CHAPTER IV - Differential Equations and the Paths of the Souls
III – The Graphs of Differential Equations
In order to solve a system of differential equations, one first uses algebra to compute the eigenvalues and eigenvectors of the matrix. Once one finds these values, one may write down the straight-line solutions of the system. These straight-line solutions form what we call a mathematical graph or a phase portrait. As we shall see the sign of the eigenvalue plays an important role in determining the behavior of the corresponding mathematical graph.
1 – What Mathematical Graphs Represent in Spirituality
Graphs allow us to visually analyze and interpret mathematical models. Spiritually speaking, they are metaphoric representations of the spiritual paths of the souls of human beings. They make the way leading to the Origin manifest. A simple example of the correlation between graphs and spirituality would be the three stages of Islam:
These three spiritual levels are manifested in geometry as the three most important dimensional graphs:
- The two-dimensional graph
- The three-dimensional graph
a – The Straight Line and the Divine Law
The straight line is a one-dimensional graph and as such is the most basic graph in geometry. A straight line is always the shortest distance between two points. This mathematical rule is a manifestation of the following spiritual rule: the shortest path that lies between the Divine Presence and mankind is the Straight Path (Sirat al Mustaquim in Arabic). The concept of the straight line is one of foundations of Islam.
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.
THE PRINCIPLE OF ETERNITY
Bridging the Gap between Spirituality and Science
At this level of spirituality, one must completely focus on the strict and straight application of the heavenly rules embodied by the divine law. However, limiting our knowledge to this level does not provide any spiritual elevation as the variables (i.e., the seekers on the path) that are located on the line may only move right or left but never up or down: spiritual elevation is indeed not possible.
b – The Two-dimensional Graph
The two-dimensional graph is very common in science. It has an x-axis and a y-axis. This graph is the manifestation of the second level of religion: the spiritual path (Sufism, asceticism, etc). The x-axis represents the straight path mentioned earlier the y-axis represents spiritual elevation. The variables (or seekers) may therefore move left or right but also up or down.
It is important to note that the divine law, which is represented by the straight line or x-axis, is not lost; rather, it is an essential component of the graph. Therefore, the divine law of a religion (i.e., the set of rules transmitted by the Holy Quran, the Psalms or the Thorah) must complement the spiritual exercises of the seeker.
c – The Three-Dimensional Graph
This graph is the manifestation of the last level of Islam: Reality (Haqiqa). The graph is made up of the two previous lines (horizontal for the straight path and vertical for the spiritual path) and a new straight line that provides depth (manifested as the z-axis) to the graph. The three-dimensional graph allows the variable (or seeker) to move on a straight line, receive elevation and advance in depth.
As we all know our physical world appears in three dimensions as none of us is confined in a two-dimensional world. Thus, the reality of our existence on earth lies within the three levels of any revealed religion such as Islam, Christianity and Judaism.
2 – Graphs of Differential Equations
a – The Three Types of Graphs
When the eigenvalues are real and distinct, as in the case of our study, there exist three types of mathematical graphs that describe the straight-line solutions of a set of differential equations (note the sequence of three levels manifesting itself again in science; see chapter II: Secrets of Odd Numbers for more details). These three graphs are:
The sign of the pair of eigenvalues determine the classification of these mathematical graphs.
b – Spiritual Interpretation of the Graphs of Differential Equations
The graphs of linear systems of differential equations (source, saddle and sink) represent the spiritual paths of the three types of human souls (prophets, saints and normal individuals, respectively).
- Source = Spiritual paths of prophets
- Saddle = Spiritual paths of saints
- Sink = Spiritual paths of normal individuals
As for the two eigenvalues, they are manifestations of the two types of souls living in every human beings:
Therefore, we have:
λ1 + λ2 = complete human soul
3 – Detailed Study of the Source
a – Mathematical Introduction
The origin of the graph is said to be a source if and only if the two eigenvalues of the linear system are positive, that is:
0 < λ2 < λ1
In this case:
- All the solutions of the linear system (except the equilibrium solution) go to infinity.
- The graph is said to be unstable
- Most solution curves leave the origin (0,0) in the direction of the λ1 eigenvector.
Therefore, when the two eigenvalues are positive, the solutions curves go from the origin to infinity. Remember, the movement is important because graphs represent the spiritual trajectories of our souls.
A source is said to be unstable because there are initial conditions arbitrarily close to the equilibrium point whose solutions still move away from the origin.
Also, note that the directions of the solutions tend to follow that of the vector representing the highest value, which is the λ1 eigenvector in this case.
Below is the graph of the source. The arrows represent the solution curves that are moving from the origin to infinity. As we shall see this diverging motion is the manifestation of the spiritual trajectories of the prophets’ souls as they move from the Divine Presence to the infinite creations to convey the heavenly messages.
b – Spiritual Interpretation of the Source
Every human being possesses a universal soul that is made up of a spiritual part (manifested as λ1) and an earthly one (manifested as λ2). However, the souls of the prophets are different from the souls of the rest of human beings in many aspects. For instance, prophets cannot commit sins. Thus, both their spiritual souls and their earthly ones dwell in the higher realm of heaven as they have a direct and intimate relation with the Divine Presence. Their souls are indeed entirely heavenly. This is manifested in differential equations by the fact that the two eigenvalues of the source (λ1 and λ2) are both positive. Prophets carry heavenly messages that need to be transmitted to their communities. They are coming from the Divine Presence and moving towards creation. The movement described by their spiritual paths is therefore from the origin (0,0), which represent the Divine Presence, to the infinite creations of God.
4 – Detailed Study of the Saddle
a – Mathematical Introduction
The graph is said to be a saddle when there is one positive eigenvalue and one negative eigenvalue, that is:
λ2 < 0 < λ1
In this case:
- There are two lines of solution lines on the graph.
- One solution line converges towards the origin (0,0).
- The second solution line moves away from the origin (0,0).
Therefore the graph of a saddle describes two opposite movements at the same time: towards the origin and away from it.
b – Spiritual Interpretation of the Saddle
The graph of the saddle is a direct manifestation of the spiritual paths of the saints’ souls. The solution curves that converge towards the origin represent the saints’ earthly souls which are moving from creation to the Divine Presence. The solution curves that move away from the origin represent the saints’ spiritual souls which are moving from the Divine Presence to the communities. Saints are human beings who successfully established a perfect balance between the spiritual world and the material one. They have secured an eternal connection with heaven while maintaining a connection with the temporal world for the sake of their communities.
In summary, the spiritual path of a saint is dual in nature:
- His spiritual soul inherits from the reality of the prophets’ souls and therefore moves from the Divine Presence to creation. As the keeper of the keys to the manifestation of the divine reality in every human being he must attract the receptive souls.
- His earthly soul inherits from the reality of normal individuals and therefore moves from creation to the Divine Presence. His soul is constantly busy with spiritual practices that aim at separating the self from the worries of the temporal world and ascending to the realm of spirituality.
Mathematically speaking we have:
- λ1 is positive <=> solution curves tend away from the origin.
- λ2 is negative <=> solution curves tend toward the origin.
5 – Detailed Study of the Sink
a – Mathematical Introduction
The graph is said to be a sink whenever the two eigenvalues are negative, that is: λ2 < λ1 < 0 In this case:
- All solutions tend to the origin as time increases.
- Most tend to the origin in the direction of the highest eigenvector λ1.
- The graph is said to be stable.
The sink is basically the opposite of the source. However, both share a crucial similarity: the directions of the solution curves tend to follow that of the highest eigenvector λ1.
The graph of a sink is stable because the points that are close to the origin lean toward the equilibrium point as time increases.
b – Spiritual Interpretation of the Sink
Generally speaking, most human beings evolve in a material world filled with sins, temptations and selfishness. Therefore, both their spiritual and earthly souls fight to maintain a healthy balance. Mathematically speaking, this sort of spiritual imbalance is manifested by the fact that both λ1 (spiritual soul) and λ2 (earthly soul) have negative values such as:
λ2 < λ1 < 0
Normal individuals need to establish a connection with the Divine Presence. The path of normal people is therefore from the material world to the Origin. Prophet Muhammad mentioned this trajectory when he says:
“Oh God, You are my destination and Your good pleasure is what I seek.”
“Illahi anta maksudi wa riddaka matlubi.”
Here, Prophet Muhammad (SAW) shows the way for all humanity to follow: to leave the temporal world and to return to the Origin.
Note the path of the solution curves of the sink tends to the origin in the direction of the highest eigenvector λ1. The vector λ1 “shows” the way for all the solution curves to follow. This vector is the eigenvector that is closer to the number 0, which represents the origin.
Therefore, on the journey that leads to the Divine Presence, all human beings (represented by λ2) have to follow the way of the one that is higher (represented by λ1) and who acts as a guide.
This obedience is expressed in the Holy Quran:
“Obey Allah and obey the Messenger
and those vested with authority from among you.”
“Ati Allah wa ati’u Rasul wa ulil umri minkum’
Holy Quran 4:59
6 – The Graphs of Differential Equations Hidden in a Holy Verse
The three main types of graphs that describe the straight-line solutions of a set of a differential equations are mentioned in the following holy verse:
“To God we belong and unto Him we shall return”
“Inna Lillahi wa inna ilayhi raji’un”.
Holy Quran 2:155-157
- The first part of the verse “To God we belong” describes the spiritual path of the prophets. They are moving from God to His infinite creations. This part describes the graph of the source whose solution curves move from the origin to infinity.
- The second part of the verse “unto Him we shall return” describes the spiritual path of normal individuals. They are moving from the temporal world to the Divine Presence. This part describes the graph of the sink whose solution curves move from infinity to the origin.
- The entire verse “To God we belong and unto Him we shall return” describe the spiritual path of the saints. They are at the same time moving from the origin to the infinite creations (“To God we belong”) and from the temporal world to the Divine Presence (“to Him we shall return”).
Note the sequence of spiritual events we discussed in the 2nd chapter (Beauty, Majesty, Beauty) is made manifest in this holy verse. The sequence is mathematically manifested as:
+ - +
The first positive sign represents the first movement of the holy verse (“To God we belong”). This first movement is positive because the prophets’ spiritual and earthly souls are both disconnected from the temporal world and connected to the Divine Presence.
The negative sign represents the second movement of the holy verse (“unto Him we shall return”). This movement is negative because the souls of normal individuals are clouded by negative energy which is generated by their egos.
The last positive sign represents the movement of the entire holy verse (“To God we belong and unto Him we shall return”). This encompassing movement is positive because the reality of sainthood is a confirmation and a continuation of that of the prophets as they follow in their footsteps.