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Uniformities are/not used to simplify our lives.

Golden ratio

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In the world of art, architecture, and design, the golden ratio has earned a tremendous reputation. Greats like Le Corbusier and Salvador Dalí have used the number in their work. The Parthenon, the Pyramids at Giza, the paintings of Michelangelo, the Mona Lisa, even the Apple logo are all said to incorporate it.

It's bullshit. The golden ratio's aesthetic bona fides are an urban legend, a myth, a design unicorn. Many designers don't use it, and if they do, they vastly discount its importance. There's also no science to really back it up. Those who believe the golden ratio is the hidden math behind beauty are falling for a 150-year-old scam.

First described in Euclid's Elements 2,300 years ago, the established definition is this: two objects are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The value this works out to is usually written as 1.6180. The most famous application of the golden ratio is the so-called golden rectangle, which can be split into a perfect square, and a smaller rectangle that has the same aspect ratio as the rectangle it was cut away from. You can apply this theory to a larger number of objects by similarly splitting them down.

The golden ratio is always going to be a little off.

In plain English: if you have two objects (or a single object that can be split into two objects, like the golden rectangle), and if, after you do the math above, you get the number 1.6180, it's usually accepted that those two objects fall within the golden ratio. Except there's a problem. When you do the math, the golden ratio doesn't come out to 1.6180. It comes out to 1.6180339887... And the decimal points go on forever.

"Strictly speaking, it's impossible for anything in the real-world to fall into the golden ratio, because it's an irrational number," says Keith Devlin, a professor of mathematics at Stanford University. You can get close with more standard aspect ratios. The iPad's 3:2 display, or the 16:9 display on your HDTV all "float around it," Devlin says. But the golden ratio is like pi. Just as it's impossible to find a perfect circle in the real world, the golden ratio cannot strictly be applied to any real world object. It's always going to be a little off.

It's pedantic, sure. Isn't 1.6180 close enough? Yes, it probably would be, if there were anything to scientifically support the notion that the golden ratio had any bearing on why we find certain objects like the Parthenon or the Mona Lisa aesthetically pleasing.

But there isn't. Devlin says the idea that the golden ratio has any relationship to aesthetics at all comes primarily from two people, one of whom was misquoted, and the other of whom was just making shit up.

The first guy was Luca Pacioli, a Franciscan friar who wrote a book called De Divina Proportione back in 1509, which was named after the golden ratio. Weirdly, in his book, Pacioli didn't argue for a golden ratio-based theory of aesthetics as it should be applied to art, architecture, and design: he instead espoused the Vitruvian system of rational proportions, after the first-century Roman architect, Vitruvius. The golden ratio view was misattributed to Pacioli in 1799, according to Mario Livio, the guy who literally wrote the book on the golden ratio. But Pacioli was close friends with Leonardo da Vinci, whose works enjoyed a huge resurgence in popularity in the 19th century. Since Da Vinci illustrated De Divina Proportione, it was soon being said that Da Vinci himself used the golden ratio as the secret math behind his exquisitely beautiful paintings.

One guy who believed this was Adolf Zeising. "He's the guy you really want to burn at the stake for the reputation of the golden ratio," Devlin laughs. Zeising was a German psychologist who argued that the golden ratio was a universal law that described "beauty and completeness in the realms of both nature and art... which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical."

He was a long-winded guy. The only problem with Zeising was he saw patterns where none exist. For example, Zeising argued that the golden ratio could be applied to the human body by taking the height from a person's navel to his toes, then dividing it by the person's total height. These are just arbitrary body parts, crammed into a formula, Devlin says: "When measuring anything as complex as the human body, it's easy to come up with examples of ratios that are very near to 1.6."

In my own work, I can't ever recall using the golden ratio.

But it didn't matter if it was made up or not. Zeising's theories became extremely popular, "the 19th-century equivalent of the Mozart Effect," according to Devlin, referring to the belief that listening to classical music improves your intelligence. And it never really went away. In the 20th century, the famous Swiss-French architect Le Corbusier based his Modulor system of anthropometric proportions on the golden ratio. Dalí painted his masterpiece The Sacrament of the Last Supper on a canvas shaped like a golden rectangle. Meanwhile, art historians started combing back through the great designs of history, trying to retroactively apply the golden ratio to Stonehenge, Rembrandt, the Chatres Cathedral, and Seurat. The link between the golden ratio and beauty has been a canard of the world of art, architecture, and design ever since.

In the real world, people don't necessarily prefer the golden ratio.

Devlin tells me that, as part of an ongoing, unpublished exercise at Stanford, he has worked with the university's psychology department to ask hundreds of students over the years what their favorite rectangle is. He shows the students collections of rectangles, then asks them pick out their favorite one. If there were any truth behind the idea that the golden ratio is key to beautiful aesthetics, the students would pick out the rectangle closest to a golden rectangle. But they don't. They pick seemingly at random. And if you ask them to repeat the exercise, they pick different rectangles. "It's a very useful way to show new psychology students the complexity of human perception," Devlin says. And it doesn't show that the golden ratio is more aesthetically pleasing to people at all.

Devlin's experiments aren't the only ones to show people don't prefer the golden ratio. A study from the Haas School of Business at Berkeley found that, on average, consumers prefer rectangles that are in the range of 1.414 and 1.732. The range contains the golden rectangle, but its exact dimensions are not the clear favorite.

The designers we spoke to about the golden ratio don't actually find it to be very useful, anyway.

Richard Meier, the legendary architect behind the Getty Center and the Barcelona Museum of Contemporary Art, admits that when he first started his career, he had an architect's triangle made that matched the golden ratio, but he had never once designed his buildings keeping the golden ratio in mind. "There are so many other numbers and formulas that are more important when designing a building," he tells me by phone, referring to formulas that can calculate the maximum size certain spaces can be, or ones that can determine structural load.

There are so many other numbers and formulas that are more important when designing a building.

Alisa Andrasek, the designer behind Biothing, an online repository of computational designs, agrees. "In my own work, I can't ever recall using the golden ratio," Andrasek writes in an email. "I can imagine embedding the golden ratio into different systems as additional 'spice,' but I can hardly imagine it driving the whole design as it did historically... it is way too simplistic."

Giorgia Lupi of Accurat, the Italian design and innovation firm, says that, at best, the golden ratio is as important to designers as any other compositional rule, such as the rule of thirds: maybe a fine rule-of-thumb, but one that good designers will feel free to reject. "I don't really know, in practice, how many designers deliberately employ the golden ratio," she writes. "I personally have never worked with it our used it in my projects."

Of the designers we spoke to, industrial designer Yves Béhar of Fuseproject is perhaps kindest to the golden ratio. "I sometimes look at the golden ratio as I observe proportions of the products and graphics we create, but it's more informational than dogmatic," he tells me. Even then, he never sets out to design something with the golden ratio in mind. "It's important as a tool, but not a rule."

Even designers who are also mathematicians are skeptical of the golden ratio's use in design. Edmund Harriss is a clinical assistant professor in the University of Arkansas' mathematics department who uses many formulas to help generate new works of art. But Harriss says that the golden ratio is, at best, just one of many tools at a mathematically inclined designer's fingertips. "It is a simple number in many ways, and as a result it does turn up in a wide variety of places..." Harriss tells me by email. "[But] it is certainly not the universal formula behind aesthetic beauty."

The Sacrament of the Last Supper, 1955, Salvador Dali

If the golden ratio's aesthetic merit is so flimsy, then why does the myth persist?

Devlin says it's simple. "We're creatures who are genetically programmed to see patterns and to seek meaning," he says. It's not in our DNA to be comfortable with arbitrary things like aesthetics, so we try to back them up with our often limited grasp of math. But most people don't really understand math, or how even a simple formula like the golden ratio applies to complex system, so we can't error-check ourselves. "People think they see the golden ratio around them, in the natural world and the objects they love, but they can't actually substantiate it," Devlin tells me. "They are victims to their natural desire to find meaning in the pattern of the universe, without the math skills to tell them that the patterns they think they see are illusory." If you see the golden ratio in your favorite designs, you're probably seeing things.

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The Golden Ratio: Design's Biggest Myth | Co.Design

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In the world of art, architecture, and design, the golden ratio has earned a tremendous reputation. Greats like Le Corbusier and Salvador Dalí have used the number in their work. The Parthenon, the Pyramids at Giza, the paintings of Michelangelo, the Mona Lisa, even the Apple logo are all said to incorporate it.

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Is the spontaneous perception of connections and meaningfulness of unrelated phenomena. The term was coined by German neurologist and psychiatrist Klaus Conrad (1905-1961). Conrad focused on the finding of abnormal meaning or significance in random experiences by psychotic people.

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Apophenia is the spontaneous perception of connections and meaningfulness of unrelated phenomena. The term was coined by German neurologist and psychiatrist Klaus Conrad (1905-1961). Conrad focused on the finding of abnormal meaning or significance in random experiences by psychotic people. The term has found a place outside of psychiatry and is used to describe the natural tendency of human beings to find meaning and significance in random, coincidental, or impersonal data. Apophenia may be described as the tendency to find personal information in noise, e.g., happening upon an open safety pin and seeing the arms as a sign indicating the time your son committed suicide.

Pareidolia is a type of illusion or misperception involving a vague or obscure stimulus being perceived as something clear and distinct. For example, in the discolorations of a burnt tortilla one sees the face of Jesus. Or one sees the image of Mother Teresa in the folds of a cinnamon bun or Vladimir Lenin in the soap scum of a shower curtain.

Apophenia and pareidolia can occur simultaneously as in the case of seeing a birthmark pattern on a goat as the Arabic word for Allah and thinking you've received a message from god. Likewise, not only seeing the Virgin Mary in tree bark but believing the appearance is a divine sign brings together apophenia and pareidolia. Seeing an alien spaceship in a pattern of lights in the sky is an example of pareidolia, but it becomes apophenia if you believe the aliens have picked you as their special envoy. Seeing Satan in the smoke of a burning building slips from pareidolia to apophenia when the viewer starts thinking that Satan is giving the world a sign that he is alive and well.

Under ordinary circumstances, apophenia provides a psychological explanation for many delusions based on sense perception. For example, it explains many UFO sightings, as well as the hearing of sinister messages on records played backwards. Pareidolia explains Elvis, Bigfoot, and Loch Ness Monster sightings. Pareidolia and apophenia explain numerous religious apparitions and visions. And they explain why some people see a face or a building in a photograph of the Cydonia region of Mars.

Under clinical circumstances, some psychologists encourage pareidolia and apophenia as a means to understanding a patient, e.g., the Rorschach ink blot test.

According to Peter Brugger of the Department of Neurology, University Hospital, Zurich, "The propensity to see connections between seemingly unrelated objects or ideas most closely links psychosis to creativity ... apophenia and creativity may even be seen as two sides of the same coin." Some of the most creative people in the world, then, must be psychoanalysts and therapists who use projective tests like the Rorschach test or who see patterns of child abuse behind every emotional problem. Brugger notes that one analyst thought he had support for Freud's notion of penis envy because more females than males failed to return their pencils after a test. Another spent nine pages in a prestigious journal describing how sidewalk cracks are vaginas and feet are penises, and the old saw about not stepping on cracks is actually a warning to stay away from the female sex organ.

Soon after his son committed suicide, Episcopalian Bishop James A. Pike (1913-1969) began seeing meaningful messages in such things as a stopped clock, the angle of an open safety pin, and the angle formed by two postcards lying on the floor. He thought they were conveying the time his son had shot himself (Christopher 1975: 139).

Brugger gives examples of pareidolia and apophenia from August Strindberg's Occult Diary, the playwright's own account of his psychotic break:

He saw "two insignia of witches, the goat's horn and the besom" in a rock and wondered "what demon it was who had put [them] ... just there and in my way on this particular morning." A building then looked like an oven and he thought of Dante's Inferno.

He sees sticks on the ground and sees them as forming Greek letters which he interprets to be the abbreviation of a man's name and feels he now knows that this man is the one who is persecuting him. He sees sticks on the bottom of a chest and is sure they form apentagram.

He sees tiny hands in prayer when he looks at a walnut under a microscope and it "filled me with horror."

His crumpled pillow looks "like a marble head in the style of Michelangelo." Strindberg comments that "these occurrences could not be regarded as accidental, for on some days the pillow presented the appearance of horrible monsters, of Gothic gargoyles, of dragons, and one night ... I was greeted by the Evil One himself...."

Brugger's research indicates that high levels of dopamine affect the propensity to find meaning, patterns, and significance where there is none, and that this propensity is related to a tendency to believe in the paranormal.

In statistics, apophenia is called a Type I error, seeing patterns where there are none. It is highly probable that the apparent significance of many unusual experiences and phenomena are due to apophenia and pareidolia, e.g., ghosts and hauntings, EVP, numerology, the Bible code, most forms of divination, the prophecies of Nostradamus, remote viewing, and a host of other paranormal and supernatural experiences and phenomena.

One particular area where apophenia plays a significant role is in the subjective validation that occurs in a cold reading by an alleged psychic, astrologer, palm reader, tarot reader, or medium. Readers throw out words and expressions, sitters find meaning and significance in them. Even if the words are random and impersonal, sitters strive to find personal meaning in them.

It is well known that apophenia and pareidolia are related to the human instinct to see patterns and find meaning; both are rooted in our evolutionary history. Our species has stupendous pattern-recognition abilities, so stupendous, in fact, that we often see patterns where there are none. We've evolved to find meaning in patterns and infer causal relationships from coincidences. "Our understanding of our world is systematically biased to perceive meaning rather than randomness and to infer cause rather than coincidence. And we are usually completely unaware of these biases" (Chabris and Simons, The Invisible Gorilla). Consider how automatic some of these brain processes are:

....visual areas of your brain can be activated by images that only vaguely resemble what they're tuned for. In just one-fifth of a second, your brain can distinguish a face from other objects like chairs or cars. In just an instant more, your brain can distinguish objects that look a bit like faces, such as a parking meter or a three-prong outlet, from other objects like chairs. Seeing objects that resemble faces induces activity in a brain area called the fusiform gyrus that is highly sensitive to real faces. In other words, almost immediately after you see an object that looks anything like a face, your brain treats it like a face and processes it differently than other objects. (Chabris and Simons, The Invisible Gorilla)

Add a little religious or political zeal to the brain's natural disposition to recognize faces in just about anything with a shape and a few shadows and you've got the recipe for a dozen tortillas with the "face of Jesus" imprinted on them or a single toasted cheese sandwich that reminds people of President Obama.

It seems likely that our associations of shapes, lines, and shadows are related to our current desires, interests, hopes, and obsessions, even though associative thinking is rooted in our evolutionary history. Most people recognize illusions for what they are, but some become fixated on the reality of their perception and turn an illusion into a delusion. A little bit of critical thinking, however, should convince most reasonable people that a potato that looks like the Hindu god Ganesh, a cinnamon bun that looks like mother Teresa, or a burnt area on a tortilla that looks like Jesus are accidents and without significance. It is more likely that the Virgin Mary one sees in the reflection of a mirror or on the floor of an apartment complex or in the clouds has been generated from one's own imagination than that a person who has been dead for 2,000 years should manifest herself in such a mundane and useless fashion.

Since most of us are innumerate, we often find significance in purely coincidental events. If you think of all the pairs of things that can happen in a person's lifetime and add to that our versatility at finding meaningful connections between things in ambiguous situations, it seems likely that most of us will experience many meaningful coincidences, but we are the ones who give them meaning. Given the fact that there are billions of people and the possible number of meaningful coincidences is millions of billions, it is inevitable that many people will experience some very weird and uncanny coincidences every day. Put another way, with a large enough sample size, just about any possible weird coincidence will happen. This is sometimes called the law of truly large numbers. Those who believe that these meaningless coincidences are truly meaningful call it synchronicity, an expression coined by Carl Jung.

further reading

Brugger, Peter. 2001. "From Haunted Brain to Haunted Science: A Cognitive Neuroscience View of Paranormal and Pseudoscientific Thought," Hauntings and Poltergeists: Multidisciplinary Perspectives, edited by J. Houran and R. Lange. McFarland & Company, Inc. Publishers.

Chabris, Christopher and Daniel Simons. 2010. The Invisible Gorilla: And Other Ways Our Intuitions Deceive Us. Crown.

Christopher, Milbourne. 1975. Mediums, Mystics & the Occult. Thomas Y. Crowell Co.

Leonard, Dirk M.A. and Peter Brugger, Ph.D. 1998. "Creative, Paranormal, and Delusional Thought: A Consequence of Right Hemisphere Semantic Activation?" Neuropsychiatry, Neuropsychology, and Behavioral Neurology, Vol. 11, No. 4 pp. 177-183.

Reed, Graham. 1988. The Psychology of Anomalous Experience : A Cognitive Approach. Prometheus Books.

Sagan, Carl. 1995. The Demon-Haunted World - Science as a Candle in the Dark. Random House.

Schick, Jr., Theodore and Lewis Vaughn. 2010. How to Think About Weird Things. Mayfield Publishing Company.

Zusne, Leonard and Warren Jones. 1990. Anomalistic Psychology: A Study of Magical Thinking 2nd edition. Lawrence Erlbaum Assoc.

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Apophenia is the spontaneous perception of connections and meaningfulness of unrelated phenomena. The term was coined by German neurologist and psychiatrist Klaus Conrad (1905-1961). Conrad focused on the finding of abnormal meaning or significance in random experiences by psychotic people.

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Counterclaim - no uniformity in art

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Researching the tenuous link between project management and art - between rigorous systems of thought and more free-flowing ones, really - I came across the work of Briggs and Collings, and loved it. [ link]
Biggs conceives the colour, Collings applies it. Our form is a series of triangles in a grid. We use it not as an idea in itself, or as a quote from life (wallpaper or golfing sweaters, for example), but as an inert container for colour. Meaning is everywhere, it looms up in spite of ourselves, but we aim to examine what the world looks like if you remove familiar forms with their confusing narrative resonances, and substitute a neutral structure. One that supports a complex, changeable colour harmony - a landscape painting without the landscape. Painting is in crisis - what is the point of an antiquated form in a world where representation is ubiquitous? Our answer comes from looking at its origin, both in the history of art and as the by-product of art's original intentions: beauty. As with all our recent work, the title comes from 'Genesis', whose origin myths are integral to Christian, Jewish and Islamic history. (In this case, 'Genesis 1:16': 'God made two great lights - the greater light to govern the day, and the lesser light to govern the night...'). These tales inform early Western depiction, which is the starting-point for our visual enquiry. [link] Interview: In 1951 Kenneth Martin wrote: 'The square, the circle, the triangle, etc., are primary elements in the vocabulary of form, not ends in themselves. In a rigorous form of art which uses as figures such formal elements, a complete pictorial expression can be achieved.' Or would you say that your starting point is inspired, more directly, by existing man-made architecture or structures in nature - your observations of the real world? MC + EB: If Kenneth Martin meant geometry is the vehicle, then we agree. For us, the grid is the thing by which we limit our visual enquiry so there's a chance of finding something out. It's the visually neutral basis from which we create many patterns or configurations. These are all simultaneously present in the one object. But only one is seeable at a time. We try to create multiple visual orders. The grid isn't really anything, but bringing out multiple visual rhythms is important, and producing patterns that constantly change. Our paintings are not immutable statements. First, a pattern seems to be put together in a certain way, and then another configuration appears, then another, and so on. We don't take on a vast field of enquiry. We don't feel we could discover anything substantial with too broad a scope, so we've decided on a narrow focus. Every painting is a proposal. We ask 'Is this true?' 'Does this seem accurate?' Sometimes we discover the answer is no, but sometimes, the limitations produce something really surprising, and we think, new. EB: Artists have put down many of their visual tools; I want our work to offer a tangible demonstration of what it might mean to have skill back. A lot of contemporary art is a like a geography lesson from a liberal-minded teacher with a bent for social studies. That's fine, but it seems to me that de-skilling is what happens to a trade before the axe falls, because the trade has been complicit with its own demise.

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A system that does not produce uniformity

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Researching the tenuous link between project management and art - between rigorous systems of thought and more free-flowing ones, really - I came across the work of Briggs and Collings, and loved it. [link] Biggs conceives the colour, Collings applies it. Our form is a series of triangles in a grid.

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Religion is based on assumptions; people belief in God without actual proof of his existence.Then, for instance, knowledge is spread through the Bible.

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The beauty of art - pattern

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Pattern is such a universal term that trying to define it may seem redundant. Nonetheless, definitions are usually the ones that keep hidden knowledge and cyphered information, even when they give an impression of commonplaceness. That is why we'll describe the basic term in order to determine the role of pattern in art and architecture, and see how the context affects the way we perceive it. A pattern is described as a " discernable regularity" by Wikipedia and as an " arrangement of repeated or corresponding parts " by dictionary.com. As you can already imagine, the applicability of patterns in visual art is probably as old as art itself. We only need to visualize a temple from ancient Greece, or any of its decorative elements, to give ourselves a great example of discernable regularity - both the facade, the plan and the decorations are related to one another and forming easily recoginzable patterns.

Pattern In Nature

Most of us imagine somewhat regular patterns when we hear the word. The association usually makes us think of garments , tiles and similar designed products, all of which are man-made and artificial, even when they tend to imitate occurrences that come from nature. Still, it's not wrong to say that nature has a rhythm of its own, even though recognizing its pattern takes deeper analysis and consideration. Although nature may seem, well - natural, its spontaneity is not necessarily mathematically unfathomable. Nature often follows the pattern of fractals, which were explained by the famous mathematician Benoit Mandelbrot in the 20th century. Fractals resemble " discernable regularity", but they include the possibility of scaling, mirroring and rotating elements, which makes the pattern much more complex and harder to comprehend than the simple geometric arrangements designed by humans. Nature's logic can be seen in the way trees branch into smaller parts that visually resemble their parent branches, or through tessellations that appear on flower petals, in foams and honeycombs. However, natural patterns are rarely entirely consistent, but much more due to the imprecise symmetry of the elements than the irregularity of the process.

How Patterns Are Translated From Nature

Paradoxically, the imperfection of nature has been desperately imitated by humans for centuries, with a brief moment of reverse logic in the 20th century, when the Modernists decided to make the most honestly industrialized objects and artworks and to celebrate the power of machines, i.e. the power of humans. On the other hand, the idea to translate patterns from nature often resulted in producing grids and arrays that would never occur naturally. Gifted craftsmen were able to produce visually appealing combinations of floral or nature-derived elements that served as decorations or embellished the garments used for clothes and furniture. But more often than not, these patterns couldn't escape the fact that they were shaped by a hand that is guided by a human mind - reductive and prone to disentangle. That's why William Morris' patterns are determinable at first glance, even though they mimic natural shapes. The other way to look at nature was to try to learn from its patterns - the tessellations, fractal derivations and flows, which was most cautiously studied through architecture. Still, one of the first architects who truly seemed to achieve the arbitrariness of nature was Gaudi, whose approach to architectural elements was sculptural rather than pragmatic, albeit underpinned by mathematics. Nature was generally examined throughout the entire Art Nouveau period in various ways, but it was still very clear that its patterns are vastly inimitable, at least visually.

A Real Man-Made Kind of Repetition

The 20th century was a period in which the most ordinary patterns were used and made as simple as possible, especially in architecture and precisely in order to make living as simple, as universal as possible. It was the age of the grid, in which several attempts were made to work out a universal language of visual data. De Stijl architects and artists had a vision to translate all meanings into visual patterns, using three elementary colors and a grid that articulates them. Bauhaus and the future Modernist movements that were more or less derived from it were all guided by the aesthetics that stemmed from the shapes and forms produced by the industry. In Modernism, patterns were used not only for facades or structural elements, such as columns or lintels, but for entire blocks and living units. Along with the age of mass production, buildings became too big not to be formulated into units, and so flats and offices were massively arranged into simple patterns. Le Corbusier used the term " machine for living in ", and oh how descriptive he was.

Pattern In Art

But regularity does not necessarily mean orthogonal assembling, nor is the orthogonal array necessarily boring. Visual artists have been deploying patterns in numerous ways with an awareness of the effect that it might have on the spectator. Namely, as we already discussed, human mind is somehow strangely satisfied when able to discern a certain rhythm (which is why we get so easily addicted to pop songs). Still, the repetition is only rarely made too apparent or left unaccompanied by at least one "irregularity" (which is why we get addicted to pop music and not techno). All joking aside, music truly is a good reference for analyzing patterns in art, since it could not exists without tact and rhythm - except for jazz perhaps, which only naturally occurred in the same moment as all the other revolutionary avant-garde movements in art. If we go by analogy, we could associate jazz with Dada and its complete detachment from any kind of pattern and order; and we could try to relate Hirst's spot paintings to happy electronic music, or as the artist put it - " To create that structure, to do those colours, and do nothing. I suddenly got what I wanted. It was just a way of pinning down the joy of colour." In between the two extremes that we have just listed, there's a myriad of artworks that use patterns here and there - in the background, for illusive optical effects , as a guideline for a good composition, or instead of using color and shade - well, even Pointillism could be considered to follow a certain pattern in terms of technique. Patterns were also widely used in applied arts, mostly for stained glass and tapestry in the past, and for graphic design today. Regardless of the discipline, pattern is always used either as mere decoration, for its pictorial qualities, or as a method that incorporates structure into the pattern, when the process behind its use is a bit more intellectual.

Editors' Tip: Pattern Design A beautifully illustrated guide to the use of decorative patterns in art and design explains how " the use of pattern in design is no haphazard matter, but a disciplined activity in which the artists must impose a pleasing order and structure on the whole to achieve an aesthetically satisfying end product ". The author Lewis Day imagined the book as a guide to artists who wish to explore the power and the potential of different art patterns and repeated ornaments, providing an in-depth insight into the geometric backgrounds, but referring to nature and anatomy as well.

Featured image: A girl in front of one of Damien Hirst's spot paintings, image via Observer; Agnes Martin exhibition view at Tate Modern, image via Artsy; Frank Stella, Damascus Gate, 1970; Andy Warhol Soup Cans via Time. All images used for illustrative purposes only.

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Pattern is such a universal term that trying to define it may seem redundant. Nonetheless, definitions are usually the ones that keep hidden knowledge and cyphered information, even when they give an impression of commonplaceness.

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induction reasoning :)
Also can be connected to art

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How we generalise about things based on specific experiences or examples?

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In his Treatise on Human Nature (1739), the philosopher David Hume argued that it is not reasonable to believe that the sun will rise tomorrow.

I wonder if Hume was always such a gloomy character. I for one wouldn't be inclined to invite a friend to my party if he would preach to everyone that they should give up the belief that the sun will rise tomorrow. That'd kill the party mood.

But is Hume right? Here's why he thinks it's not reasonable to believe the sun will rise tomorrow.

Let's first start asking the question: why do we believe the sun will rise tomorrow? Well, you might say, it has risen every day until now. Therefore, it is reasonable to believe that the sun will rise tomorrow as well.

But Hume doesn't think so. He argues that induction cannot lead us to this conclusion.

What is induction?
Suppose you find a raven. It is black. You look for another raven. It is also black. You go on a trip through the country, looking out for as many ravens you can find. All ravens you see are black. At the end you've seen many ravens and all of them are black. Based on these observations, you draw the conclusion: the next raven I'll see will be black. Is this a valid inference?

An inference is valid if (and only if) the truth of all premises guarantees the truth of the conclusion. In this case, there are many premises:

p1. Raven no. 1 is black
p2. Raven no. 2 is black
p3. Raven no. 3 is black
...
pn. Raven no. n is black
------
conclusion: Raven no. n+1 will be black.

But do these premises guarantee the conclusion? No. Even if you've seen 23786 black ravens, and none of another colour, it is still possible that the next raven is an albino raven, and therefore white. So the conclusion is not valid...

...unless you add another premise. The key premise you need to add to make the inference valid is: the future resembles the past. So you get:

p1. Raven no. 1 is black
p2. Raven no. 2 is black
p3. Raven no. 3 is black
...
p n. Raven no. n is black
and
p m. The future resembles the past.
------
conclusion: Raven no. n+1 will be black.

But remember that the conclusion only follows if the premises are true. Is the premise 'the future resembles the past' true? Well, you might say, the future has so far resembled the past, because during my entire trip, the next raven was black all the time. But then you get the same problem:

p1. At t1, the future turned out to resemble the past.
p1. At t2, the future turned out to resemble the past.
...
p n At t n, the future turned out to resemble the past.
-------
conclusion: The next time, the future will also resemble the past.

This is, again, an induction, so you'll need 'the future resembles the past' as a premise, again. But that leads to a fallacy of circularity: one of the premises is the same as the conclusion. Circularity doesn't prove anything, so we can't say with certainty that it's true that the future will resemble the past. Therefore, we can't use it in our induction about the ravens, and therefore we can't conclude that the next raven will be black.
But what have these ravens to do with the sunrise?
According to Hume, the same applies to the sun. Just because it has risen every morning, doesn't mean we can conclude that it will rise again tomorrow.
Why should I care?
Science, and indeed our daily conduct, would be impossible if we didn't rely on inductive reasoning. So what do we do now? We have a problem!
Resources
This video explains the above in a slightly different way.

More info on Hume's philosophy can be found in this episode of BBC's In Our Time.

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Will the sun rise tomorrow? The problem of induction - 01/02/2016

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In his Treatise on Human Nature (1739), the philosopher David Hume argued that it is not reasonable to believe that the sun will rise tomorrow. I wonder if Hume was always such a gloomy character. I for one wouldn't be inclined to invite a friend to my party if he would preach to everyone that...

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It supports the claim
One of the definition

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To understand is to perceive patterns. - Isaiah Berlin at Lifehack Quotes

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To understand is to perceive patterns. - Isaiah Berlin at Lifehack Quotes

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http://www.philosophybasics.com

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Any studies we learned in psychology classes.

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Limitation/mistakes of Apophenia - science

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Limitation/mistakes of Apophenia - science

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After a run of red lights, surely a green one is due, or is that a red flag?

How to recognise this tactic

This happens when you convince yourself, or someone tries to convince you, that some data reveal a significant pattern when really the data are random or meaningless. Some examples: seeing religious symbols in toasted bread or believing you are on a winning streak when gambling.

Why do people use this tactic?

This is one of those tactics we use on ourselves - a cognitive bias which prevents us from thinking rationally. We have all evolved the ability to detect patterns and regularities in our environment. Our ancestors used patterns they found in the seasons, in animal behaviour, in the earth's crust, in materials, to help them survive. It is the major task of science to uncover patterns. Unfortunately, we often succumb to wishful thinking and 'discover' patterns that we want to be there, but are not. That's apophenia. And scammers are quite prepared to exploit it.

What's wrong with this tactic

Apophenia leads you to believe, wrongly, that you have evidence to support a position when you don't. You believe you can continue to gamble because you're on a winning streak, or that Mars is inhabited because some observers see canal-like patterns on its surface. It can lead you to ignore evidence that falsifies your position, or that supports a contrary position.

What to do when confronted by this tactic

Since it's usually wishful thinking that leads us to find non-existent patterns, we need to guard against it first. Look for the signs, and guard against the temptation to dismiss evidence that doesn't support your wishes. Beware of those who try to exploit your tendency for wishful thinking and discipline yourself to accept only conclusion that are supported by real-world evidence.

Variations and related tactics

Michael Shermer calls this cognitive bias ' patternicity '. It comes in several forms:

Pareidolia - finding shapes, such as faces, in things like clouds, geological features and slices of toast.

The gambler's fallacy - believing that past random events can influence the probability of future ones, for example, that a flipped coin is more likely to show heads after a run of tails.

The clustering illusion - believing that the clusters that are always found in random data actually indicate something meaningful, for example, that a run of wins in a game of dice means you are on a winning streak.

In science, apophenia is related to what's known as a Type I error, in which a test seems to show that two variables are related when, in fact, they are not. It can also contribute to confirmation bias , in which an investigator deliberately looks for evidence which supports a favoured model and avoids evidence which refutes it.

Apophenia is also related to the Texas sharpshooter fallacy, in which a person zooms in on an apparent pattern in the midst of a sea of data, and claims that this is the key to a significant issue (as in the fable of the Texas sharpshooter who fires random shots at a wall, then draws the bullseye around the tightest cluster of bullet holes).

Examples

On the political side of the debate over climate change, many fall into the trap of using current weather patterns to support their positions - a cold spell, for instance, is quoted as evidence that the earth is not warming. Climate scientists are reluctant to claim that any particular event is due to global warming because they are aware of the dangers of the clustering illusion. Before any such pattern can be held up as real evidence, a convincing argument must be presented. James Hansen and colleagues have recently published an analysis (here and here) which, they claim, shows that recent weather events are consequences of global warming.

Here's how difficult it is to decide whether the clustering illusion is at work or not. In the early 2000s, it became apparent that there was an unusually high occurrence of breast cancer among female employees at the ABC studios at Toowong in Queensland. In 2007, an expert panel found that the rate was six times higher than the rate of breast cancer in Queensland, and that there was only a 1 in 25 chance (estimated p value of 0.04) that this could have occurred by chance. It found a correlation between breast cancer occurrence and length of service at the studio, but could find no evidence that the cancer was due to any factor related to the site or to genetic or lifestyle factors of the employees. A 2009 study investigated breast cancer rates in ABC employees across Australia and found no increased rates in any other site. The ABC abandoned the Toowong studios and now operates from new facilities several kilometres away. Was the cancer cluster a statistical artefact or was there some yet unidentified cause?

In the early 1900s, German meteorologist Alfred Wegener noticed a pattern in the earth's continental shapes. The edges of the continents appeared to fit into each other, like pieces of a jigsaw. This led him to believe they had once been joined together and he proposed his model of continental drift. Although this model explained many observations, it was not accepted by earth scientists because Wegener was unable to come up with a mechanism that could account for continents moving. Discoveries in the 1950s and 1960s revealed a mechanism, and Wegener's idea became incorporated into the Plate Tectonic model.

Further reading

Pareidolia in Politics: The Face of Faith's Corrupting Influence by Jeff Schweitzer at The Huffington Post.

You can add to the list of examples by leaving a comment.
Schiaparelli's drawing is from Wikimedia Commons.
This is one of ScienceOrNot's Science red flags. See them all here.

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After a run of red lights, surely a green one is due, or is that a red flag? This happens when you convince yourself, or someone tries to convince you, that some data reveal a significant pattern when really the data are random or meaningless.

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Altmetric: 107

More detail

Article

Nature Human Behaviour 1, 748-756 (2017)

doi:10.1038/s41562-017-0207-1

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Published online:

Results

Depression index distribution

To examine differences in attitudes towards economic inequity, we first measured the social value orientation10 of 343 participants using the 8 (trials) triple-dominance measure task (see Supplementary Methods). Next, we quantified the depression tendency of the participants identified as prosocial or individualist using the BDI questionnaire24. Figure 1a shows the distribution of the depression index for prosocials (defined as those who dislike economic inequity and maximize joint outcome) and individualists (defined as those who prioritize self-reward regardless of the other). This was significantly different between the two groups (bootstrap Kolmogorov-Smirnov test, P = 0.0045), with middle-to-high scores more frequent in prosocials. Differences in the scores for each question are shown in Fig. 1b, in which 'guilty feelings' is higher for prosocials (Wilcoxon rank sum test, P = 0.022). Such a distributional difference was not seen in terms of gender (male versus female, bootstrap Kolmogorov-Smirnov test, P  = 0.80).

Fig. 1: Tasks and responses.

a, Probability density plots of BDI scores were different between prosocials (solid line, n =  174) and individualists (dashed line, n = 59) ( P = 0.0045, bootstrap Kolmogorov-Smirnov test; number of bootstrap sampling = 100,000). The proportion of low BDI scores was higher in individualists, while prosocials exhibited a higher proportion in the middle-to-high score population range. b, Differences in scores for each BDI question between prosocials ( n = 174) and individualists ( n = 59). Scores of the question on guilty feelings (question 5) were significantly different ( P = 0.022, Wilcoxon rank sum test). c, Task design of the ultimatum game. Each participant took the role of responder. d, Mean rejection rates to unfair offers were significantly different ( P = 0.034, Kolmogorov-Smirnov test) between prosocials (black, n = 59) and individualists (grey, n = 35). The error bars represent standard errors. e, We identified a significant difference in amygdala and hippocampus activation between prosocials ( n = 59) and individualists ( n = 35) in correlation with disadvantageous inequity (multiple regression, P < 0.05 small volume corrected). The Montreal Neurological Institute (MNI) coordinates of the peak position were −26, −16, −18 (peak positions are also presented in Table 1). f, The beta values (representing the strength of correlation with inequity) at the peak position in the left amygdala and hippocampus ( e) were positive in prosocials (black) and negative in individualists (grey) and the difference was significant ( P = 8.2 × 10 −5, Kolmogorov-Smirnov test). The error bars represent standard errors.

Differential neural activity between prosocials and individualists

To identify the neural substrates underlying the differences between prosocials and individualists with respect to inequity, of the original 343 participants, 59 prosocials and 35 individualists underwent fMRI while playing an ultimatum game22 (see Methods). In this game, proposers made a series of offers about money sharing to a responder who then decided whether to accept or reject each offer. If the responder accepted an offer, money was distributed as proposed; otherwise, neither side received any money. In our experiments, participants were asked to take the role of responder (Fig. 1c). Consistent with the definition of prosocials as people who like to minimize reward difference, prosocials exhibited a higher rejection rate of unfair offers in comparison with individualists (Fig. 1d, Kolmogorov-Smirnov test, P = 0.034) and the rejection rate for fair offers was comparable between prosocials and individualists (Kolmogorov-Smirnov test, P =  0.94).

We conducted a general linear model analysis of the fMRI data using SPM12 ( http://www.fil.ion.ucl.ac.uk/spm). In the first each subject-level analysis, we separated offer events intro thee types-unfair (disadvantageous), fair and unfair (advantageous)-and included disadvantageous inequity (hereafter referred to as 'inequity' and defined as the difference between the self and other reward when the reward for the other was larger, otherwise 0; see also Methods) as a regressor associated with offer presentation events. In the second across-subject analysis, we contrasted prosocials and individualists in terms of inequity using multiple regression, which incorporated social value orientation (prosocial or individualist), gender (male or female), age, income level (discretionary money per month) and socioeconomic status (social hierarchy) obtained through a post-experimental questionnaire (see also Supplementary Methods).

We found that activity in the amygdala and hippocampus correlating with inequity was higher in prosocials than individualists ( P < 0.05 small volume corrected; Table 1 and Fig. 1e; this key activity was also significant with P < 0.05 family-wise error (FWE) corrected for the whole brain), consistent with previous reports11,12,13. At the peak voxel of this inequity-correlated amygdala and hippocampus response, prosocials exhibited positive correlation, while individualists exhibited negative correlation (Fig. 1f, Kolmogorov-Smirnov test, P = 8.2 × 10 −5). This differential activity remained the same even when we equalized the number of prosocial and individualist participants (that is, the first 35 prosocials and 35 individualists; supplementary Fig. 1; see also Methods). However, we did not find any inequity-correlated activity common to prosocials and individualists, nor was a correlation revealed by the opposite contrast of individualists minus prosocials. In addition, other than social value orientation, socioeconomic status produced significant inequity-correlated activity in the caudate nucleus ( P < 0.05 small volume corrected; Table 1), consistent with a previous study25 reporting the involvement of the striatum in inequity processing in terms of the relative status of people. We did not find any other significant activity for any event that correlated with gender, age or income level (even at a very weak threshold P  < 0.005 uncorrected). Based on these analyses, we were able to exclude gender, age, income level and socioeconomic status from the potential confounders of the differential activity of the amygdala and hippocampus.

We contrasted unfair and fair offers and obtained stronger activity in the insula and anterior cingulate cortices for unfair (disadvantageous) offers than fair offers (Fig. 2 and Table 2, P < 0.05 FWE corrected), which is highly consistent with previous reports11,23. However, the other contrast-unfair (advantageous) minus fair offers-did not yield a significant result. In addition, we contrasted rejected and accepted offers using a separate general linear model in which all other contrasts were kept the same. Again consistent with previous reports23, we found higher activity in the anterior insula for rejected offers than accepted offers (Supplementary Fig. 2 and Supplementary Table 1 (activity differences between rejected and accepted trials induced by offer presentations)).

Fig. 2: Basic brain activity during the ultimatum game.

a, Brain responses to unfair (disadvantageous) offers were compared with fair offers in all participants ( n = 94). Peak positions are presented in Table 2 (two sample t-test, P < 0.05, FWE corrected). For display purposes, an image with P < 0.001, uncorrected, is shown. The MNI coordinates of the peak position were −28, 24, 8. P FWE = 5.0 × 10 −3. b, c, Correlations between the present BDI scores and beta values of each single voxel in the amygdala and hippocampus clusters were computed and the maximally negative and positive correlations were identified in the hippocampus ( b) (MNI coordinates −30, −16, −18; P = 1.8 × 10 −3, R  = 0.32) and amygdala ( c) (MNI coordinates 12, −8, −16; P = 0.046, R  = 0.21).

Correlations with BDI score

To investigate whether behavioural indices of the participants could predict their BDI index, we plotted the total BDI score of each participant against their number of prosocial choices in the 8 triple-dominance measure task, which was used to determine social value orientation (Supplementary Fig. 3a, b), and the rejection ratio in the ultimatum game (Supplementary Figs. 3c and 4d). We found that these basic behavioural indices could not predict the BDI score. In addition, neither income level (discretionary money per month) nor socioeconomic status (social hierarchy) was correlated with the BDI score (Supplementary Table 2). There was no significant gender difference in terms of the BDI score either (Kolmogorov-Smirnov test, P  = 0.72).

Next, we evaluated the Pearson correlation coefficients between the total BDI scores of the participants and their beta values in each voxel in correlation with inequity. We found negative correlations in the hippocampus (Fig. 2b) and positive correlations in the amygdala (Fig. 2c), but no significant difference between the correlation coefficients of prosocials and individualists (analysis of covariance, P = 0.73, Supplementary Fig. 4). Based on these voxel-level correlational observations, we further hypothesized that the amygdala and hippocampus response patterns to inequity might predict present and future depression indices.

Predicting present BDI scores from brain response patterns

We then examined whether the amygdala and hippocampus response patterns to inequity could predict total BDI scores. To analyse multi-voxel activity patterns, we developed a model-based multi-voxel pattern analysis method (Fig. 3a), which focused on the voxel response pattern specific to the inequity when conducting a prediction. This method was based on the sparse Bayesian kernel-based leaning algorithm and had two remarkable characteristics. The first was the model-based feature: the input to the prediction system was the beta values (that is, linear coefficients) of brain activity in correlation with inequity. This property allowed us to focus on amplitudes of inequity-correlated brain activity for prediction and not only on a simple increase or decrease in activity. The second was the sparseness achieved by sparse Bayesian methods. This property allowed us to automatically select a limited number of participants who were informative for prediction and was particularly useful for high-dimensional noisy data such as fMRI signals (see also Supplementary Methods). We defined functional regions of interest (ROI) as the clusters shown in Table 1.

Fig. 3: Prediction of the present and one-year change ﻿in BDI scores.

a, Conceptual scheme of the prediction method of BDI scores using multi-voxel pattern analysis. For each participant ' i', beta values for inequity were computed for all voxels in the ROI and principal component scores were used as input, X i, to the kernel function, which quantifies the similarity of X i with new data, X t. Y i is a target value representing a present or future BDI score. Y t is predicted as the weighted sum of w i × K( X i, X t). The optimization of w i was conducted using SparseBayes version 2 software ( http://www.miketipping.com), where the majority of w i is estimated to be 0, which allows us to select participants important for the prediction. b- d, Prediction of present BDI scores from brain response patterns induced by inequity. The present BDI scores were predicted using the inequity-correlated beta values in b (left amygdala and hippocampus, c (right amygdala) and d (right insula and caudate) and plotted against actual BDI scores. Generalization was evaluated by the leave-one-out cross-validation procedure ( n = 73) and linear regression indicated that prediction of the present total BDI scores was possible in the amygdala and hippocampus regions, but difficult in the insula region. e- g, Prediction of one-year change in BDI scores from brain response patterns induced by inequity. e, The distributions of present (first) BDI scores for prosocials ( n = 47, solid line) and individualists ( n = 26, dashed line) were different (bootstrap Kolmogorov−Smirnov test, P = 0.017). BDI score changes were predicted from the left amygdala and hippocampus ( f) and the right amygdala ( g) ROIs and plotted against the observed changes in BDI scores. Generalization was evaluated by the leave-one-out cross-validation procedure ( n  = 73) and linear regression indicated that prediction of the present BDI scores was possible.

We compared actual and predicted (by the leave-one-out method) total BDI scores for each participant. The total BDI score predicted by the response pattern to inequity in the left amygdala and hippocampus ROI had a positive correlation with the actual BDI (Fig. 3b, slope = 0.24, R = 0.48, P = 1.5 × 10 −5), demonstrating predictability of the present depression index. A similar prediction was also possible by the response pattern to inequity in the right amygdala (Fig. 3c, slope = 0.15, R = 0.36, P = 1.6 × 10 −3). Importantly, this prediction was also possible even if we split the left amygdala and hippocampus ROI into separate amygdala and hippocampus measures (amygdala: slope = 0.20, R = 0.42, P = 2.2 × 10 −3; hippocampus: slope = 0.19, R = 0.47, P  =2.8 × 10 −5). We also found that the response pattern in the caudate nucleus to inequity (detected by the socioeconomic status) could weakly predict the present BDI scores (Supplementary Fig. 5, slope = 0.14, R = 0.31, P =7.5 × 10 −3). However, such a prediction was not possible for the other region in Table 1 (that is, the insula, Fig. 3d, slope = 0.010, R = 0.081, P  = 0.50).

Prediction from the response pattern induced by the proposer's face presentation (slope = 1.4 × 10 −5, R =1.1 × 10 −4, P = 0.99) or offer presentation (slope = 8.8 × 10 −3, R =0.075, P = 0.53) was impossible (Supplementary Fig. 6, see also Methods). Indeed, we found that the slopes of the regression lines were different between the inequity condition (Fig. 3b) and the face or offer presentation condition (Supplementary Fig. 6, analysis of covariance; interaction of linear regression, inequity versus face, P ≈ 0 and inequity versus offer, P ≈ 0). Finally, we also tried to predict the BDI score from brain activity patterns in the insula and anterior cingulate cortex identified by the (unfair (disadvantageous) - fair) contrast (Fig. 2a) and found that prediction was not possible (insula: slope = 4.4 × 10 −3, R =0.053, P = 0.66; anterior cingulate cortex: slope = −0.011, R =0.011, P  = 0.34).

Predicting long-term changes in BDI scores from brain response patterns

We next considered whether the amygdala and hippocampus response pattern to inequity could also predict the future depression index. We addressed this issue by testing whether the inequity response pattern could predict a one-year change in the total BDI score after the same group of participants ( n = 73) underwent BDI one year later (see also Supplementary Methods). The distributions of the second total BDI were different between prosocials and individualists (Fig. 3e, bootstrap Kolmogorov-Smirnov test, P = 0.017), consistent with the results from one year before. However, the distribution of the change in total BDI score was not different between the two groups. This finding indicates that distinctions in prosocials and individualists cannot predict BDI changes (Supplementary Fig. 7, bootstrap Kolmogorov-Smirnov test, P = 0.79). In addition, there was no single question in the BDI questionnaire that showed a significant increase or decrease after one year. There were no significant differences in the first and second fMRI participants in income level, socioeconomic status or BDI score (Table 3). Importantly, there was no correlation between income level and socioeconomic status and BDI score one year later (Supplementary Table 3) or the change in BDI scores (Supplementary Table 4). All these results indicate the difficulty in predicting BDI changes behaviourally.

Prediction of the one-year change in BDI score was conducted using the same sparse Bayesian kernel-based multi-voxel pattern analysis method used for the present case. Figure 3f, g shows plots of the observed ( x axis) and predicted ( y axis, by the leave-one-out method) changes of the total BDI score for each participant (from the left amygdala and hippocampus and the right amygdala, respectively). These results show that the predicted change had a positive correlation with the observed change (slope = 0.23, R = 0.36, P = 1.9 × 10 −3 and slope = 0.11, R = 0.30, P = 9.5 × 10 −4, respectively). However, the prediction was not possible for other areas in Table 1; for instance, the insula and caudate nucleus (insula: slope = 0.047, R =0.051, P = 0.67; caudate nucleus: slope = −0.014, R =0.44, P = 8.7 × 10 −5). Again, it was difficult to predict changes in the BDI score from the response pattern induced by the proposer's face presentation (slope = 0.054, R =0.16, P = 0.19) or offer presentation (slope = 0.067, R =0.25, P = 0.035, Supplementary Fig. 8) and the slopes of the regression lines were significantly different (interaction of linear regression, inequity versus face, P =0.032; inequity versus offer, P  = 0.035). Thus, the amygdala and hippocampus response pattern to inequity was found to predict not only the present but also the future depression index.

Differences between prosocials and individualists

We have so far focused on (disadvantageous) inequity in our analysis. However, the definitions of prosocials and individualists (prosocials dislike bidirectional (disadvantageous + advantageous) inequity; individualists prioritize self-reward) made us hypothesize that bidirectional inequity might have different effects on the two groups. Therefore, we examined whether bidirectional inequity (hereafter referred to as 'absolute-value inequity' and defined as the absolute value of the difference between the self and other rewards) and disadvantageous inequity result in different predictions of the depression index.

Consistent with disadvantageous inequity (Fig. 1e), the same general linear model analysis showed that amygdala and hippocampus activity correlated with absolute-value inequity more in prosocials than in individualists (Fig. 4a and Table 4). We then compared the predictions of the present BDI score based on disadvantageous and absolute-value inequity. Predictions of the present BDI score based on disadvantageous inequity (using the same ROI as in Fig. 3b) are separately displayed for prosocials and individualists (Fig. 4b). The estimated slopes were positive for both groups and no significant differences were seen (interaction in linear regression, P = 0.36), indicating that the sensitivity to disadvantageous inequity in the amygdala and hippocampus is important for the prediction in prosocials and individualists. In contrast, the same analysis based on absolute-value inequity revealed that prediction of the depression index was possible in prosocials, but not in individualists (Fig. 4c, d). The difference in the slopes of these two regression lines was significant (interaction of linear regression, P  = 0.019). These results indicate that advantageous inequity (when reward for the self is larger than reward for the other) has effects only on prosocials.

Fig. 4: Prediction of the present BDI scores from absolute-value inequity responses.

a, Whole-brain general linear model analysis conducted in correlation with the absolute value of the difference between self and other rewards. We identified a significant difference in amygdala and hippocampus activation between prosocials ( n = 59) and individualists ( n = 35) in correlation with absolute-value inequity (multiple regression, P < 0.05 small volume corrected). The MNI coordinates of the cluster peak were −26, −16, −18. P svc = 1.2 × 10 −3 (see also Table 4). The displayed peak cluster was used as the ROI for prediction. b, Prediction of the present BDI scores based on (disadvantageous) inequity was separately conducted in prosocials (left, n = 47) and individualists (right, n = 26) and linear regression showed that both predictions were possible. c, Present BDI scores were predicted based on absolute-value inequity responses. The prediction was possible in prosocials (left), but not in individualists (right). d, The difference in the slopes of the regression lines between prosocials and individualists was significant for absolute-value inequity (linear regression, P = 0.019), but not for (disadvantageous) inequity (linear regression, P  = 0.36). NS, not significant.

Discussion

We have discussed in this paper the relationship between brain response patterns induced by inequity and the depression index in a non-clinical population. We first showed that the distribution of the depression index (that is, the BDI score) was different between prosocials and individualists, who also showed different responses to BDI questions, such as guilty feelings, and differential inequity-correlated fMRI activity in the amygdala and hippocampus during an ultimatum game. We found voxel-level positive and negative correlations between BDI scores and inequity-correlated activity in the amygdala and hippocampus, respectively (Fig. 2b, c). We further showed that the response pattern in the amygdala and hippocampus induced by the inequity could predict not only the present but also the future depression index using a model-based multi-voxel pattern analysis method (Fig. 3). Separate analyses based on disadvantageous and absolute-value inequity revealed that the response to inequity was capable of predicting responses by both prosocials and individualists, but the prediction in response to absolute-value inequity was only effective in prosocials (Fig. 4).

The amygdala and hippocampus have been consistently identified as key brain structures in terms of major depression. Functional activity in the amygdala and hippocampus is different for emotional stimuli14,15 in depression patients compared with normal controls, as is the anatomical volume of the two structures16,17,18,26. Guilty feelings and self-criticalness, which are often observed in patients with depression have been associated with changes in amygdala and hippocampus activity27,28. These functional and anatomical changes in patients with depression have been partly linked with the increased cortisol levels produced by the HPA axis. The HPA axis produces a cascade of hormones including the primary stress hormone cortisol under the influences of excitation input from the amygdala and feedback inhibition from the hippocampus19. Several recent fMRI studies have reported that activity in the amygdala is correlated with the way in which prosocial people dislike inequity11,12,13. These results may suggest that in the present study, amygdala activity represents the stress-induced response induced by economic inequity and hippocampus activity represents the feedback inhibition response. This view is consistent with the voxel-level positive and negative correlations of the amygdala and hippocampus with BDI scores. Other than the amygdala and hippocampus, the inequity response pattern in the caudate nucleus, which was also implicated in inequity processing13,25, could predict the present BDI but not the one-year change in the BDI. Recent studies have started to show that experiences affect mood and mood in turn affects subsequent decision-making and experiences29,30,31. The amygdala and hippocampus, together with the caudate nucleus, may well be involved in such a two-way interaction mechanism that connects mood and decision-making and can therefore be used to predict the dynamics of an individual's mood.

Accurate prediction of the present and future depression index was not possible using only basic behaviours and the status of the participants, such as the number of prosocial choices in the 8 triple-dominance task and the rejection rate in an ultimatum game, gender income level and socioeconomic status. It might be puzzling at a glance that although prosocials exhibited a higher proportion of middle-to-high BDI scores at the population level, individual BDI scores could not be predicted using the prosocial versus individualist distinction. As an explanation, we propose that the amygdala response to inequity lies at the core of both the BDI dynamics and social value orientation, but the behavioural link between BDI dynamics and social value orientation is weaker than the multidimensional neural link between BDI dynamics and the amygdala response to inequity.

We also showed that amygdala and hippocampus activity patterns for face presentation and offer presentation could not predict present or future BDI scores (Supplementary Figs. 6 and 8). Although these data suggest that brain response patterns in the amygdala and hippocampus to inequity measured by fMRI may be a better predictor of the depression index than behavioural and status information, this does not necessarily mean that inequity-driven brain activity is th﻿e only information in the brain that can be used to predict BDI scores. For instance, brain response patterns to other emotional stimuli, such as fearful faces, may also predict the future depression index. Identifying which kind of brain response is most effective for predicting the future change in the depression index is crucial for basic understanding of human social stress and the development of clinical methodologies.

Finally, we demonstrated that the amygdala and hippocampus response pattern induced by absolute-valued inequity could predict the depression index in prosocials but not in individualists (Fig. 4c). This result marks a sharp contrast with disadvantageous inequity, which was predictive of the depression index for both groups (Fig. 4b). This difference coincides well with the definitions of the two groups, as prosocials prioritize equity and thus minimize differences in the self and other interests, while individualists prioritize self-interest. Related to this distinction, previous clinical studies have proposed that people who worry about others' perspectives are more prone to depression20,21 and this is consistent with our observation that prosocials showed a higher proportion of middle-to-high BDI scores at the population level even though the prosocial versus individualist distinction did not predict individual-level BDI scores. In addition, a recent resting-state fMRI study reported that a neural network that includes the amygdala and hippocampus defines a subcategory of depression patients. Thus, the amygdala and hippocampus response pattern described in the present study may be useful for understanding premorbid personalities and different types of depression32.

In summary, this study revealed that amygdala and hippocampus response patterns to inequity can predict both the present and future depression index for a non-clinical population, providing behavioural and neural evidence of a biological link between economic inequity and the depression index. In the future, not only large-scale but also long-term clinical (and cross-cultural) studies with higher temporal and spatial resolutions are needed to clarify how interactions between the amygdala and hippocampus lead to depression symptoms under economic inequity and which response patterns can be recognized as warning signals of depression.

Methods

Ultimatum game

We used the ultimatum game22 to examine brain responses to inequity. As shown in Fig. 1c, after the name and face of a proposer were displayed for 1 s, the participant was asked to decide whether to accept or reject the offered division of ¥500 (equivalent to US $5) and respond by a button press within 1 s after a beep (otherwise, the reward for the trial became ¥0). Base offers were one of ¥350-150, ¥300-200, ¥250-250, ¥200-300, ¥150-350, ¥100-400 or ¥50-450 for the participant and proposer (each base offer appeared eight times in one session in random order). Therefore, one session comprised 56 trials. We did not include ¥400-100 or ¥450-50 splits, not only because we confirmed that very few participants rejected them in our pilot experiment but also because we wished for the scan time to remain short for participants' concentration. In our version of the ultimatum game, the base offers were fluctuated in each trial by adding a uniform random number ranging from ¥−25 to ¥25. This modification was introduced to maximize the participants' involvement in the task. The 'unfair (disadvantageous)' condition was defined as trials in which the reward for other was larger than the reward for self (that is, ¥200-300, ¥150-350, ¥100-400 and ¥50-450 base offer trials). The disadvantageous inequity was the amount of the reward for the other minus the reward for self when positive, otherwise it took a value of 0. On the other hand, when we considered absolute-value inequity, we used the absolute difference between the self and other rewards in all trials. Participants were instructed that their task performances determined the amount of reward. In addition, the name and face were utilized once for each individual proposer and the task was a sequential one-shot ultimatum game. One-fourth of the distributed money plus a basic honorarium (¥3,000) were provided to the participants. All faces were taken from the Advanced Telecommunication Rearch Institute facial expression database33 and had neutral facial expressions. Participants were informed that the proposers were students of other universities nearby and their rewards were determined by the participants' choices in the scanner. No participants raised questions about this experimental setting in the post-experimental interview. One trial lasted 21 ± 2 s, and the total time of a session was 1,176 s.

Prediction of the BDI score

For participant ' i', beta values in correlation with (disadvantageous or absolute-value) inequity extracted from all voxels in the ROI were processed by principal component analysis ( X i). The actual BDI score (or its change) for participant ' i' is denoted as Y i. The i-th input to the prediction system of n participants was X i and Y i. The task of the system was to make a prediction ( Ŷ i) for the target participant ' t' that minimized the expectation of error ( Y t - Ŷ t), which we achieved using the sparse Bayesian methodology34; that is, maximization of the posterior probability of parameters and hyper-parameters given the data. In the present study, we used a kernel-based model35. When we conducted the predictions based on a proposer's face presentation and offer presentation, we used beta values in correlation with the face presentation and offer presentation regressors, respectively.

Bayesian kernel method

In the Bayesian kernel method36, the prediction Ŷ t of Y t was modelled by a weighted sum of ( n − 1) kernel functions:

Ŷ t = ∑ i w i K ( X i, X t)

Here, kernel function K( X i, X t) measures the similarity between the i-th participant's brain activity pattern, X i, and target brain activity pattern, X t. Weights, w i, are adjustable parameters and estimated so as to minimize the expectation of error ( Y t- Ŷ t) in a Bayesian manner. This model is linear in its parameters, which allows for efficient computation. At the same time, the kernel function can be non-linear and represent complex functions. We subtracted baseline symptom by including a constant in the regression model.

The key to the sparse Bayesian methodology is the definition of a hyper-parameterized prior over the model parameters w (set of w i) of the form:

p (w | α)∝ ∏ i=1 n α i 1 2 e x p (− α i 2 w i 2 )

This type of prior favours models that fit the data well and are also sparse. In other words, the prior locates most of its probability mass at w i = 0, which means in the present study that only a small number of participants take a non-zero value of w i and are used for the prediction. The sparse Bayesian method computes the posterior distribution over the hyper-parameters α i given the data and returns their most probable values by maximizing the marginal likelihood function, which results in a posterior distribution over the parameters w i, many of which are set to 0. In the current implementation, optimization was conducted using SparseBayes software version 2.0 ( http://www.miketipping.com/sparsebayes.htm) on MATLAB and we computed K based on the linear spline kernel K( X i, X t) proposed by Vapnik et al.37 for X i and X t. Hyper-parameters of the kernel were determined by the tenfold cross-validation procedure so that the mean squared errors between the observed and predicted changes in the BDI score were minimized (Supplementary Fig. 9). The goodness of the generalization in prediction (that is, Fig. 3) was evaluated by the leave-one-out cross validation procedure34.

Participants

The ethical committees of the National Institute of Information and Communications Technology and Tamagawa University approved this study. Informed consent for the behavioural and fMRI experiments was obtained from all participants before the experiments started.

Code availability

The MATLAB codes that support the findings of this study are available from the corresponding author upon request.

Data availability

The preprocessed fMRI and BDI data that support the findings of this study are available from the corresponding author upon request.

Additional Information

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

References

36.
Tipping, M. E. Sparse bayesian learning and the relevance vector machine. J. Mach. Learn. Res. 1, 211-244 (2001).

Download references

Acknowledgements

This work was supported by Core Research for Evolutional Science and Technology (CREST), Japan Science and Technology Agency (JST), the Center of Innovation at Osaka University and Grant-in-Aid for Scientific Research (KAKENNHI) (17H06314 and 26242087). We are grateful to S. Tada and T. Haji for technical assistance and P. Karagiannis for editing an early version of the manuscript. The funders had no role in study design, data collection and analysis, decision to publish or preparation of the manuscript.

Author information

Affiliations

1. Center for Information and Neural Networks, National Institute of Information and Communications Technology, Osaka 565-0871, Japan

Toshiko Tanaka

& Masahiko Haruno

2. Brain Science Institute, Tamagawa University, Tokyo 194-8610, Japan

Toshiko Tanaka

& Masahiko Haruno

3. NHK (Japan Broadcasting Corporation), Tokyo 150-8001, Japan

Authors

1. Search for Toshiko Tanaka in:

2. Search for Takao Yamamoto in:

3. Search for Masahiko Haruno in:

Contributions

M.H. and T.Y. designed the study. T.T. conducted the experiment. T.T. and M.H. analysed the data. T.T. and M.H. wrote the paper.

Competing interests

The authors declare no competing interests.

Corresponding author

Correspondence to Masahiko Haruno.

Electronic supplementary material

1. Supplementary Figures 1-9, Supplementary Tables 1-5, Supplementary Methods

2. Life Science Reporting Summary and Reporting Summary for MRI

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Brain response patterns to economic inequity predict present and future depression indices

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Widening economic inequity has been suggested to associate with depression. However, little is known about the underlying neural mechanisms of this link. Here, we demonstrate that functional magnetic resonance imaging activity patterns in the amygdala and hippocampus induced by the inequity between the self and other rewards during an economic game can predict participants' present and future (measured one year later) depression indices.

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You experience the following on a Monday: The alarm clock does not wake you up and you are now late for work. The cat has peed on the couch. The coffeemaker is making strange noises and doesn't seem to be working. The kids are fighting with each other. It's raining. And on top of everything else, the car won't start. What do you conclude? One or two of these minor irritants would seem insignificant and unmemorable. Once the list grows, however, it begins to take the shape of a plot-a plot of unseen forces perhaps conspiring in a meaningful way against you.

Our brains are pattern-detection machines that connect the dots, making it possible to uncover meaningful relationships among the barrage of sensory input we face. Without such meaning-making, we would be unable to make predictions about survival and reproduction. The natural and interpersonal world around us would be too chaotic. In the above example, if I draw conspiratorial conclusions (i.e., seeing a pattern where none really exists), I am making what statisticians would call, a Type I error, also called a false positive.

What if you experience the following upon entering your house alone at night: The front door is left open. Household objects are strewn everywhere (you left the house neat and clean just a few hours ago). Your computer is missing. There are faint but unrecognizable smells. You can hear someone talking. In all likelihood, we would not only draw conclusions about has occurred but we will have a palpable, physiological response as well. Nature ensures that we are prone to seeing patterns rather than missing them. The Type II error, seeing no pattern where a pattern exists, turns out to be more dangerous. Far better-from a Darwinian perspective-to erroneously interpret danger where none is present than miss out on important cues that put our survival on the line. There is cognitive efficiency built into this equation: quick reactions depend on a cost-benefit ratio that favors safety and survival.

So, when our pattern-recognition systems misfire, they tend to err on the side of caution and self-deception. The experience of seeing patterns or connections in random or meaningless data was coined apophenia by the German neurologist, Klaus Conrad. He originally described this phenomenon as a kind of psychotic thought process, though it is now viewed as being a ubiquitous feature of human nature. Science historian Michael Shermer has called the same phenomenon patternicity. Shermer has pointed out that our brains do not include a "baloney-detection network" that would allow us to distinguish between true and false patterns.

Examples of apophenia, or patternicity, are everywhere. Many people perceive faces in seemingly random places-such as in clouds, in patterns of dirt left on cars, or on the moon. We take such patterns a step further by ascribing meaning to them. People have seen the images of Jesus and Mary inside a halved orange; or the face of Jesus on a piece of toast. Sometimes such objects are then worshipped or granted a sacred status. Some forms of apophenia have to do with sequences of behavior-such as the gambler's fallacy or other misperceptions of probability (this is illustrated most simply by sequential coin tosses where one might erroneously believe that after 5 tosses of heads, the probability of getting tails would somehow be higher than 50%). Apophenia also surfaces in the more complex patterns of our interpersonal world. Conspiracy theories, such as the belief that the twin towers of 9/11 were destroyed in a controlled demolition perpetrated by the government, are confabulations based on misperceived patterns. Such fallacious reasoning also has potentially adverse social consequences. For example, despite a paucity of evidence showing a causal connection, many parents do not vaccinate their children because they believe such vaccinations cause autism.

While it's tempting to view apophenia as merely a defect in our cognitive processing capacities (i.e., something that must be overcome or defeated), it might be useful for us to view this tendency as being an ironic, even amusing aspect of our nature. We are fooled by optical illusions-apophenia of the visual cortex-but we don't take such cognitive errors personally. Magic shows are often enjoyable precisely because we know that we are being tricked. If we embraced our vulnerability to cognitive errors, we would not be so easily caught off guard.

What does psychoanalysis (and psychotherapy more generally) add to the conversation about apophenia? One immediately thinks of free association-a clinical tool that specifically focuses on meaning generated from word associations. Rather than merely viewing apophenia as a kind of unfortunate side effect of our cognitive architecture, psychoanalysis pushes us to look at meaning where it seems least obvious. In this way, patternicity is the point, not the problem. Good novelists understand this, of course, and depend on the suspense and anticipation that seemingly unrelated associations create in readers. In an excellent essay on the subject, the writer Christopher Moore said, "A mild case of apophenia is a novelist's secret weapon that brings readers and literary success. We spend our working days seeing spontaneous connections between unconnected events, people, and lives, and weaving meaning into those connections." In psychotherapy, we co-construct meaning with our clients in an active process of making sense of interpersonal noise and randomness. The journey of psychotherapy can take on a narrative quality and often depends-like good storytelling-on a recognizable, coherent plot. Of course, our desire for patternicity may underlie larger questions and rituals of meaning-making. Our relentless detection of patterns is part of our larger search for meaning. Our greatest challenge may be learning to bear incoherence.

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You experience the following on a Monday: The alarm clock does not wake you up and you are now late for work. The cat has peed on the couch. The coffeemaker is making strange noises and doesn't seem to be working. The kids are fighting with each other. It's raining.

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There's nothing that that is perfectly circular in the whole universe. We assume that planets and stares are perfectly circular, but they aren't

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Do perfect circles exist in our universe? | Stuff to Blow Your Mind

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Our universe is devoid of perfect circles.

Think about that for a moment.

Certainly, circles and spheres are all around us. We see them gazing back at us in the mirror, but our pupils fall short of perfection.

The Great Circles

At the macro-level we see them in the suns and planets. The even distribution of gravitational forces draws matter into spherical shapes, but the centrifugal force of rotation causes spheres to bulge out at the equator. According to Clark Planetarium director Seth Jarvis, we're talking a barely noticeable 0.3 percent oblate bulge at Earth's equator, but a hefty 10 percent with Saturn.

Even our sun, which boasts incredible mathematical roundness, bulges out roughly 10 kilometers at its equator. Many scientists predict the event horizon of a black hole would constitute a perfect circle or sphere, but we've yet to prove that out -- and as this Physics Forum thread illustrates, not everyone is convinced we'd find perfection there either.

And there's more: According to Stephen Hawking (as summarized in this Daily Galaxy article), "quantum effects around the black hole (may) cause space-time to fluctuate too wildly for a sharp boundary surface to exist." The precision of the circle erodes like a shape drawn in a sandy beach.

The Small Circles

At the micro-level, we see the near-perfect roundness of the electron particle -- and its imperfection factors into our best theories regarding the physical nature of the universe. Simply put, if improved measuring techniques prove electrons to be too perfectly round, then we're forced to cast out some of our theories proposing particles beyond those accounted for in the Standard Mode. But even these particles fall short.

Man-made Circles

Artificial spheres and circles also fail to achieve perfection, whether you're considering a hand-drawn circle on a blackboard or the quartz gyroscopic rotors built for NASA's Gravity Probe B spacecraft. Those quartz gyros, incidentally, stand as the most perfect man-made spheres ever created, landing less than three ten-millionths of an inch from perfection.

Mathematical Perfection

Mathematically speaking, a circle is the set of points in a plane that are equidistant from a given point. For a circle to be perfect, you'd need all those points in the circle's circumference to match up exactly. And for all those points to match up exactly you'd need this precision to remain constant no matter how closely you looked: the particles, the cells, the atoms... And are these "points" stationary or are they in motion? The maddening search for perfection simply breaks down.

Only in the abstract world of pure mathematics can we find our perfect circle -- a world of points and infinitely-thin lines with no room for particle inconsistencies or spherical oblateness.

Forms From Beyond

The situation brings to mind Plato's Theory of Forms. We live in the material realm, it states, but beyond our plane exists an immaterial realm of ideal forms. You can think of these ideal forms as the absolute perfection of a given thing, a truth that cannot be manifested in our universe. All we can do is echo it.

In our world there is no true beauty, but we have an innate understanding and longing for the true form of beauty as it exists beyond the limits of our reality. There's no true justice here, but we have a sense of it because the unreachable ideal exists in the realm of forms.

The Theory of Forms applies to chairs, apples, fears, sex, art -- everything we can comprehend and long for, really. For each there is a godlike ideal beyond our worldly grasp, residing in a pantheon of other awesome and terrible forms.

The circle is but one of them, its perfection impossible in our imperfect world.

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Do perfect circles exist in our universe?

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Our universe is devoid of perfect circles. Think about that for a moment. Certainly, circles and spheres are all around us. We see them gazing back at us in the mirror, but our pupils fall short of perfection. At the macro-level we see them in the suns and planets.

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According to Dr Shermer, science is the best tool ever devised to determine whether or not a belief matches reality.

Compare human science to math or natural science

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Saying art is a type of uniformity
The quote supports the uniformity can show the beauty of art.

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Scientists assume that the natural world has certain constants and in order to know how nature works, a scientist will construct various hypotheses to test against these constants in order to gather and apply EVIDENCE or DATA

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What Is A Law Of Nature?

The laws of physics are generally looked upon as a paradigm of exactitude. Therefore one would naturally take it for granted that probably no other science would be able to give such a clear and definite answer when asked what is meant when we speak of a law of nature.

What is a Law of Nature? The answer does not really seem to be very difficult. When man's higher consciousness first awakens he finds himself in an environment whose changing elements are of the highest significance for his weal or woe. Experience -first the unsystematized experience of his daily struggle for life and afterwards the experience derived from systematically and rationally planned scientific experiments-shows him that the natural processes which take place in his environment do not follow one another in an arbitrary, kaleidoscopic manner, but that they present a notable degree of regularity. He eagerly strives to become acquainted with the nature of this regularity, because such knowledge will be of tremendous advantage to him in his struggle for life. The order of nature thus perceived by man is of the same type throughout. Certain features in the succession of natural events always and everywhere show themselves to be connected with certain other features. Of special biological significance is that case in which the one group of characters precedes the other group in time. The circumstances preceding a certain happening (A) which is often observed in nature, divide themselves into two typical groups: (1) circumstances that are always present - the invariable, and (2) those which are only sometimes present - the variable. When it is further discovered that conversely the unchanging group is always followed by A, this discovery gives rise to the statement that this invariable group of circumstances is the cause that brings about the phenomenon A. Thus, hand-in-hand with the discovery of special regular connections, we come to the idea of a general necessary connectedness between one phenomenon and others as an abstraction from the mass of connections as a whole. Above and beyond our actual experience, the general postulate is laid down that in those cases in which we have not yet succeeded in isolating the causal source of any specific phenomenon, such a source must surely exist -in other words, that every natural process or event is absolutely and quantitatively determined at least through the totality of the circumstances or physical conditions that accompany its appearance. This postulate is sometimes called the "principle of causality." Our belief in it has been steadily confirmed again and again by the progressive discovery of causes that specially condition each event.

Now, what we call a "law of nature" is nothing else than any one of the regularities observed in natural occurrences, insofar as it is looked upon as necessary, in the sense of the above-mentioned postulate. Is there still some obscurity here, some occasion for doubt? And, if so, where? Since about the actual facts there can be no doubt whatever, the only questionable feature is the justifiability or universal applicability of the causal postulate.

Within the past four or five decades physical research has clearly and definitely shown - strange discovery - that chance is the common root of all the rigid conformity to Law that has been observed, at least in the overwhelming majority of natural processes, the regularity and invariability of which have led to the establishment of the postulate of universal causality.

In order to produce a physical process wherein we observe such conformity to Law innumerable thousands, often billions, of single atoms or molecules must combine. (For professional physicists I may say here in parenthesis that this is also true of those phenomena in which, as we often say to-day, the effect produced by a single atom can be successfully studied; because in truth the interaction of this atom with thousands of others determines the observed effect.) In a very large number of cases of totally different types, we have now succeeded in explaining the observed regularity as completely due to the tremendously large number of molecular processes that are cooperating. The individual process may, or may not, have its own strict regularity. In the observed regularity of the mass phenomenon the individual regularity (if any) need not be considered as a factor. On the contrary, it is completely effaced by averaging millions of single processes, the average values being the only things that are observable to us. The average values manifest their own purely statistical regularity, which they would also do if the outcome of each single molecular process were determined by the throwing of dice, the spinning of a roulette wheel or the drawing of sweepstake tickets from a drum.

The statistical interpretation of the laws is illustrated in the simplest and clearest manner by the phenomena of gases, from which, by the way, the new ideas started. In this case the individual process is the collision of two gas molecules, either with one another or with the wall of the container. The pressure of the gas against the walls of the container was formerly attributed to a specific expansive force of matter in the gaseous state; but according to the molecular theory it is due to the bombardment of the molecules. The number of collisions per second against one square centimeter of the surface of the wall is tremendous. For atmospheric pressure at zero degrees Centigrade it runs into twenty-four figures (2.2. X 10"). Even in the most complete terrestrial vacuum and for only one square millimeter and only one-thousandth of a second the number still runs into a figure of eleven places. Besides giving a complete account of the so-called gas laws, that is, of the dependence of pressure on temperature and volume, the molecular theory also explains all other properties of real gases, such as friction, heat conduction, diffusion - and this purely statistically, as a consequence of the molecules being exchanged between different parts of the gas by individual processes of the utmost irregularity. In performing the corresponding calculations and discussing the relevant considerations we generally assume the validity of the mechanical laws for the single happening, the collision. But it must be stated that this is not at all necessary. It would be quite sufficient to assume that at each individual collision an increase in mechanical energy and mechanical momentum is just as probable as a decrease, so that taking the average of a great many collisions, these quantities remain constant in much the same way as two dice cubes, if thrown a million times, will yield the average 7 whereas the result of each single throw is a pure matter of chance.

From what has been said it follows that the statistical interpretation of the gas laws is possible, perhaps also that it is the most simple; yet we cannot conclude that it is the only possible interpretation. But a crucial test is furnished by the following experiment. If the pressure of a gas is really only a statistical average value it must be subject to fluctuations. These must become all the more obvious the more the number of cooperating elementary processes is reduced by reducing (1) the surface against which the pressure is exerted and (2) the inertia of the body which experiences the pressure, in order to allow a prompt reaction to a fluctuation that occurs within a short period of time. Both these conditions can be attained by suspending tiny, ultra-microscopical particles in the gas. These actually show a zig-zag movement of extreme irregularity, long known as the Brownian movement, which never comes to rest and agrees in all particulars with the theoretical predictions. Although the number of molecules which hit the particle during a measurable period of time is still very large, it is yet not large enough to produce an absolutely uniform pressure from all sides. Through a chance preponderance of the impact in a chance direction, which changes quite irregularly from moment to moment, the particle will be driven hither and thither on quite an irregular path. Here, therefore, we see a law of nature - the law of gas pressure - losing its exact validity in proportion as the number of cooperating individual processes decreases. One cannot easily imagine a more convincing proof of the essentially statistical character of at least this law.

I could here mention numerous other cases that have been experimentally and theoretically investigated, such as the uniform blue of the sky, which results from entirely irregular variations of atmospheric densities (consequent upon their molecular constitution), or the strictly law-governed decay of radioactive substances which results from the disintegration of the individual atoms, whereby it appears to depend entirely on chance whether an individual atom will disintegrate immediately or to-morrow or in a year's time. But however many examples are considered, they scarcely suffice to render our belief in the statistical character of physical laws as certain as does the fact that the Second Law of Thermodynamics, or Law of Entropy, which plays a role in positively every physical process, has clearly proved to be the prototype of statistical law. Although this matter would justify a closer examination, on account of its extraordinary interest, I must confine myself here to the very cursory remark that empirically the Law of Entropy is very intimately connected with the typical one-directional character of all natural processes. Although the Law of Entropy by itself is not sufficient to determine the direction in which the state of a material system will change in the next instant, it always excludes certain directions of change, the direction exactly opposite to the one which actually occurs being always excluded. By virtue of the statistical method the Law of Entropy has taken on the following content: namely, that every process or event proceeds from a relatively improbable--that is to say, more or less molecularly ordered-state to a more probable one--that is to say, to a state of increasing disorder among the molecules.

In regard to what I have said up to now there is no essential difference of opinion among physicists. But the case is otherwise in regard to what I shall have to say from now on.

Although we have discovered physical laws to be of a statistical character, which does not necessarily imply the strictly causal determination of individual molecular processes, still the general opinion is that we should find the individual process - for instance, the collision of two gas molecules - determined by rigid causality, if we could trace it. (In a similar way the result of a game of roulette would not be something haphazard for any one who could measure exactly the impetus given to the wheel, the resistance of the air, the friction on the axis, etc., etc.) In some cases, among which is also the one of colliding gas molecules, it is even claimed that quite definite features of the individual process can be ascertained; viz., the conservation of energy and momentum at every single impact, not merely in the average.

It was the experimental physicist, Franz Exner, who for the first time, in 1919, launched a very acute philosophical criticism against the taken-for-granted manner in which the absolute determinism of molecular processes was accepted by everybody. He came to the conclusion that the assertion of determinism was certainly possible, yet by no means necessary, and when more closely examined not at all very probable.

As to the non-necessity, I have already given my opinion; and I believe with Exner that it can be upheld, even despite the fact that most physicists claim quite definite characteristics for the elementary laws which they postulate. Naturally we can explain the energy principle on the large scale by its already holding good in the single events. But I do not see that we are bound to do so. In like manner we can explain the expansive force of a gas as the sum of the expansive forces of its elementary particles. But this interpretation is here decidedly incorrect, and I do not see why there it should be looked upon as the only possible one. I may further remark that the energy-momentum theorem provides us with only four equations, thus leaving the elementary process to a great extent undetermined, even if it complies with them.

Whence arises the widespread belief that the behavior of molecules is determined by absolute causality, whence the conviction that the contrary is unthinkable? Simply from the custom, inherited through thousands of years, of thinking causally, which makes the idea of determined events, of absolute, primary causalness, seem complete nonsense, a logical absurdity.

But from what source was this habit of causal thinking derived? Why, from observing for hundreds and thousands of years precisely those regularities in the natural course of events which, in the light of our present knowledge, are most certainly not governed by causality; or at least not so governed essentially, since we now know them to be statistically regulated phenomena. Therewith this traditional habit of thinking loses its rational foundation. In practice, of course, the habit may safely be retained, since it predicts the outcomes satisfactorily. But to allow this habit to force upon us the postulate that, behind the observed statistical regularities, there must be causal laws, would quite obviously involve a logically vicious circle.

Not only are there no considerations that force this assumption upon us, but we should realize, still further, that such a duality in the laws of Nature is somewhat improbable. On the one hand we should have the intrinsic, genuine, absolute laws of the infinitesimal domain; while on the other there would be that observed macroscopic regularity of events which in its most essential features is not due to the existence of the genuine laws but is determined rather by the concept of pure number, the most translucent and simple creation of the human mind. Clear and definite intelligibility in the world of outer appearances, and behind this a dark, eternally unintelligible imperative, a mysterious Kismet! The possibility that this may be in reality the case must be admitted; but this duplication of natural law so closely resembles the animistic duplication of natural objects, that I cannot regard it as at all tenable.

It must not be supposed, however, that I consider it a simple and easy matter to carry through and defend this new, a-causal (i.e., not necessarily causal) point of view. The ruling opinion to-day is that at least the laws of gravitation and electro-dynamics are of the absolute, elementary type, that they also govern the world of atoms and electrons and are perhaps at the basis of everything as the primary and fundamental Law. You are all familiar with the amazing success of Einstein's gravitation theory. Must we conclude from this that his gravitational equations are an elementary law? I hardly think so. In no case of a natural process is the number of single atoms which must cooperate in order that an observable effect may be produced so vast as in the case of gravitational phenomena. This would explain, from the statistical point of view, why we can attain such extraordinary accuracy in forecasting movements of the planets centuries ahead. Moreover I shall not deny that Einstein's theory yields powerful support to the belief in the absolute validity of the energy and momentum principles. With reference to the particle, these principles actually involve nothing more than a tendency towards absolute perseverance. For Einstein's gravitation theory is not really anything more than the reduction of gravitation to the law of inertia. That under certain conditions nothing changes is surely the simplest Law that can be conceived, and hardly falls within the concept of causal determination. It may after all be reconcilable with a strictly a-causal view of Nature.

In contradistinction to gravitation, the laws of electro-dynamics are quite generally applied to-day to processes within the atom itself, and indeed with amazing success. These positive results will be considered the most serious objection that can be advanced against the a-causal view. The space at my disposal does not allow of my going further into this question. I must confine myself to the following general remark, which at the same time briefly sums up the conclusions we have reached -

Exner's assertion amounts to this: It is quite possible that Nature's laws are of thoroughly statistical character. The demand for an absolute law in the background of the statistical law - a demand which at the present day almost everybody considers imperative - goes beyond the reach of experience. ¹ Such a dual foundation for the orderly course of events in Nature is in itself improbable. The burden of proof falls on those who champion absolute causality, and not on those who question it. For a doubtful attitude in this respect is to-day by far the more natural.

The electro-dynamic theory of the atom appears unsuited to furnish the proof, because this theory itself is universally recognized to be suffering from serious intrinsic incoherences which are often felt to be of a logical character. I prefer to believe that, once we have discarded our rooted predilection for absolute Causality, we shall succeed in overcoming these difficulties, rather than expect atomic theory to substantiate the dogma of Causality. This was Schrodinger's Inaugural Address at the University of Zurich, December 9th, 1922. This address was not printed on the occasion of its delivery. Some time afterwards the development of quantum mechanics brought Exner's ideas into the foreground of scientific interest, without, however, Exner's name being mentioned. The text here follows the original manuscript from which the address was read. (Source, Schrodinger, 1935, Science and the Human Temperament, Chapter VI.) ¹ The reference is to Exner, Vorlesungen uber die Physikalischen Grundlagen der Naturwissenschaften, chapter 89, der Zufall (Chance), p.677, (Deuticke, Leipzig/Wien, 1922)
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What Is A Law Of Nature? The laws of physics are generally looked upon as a paradigm of exactitude. Therefore one would naturally take it for granted that probably no other science would be able to give such a clear and definite answer when asked what is meant when we speak of a law of nature.