Caravaggio’s Judith Beheading Holofernes


Caravaggio’s circa 1598-9 145 cm × 195 centimeters oil painting on canvas ‘Judith and Holofernes’ is one of the early examples of his dramatic and visceral religious paintings that helped usher in the Baroque period.  The painting is Caravaggio’s interpretation of the Biblical  story about how the widow Judith saved her people, the Israelites, by seducing and getting drunk the Assyrian general Holofernes before decapitating him with a sword.  The painting is a reflection of Caravaggio’s strong personality and revolutionary artistic vision and the Catholic church’s then new aesthetic philosophy.

Caravaggio was a strong-willed, independent person with a  tempestuous and often violent personality. He led a life of both acclaim and trouble.   He was jailed numerous times, killed a man in a brawl, was badly injured in another brawl, had a death sentence put on his head by the Pope and spent his last years on the run, though painting all along the way.  He sued other artists who he felt copied his style and was said to be difficult to get along with.  His strong and dark personality is shown in his art, and, as his life became more desperate, his paintings became more and more sensational.

Caravaggio and Baroque painting went against the Renaissance tradition.  As exemplified by Botticelli’s humanists works, Renaissance paintings idealized its subjects and were based in gracefulness, harmony, symmetry and order.  Caravaggio didn’t idealize his subjects, but instead made them realistic, often showing bruises, scratches and wrinkles.  He was influenced by the realism of Northern European art.  Further, his religious scenes were sensational, immediate and often imbalanced, showing realistic action.   His scenes were snapshots from the most intense moments of the events.  His models were people from the streets, and the model Judith likely was a well-known prostitute.  Unorthodoxically, he did not make preliminary sketches, but painted directly from the models, drawing outlines in the paint with the end of his brush as he needed.  This technique and that his figures were so realistic and unidealized was controversial to many fellow artists.

Caravaggio lived at an opportune time for his artistic vision.   The Catholic church was changing their aesthetic philosophy, wanting religious paintings that emotionally connected with the parishioners.   Further there was much construction going on, with a need for many murals and artists.   

Caravaggio’s paintings were a sensation at the time due to their realism, immediacy and emotional directness.  Caravaggio had great success, but not without controversy.  The church enjoyed his paintings, but sometimes felt he went overboard in the violence and realism.  Still, even the paintings that were rejected were bought up by wealthy collectors.  Often called Caravaggisti or Caravagesques, other artists mimicked his style, and Caravaggio was an influence on such great painters as Rubens and Rembrandt.

Caravaggio was famous for use use of chiaroscuro and tenebrism, which helped with the drama and emotions of his subjects.   Chiaroscuro uses shading to give the figures a three-dimensional effect.  Tenebrism makes large areas  black, which focuses the viewer’s attention of the desired figures and action.  Along with the realistic and often gritty depictions of humans, these two techniques brought the scenes to life and made the audience emotionally connected to the familiar Biblical scenes.  Even today’s audiences are emotionally connected to the scenes.

Judith Beheading Holofernes has for centuries been a regular subject for artists, including in paintings, sculptures and even stained glass.  It was a subject long before Caravaggio and is still today.

Though Judith is considered by Christians to be an important and brave heroine, her portrayal in art had varied and developed over time before Caravaggio’s treatment.   She was originally portrayed as an entirely  wholesome Mary figure, sometimes praying, but was later developed into more of an ambiguous Eve– a heroic but somewhat fallen and sexualized figure.  Still, many of the art works before Caravaggio, and even after, showed her as stoic and aloof,  removed from the dirty deed.  The scenes are often cold, bloodless and sanitized in a highly stylized way.  They come across more as icons than realistic depictions.   

Caravaggio_Judith_Beheading_HolofernesIn Caravaggio’s version, he makes the scene immediate and dramatic using many techniques.   Unlike many before or after after the decapitation versions, he shows the moment of the decapitation, the sword half through the neck and blood spurting onto the white sheets.   Holovernies is screaming out, his body contorted, one hand desperately clutching the sheets.   Judith’s face shows both determination and disgust at what she is doing, perhaps having mixed feeling.  Behind her, her maid is full of anger and vengeful bloodlust, firmly holding the bag for the head.   

Judith is by definition the center of the story and scene.  She is the first name in the title and the protagonist of the Biblical story, something the audience of the time and was well aware of.    In the painting, she is shown as the central figure in that she she is most brightly lit, the only one entirely (laterally) in the scene, raised above the others and she is the one holding the weapon and performing the deed.  However, Caravaggio does include the others as key characters, which was often not the case for this subject.   In many earlier and later paintings of the scene, Judith was nearly the entire focus, with the severed a minor ornament or afterthought.  In some depictions you have to look hard to find the severed head.  Caravaggio has all three as integral actors of the action scene.  It is not a posed shot, but a snapshot of ongoing action.

The large size of the painting gives the scene power and presence, and the horizontal length makes it an action scene.  

This chiaroscuro and tenebrism makes the scene jump out at you and focuses on the action.  The surrounding darkness gives it a dark mood, and also give the theme of Judith sneaking in from the dark.  Western viewers tend to read text and pictures from left to right, so we see Judith and her maid sneaking into kill him.  That Holofernes is not entirely in the scene and the maid is cropped as she is entering the scene give the sense of action and movement, like a photographic snapshot.

Unlike the balanced Renaissance paintings, the figures are not balanced.  This makes it seem like a realistic scene, not unlike one of Caravaggio’s real life brawls, with the two women rushing in and ganging up on the single foe.

There is a shallow dark background  and a spotlight on the action.  This focuses attention and give drama, like actors spotlit on a darkened stage.   

In the inky background is a blood red cloth, a billowing banner of victory and glory, but also of bloody murder.  Judith herself is draped in both virginal white and red, symbolizing the Eve-like dichotomy.  In an earlier version, Judith is bare breasted, suggesting she had just left the bed and slept with the General.  She is strong and determined, with her sleeves rolled up, but she also keeps the dirty deed at arm’s length, a sign of her distaste and repulsion for the job.  The mixed emotions and symbols makes her complex and ambiguous, unlike the many other single-minded, one-dimensional  caricatures of her in earlier paintings.

It is noteworthy that many believe that Caravaggio saw the public beheading of Beatrice Cenci, a young woman who helped killed her insenstous father in a controversial and sensational case.  This historical footnote not only accounts for the realism in the beheading of Holofernes but perhaps says that there may have been mixed emotions about beheading.   Many at the time thought the beheading of Cenci was unjust, as she was a victim too, but the Pope said she must be killed.

People today value this painting for many reasons.  As with the original audiences, we enjoy it as an as immediate and visceral telling of a famous Biblical story.    It is fascinating to compare it to other depictions of the event, before and and after.  There are many takes on this story, showing many different views of Judith.   It is valued as a landmark in art, a departure from the Renaissance and the beginning of the Baroque period.   It is valued as a great influence on later art, including today’s.  The work is imitated, often satirically, today.   You see the tenebrism and the spotlight effect in many of today’s movies and photography.

Wikipedia articles on Caravaggio, Beatrice Beheading Holofernes and Beatrice Cenci

Sandro Botticelli’s Primavera


Sandro Botticelli’s Primavera, or Allegory of Spring, is a famous large (over 6 x 8 feet) 15th century artwork commissioned by the Medici family, Botticelli’s common patron and the major patron and  influence of Florence Italy’s Renaissance art.  Botticelli’s work falls into the early Renaissance period and he was a pioneer in the use of Pre-Christian Greek and Roman mythology in the era’s work.

The exact history of the painting is unknown.  The exact date it was made is not known (Wikipedia says “circa 1482”) and its current name was assigned long after Botticelli’s death.  However, according to and other sources, “it was probably created for the marriage of Lorenzo di Pierfrancesco (a cousin of the powerful Lorenzo the Magnificent Medici).”  Many believe this was made for his bedroom and this would mean it would have been the painting first seen by his wife on their wedding night.

The exact meaning of the painting is also unknown, but most scholars believe it is about spring, love and marriage.  Scholars generally believe the figures are as follows: On the far left is Mercury, the Roman God of May in his identifying winged shoes.  It is ambiguous what he is doing but he may be dissipating the clouds of winter. On the far right is the god of wind Zephyr, who captured the nymph Chloris, to his left, and forced her to be his wife.  Afterward, he felt bad about this and made her into Flora, the goddess of Spring.  Flora is shown to the left of Chloris, in a floral dress and spreading out flowers.  To the right of Mercury are the Three Graces, Chastity, Beauty and Love.  At the top center is Cupid, and his arrow of love and marriage about to strike.  Except for Mercury, the figures are barefooted in a natural setting full of plants and flowers.

At the physical and figurative center is Venus.  Venus was the Roman goddess of love, sex, beauty and fertility.  She also resembles a Mary figure, both in pose and with Mary commonly depicted in red.  She is shown as the figurative central figure as she is placed in the center, raised above others, is framed by the foliage, and has Cupid, her son, directly above her head.  Further, while the others looking in various directions doing various often playful things, Venus is looking forward with a motherly expression.  She appears as a motherly figure overlooking, tempering and approving the goings on and the themes of the painting.

As an artwork for the bedroom of newlyweds (if that is the case), the painting is an enjoyment for the eyes, a furnishing for the room and a symbol of love, youth, spring and marriage.  The painting is a secular ‘private’ painting rather than a church  one, and, though overseen by a semi-Mary figure, it is filled with ‘pagan’ gods and goddess.

To modern minds, Zephyr forcing Chloris to marry him is interesting theme.  Back then, women didn’t get to chose their husbands.  However, the story says that being forced to marry a Medici had its perks of glory and riches. Chloris is made into a beautiful and celebrated goddess.  This could also be seen as an apology or justification to the wife.  Today’s feminists will have a lot to say about this.

Botticelli-primaveraWhen I first looked at the painting I found it formal, somber and serious, as the people seemed so serious with unsmiling expressions and the overall painting was literally dark.  However, after reading more, I realized it was the opposite.  The seeming seriousness is a relative thing to my modern sensibilities.   Back then, they likely didn’t show bouncing people and overly smiling faces as they do in modern magazine ads ads televisions situation comedies.  Upon my closer and second look, the faces are very human, fresh, young, beautiful and the whole scene and is natural and relaxed.  Upon close inspection of Flora’s face, she has partially open mouth that expresses a sexuality and having fun.  Except for Venus, the figures seem playful or at least action filled.  They are facing various directions, the Three Graces are dancing in flowing dresses, Zephyr is shown in amorous pursuit of Chloris, Flora is spreading flowers.  Except for Mercury and Venus, the figures are barefoot in the grass.  Further, the way the figures are spread is informal compared to the many formally structured Christian paintings, including those of Botticelli.  In Primavera, the people are doing their own things and the whole thing resembles a dance or celebration.

The painting was meant as a feast for the eyes, a glorification of youth and spring and love.  The Roman mythology is opposed to the stern, formal Christian aesthetic.  Today’s audiences appreciate how sensuous Botticelli’s faces are, how beautiful and modern they look. It strikes them how they look like real, modern people.

I learned that the physical darkness I commented on  is because the painting has darkened over time, a common occurrence with varnished old paintings.  Revarnishing old paintings to reveal the original tones and colors is done by conservators, though they sometimes choose not to do it for certain priceless paintings.  I was told by a recent American visitor to the Louvre that they have chosen not to revarnish the Mona Lisa.

The painting is a famous example of  humanism, which is  closely associated with the Italian Renaissance, Humanism is a departure from early Christian spiritual-centric view.  Humanism placed emphasis on humans as individuals, capable of reason and rationality, and was a forerunner of modern rational thought and science.  It produced such thinkers as Leonardo da Vinci, Machiavelli, Copernicus and Galileo. It is shown in art by realistic secular depictions of humans and the use of Pre-Christian Roman and Greek myth (‘pagan gods.’)  Before humanist influence, Primavera would not have been approved of by the church.  The secular humanism is also symbolized in that it was made for a Medici, (it is believed) as a furnishing for his bedroom.

Venus is also commonly used a symbol of humanism, and this painting shows that humanism, human individuality and reason, are central to all the things going.

The orange grove was a symbol of the Medicis and most of the models for the gods and goddesses in this painting were Medici or other higher ups in Florence society.  The painting was designed for and shows the Medici world and society.  Venus oversees a Medici world.  

This whole painting is an expression and glorification of the Medici humanist philosophy and aesthetic. While Christian, the Medici family were secular leaders and bankers with early persuits.  The painting has a semi-Madonna, but is about their beliefs in individuality, rationality and the enjoyment of life  This is what they felt, but the painting may also also an apology or justification of their secular ways.  The use of an overseeing Madonna may be seen as justification of their earthly ways.  Even the Virgin Mary approves and watches over in a motherly manner human individuality and reason.

The painting is a tempera on wood panel, called a panel painting.  Tempera was a standard form of paint and painting before oil painting.  Wood was the common backing before canvas.  Some have commented on the subtilty of the colors and details of Primavera.  This is the nature of tempera painting.  The paint is  translucent and quick drying, and the artist paints by building up the details and colors painted line by painted line.  This leads to gradual gradations in color changes, so the details of a face lack the great contrast of an oil painting or acrylic and leads to the delicacy that Botticelli’s paintings were known for.  The paint itself on a tempera painting is physically flat, as opposed to the common raised  relief surface of oil paintings.  Tempera paint goes hand in hand with Botticelli’s delicate people.

What is interesting about this humanist painting is that Botticelli later became deeply religious under the influence of the strict Dominican friar Girolamo Savonarola and renounced his humanist paintings.  It has been said that he may have burned many in the Bonfire of the Vanities. He gave up the pre-Christian and secular themes and turned to devout Christian themes.  His later Christian paintings were more formal and orderly, lacking the relaxed ‘fun’ look of his pagan paintings.   

The painting is valued for many reasons.  It is enjoyed as an artwork showing Botticelli’s talents, as a lush display celebrating ever popular themes  love, spring, youth and beauty. The painting is valued as an example of humanism and Medici thought and influence. In context of Botticelli’s works and life, it shows the battles between religious and secular power and thought, both in Florence history Italy and within one person’s mind.  It is valued as a physical and historical artifact, showing the types and paint and backing.  Dendrology is used in the study of panel paintings.   It is valued as a personal artifact of a famous family.  People are fascinated by things owned by famous people, even if it’s a robe or a chair.  It is also valued as a puzzle.  Humans are drawn to ambiguous stories and unsolved mysteries.  This is why this, and other paintings, still fascinate audiences and are studied endlessly.



Wikipedia articles on Botticelli’s Primavera, Botticelli and Girolamo Savonarola

“La Primavera by Botticelli”  at

“La Primavera by Botticeli” at

Sandro Bitticelli at history101link

Articles on Botticelli, the Medicis and Girolamo Savonarola in the New World Encyclopedia

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.

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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.

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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.

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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.

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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.

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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.

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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.

Mirages: Not Incorrect Views of Reality, Just Different


Commonly associated with nature, mirages are visual illusions where what we see is correct, but abnormal. Mirages in nature are most commonly caused by unusual bending of light under unusual air conditions. The view can be so abnormal that the viewer ‘can’t believe his eyes.’

The most famous mirage is when it erroneously appears as if a pool of water is in the desert. More than a few thirsty wanderers have found nothing but disappointment ahead. The above pictured water in the road is the same type of mirage. Another related mirage is when sailors see an upside down ship in the sky. Enough to convince a pirate to swear off the hooch

These three particular mirages happen when there are abnormal layers of hot versus cold air that cause the light to refract, or bend, from its usual course. This bending causes an object to appear in an unexpected place. In the desert and highway a piece of the blue sky appears below the horizon, and is wrongly interpreted to be water. At sea a ship is bent upwards so it appears to be in the sky air.

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A mirage is called a superior mirage where the object appears above where it normally appears (boat in sky). An inferior mirage is when the object appears below the where it normally appears (sky in desert).

The inferior mirage happens when there is hot air near the ground. It shouldn’t surprise that inferior images commonly happen when the ground surface is hot (desert, summer highway).

A superior mirage happens when there is cold air near the surface. They commonly appear in the arctic and over frozen water.

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Sunrise mirage. One of the most striking superior mirages is a sunrise mirage. These are seen over frigid areas, such as frozen lakes and seas. The light of the sun is bent upwards along the earth’s curved surface making the sunrise appear earlier than normal. The sun is also distorted. Sometimes two suns are seen at once, one superimposed over the other.

sunrise mirage over a frozen Minnesota lake

sunrise mirage over a frozen Minnesota lake

This mirage was noticed centuries ago by Western explorers stranded in the arctic over the winter. That far north there is no sun 24 hours a day for much of the winter. The explorers were surprised when the first sunrise of the season appeared days before it was supposed to. It wasn’t until centuries later that experts realized the explorers had witnessed this mirage.

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Water can bend light just as air can, the light bending from air to water or water to air (or air to water to air, etc). A hardboiled egg distorts from normal appearance in a glass of water. The experienced spear fisher knows to spear to the side of the image of the fish or he will miss. Stones appear to ripple and wave in a crystal clear brook. One can study and demonstrate how mirages work with a drinking glass.


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The mirages aren’t wrong views of an object, just different. Our normal vision involves distortions, including to color, details and angles, so one can hardly claim our normal vision is perfect and anything different imperfect. When they wish a better look, people with 20/20 vision intentionally distort their vision with magnifying glasses, binoculars, periscopes, video cameras and sunglasses.

Sailor looking through a submarine periscope

Sailor looking through a submarine periscope

When you view a bird through binoculars the lens distorts the light to make the bird appear larger and more detailed. You don’t consider the binocular view of the bird wrong. You consider it to be more reliable than your naked eye view (“I thought it was a hawk, but it’s just a crow.”) A submarine’s periscope bends light via mirrors so a sailor can see above water. The sailor doesn’t consider the view make believe. He considers it a view of reality.

Humans classify views as mirages when they are abnormal and mysterious (at least to the viewer). There are many brilliant atmospheric effects that aren’t considered mirages, as they are well understood. Little is more magnificent than a rainbow, but they are frequent and people know there is a scientific explanation. Fog, snow, sunsets and seeing our reflection in puddles would be considered astounding if they weren’t common events.

That thousands of pounds of bright white snow changed into grass in one (hot) weekend doesn’t cause you to write to Ripley’s Believe It Or Not. You are well aware heat melts snow and underneath the snow is grass. You mowed that grass a few months ago. Ripley himself likely had this occur on his lawn numerous times. The changing of the season is impressive, but only a mirage to folks who have no memory of it.

After waking up in the morning and seeing the season’s first blanket of snow, my very young sister turned to my dad and said, “Daddy, how’d you do that?”

When people move to new geographies they often experience new weather phenomena. When I moved to Seattle, I experienced unusual (to me) night lighting effects caused by Puget Sound and clouds. One night I thought there was a large fire on the other side of the sound. I later found out it was the lights of a distant hill-hidden town reflecting off of low clouds. This created a low, fiery glow. I see this lighting and it no longer fazes me. The first time I saw it, it was a mirage. Now it’s town light reflecting off of low clouds.

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You can’t trust water. Even a straight stick turns crooked in it.” — W.C. Fields