Math and Spirituality: Connections History and Lessons Learned
Chapter Four
There is no symbol of spiritual significance that shows up more than the pentagram.
The pentagram is a figure constructed from a regular pentagon, a 5 sided shape with all sides of equal length. The pentagram is made by drawing the diagonals of the pentagon, drawing lines to connect all of the corners of the shape that are not already directly connected.
Pentagrams are Everywhere
It is curious how frequently this symbol shows up. Again and Again. Across time and across civilizations. The earliest recorded instance of the pentagram shows up around 3500 B.C. inscribed in Sumerian pottery.
The symbol can be seen in ancient Greece, it can be found in ancient Jerusalem, it pops up in England during the Middle Ages, and it is the foundation of Da Vinci’s “Vitruvian Man” an influential art piece from the Renaissance. In modern times the symbol crops up everywhere in fiction and modern occultism.
Depictions of pentagrams are associated with cults and satanism. You can find the 5-pointed star littered throughout the horror and thriller genres.
Pentagrams are everywhere. But that fact alone might not be a groundbreaking revelation. To be a little skeptical here, a star is a pretty simple shape to draw. So maybe it is not so crazy to think that people just like drawing star shapes. And that would explain why the pentagram keeps showing up.
The groundbreaking revelation comes from asking a slightly different question. Instead of asking “Why does the pentagram show up so consistently?” We should ask “Why does the pentagram show up in such consistently spiritual contexts?”
No other shape can compare. Why weren’t people etching trapezoids onto their church walls or making octagons the symbols of their sacred orders? It is simply too much of a pattern to be mere coincidence.
Examples Throughout History
Here is a by-no-means exhaustive list of spiritual meanings assigned to the pentagram by different peoples. This list of examples could fill a dozen textbooks so I would advise you to do some exploring on your own if any of these pique your interest.
In ancient Greece the pentagram was a symbol used to represent their creation myth (in a similar vein to our last chapter). They represented the 5 “niches” or “regions” that the original gods created. The pentagram was the official seal of the holy city of Jerusalem. In Renaissance era England the pentagram symbolized the five wounds of Christ suffered during the crucifixion. And the Vitruvian man, purposefully drawn in the shape of a pentagram, is regarded as the most famous drawing in the world. It has been interpreted by many as having “secret” or “hidden” spiritual meaning.
History cannot trace the significance of the pentagram all the way back to its first sighting in 3500 B.C. Sumeria. However, in the 6th Century B.C. the construction of the Pentagram led to an incredible discovery, one that an ancient order of mathematicians chose to hide. Its discoverer was exiled, cast out to sea to die. And the leader of this order, surprisingly, is someone that you may have heard of.
Pythagoras of Samos and the Invention of Math
Pythagoras is famous in modern times for his triangle based theorem, the Pythagorean Theorem. Funnily enough there is a lot of historical evidence that Pythagoras was not the first to discover the famous theorem that is named after him.
But his contributions to society remain vast. Pythagoras of Samos was said to have gained his knowledge and wisdom through travel. In 6th Century B.C. he traveled from Greece to Babylon and ended up settling in Croton, a city that would now be located in Southern Italy.
Pythagoras was the first person who decided that mathematics should be studied as “free teaching”. Mathematical problems had of course been solved prior. But problems had been solved within the context of practical issues, necessities or sacred duties.
Humans were solving math problems to build river dams and needed to do math to carve large circular stones for altars. But Pythagoras was the first person who decided to study mathematics for its own sake. There did not need to be a practical, current problem to solve. Pythagoras began exploring mathematical problems and proving mathematical truths, just because he found it interesting.
So with this in mind it is fair to say that Pythagoras invented the field of mathematics. Mathematics as a disicipline and a field of study began with Pythagoras. Once mathematics as a subject was invented, something curious happened. A new religion sprang up known as Pythagoreanism.
For emphasis. Once it was discovered that math could be proven and discovered, a religion immediately followed.
The Pythagorean Religion
So what were the Pythagoreans like? There is not a ton of surviving information from the religion itself but we have reason to believe that they embraced some ideas that would be considered quite progressive. Pythagoreans preached vegetarianism and defended the right for women to learn math and philosophy.
The intellectual achievements of the religious group are well known and celebrated. Pythagoreans created the field of mathematics known as Number Theory, a field of study concerned with the natural numbers, everyday numbers like 2, 3 and 4. They also developed the study of music theory, founding the belief that music could be expressed by numbers and their ratios to one another.
Music was beautiful because it could be expressed mathematically. And this belief went beyond music or other pleasures. The Pythagoreans started studying mathematics and they began to feel greatly that numbers were the building blocks of our cosmos. Deeply embedded in the spiritual philosophy of the Pythagoreans was the idea that everything in the universe could be expressed neatly by numbers.
It was devastating when they found out that this was not the case.
Properties of the Pentagram
How did they find out? They were exploring the mathematical properties of the pentagram.
The Pythagoreans were trying to understand the ratio between the sides of the pentagon and their diagonals. By drawing the diagonals, the pentagram is constructed within.
Nowadays, if we wanted to figure out a ratio between two values we would divide them. However the mechanism we use for division was not yet used during the times of the Ancient Greeks.
Back then they used a method known as anthyphairesis (ann-the-fair-uh-sis). Which translates roughly to “repeated subtraction”.
Let’s take an example. Say I wanted to find the ratio of 162 to 27 through the method of anthyphairesis
First we subtract 27 from 162
162-27 = 135
Then we repeat this subtraction until we get a number lower than 27
135-27= 108
108-27= 81
81-27= 54
54-27 = 27
27-27 = 0
This result of 0 means that 27 divides evenly into 162, and if you were to check 162/27= 6.
A number other than 0 would imply that the numbers do not divide evenly into one another and the result would give a remainder.
A Devastating Cognition
This process of anthyphairesis was used to investigate a certain ratio. The Pythagoreans wanted to know the ratio of the length of a side of the pentagon to the ratio of one of its diagonals.
As they began their method of repeated subtraction, something slowly became clear. This method of repeated subtraction would go on endlessly. The process would never end.
In their investigation of the properties of the pentagram the Pythagoreans had discovered irrational numbers. An irrational number is a number that cannot be expressed as the ratio of two integers. ∏, which we discussed in prior chapters is an irrational number. √2 is an irrational number as well.
This discovery is a significant one when it comes to the advancement of mathematics as a field. However the concept of irrational numbers did not sit well with the Pythagoreans. Their belief system centered around numbers. The very order of the universe could be expressed neatly by numbers! This discovery of irrational numbers spat in the face of the very core of their beliefs.
The mathematician who discovered the irrational numbers was named Hippasus of Metapontum. The members of the Pythagorean order decided that they would keep this discovery a secret, as it sat in contradiction to their established beliefs. Hippasus decided to reveal his secret discovery. And he was banned from the order. Legend says they tied him to a raft while he was asleep and pushed him out to sea to die.
There is a long and storied tradition of mathematicians who died in peculiar and dramatic fashions. Hippasus of Metapontum did receive some vindication however. After the initial overreaction to the discovery of irrational numbers, the Pythagoreans decided to embrace the discovery as truth. The understanding of irrational numbers became an intellectual rite of passage.
The Pythagoreans adopted the Pentagram as the symbol of their holy order, as an homage to the irrational number. A nod to the notion that the nature of the universe could not be defined and understood so easily.
In Closing
The meaning that the Pythagoreans assigned to the Pentagram persists throughout history. If one wants to understand the nature of all things with their mind, they will encounter pure chaos. In our journey as spiritual beings we encounter things that we must accept as true, even though we cannot understand these things. In our lives there are phenomena that cannot be defined. We brush up against the infinite and lack the ability to describe it.
The spiritual significance of the pentagram has been assigned and redefined by many people throughout history. However, I believe that its pervasiveness truly boomed once it became the symbol of the Pythagoreans. This spiritual order, the very people who founded mathematics as a field of study, believed that mathematics was the key to defining and ordering our universe.
But no such well-ordered and well-defined universe exists. At least it is not the one that we all live and breathe in. The Pentagram exists as acknowledgement of this fact. And it is the very mathematical properties of this shape that showed us that even with the best tools available, we would be incapable of defining our world in a simple and elegant way.
As science and technology advances throughout the centuries humanity continues to discover more and more complexity in our universe. This complexity remains forever out of our grasp. And the pentagram keeps showing up in art, religion and culture. A reminder hidden in plain sight, of the elaborate nature of our universe.
Next Time:
The Differing Levels of Infinity
