithmetic, as above.

It is very, very, simple indeed. The Primary Algebra contains the essence of logic. Now think about what happens when a person perceives conventionally. Instead of allowing sensory impressions to arrive, and allowing the wheat field, crows or whatever to emerge in their mind, they drag each object onto their whiteboard, and as described above they end up being forced to use a mark (in other words, do something) for TRUE (or existence), and no mark for FALSE (or negation of that which exists). In terms of the Primary Algebra, they drag each object across a boundary, and so change its sign from TRUE to FALSE and vice versa. The algebraic description of the dragging contains exactly the same situation as the cognitive processing description. In the very act of seeing something, conventional perceivers must, inherently, sign-flip.

We can see the result of this sign-flipping that is introduced at the lowest possible level in the simplest possible logical operation above the sign-flip itself. A logical OR is really just like the word "or" is used in normal conversation. If one thing, or the other, or both are true, then the proposition is true. For example, "If it is Friday or I am celebrating, I go out". OR is sometimes drawn as a "truth table", like this:

Recall the seemingly abstract point about the logical asymmetry of the universe, made above. Here we can see that it is not abstract at all. The inputs to OR are symmetrical, but the outputs are not. The TRUE input dominates. What is concealed in our understanding of the simplest logical relationship between two values is the Ghost Not delusion that it is the logical TRUE that is dominant in our universe, not the logical FALSE. Or in more conversational language, the idea that if you have a pipe next to a sign saying "Ceci n'est pas une pipe", the conclusion to be drawn from the situation is that it is a pipe and there is some strange paradox in play, rather than that it is not a pipe, which the naive (or artistic) mind might cheerfully accept - perhaps it is made of chocolate.

This is all very philosophical, but is there any practical evidence that our understanding of logic is warped even at this low level? Yes there is! If you pick up any book on electronic engineering, you will find that actual electronic switching circuits are built out of basic components called NOR gates and NAND gates. These terms mean an OR with a NOT around it, and an AND with a NOT around it. It might seem rather odd behaviour for electronic engineers to make every component with an extra layer just to invert the output, with the result that all their logical theorems will be the wrong way around - which would be inconvenient as well as cost more to manufacture. But that is not the situation - they don't fit an extra layer at all. What nature actually provides is an operation that we call "OR with a NOT around it", that we cannot bring ourselves to call fundamental.

Slightly more complex than the OR operation is the XOR - eXclusive OR - operation. This means if one thing is true or the other, but not both, the proposition is true. For example, if I have got a left sock, XOR I have got a right sock, I have got an odd sock. If I have zero or two socks, I do not have an odd sock, but if I have one but not the other, I do have an odd sock. The truth table is:

We are now complicated enough that the usual symmetry of the universe has emerged. But this just means the Ghost Not is more deeply hidden. Here we can expose it in the very structure of the English language. In English (and putting some brackets in to make the translation to the Primary Algebra easier), we can express XOR as:

(a OR b) AND (NOT(a AND b))

If you take the mark to indicate true (as Ghost Not perception does), you can make an OR in the Primary Algebra like this:

a b

and you can make the AND operation like this:

So we can express XOR as:

Notice the three nested NOTs in the right of the expression! It isn't obvious in the English, but the English way of describing this relationship is very inefficient. We can lose two of the nested NOTs, giving us:

Which in English reads:

(a OR b) AND ((NOT a) OR (NOT b))

The more complex English version is the simpler version in the Primary Algebra, and vice versa. Normal conversational English is logically contorted! By looking at the expression we ended up with, we can understand the problem in terms of the Ghost Not. It derives from the need to expressly move each perceived thing from the reality into the internal whiteboard before shoving it around on the whiteboard to produce conclusions. In the more complex English version, both a and b appear in the left and right hand parts of the AND. If a is TRUE on one side of the AND it will be TRUE on the other side of the AND too. Because the universe always knows what value a holds and it is always the same. The same is true of b as well. We don't care at all which the true one is. The a's and b's will actually take care of themselves, so if one of them manages to render the left side of the AND TRUE, it will contribute a FALSE on the right automatically, because of the mark around it. The viewer doesn't need to do anything on their internal whiteboard to make it so. So all we then care about is if the other value is able to produce a TRUE when NOTed, meaning that only one of the values is TRUE. The difficulty is that people are unwilling to acknowledge that the universe has a real state which is self-consistent, even if it isn't represented on their whiteboards.

Now a more complex one, but still in the engine room of logical thinking. Consider two statements:

1. Some liontamers are women.

2. Some women are redheads.

The question is, are there any redheaded liontamers? The usual answer is to say "Undefined", but that is not correct. As with the XOR operation, the problem is that anything not done explicitly on the internal whiteboard is not regarded as real. People consider the problem in one of two ways, determine that both are unsuitable, and so say the question cannot be answered. The two ways are:

The inability to make any answer to the question comes from the insistence that any thing or relationship not represented "omnisciently" on the internal whiteboard does not exist. In fact, the true situation as given in the problem can be drawn in a way that no-one is ever taught to draw a Venn diagram - a way that recognises that the statement of the problem hides the relationship between liontamers and redheads behind the set of women:

So the correct answer is "Possibly." It is different to "Undefined", because "Undefined" says nothing, whereas the problem as given does not preclude there being a set of redheaded liontamers. Which is information that "Undefined" loses.