cognitive psychology


While identifying facts and making accurate perceptions are important parts of the human function and survival, the human mind is not entirely about this or perhaps even mostly about this.

To survive and function, the human must do other things such as act and guess in ambiguous and mysterious situations. Many of these functions are not about identifying facts and assessing truth, but making speedy and practical decisions. In fact, humans are in part hard wired to make speedy intuitive decisions in the face of lack of knowledge.

As an example I use way too often, avoiding instant danger is often about how to react to the unknown and unknowable. If a mysterious large shape is moving quickly at you, taking the time to accurately identify the shape (‘gathering the facts’) is the opposite of what you need to do. Get out of the way right now, then worry about identification later. If it turns out to be nothing harmful, say just a shadow, no big deal other than you might look a bit foolish. If it turns out to be a boulder or falling board, you’ve saved yourself from harm or worse. And this is the natural and automatic subconscious self-preservation instinct of humans.

This is just one example of how truth finding is not always the priority of the mind and in fact can get inhibit function. Survival is commonly said to be about erring on the side of safety– as it takes only one time being hit by a speeding car or falling off a cliff to be dead. The key word there being ‘erring.’ In this case, the mind is designed to err.

The human mind has limited capacity and capabilities, and human function can be inhibited by too much information including facts and truths (see How Humans Use False Information and Made Up Beliefs to Produce Personal Achievement). If your task is to move across a room, trying to identify and learn the history and “truth” of everything and everyone in the would lead to you dying of old age before you reached the other side. Humans must actually block out and distort information in order to function. In order to read a complicated passage or do math, people cover their ears to block out noise or tell others around them to quit talking.

And don’t forget that humans are social animals and functioning, thriving and surviving involves interaction with people and other animals that are full of cognitive biases, delusions, limited information and viewpoints, emotions, selfish motives, social politics and order, subjective tastes and irrational drives. Humans survived and thrived as a species because they work as social groups.

Early economists made the fatal mistake of basing their models on the assumption that humans act entirely rationally when making economic decisions. Later economists realized the models had to be thrown out, because they learned that humans do not act entirely rationally when purchasing, selling, investing, valuating and saving.

In short, the commonly voiced sentiment that the human is by nature a truth seeker, and that is its key function, is highly debatable.

Wassily Kandinsky’s Composition VI


Wassily Kandinsky was one of the first artists to make completely abstract paintings.  His 1913 oil painting on canvas Composition VI is an example of his non-representational works.  

Kandinsky is another step in the progression from the previous artists Constable, Renoir and Boccioni.  Constable, Renoir and Boccioni used recognizable figures and scenes, but used new and often abstract techniques to express perception, feelings and ideas.  Kandinsky took it to the next level, dropping recognizable things and trying to express his ideas and feelings entirely through pure colors, shapes, lines, marks and composition. Not only did he want to express his ideas in non-representational art, he felt that was the only way to do it.  

Kandinsky was a devout Orthodox Russian and felt colors and other qualities not only affected the emotions and aesthetic experience but resonated with the soul.  He aspired to have a communion between art, artist and audience, and the act of painting was an emotional and spiritual experience for him.

Europe was in a time of turmoil, of industrial, artistic and philosophical change, and the upcoming World War I.  This affected many artists, including Boccioni, who saw the world as changing and saw the world in a new light.  They tried to express and address the changes in new aesthetic languages.  The old languages were outmoded, artifacts of a world that no longer existed.  A new world needed new languages.  Many artists were as much philosophers and theorists as artists.  As a deeply religious person who saw things differently that the average person, Kandinsky felt he was a prophet bringing people to a new art, language and era. He felt that today’s avant garde understood by few was tomorrow’s common knowledge, and he was a leader bringing a new knowledge and rebirth out of the day’s turmoil.

In previous painting essays I have discussed how neurobiologists believe humans naturally and neurologically react to basic sensory qualities, including colors, symmetry (or lack thereof), shapes, angles, etc., and this is a major part of our aesthetic and artistic experience, along with how we perceive the natural world.  These are the things that Kandinsky worked in.  He was trying to communicate entirely this way, using what he felt were natural inborn reactions to basic qualities,  He wanted to make an art that communicated beyond culture and historical perspectives.  

While Kandinsky was concerned with resonating on an emotional, aesthetic level, he was a theorist and academic and his works involved much academic study, research and planning.  He was a law professor before turning to art, taught art design courses and was a theorist who wrote extensively on how colors, shapes etc created meaning.  He experimented with and tested how colors and other basic qualities resonated with him and, he hoped, resonated with others (Though this shows how his theories and color rules were in part subjective to him.  Whether or not they apply to everyone’s mind and eyes is debatable.).  Composition VI involved six months planning.  It also was finished with him repeating a mantra (‘flood’) to get over a mental block and free his subconscious.  This illustrates how his work was both academic and intuitive.

He compared visual art to music and wanted his art to be like music.  He said music is abstract yet evokes specific ideas and emotions.  And this is true.  Combinations of notes and instruments can communicate ideas such as speed, physical landscape, danger, drama, evil, happiness, joy.  Music can be sad and it can be funny.  And much of this reaction to sounds in inborn in humans. Our reactions to thunder and songbirds, loud low notes and soft high ones are natural.  This is what Kandinsky was trying to achieve through colors, shapes and visual composition.

Kandinsky had synesthesia, where people see or strongly associate colors with musical notes, tastes, other.  He saw colors when he heard notes and heard notes when he painted color.  In fact he associated specific colors with specific notes.  This clearly was an influence on his art and theories, and other artists have had synesthesia including Duke Elliott, Vladimir Nabokov and Frank List.   

Beyond just emotions, he was trying to express complex ideas and stories. Composition VI is about the apocalypse, a giant flood and rebirth– depicted all at once! As with Boccioni, he was trying to show many, often conflicting and juxtaposing ideas and qualities in one work.  The apocalypse and rebirth are conflicting– one is about turmoil, violence and terror, while the other is about peace and happiness.  His work is like trying to depict the “calm before the storm,” but with the calm and storm happening at the same time.  In writing about Composition VI, Kandinsky said how there were many different feelings, conflicting and juxtaposing emotions from the different colors and shapes.  It is telling a complex story with many parts and details.

This painting was a challenge for me– in part because it is so complex, busy and non-representational.  However, when I looked at his other paintings, I did get different reactions and feelings from them, and saw how colors and shapes evoke aesthetic feelings and aesthetic responses.  And Composition VI is supposed to be complex.  It is is supposed to take thought and examination.  Kandinsky even said the painting had three centers.  He also said the meaning and feelings one gets from it change as you look at the different parts and move closer to the painting.  This is comparable to Boccioni’s sculpture where its form changed as you walk around it.  

One way the work is harder to understand than Boccioni’s is there is no clearly recognizable representational anchor.  It is, after all, non representational.   Boccioni’s sculpture was complex and abstract, but it clearly was a running figure. Your reading and interpretations started from this anchor and easy to understand theme. Though once I read about Composition Vi being about the apocalypse, flood and rebirth, that was something from which I could start.  And in fact, once this theme is known, many visitors to a museum may enjoy discussing how it shows this and if it shows it well.  Boccioni is also better understood and appreciated when you know his philosophy and what he was trying to thematically express in his art

The painting certainly perplexed me at first.  But looking at his other paintings and watching a youtube video where his paintings were shown with Schoenberg music gave me a better handle.  I like some abstract art, especially music.  This includes Schoenberg, Gyorgi Ligeti and noise music (try listening to the drone metal band Sunn 0))).  It can greatly resonate with me.  The Kandinsky didn’t, but that’s all subjective and I do understand and appreciate what he was trying to do.  I also have bad color vision, which won’t help.

The painting is valued as an artwork and a new artistic movement.  Further Kandinsky studied and showed how we communicate and perceive things, including emotionally and aesthetically,  through basic qualities.  This helps teach us how all art works, including centuries old representational works.  Even representational works use qualities of color, angle, shape to express ideas and emotions.  And art involves reactions that are beyond the literal and conscious.  Kandinsky’s works and theories help us look at past art, but also how we look at the world and how artist can do things in their own works even when representational.

Though I didn’t entirely get into Kandinsky, some of his works remind me much of Hieronymous Bosch’s works, which I do like a lot.  Bosch’s great paintings on similar religious themes used identifiable if fantastical creatures, but were complex and busy stories like Composition VI and produce visceral, sublime reactions from the audience.  Bosch speaks to the subconscious. He had a way with composition and details that to me is Kandinsky-esque.  And, as mentioned, really all art communicates to us using these devices.  All art connects to the subconscious and communicates feelings and ideas that are beyond the conscious.  That is what is the art.

Umberto Boccioni’s ‘Unique Forms of Continuity in Space’

.  boccioni_unique_forms_of_continuity_in_space_1913

Unique Forms of Continuity in Space is a famous large bronze sculpture by Italian futurist artist Umberto Boccioni.  Boccioni made the plaster sculpture in 1913, with the bronze examples seen in museums being cast from the plaster sculpture or from other bronze castings.  

The work is a part of the short lived but influential futurist movement. Originating in early twentieth century Italy, futurism was an artistic, philosophical and social movement that emphasised modernity, constant change and movement into the future, technology and inventions, speed, youth and violence.  It said it wished to destroy Italy’s old artistic and social past (Though I would contend it merly built on it).  It lauded such modern technology as cars, airplanes and the industrial city.  One futurist said the car was more beautiful than an ancient Greek statue.  The futurists glorified war and welcomed World War I as a way to cleanse society of its old ways.  It was nationalistic, aspiring to make Italy modern and victorious, bring it to a glorious future. It is telling that Boccioni and the movement died with World War I.

Unique Forms of Continuity in Space

Along with issuing numerous manifestos, including by Boccioni, futurism wished to express its ideas in new, original artistic ways.  It encourage originality, rebellion against traditional styles and even good taste, and dismissed traditional art criticism.  It used many mediums, including painting, sculpture, architecture, theater, film and fashion. This not only showed that they wanted to dominate all aspects of society and life, but that their ideas couldn’t be captured by just one medium.  Boccioni’s works were in painting, literature (futurism manifestos) and sculpture, and he was the first to freely use different materials in his sculptures.

In Unique Forms of Continuity in Space, Boccioni wanted to express this philosophy, including of constant change, movement, speed and moving in the future.  

To the casual viewer, there may be mystery to and head-scratching about the abstract parts of the sculpture, as there is to much abstract art, but the theme of movement and speed are clear. The body and legs positions and the flames, wind or wing symbols on the back of the legs visually and symbolically represent speed and movement.  As a test, I showed the picture of the sculpture to my 81 year old dad with dementia and he said it looked like a running figure. Color and texture express ideas and evoke aesthetic responses, and the shiny bronze evokes to the average person modernity, invention and futuristic technology.   Some call it a superman.

This shows that Boccioni was not completely breaking from the past and past art as he supposedly aspired to, but was building on it.  He incorporated traditional representational forms, recognizable symbols and medium (bronze).   I would call his sculpture a response, or furthering, of traditional art, rather than a break.  Art, even rebellious art, requires shared language the audience can understand.  The sculpture bends and expands the language, but does not get rid of it.


Rodin’s The Walking Man

It has been said that the sculpture alludes to Rodin’s The Walking Man, which shows a more recognizable and traditional depiction of a man walking, though without head and arms.  Rodin focused on part of the person, breaking away from the academic tradition of showing a full figure.  He also produced an unidentified person, where traditional sculptures usually depicted specific and famous people.  Boccioni can be seen as an extension of Rodin’s work.  Michelangelo’s sculptures showed stationary known people in perfect detail, Rodin showed a walking anonymous and more impressionistic man and Boccioni showed a running, even more abstract figure.  The progression is clear.

Boccioni was familiar with and influenced by the impressionists.  His early paintings have an impressionistic look and even his later abstract paintings use bright impressionism colors.  The wings on the legs and the ‘blurry’ abstraction on Unique Forms of Continuity in Space is a very impressionistic way to depict movement and change.  Interestingly, he dismissed impressionism, saying: “While the impressionists paint a picture to give one particular moment and subordinate the life of the picture to its resemblance to this moment, we synthesize every moment (time, place, form, color-tone) and thus paint the picture.”  I bet the impressionists would say they were evoking movement and thus time.  No doubt Boccioni’s rhetorical rejection of impressionism was in part because of their quaint, quiet, mundane life topics– something he definitely was rebelling against.

culpture is often a reaction to or commentary on previous works, so to understand a work you must know the history of sculpture.  So comparing Boccioni’s works to previous works, including impressionistic paintings, is important to understanding them.  Understanding the futurism philosophy is also essential.  Even a reaction or rebellion can only be understood by known what it was reacting to and rebellion against.  And rebellions often retain many of the qualities and methods of what they are rebelling against.  As I mentioned, Boccioni was both rebelling against impressionism and using impressionistic visual techniques.  

Beyond depicting movement, the sculpture intended to express other less tangible and more theoretical  futurism ideas, including the futurism idea of ‘’universal dynamism.’  Universal dynamism says that objects are not separate from each other and that an object is not separate from its surroundings.  A futurism manifesto said “The sixteen people around you in a rolling motor bus are in turn and at the same time one, ten four three; they are motionless and they change places … The motor bus rushes into the houses which it passes, and in their turn the houses throw themselves upon the motor bus and are blended with it.”   Boccioni wrote about his art “Let us fling open the figure and let it incorporate within itself whatever may surround it.”  This explains much of the abstraction of the sculpture.  It is not about a constant, single figure, but a changing figure and it’s surroundings, all in one and one in all.  If the traditional viewer wants a single, unchanged, easily identifiable image, they miss the entire point and are exactly what the futurists were rebelling against.  It is not supposed to be one figure, have one identity. After all, it is titled ‘Forms’ not ‘Form.’  

People have commented that the sculpture changes significantly as you walk around it.  This represents the theme of constant change, in particular in relationship to movement and time.  As the viewer moves, both physically and in time, the work changes.


Boccioni and the futurists adopted techniques of the cubists, and Boccioni’s later paintings are clearly influenced by cubism. Cubism expressed things in new, radical ways.  It addressed the age old problem of depicting three dimensions in two dimensional space by showing objects from multiple angles at once.  It also used abstract techniques to show such things as movement and the passage of time.  For example, to depict movement and the passage of time a cubist painting might show a series of still images of a figure in different places.  Boccioni’s paintings and sculpture are similarly abstract and ‘different’ to casual eyes because they tried to show many things all at once.  It attempts to show different perspectives, including from physical and time changes, in one solid piece.

There are many different ways of depicting things such as movement and time in a still work of art, usually using symbols and mimicking natural cues recognized by the audience.  Literature describes it in letter symbols, demonstrating how humans use symbols to perceive things in their minds.  A human can ‘see’ a sunrise and ‘hear’ sounds in black letters on a white sheet of paper.  The Bayeux tapestry showed a series of figures and events along a long time line, which is similar to how cubists sometimes expressed movement and time.  A still painting or photo expresses movement by body and object position, cropping and effects such as blurriness and color changes.  Cartoons often use lines to express movement.  None of these show actual movement, but evoke it in the viewers’ minds.  Art perception is about imagination, and, in part, the audience looking at painting, sketch or snapshot just assume it is a snapshot in time.  There are countless other ways to depict and express movement, including by incorporating real movement: a mobile, play, the changing series of still images of motion picture film.


Wikipedia articles on Boccioni, futurism, the sculpture

Khan academy website

Tate museum website article on futurism


Swarm Intelligence

13512094_10206571386658943_4948230813354131650_nSwarm intelligence is where large groups of animals exhibit a group intelligence and capability much larger than any of the individual animals exhibit or are even aware of. Examples include small fish and birds that unconsciously and instinctually form large groups that protect themselves from predators (essentially forming one large animal), ant groups that gather food in long lines and termites that build giant, intricate homes. Each of the animals does a very simpleminded task in its own immediate surroundings (a fish in a school will swim a certain distance from surrounding fish) and is unaware of the groups’ overall structure and capability.

Humans exhibit swarm intelligence, such as in economics and mobs. Computer scientist study swarm intelligence to make crowded areas, such as airport terminals and commercial transportation routes, function more efficiently.

The topic of swarm intelligence begs the question of if there are swarm intelligences and group functions the human species are doing that they are not unaware of.

It also begs the question of if individual consciousness, or consciousness itself, is as important as humans say it is. We could be, in fact are, doing things higher and more intelligent than we, both as individuals and groups, are conscious of. Consciousness and awareness are things humans traditional aspire to, greatly value, but perhaps human consciousness of things is nothing more than a quaint and relatively minor quality in the big picture of group intelligence, group function, group minds and beyond.

The Unique Subjective Experience

(Excerpted from the book Noise Music: Cognitive Psychology, Aesthetics and Epistemology)

Subjectivity is a constant and integral part of the human experience. Love, lust, like, dislike, taste, smell, views about beauty and ugliness and art. How you view this paragraph and this book involves subjectivity— your taste about the writing style, word choice, chapter subjects and length, book cover.

By definition, a subjective experience is a product of the individual’s mind. While real and often profound, the subjective experience cannot be objectively measured by others. When someone is listening to music, the music’s note, pitch, speed, volume and the listener’s ear vibration and heartbeat can be measured by scientific instruments, but the listener’s aesthetic experience cannot. This experience is experienced by the listener alone. Even if asked to, the listener could not fully translate the experience to others, in part because it is beyond words.

It’s doubtful that two people have the same subjective perceptions. People may have similar, but not identical perceptions. People regularly like the same song but perceive it differently. It’s common for best friends to like a movie, but one likes it more than the other or for different reasons.

* * * *

A large range of things determines a person’s subjective perception and experience. This includes genes, education, culture, where and when born, personal experiences, upbringing, travel, family make up and personalities, friends, acquaintances, natural temperament, mental abilities, physiological abilities (quality of eyesight, hearing, smell), talents, language, mood, health, hobbies and work.

Little things influence, such as what toy one had as a six year old and what tea grandmother drank. While walking in a foreign land, the scent of jasmine tea can bring back a rush of memories. The appearance of the toy in a movie will alter one’s emotional reaction and interpretation of the move. It may have been chance that the movie viewer’s parents bought that toy, making his movie interpretation a result of chance. It’s not just the tea and a toy, but millions of little things that influence, including from forgotten events.

If a bird watcher and a rock collector go for a walk together in the park they may have equally grand times, one due to the birds in the trees and the other due to the rocks on the ground. Though they were side by side, they will give decidedly different descriptions of the walk.

Do you dislike a name simply because it was the name of someone you couldn’t stand?

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Even when they experience similar feelings people will usually have these feelings under different circumstances, if only slightly different. People will be artistically excited, but for different works of art or when interpreting differently the same work of art. People have similar feelings of romantic love, but for distinctly different people— different looks, personality, culture, interests, sex, race. The emotional states may be alike, but the objects of desire are not.

* * * *

You cannot separate your biases from your perception, because it is those biases that help create the perception. Without those biases, you would have a different perception. Even that childhood toy affected the movie goer’s perception thirty years later.

* * * *

Humans believe they receive important objective insights, including cosmic truths, through strong subjective experiences— such as through the sublime experience of art, epiphany of music, nature, love, lust, religious experience. The psychological power of these experiences is considered verification of the ‘truths.’

A question is whether these experiences involve genuine insight into external reality or are merely strong biological reactions. Love and lust themselves, after all, are standard genetic, hormonal reactions. Psychological reactions to certain sounds, such as in powerful music, involve genetics.

The reactions to high delicate notes (such as from song birds or a pop song) and low booming notes (distant thunder, the start of Beethoven’s Fifth Symphony) have been shared by humans for thousands and thousands of years. You and your ancient ancestor have remarkably similar psychological reactions to the sound of a songbird and the sudden deep roar of a bear. It’s no coincidence that church music uses delicate high notes to invoke heaven in the audience, and the loud, deep bass of the organ to invoke power and awe.

It’s not coincidence that horror movies use discordant notes. The director knows audiences find the sounds scary and creepy.

In the famous 1960 Psycho shower scene, the sharp, grating, discordant musical notes invoke violence, evil, something gone horribly wrong. They sound similar to someone scratching a chalkboard, one of the most despised sounds to humans.

It can never be known to the experiencer that an epiphany made through a strong psychological experience is anything more than a genetic reaction. If there is insight into the external, the insight is shaped by the experiencer’s subjectivity, and what parts of the insight are objective and what parts subjective is unknowable.

Even if important insights into the universe are gained they still are in subjective format. For example, if your epiphany comes through your experience of art, your experience of art is personal and different than that of others. Not only is your ‘insight’ intrinsically tied to your subjective views, you likely would not have had the insight at that same time, place or format, or at all, if you had different aesthetic views.

* * * *

Humans use aesthetic rules for defining truths, including what is good and evil, what is moral and immoral. Common rules include conditions of beauty, symmetry, color, tone (light versus dark), fashion and order.

Even if the rules were valid, it would mean truth is subjective. If truth is beautiful, your definition of what is beautiful differs from others’ definitions. Further, an individual’s perception of beauty changes with time and experience. A culture’s perception of beauty changes with time. Compare the depictions of the desirable feminine body from 1450, 1850, 1950 and this year.

Cultural definitions of ‘objective truth’ are formed by cultural sensibilities, including fashion, politics, gender, race, beauty, geography, self interest, desire for social order, etc. There is no indication these are identifiers of objective truth, or are even related, but they are still used as criterion.

* * * *


To humans, simplicity is that which is simple to them. Simple matches one’s sensibilities, knowledge, intuition and expectations. If it didn’t, it wouldn’t be simple. What may be simple to one human may not be to another. What may be simple to humans may be simple only to humans.

Simplicity has long been used by humans to define supposedly absolute things such as cosmic truth, goodness, beauty, logic and purity. There are a number of problems with this. One is there is no proof that cosmic truths, for example, are simple. Another problem is simplicity, and thus what is defined as cosmic truth, is in the eye of the beholder.

Normal, even nonconscious thinking involves simplification, translating complex information into something understandable. Conceits are simplifications.

Your visual perception involves simplification– interpreting a complex scene, grouping and labeling the objects according to your experience, focusing on what you seem to recognize and ignoring what you don’t. Visual illusions and mirages shown throughout this book involve simplification. The scene or graphic is translated by the viewer into something understandable, an understandable translation that happens to be wrong. This alone proves that simplicity is not proof of truth, and that truth isn’t always simple. Lies are often simpler than truths.

Simplicity, of course, has many practical uses. Scientists strive for simplicity in theories and testing. A scientific theory that is needlessly complicated will needlessly confuse students and seasoned scientists alike. Needlessly muddled theories are harder to test, study, correct and understand. In our daily life, good verbal communication requires simplicity, including using words, phrases and language the listener understands. If a traveler speaks only English, it does them no good for you to give road directions in Spanish. Road directions in Spanish may be simple to a Spanish speaker, but it’s complicated to someone who doesn’t know the language.






Numeration Systems and Psychology

Looking at different historical numeration systems demonstrates how language and grouping systems profoundly effect human thinking, perception and function, and how the the system you ‘naturally’ use to perceive the universe isn’t the only way.

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In some Western Hemisphere high rise buildings there are no thirteenth floors. Well, there are thirteenth floors, but the floors are labeled 10, 11, 12, 14, 15 to give the superficial appearance of having no thirteenth floors. The building owners know many have a superstition against the numeral thirteen and it’s easier to rent an apartment or office if it’s called ‘fourteen.’

In Korea and Japan where four is considered unlucky as it’s the sign of death, some buildings ‘omit’ the fourth floor.

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Our base-10 numeral system

The common modern human counting system— the one you and I use– is based on ten, and is referred to as base-10. It uses 10 different numeral symbols (0,1,2,3,4,5,6,7,8,9) to represent all numbers, and many popular groupings are divisible by ten: 10, 20, 100, 300, 10,000, century, decade, top 10 lists, golden anniversary, etc.

Our base-10 system is based on the number of digits on a human’s hands: eight fingers and two thumbs. As with today, many ancient humans found fingers and thumbs convenient for counting and it seemed only natural to base a counting system on the 10 digits.

While the base-10 is a good system and has served us well, ten as the base was a somewhat arbitrary choice. Our numeral system could have been based on 3, 8, 9, 11, 12, 20 or other number. Instead of basing it on the total digits on a pair of hands, it could have been based on the points of an oak leaf (9), the sides of a box (6), the fingers on a pair of hands (8). These different base systems would work. Some might work as well or better than our base-10 system. Nuclear physicists and tax accountants could make their calculations using a 9 or 11-base system. Once you got used to the new system, you could count toothpicks and apples just as accurately as you do now.

Quick comparison: counting with base-10 versus base-8

The above pictures compare counting with a base-10 system based on the ten digits of the hands (fingers + thumbs), and with a base-8 system based on just the eight fingers (thumbs not used). Notice that the base-8 system, not using the thumbs, is missing two numeral symbols: 8 and 9.


This comparison picture shows how assorted designs (top row) are counted with the base-10 and with the base-8 systems. As base-8 omits the two symbols 8 and 9, ‘10’ comes sooner when counting in base-8. In one numeration system, the cat is ‘9’ and in the other is ’11.’ As you can see, the real value of 10, amongst other numeral symbols, is not an absolute. It depends on what base is being used.

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Another example of counting with different bases

 base-5  base-8  base-9  base-10  symbols
 0  0  0  0  $
 1  1  1  1  #
 2  2  2  2 @
 3  3  3  3  !
 4  4  4  4  %
 10  5  5  5  ^
 11  6  6  6  &
 12  7  7  7  *
 13  10  8  8  )
 14  11  10  9  _
 20  12  11  10  +
 21  13  12  11  =
 22  14  13  12  –
 23  15  14  13  <
 24  16  15  14  >
 30  17  16  15  ?
 31  20  17  16  “
 32  21  18  17  ;
 33  22  20  18  ‘

The following table illustrates how you can count symbols (far right column) using the base-10, base-9, base-8 and base-5 systems. If you wish, the symbols can represent physical objects like fruit or cars or plants. In this table the symbols are constant, while the numeral systems create different numeral labels for the symbols (or fruit or cars or plants). For those who consider ‘13’ unlucky, notice that each counting system labels a different symbol as being 13.

* * * *

This counting stuff is not idle abstraction. Civilizations have used and use different numeral systems.

The Yuki Indians of California used a base-8 numeral system. Instead of basing their system on the digits on their hands, they based it on the spaces between the digits.

The Ancient Mayans used a base-20 system, as they counted with the digits on their hands and feet. They lived in a hot climate where people didn’t wear closed toe shoes.

Today’s computer scientists use 2, 8 and 16-base systems. For some mathematical work base-12 is more convenient than base-10. For this base-12 system they usually use the normal 0,1,2,3,4,5,6,7,8,9 numerals and add the letters a and b to make twelve (0,1,2,3,4,5,6,7,8,9,a,b). It goes without saying that these mathematicians, often university professors and researchers, are using this system to perform higher levels of calculations than you or I perform in our daily lives. They aren’t counting change at the grocery store.

Our normal lives show the vestiges of ancient numeral systems. We sometimes count with Ancient Roman numerals (Super Bowl XXIV, King Richard III), letters (chapter 4a, chapter 4b, chapter 4c… Notice how this combines two different systems, standard numerals with letters) and tally marks. We group loaves of bread, inches and ounces by the dozen, and mark time in groups of sixty (60 seconds per minute, 60 minutes per hour). Counting inches and ounces by twelve comes from the Ancient Romans. Our organization of time in groups of 60 comes from the Sumerians, an ancient civilization that used a base- 60 system.

The traditional counting of bread into groups of twelve has practical convenience. At the market, a dozen loaves can be divided into whole loaves by two, three or four. Ten loaves can only be divided by two into whole loaves. Sellers and customers prefer the grouping that gives more whole loaf options, not wanting a loaf to be torn apart. This should give you an idea why feet and yards are divisible by twelve, and there were twelve pence in a shilling— you get more ‘whole’ fractions out of twelve than you do ten.

These have been just some examples of other numeral systems, as there have been a wide and varied number over history. This not only includes systems with different bases, but with different kinds and numbers of numeral symbols. In Ancient Eastern countries, physical rods were used to represent numbers. The number, position, direction and color of the rod represented a number. In Ancient Egypt, pictures, known as hieroglyphics, were used to represent numbers. One thousand was written as a lily, and 10,000 as a tadpole. The Ancient Hebrews had a similar system to ours, except they used 27 different symbols to our ten. For the Hebrews, numbers 20, 30, 40, etc each got its own unique symbol.

Ancient Egyptian numerals for 1,000 (lily flower) and one million (man with raised arms)


Tallying is an ancient basic counting system many of us use. The practical problem with this system is that numbers like 500 and 10,000 require a whole lotta tally marks. 500 requires 500 tally marks. Over history, numeral systems have changed and evolved to correct inconveniences like this. Notice we use the tally system only for simple tasks, like keeping score in a ping pong game and marking days.

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A kid’s counting system: Eeny meeny miny moe

Kids have long used counting rhymes to decide who is it. The below common rhyme does the equivalent of counting to twenty, with the last word being the twentieth word.

Eeny, meeny, miny, moe / Catch a tiger by the toe If he hollers let him go / Eeny, meeny, miny, moe

There are a few interesting things about this eeny meeny counting system. First, it is quasi base-20, not our normal base-10. Second, words are used as numerals, or as the practical equivalent of numerals. Kids could count to 20 for the same practical result, but they chose to use words. Third, while lucky 7, 10 and unlucky 13 have popular importance compared to other numerals in our base-10 system, the seventh, tenth and thirteenth words in the rhyme do not.

This is an example where a different counting system changes what numbers are perceived as important. Most kids who count with this rhyme aren’t even aware which are the seventh, tenth and thirteenth words.

Humans often say they can’t conceptualize numbers in anything but the normal base-10, but here is a base-20 words counting system that we have all used. Granted this counting system is simplistic in the extreme, used for one and only one purpose— to count to twenty (moe). You wouldn’t want to try and use it to calculate your taxes.

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Numerals and human psychology

Humans form psychological attachments and biases for the numeration systems they use. Having grown up using a particular system, and seeing all those around them using the same, many people assume their numeration is absolute and eternal. Before reading this chapter, you may not have known or thought about the existence of other systems. Your base-10 system was all you knew, the prism which you saw the universe. 10, 100 and 1000— popular products of your base-10 system— are numbers you are attracted to. Thinking in base-8 or base-7 is foreign.

It’s telling to look at how humans change their perception from system to system, and how a change of numeration system changes peoples’ perceptions of things. The perception is not just about the numeration system itself, but the things the numeration system is used to count— objects, time, ideas.

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As the earlier tables showed, a different base numeral system doesn’t change the accuracy of our calculations or the physical objects we calculate. However, if we retroactively changed our base-10 system to a non base-10 system (like say the Yuki’s base-8 system) we would change how humans perceive and react to objects and concepts.

As with the high rise buildings and the superstitious renters, the historical changes would be caused in large part by human perceptions of the numerals themselves rather the things the numerals represent. No matter what the Mexico City building owner calls the thirteenth floor, it is the same floor. If he changes the label on the elevator directory from ‘13’ to ‘9988’ or to ‘789’ or to ‘Q,’ it is the same floor with the same walls, ceiling and windows and distance above the sidewalk. The numerologist apartment seekers aren’t reacting to the floor but to the symbol ’13.’ It should not surprise that a change to the symbols, such as caused by the changing to a new counting system, will change their reaction to the floors, along with many other things.

With a large lot of stones lined up on a table, changing the numeral system has no direct effect on the amount or physical nature of the stones. With a new counting system, the stones would be the same stones, but many to most would be assigned different numeral names. While the stones are the same stones no matter what we call them, human perceptions of the stones change as the stones’ numeral names change. Under our popular base-10 system, humans consider certain numerals to be special, including 10, 100, 1000 and 13, and react accordingly to objects labeled with these names. With the new numeral representations, humans’ perception and treatment of the stones will change. If before a person avoided a stone because it was unlucky 13, in the new system a different stone would be called 13. If in the old system the stone labeled ‘100’ was singled out as special, in the new system ‘100’ would represent a different stone.

If a human is asked to count and group the stones, the grouping will change with the different counting system. In the base-10 system, it’s likely the person would make piles of 10 or 25 stones or similar standard. In an 8 or 9 base system, the number and size of the piles would be different. To someone standing across the room, the rock design would be different. Her aesthetic reaction to the formation would be different.

This shows that your numeration system isn’t just an objective observation system, but helps form how you perceive objects. Under a different system, you would perceive things differently.

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Changing numeral systems, changing history

As a numeration system changes how we perceive, organize and react to things, a retroactive change to the numeral systems would change human history. The amount and type of change can be debated, but today’s history books would read different. With a change to the standard numeration system, time would remain the same but human marking of time would change. The decade, century and millennium equivalents would be celebrated at different times. No Y2K excitement at the same time as we had. Special milestones, like current marriage 10th or 25th anniversaries, would be at different times. People who now receive 30 years of service awards might receive equivalent awards but after a different duration.

Think of all those sports championships decided in the last moments, including the improbable upsets and bloop endings. If the events took place at different times and under different numeral influenced conditions some of the outcomes would be different. If an Olympic sprint is decided by a fraction of a second, it’s unlikely the first to last place order would be identical if it took place the day before with the runners in switched lanes and running a different length race. The changes to marking of time and distance would likely result in different gold, silver and bronze medal winners over the years. If a horse race was a tie, it is unlikely the same horses would tie if the race had been run earlier or later in the day or on a different day over a different length race. Realize that the change to the numeration system would likely change the standard race distances, even if the changes were just slight.

Think of all the razor close political elections. If the elections took place at a different time, even if just a day earlier or later, it’s possible some would have different outcomes. A few of the outcomes could have been for President, Prime Minister, judge or other socially influencing position. Think of all those close historic battles that may or may not have had a different outcome if started at different times, using different size platoons and regiments and Generals who made decisions using different number biases. Napoleon Bonaparte was superstitious of 13 and made his government, social and military plans accordingly. Think of the influential or not yet influential people who died at relatively young ages in accidents, from Albert Camus to General Patton to Buddy Holly. James Dean died in a sports car crash at age 25. Would he have crashed if he started his drive at an earlier or later time? Popular perception of the actor no doubt would be quite different if we watched him grow old and bald.

The powerful nineteenth century Irish Leader Charles Stewart Parnell would not sign a legislative bill that had thirteen clauses. A clause had to be added or subtracted before it could become law. Irish law would have been different under a different numeral system.

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United States consumer prices would likely be affected by a different numeral system, if just marginally. Again, this would be due to human psychological perceptions of numerals.

Even though most current US sellers and buyers think nothing of one penny, often tossing it in the garbage or on the sidewalk, sellers regularly price things at $9.99 instead of $10, and $19.99 instead of $20. Check the newspaper ads. This pricing is purely aesthetic, intending to play on consumers biases towards numerals.

The shallowness of this 1 cent game is illustrated when it is used by stores that have a ‘give a penny, take a penny’ tray, and that it is used in many states with different sales tax rates. Most people psychologically affected by $9.99 pricing at home are also affected by $9.99 pricing when traveling by car across the country. That the daily change in sale tax charge dwarfs the one cent between $9.99 and $10, illustrates the traveler’s irrationalness.

Under a base-9 numeral system that omits the numeral ‘9,’ $9.99 and $19.99 would no longer exist, and the visually appealing “one cent below big number” pricing would land elsewhere. In a 9 digit system, it’s likely that there would be many $8.88 and $18.88 pricings in newspaper ads, and the same types of travelers would be attracted to $8.88 and $18.88 prices as they go state to state even though the taxes change state to state.

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There are a variety of intertwined reasons behind irrational biases towards numerals and numeral systems.

One reason is people form psychological attachments towards a system, its symbols and the standard groupings of objects made from the system. A three digit numeral price ($9.99) looks distinctly different than a four digit numeral price ($10.00), literally being shorter. One hundred stones grouped into 10 groups of 10 each will look different than 11 groups of 9 stones each with one left over. It’s the same amount of stones, but their physical designs look different. There’s an aesthetic aspect to how humans view symbols and groupings.

Closely related reasons are tradition and habit. If you have used our base-10 system all your life, it’s as natural to you as your native spoken language. In fact words such as nine, ten and decade are part of your daily vocabulary. If everyone you know uses this numeral system, the idea of using a different system may not have even crossed your mind before now. The idea of calculating using a base-8 or base- 11 system seems strange and even unnatural to most people because they were raised on base-10.

Another reason behind irrational biases towards numerals is the seeming, if nonexistent, absoluteness of the familiar numerals. While the true nature of time, supernatural, war, love and the cosmos are shrouded in mystery, the numerals traditionally used in representing these things seem tangible, concrete. Unlike philosophical abstractions, numerals can be written down and typed into the calculator. Even little kids can count numerals on their fingers. That folks like Isaac Newton and Albert Einstein used these same numerals seem to numerologists to indicate the numerals’ potency. Though, if asked, both scientists would agree they could have used other numeral systems to do their work, and there was nothing uniquely special about the system they adopted.

Numerals are used only as convenient notations, proverbial post-its to label objects. They have no absolute, inborn connection to the things they represent. Whether you call the animal cat or gato it’s the same animal, and whether you call a number 5, five or V, it’s the same number. Whether you count a grove of trees with a base-10 or a base- 8 system, they are the same trees. If you count and label the trees a,b,c,d,e,f,g, they are still the same trees. Numerologists incorrectly assign an absolute meaning and identity to the numerals that doesn’t exist.

Even in academia, mathematicians considered to be too enamored with the beauty of numbers at the expense of practical use are sometimes derogatorily called numerologists by applied scientists like engineers. Mathematicians are as influenced by aesthetics as the rest of us.

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Sounds Good

Many Chinese judge numbers as good or bad by what words they sound closest to. As their pronunciation of 3 sounds closest to their word for ‘live,’ 3 is considered good. Their pronunciation of 4 sounds close to their word for ‘not,’ so is often considered negative.

China is a huge country with many dialects. As numbers and words are pronounced differently in different areas, a number’s perceived goodness and badness depends on where you are. For example, 6 is considered good in some places and bad in others.





Cognitive Dissonance


The theory developed by famous American psychologist Leon Festinger, cognitive dissonance is the mental discomfort experienced by a person who has conflicts or contradictions between ideas, values, behavior and/or information at the same time. An example of this dissonance is if after deciding to go on a diet you eat a fried chicken and ice cream– there’s a contradiction between your idea and behavior that will be obvious to you. Another is if you have a blind loyalty to a public figure and news comes out that he did something bad and against his professed views.

This conflict between expectations and reality is unconformable and the person has a strong ingrained drive to try to reduce the dissonance. Humans psychologically want constancy between their expectations and reality.

There are countless ways to try to reduce dissonance– healthy and unhealthy, rational and irrational, adaptive and maladaptive, honest and dishonest. Some will deal reasonably with the conflict– such as accepting that it was a mistake to eat that bad food and vowing to get back on the straight an narrow, or making one’s views about a public figure more realistic. Others will act poorly such as trying to delude themselves that fried chicken and ice cream for breakfast really is good for you, deny or dubiously justify the facts about the public figure or, as is often the case, ‘shoot the messenger.’ Denying facts, lying to oneself and dubious justifications are common maladaptive ways to try to regain cognitive consistency.

How we deal with dissonance says a lot about us and our maturity and is a major part of our personality. We admire people who admit to their mistakes, handle well unexpected setbacks and can change their viewpoints when given new information, and we express frustration with people who stubbornly cling to false notions and who react angrily to anyone who doesn’t agree with them.

The desire to reduce cognitive dissonance explains why we all often automatically deny facts or theories that go against our ideas or beliefs, even when we later accept them.