A Pragmatic Schematic

John Venn, an English logician of the early nineteenth century,

developed an inspired little graphic symbol to help him when contemplating what various sets of things have, and do not have, in common.

Each set being compared and contrasted is represented by a shape, generally a circle or oval.  Each shape could overlap one or more of the other shapes to indicate some commonality between their contents.  Here is a fairly intricate and way-cool Venn Diagram showing the versatility of musician, Steve Miller (music video of Fly Like an Eagle):

Venn Diagrams are tools of logic, commonly used by students in middle school math classes nowadays.

The simplest Venn Diagram involves only two sets.Posited in the outer crescent shapes are those elements exclusive to only one of the sets and not found in the other.  Their differences would be indicated in either one crescent or in the other.  For, instance, we might compare sets of balls and dolls.  Ping pong balls, tennis balls, basketballs, footballs and marbles would go in one crescent, and Raggedy Anne and Andy, Barbie and Ken, matryoshka and china dolls would go in the other.

In the area of overlap are whatever elements, functions, or characteristics are shared by both sets.  Keeping to our example of balls and dolls, we would list ‘playthings’ in the center, for this is the commonality the two sets share.

What is neither compared nor contrasted remains beyond or outside the two circles.

No set can include all sets, because it cannot include itself.  That which transcends and includes all lower sets is considered a primitive class.  Sets and classes are distinguished by how sharply they are defined.  For instance, sets of balls and dolls are sharply defined.  These sets can be included in a more loosely structured class of all things that interest children.

Things to climb on, hide in, safely get wet in and which can be molded into shapes are all certainly entertaining to kids; however, being neither balls nor dolls, these are excluded from the encircled sets inside the class.

Also included in the class of things that interest children are things that require supervision, like swimming in bodies of water, and things that should not be played with at all, such as: sharp objects, poisonous critters, matches and choking hazards.

The graphic above shows the sets within a class, beyond which there is an unbounded space that ultimately transcends and includes every conceptually specified abstraction within it.  Neither confined to class nor set, the potential from which all classifications are drawn remains unlimited and unnamed.

First sentence in the Tao te Ching says:

The Tao that can be named is not the eternal Tao.

The Symbol of Relationship is Named

Says Shakespeare’s Juliet to Romeo:

What’s in a name?  That which we call a rose
By any other name would smell as sweet.

The central area of mutuality — that which joins the two circles in relationship — is shaped somewhat like an almond.  So, it is sometimes called “mandorla”, an Italian word meaning ‘little almond’.

The overlap is also known as the Vesica Piscis, which translates from Latin to ‘bladder of a fish’, due to its shape being similar to a fish’s swim bladder.  Hope the photo below does not gross you out, but having never encountered a fish’s bladder, I was curious and guessed you might be too.  Well, sure enough, the posterior bladder (right part below) does have an oval shape!

Early settlements were always situated near water.  Fish was a natural staple, quite likely more essential than almonds, though almonds were native to the Mediterranean climate of the Middle East and into India in ancient times.

Another way to make sense of this term might be a bladder, or vessel, that is shaped like a fish.

Religious and Astrological Significance

Icthyus means fish in Greek.  Icthyus as a symbol, known among Christians as the ‘Jesus fish’, is the Vesica Piscis turned sideways.  The fish’s tail fin is the short extension of each circle beyond the enclosure.

Icthys, or Jesus FishJesus is associated with the Piscean Age.  Pisces is plural of piscis; it means fishes.  As a zodiacal sign, Pisces appears as two fish, sometimes connected by a line.

Astrological Ages are about 2160 years long, due to the precession of our Earth on its axis.  Each Age is associated with the apparent position of the sun in the sky, as related to each of 12 zodiacal constellations*, each pertaining to a 30 degree span around the ecliptic through which the sun passes once per year.

*A thirteenth constellation, Ophiuchus, crosses the ecliptic between Scorpius to the west and Sagittarius to the east; however, it was not included in zodiacal signs until 2011.

The New Age refers to the Age of Aquarius, the ‘age’ that follows the Piscean Age.  There are differing opinions as to whether this Age has begun, or is soon to begin.  That discussion is beyond the scope of this site.

Jesus is often depicted within the Vesica Piscis, which symbolizes the birth passage.

His positioning within the overlap of two circles represents the Christ as both son of an Earthly Mother and son of a Heavenly Father.  Moreover, Jesus is considered by Christians to be both God and man, again connecting the dual realms of Heaven and Earth.

The mother of Jesus is Mary, etymologically related to Marina, the waters.  As such, she delivers the fish.  Star of the Sea, Madonna, Blessed Mother and Queen of Heaven, are just a few of the different names for the mother of Jesus.  By any name, she is often depicted within a mandorla/Vesica Piscis, sometimes holding the infant Jesus.

In the photo below, the interlocking circles that form the mandorla/Vesica Piscis, are part of the famous well cover at Glastonbury, England.

Legend has it that securely secreted in the depths of this well is the precious chalice used to catch the blood of Jesus as he hung on the cross.

The pope’s mitre, seen above, is in the shape of a mandorla, as is the cathedral window below.Musical Scale in Architecture of Cathedral Windows

Says Goethe:

Architecture is frozen music.

Note the remarkable relationship this symbol has with music.

The symbol of relationship created by overlapped circles has a history of usage ranging from mystical to practical.

Hidden Roots of Numbers One through Five

This sacred symbol even reveals the square roots of one through five.

Given an assigned value of 1 for the radii of both circles, the square roots of the numbers 1 through 5 are found within the rectangle whose length is delineated by the vertical diameters of the two circles and whose width is delineated by connecting the two circles at their heights and bases.  This rectangle contains two stacked squares, whose sides are 1, or √1.  The diagonal across either of these squares (depicted inside the lower square) is √2.  The vertical line extending from top to bottom of the almond shape is √3.  The length of the rectangle itself is 2, or √4.  The diagonal inside the rectangle formed by the two stacked squares is √5.

By now, we are familiar with the color-coding used throughout this site.  Green for Mother Earth, blue for Father Sky.  We also learned in the Geometry of Universal Mother and Father that Female is symbolized round and Male as straight.

The two green circles below are arranged in a precisely balanced way.  The overlapped circles produce a decidedly feminine imagery.  Masculinity is conveyed in the straight blue lines of the perspective drawing of a transparent cube aligned on its diagonal axis within.

The radius of the circles is equivalent to the side length of the cube, one unit.  The diagonal (dotted line) between the upper back to lower front vertices of the cube is equivalent to the vertical distance in the Vesica Piscis, the square root of 3.

The perfect fit of Male in Female is both geometrically significant and biologically convenient.  Enough said!

A Worthy Name

The significance of a working symbol for relatedness between two opposites can be intuited whether or not one knows all the facts behind the emblem’s origin, or its continued usage throughout history.

The Outer Crescents = Either & Or
The Central Area of Overlap = Both
Beyond or Outside the Circles = Neither

Appreciated for the mystique of religious implications, its efficiency in math and logic, its hauntingly soaring alignment with our musical scale, how it functionally puts the power of the arch in architecture, and the elegant, if not sacred, geometry of its form, this symbol surely deserves a name commensurate with its long-lived usefulness and multi-purpose strength as a symbol.

My friend, Napi, once described the Venn Diagram as a Pragmatic Schematic.

Pragmatic |pragˈmatik|
dealing with things sensibly and realistically

Schematic |skəˈmatik|
a diagram that simplifies

The name stuck, at least between the two of us.

The abbreviation for the Pragmatic Schematic (PS) is doubly apt.  We can heartily agree that it is a Profound Symbol.  And, while P. S. at the end of a letter announces an afterthought (Post Script), ‘PS’ is instead a fabulous tool of forethought!

Moving through the rest of this blog-book, PS is synonymous with the simple, two-circle Venn Diagram, and the terms Vesica Piscis, Vesica (for short) and mandorla, all refer to the pointed oval shape formed by two circles overlapping in a specific way: the circumference of each circle touches the center-point of the other.

P.S. We just couldn’t leave this page without musing on the Hourglass Nebula as photographed by Hubble in 1995.

Asides Provide Applications of PS

Pull * Transition * Push * Rest in the PS gives examples how dynamical processes can be contemplated from the logical perspective of the Pragmatic Schematic.

Body-Mind Communication shows how the PS facilitates our exploration of human scale development, both individually and collectively.