CHAPTER IV - Differential Equations and the Paths of the Souls

“Every one who is left far from his source
Wishes back the time when he was united with it.”

Jalal ad-din Rumi 
The Song of the Reed (Mathnawi I:1-3)

I – Introduction to Differential Equations

1 – Mathematical Introduction

Note: The mathematical examples discussed in this section are lifted from the book Differential Equations (Blanchard, Devaney, 2nd Edition).

Mathematically speaking, differential equations are equations containing derivatives of a function.  Differential equations are often used to describe the rate of growth of a micro-universe such as the population of a country.

An example of a differential equation would be:

In the Holy Name of God, the All-Compassionate, the All-Merciful,
With peace and blessings upon His Prophet Muhammad SAW
upon his holy family, and upon his blessed Companions.
Bridging the Gap between Spirituality and Science
This equation simply states that the sum of the first derivative of the function y and the product of x and y2 is equal to 2x.

2 – Differential Equations and the Human Souls

Spiritually speaking, differential equations provide us with some information about the different types of human souls existing in every human being.  In general, one may categorize the human souls in three distinct categories:

  • The souls of the prophets
  • The souls of the saints
  • The souls of normal individuals

The science of differential equations is a means to understanding the behavior of these three types of human souls.  Many mathematicians believe that differential equations allow us to mathematically predict the future because they provide an understanding of the rules that govern the changes that may occur in the future.  This mathematical belief is in total accordance with the spiritual interpretation of differential equations which predicts the behaviors of human souls.

3 – Differential Equations and the Principle of Eternity

a – Components of the Mathematical Model

Generally speaking, every mathematical model is made up of three “quantities”:

  • The independent variable
  • The dependent variables
  • The parameters

For instance, in the following differential equation:

  • t is the independent variable (time)
  • y is the dependent variable
  • k is the parameter

The Independent Variable

The independent variable is the entity that does not depend on any other quantity in the mathematical model.  This independent variable is just that: independent from any other variable while every variable in the model depends on it.  For instance, if one says “position is a function of time”, one means that position is a variable that depends on time.  In mathematical experiment, time is often used as the independent variable.
The Dependent Variables

These variables are the ones that entirely depend on the independent variable.  For instance position, height and depth.  They may be referred to as x, y or z.

The Parameters

Parameters are quantities that do not change with the independent variable but that can be adjusted by natural causes or by a scientist running an experiment.  For instance, if we are studying the motion of a rocket, the initial mass is a parameter.

It is important to note that the goal of a mathematical model expressed in terms of a set of differential equations is to describe the behavior of the dependent variables as the independent variable changes and as the parameters are adjusted.  In other words, everything depends on the independent variable while the parameters are adjusted to facilitate the study of the dependent variables.

b – Spiritual Reality of the Mathematical Model

The three quantities we just described correspond to the three postulates of eternity.

Manifestation of the 1st Postulate

The independent variable represents the Absolute or God’s Attribute of Eternity (As-Samad in Arabic).  The independent variable is not affected by any other variable in the plane of the mathematical graph just like God’s divine Attributes are not affected and do not depend on anything in the plane of creation.

Manifestation of the 2nd Postulate

The 2nd postulate describes the nature of spiritual mediation.  As we saw earlier in the chapter about the Muhammadan Light, the nature of the mediator (light) must be dual: it must have a permanent heavenly nature as well as a temporal earthly one.  The concept of spiritual mediation manifests itself in differential equations as the set of parameters which are quantities that possess a permanent nature (like the independent variable) but that may be adjusted if necessary (like the dependent variables).  Parameters possess indeed a dual nature: they exhibit the characteristics of both the independent variable and the dependent ones.

Manifestation of the 3rd Postulate

Creation’s complete dependence on the governing rules of the Absolute is manifested in differential equation as the perfect submission of the dependent variables towards the independent variable.

II – Eigenvalues

1 – What are Eigenvalues?

Eigenvalues are special numbers used to solve matrices.  For instance consider the following matrix A:
In order to solve the matrix A, we use a number λ (pronounced “lambda”) called the eigenvalue of A.  λ can only be the eigenvalue of A if and only if there is a vector V = (x,y) for which we have:

AV = A(X.Y) = λ (x,y) = λ * V

The vector V is called an eigenvector corresponding to the eigenvalue λ.  Let us not worry about mathematical formulas for now.  What is important to remember here is that the analysis of a set of differential equations is done through the study of the different values the number λ can assume.

2 – Spiritual Interpretation of the Eigenvalue

Spiritually speaking, the eigenvalue λ of a matrix is the manifestation of the soul that is present in every human being.  This soul, which is referred to as nafs in Arabic, constantly fluctuates between good and evil, spirituality and earthly matters, heaven and earth.  Mathematical knowledge of the eigenvalue will therefore lead to spiritual knowledge of the three types of souls we described earlier.

3 – Correlation between Mathematics and Spirituality

a – Terminology Convergence

The word eigen means “self” in German.  It is of course no coincidence that this word refers to the soul which, in Arabic, means “self” (nafs).  There is indeed divine wisdom in every sign.  As for the eigenvector, it is a vector where the vector field points to the same or opposite direction as the vector itself.  The eigenvector’s reflective nature is the reason for the term “eigen”.

b – Two Types of Eigenvalues and the Dual Nature of the Soul

The two real distinct eigenvalues of a linear system are manifestations of the two types of soul every human being possesses: the spiritual soul and the earthly soul.  These two souls are always in conflict in us as each one of them wants the opposite of what the other desires.  Our spiritual soul is always asking for more spirituality and more heavenly emotions such as love, friendship and compassion; on the other hand, our earthly soul is attracted to lust, self-interest and jealousy.

The first eigenvalue λ1 represents the spiritual soul in us.  It feeds on heavenly knowledge and spirituality.  The spiritual soul’s instinctive nature is to seek reality and to be inclined towards goodness.

The second eigenvalue λ2 represents the earthly soul which is naturally inclined towards worldly desires.  It evolves within the spheres of the material world and it determines the low characteristics of human nature.

The analysis of the pair of eigenvalues λ1 and λ2 helps us define the spiritual trajectories of the human souls through the interpretation of a mathematical graph.

Next:  III - The Graphs of Differential Equations