On the Character and Virtue of Perfect Numbers

In keeping with our presentation of mathematical ideas from the qualitative symbolic perspective, we now turn to a consideration of the perfect numbers.  A perfect number is equal to the sum of its proper divisors.  This is in contrast to a deficient number, for which the sum of the divisors is less; and a super-abundant number, whose divisors yield a sum greater than the number itself.  This classification is enlightening when we consider numbers as having a personality, a quality of character, and an ethical behavior, rather than as pure quantity.

Examples of perfect numbers are 6, 28,  and 496.  Let's see why each of these fits the definition of a perfect number to illustrate the concept:

• The proper divisors of 6 are 1, 2 and 3.  When we take the sum of these, 1+2+3 = 6.
•  The proper divisors of 28 are 1, 2, 4, 7, and 14.  Their sum 1+2+4+7+14 = 28.
•  The proper divisors of 496 are 1, 2, 4, 8, 16, 31, 62, 124 and 248.  Their sum 1+2+4+8+16+31+124+248 = 496.

The perfect numbers are extremely rare among the integers.  There are currently only 47 known perfect numbers.  The discovery of perfect numbers is inherently connected to the discovery of Mersenne primes, and the there is a famous distributed-computing project devoted to their discovery, known as the Great Internet Mersenne Prime Search (GIMPS).

The deficient numbers and super-abundant numbers are in direct contrast to the perfect numbers.  Here are some examples: 21 is deficient since the sum of its divisors is 1+3+7 = 10 which is less than 21; while 12 is super-abundant as the sum of its divisors is 1+2+3+4+6 = 16 which is greater than 12.

We begin with a quantitative phenomenon and proceed to explore the qualitative symbolism.  If a deficient number had a personality, how would it behave?  It is best described as someone who is lazy, poor, unskilled, weak, or uneducated.  If a super-abundant number had a personality, how would it behave?  It could be described as greedy, prideful, self-aggrandizing, indulging or possessive.  What about perfect numbers?  Perhaps accommodating, comforting, helpful, caring, assisting, or welcoming.

The great English Platonist Thomas Taylor writes eloquently on this classification of numbers in his Theoretic Arithmetic of the Pythagoreans:

The super-abundant numbers are such as by an immoderate plenitude, exceed as it were, by the numerosity of their parts, the measure of their proper body.  On the contrary, the deficient numbers being as it were, oppressed by poverty, are less than the sum of their parts.

Such therefore are these numbers, the former of which in consequence of being surpassed by its parts, resembles one born with a multitude of hands in a manner different from the common order of nature, or one whose body is formed from the junction of three bodies, or any other production of nature which has been deemed monstrous by the multiplication of its parts.  But the latter of these numbers resembles one who is born with a deficiency of some necessary part, as the one-eyed Cyclops, or with the want of some other member.

Between these however, as between things equally immoderate, the number which is called perfect is alloted the temperament of a middle limit, and is in this respect the emulator of virtue; for it is neither extended by a superfluous progression, nor remitted by a contracted diminution; but obtaining the limit of a medium, and being equal to its parts, it is neither overflowing through abundance, nor deficient through poverty.

With that said, Taylor then goes on to describe the virtues of the perfect numbers:

Perfect numbers therefore, are beautiful images of the virtues which are certain media between excess and defect, and are not summits, as by some of the ancients they were supposed to be.

And evil indeed is opposed to evil, but both are opposed to one good. Good however, is never opposed to good, but to two evils at one and the same time.  Thus timidity is opposed to audacity, to both which the want of true courage is common; but both timidity and audacity are opposed to fortitude.

Craft also is opposed to fatuity, to both of which the want of intellect is common; and both these are opposed to liberality.  And in a similar manner in the other virtues; by all which it is evident that perfect numbers have a great similitude to the virtues.

But they also resemble virtues on another account, for they are rarely found, as being few, and they are generated in a very constant order.  On the contrary, an infinite multitude of superabundant and deficient numbers may be found, nor are they disposed in any orderly series, nor generated from any certain end; and hence they have a great similitude to the vices, which are numerous, inordinate, and indefinite.

Thomas Taylor, 1758-1835

What we see happening with the qualities of perfect numbers is a function of balance and centering.  And this is exactly how the quantitative aspect of the number is a balance between the too-little of the deficient number and the too-much of the super-abundant number, and holds the position of a center between these extremes.

Are we then surprised that the perfect numbers are so few and far between?  How many people in this world have a harmonious balance between giving and taking?  Indeed it is a rare quality.  The qualities of the perfect numbers are to be imitated by those who are cultivating virtues, which should be everyone!

The study of the perfect numbers was initiated by the Pythagoreans, and mathematicians are still exploring their mysteries to this day.  It is still an open problem to determine if there are infinitely many perfect numbers, and whether there exists any odd perfect numbers, all the known examples being even.

The Pythagoreans, a secret society of spiritual mathematicians

 Let's explore the examples of perfect numbers and see how they come up in geometry and spiritual symbolism, beginning with the number 6.

A deep mathematical mystery is that 6 circles will exactly fit around a central circle with the same radius and whose centers intersect on the circumference, as shown here:

Six around the One

This figure is called the Seed of Life and it is the primary archetype for all Sacred Geometry, resulting in a progression into the Flower of Life and the Tree of Life, as well as sacred figures.

The fact that the sixth circle completes the figure, resulting in this mysteriously symmetrical arrangement, could be why the Book of Genesis describes the creation of the world in six days, with the seventh day being a day of rest, which would be a return to the central circle.

Notice that this pattern contains the blueprint for the Cube, one of the five regular solids known colloquially as the Platonic solids, to which Plato assigned the element of Earth.

Blueprint of the Cube

Isometric view of the Cube

The Cube is the regular solid with 6 square faces.  The 6 faces correspond to the 6 directions of space: up, down, front, back, left, and right.  This mystery is described in the Kabbalistic text Sepher Yetzirah, where God creates the Cube of Space by infinite extent of His being through the 6 directions, and then sealing them with the 6 permutations of the three-letter Divine Name IHV (Yod-Heh-Vav).  6 is the 3rd Triangular Number with 1+2+3=6, while the 6th Triangular Number is 21, since 1+2+3+4+5+6=21 and 21 is the sum of the letters IHV (10+5+6=21). 

Among the other Platonic solids, the Octahedron is the dual solid to the Cube, and has 6 vertices.  The Tetrahedron has 6 edges.

The Octahedron has 6 vertices

The Tetrahedron has 6 edges

Now let's consider 28.  There are approximately 28 days in a lunar cycle.  It is the 7th Triangular Number with 1+2+3+4+5+6+7 = 28.  This is interesting in light of Genesis 1:1 "In the beginning, God created the Heaven and the Earth" which in Hebrew has a total of 7 words and 28 letters.

Genesis 1:1 - Seven words, Twenty-eight letters
28 is a perfect number

This definitely suggests that the plan that God had in mind when creating the Heavens and the Earth was based on perfection from the very beginning.

How about 496?  It is the 31st Triangular Number, and 31 is the Divine Name El (Aleph-Lamed) attributed to the 4th Sephiroth in Kabbalah.  The 4th Triangular Number is 10, which shows up as the Tetraktys.

The Tetraktys, representing the Decad
1+2+3+4 = 10

The Hebrew word for Kingdom is Malkut, which Kabbalah uses as a name for the 10th Sephiroth.  Malkut is the Kingdom and represents the Divine archetype of the physical world.  The letters which comprise the word Malkut add to 496!  This says that the physical world is a perfect Kingdom, but we need to see it as being filled with divinity in order to realize our citizenship in this Kingdom.  Now modern physics has even discovered the significance of the perfect number 496, since in order for superstring theory to make sense, the dimension of the gauge group must be 496:

The number 496 is a very important number in superstring theory. In 1984, Michael Green and John H. Schwarz realized that one of the necessary conditions for a superstring theory to make sense is that the dimension of the gauge group of type I string theory must be 496. The group is therefore SO(32). Their discovery started the first superstring revolution. It was realized in 1985 that the heterotic string can admit another possible gauge group, namely E8 x E8.
-- Source: http://en.wikipedia.org/wiki/Number_496

The purpose of this article was to weave through the interplay of quantitative and qualitative aspects of numbers, so we can see the difference of these perspectives and yet how the qualitative aspects are derived from the quantitative properties.  This perspective has us bouncing back and forth between the left and right hemispheres of the brain, and through the harmonization of these polar opposites we find fertile ground for the growth of new ideas and insights into how our minds work and who we are spiritually, keeping with the program of Via Mathesis.

Books mentioned in this article: