The word ‘complexity’ originates from the Latin word ‘complexes’ which means ‘totality’ and signifies ‘entwined’ or ‘twisted together’. The detailed meaning of ‘complex’ is:
1. Two or more different parts or elements
2. These parts must in some way be connected or knotted together, so that it is difficult to separate them (there is a basic duality between parts, which are at the same time distinct and connected).
In mathematical terms, a complex number is written as ‘a + ib’, in which ‘a’ is a real term and ‘ib’ is the imaginary term. When I first learnt about the complex numbers as a school student, I wondered why we should imagine an imaginary number, when we have so many natural and real numbers. However, in the long run I had realized that the complex numbers are meant for representing polynomials (circles and loops) extending into different planes. The real and imaginary terms did not negate each other, in fact they complement each other to make it a ‘totality’ or ‘whole’.
These imaginary numbers exist because we encounter them here in this Universe, even if they only exist as fleeting thoughts, fragments, blue prints for something else we do understand. The imaginary numbers have a relationship to the real world. They might not be in the world of man’s common experience, especially if the experience is overly linear. But a deeper, multidimensional view of numbers – of the Universe- reveals these intricate relationships, and thus their ‘realness’. They exist because they stand in relation to that which we can see, feel and know (this concept is elaborated later in this article).
All these statements look like a philosophy, which was coined by a mathematician like Socrates and philosopher like Immanuel Kant. Science is a creative activity as much as it is a methodology and by nurturing creativity we can develop a sense of counter-intuitive which can enable us to explore these ideas further. Mathematical intuition is an interesting case. The realm of pure numbers and geometrical forms is Platonic in nature; it does not exist in the natural world and cannot be directly observed. Yet, one significant aspect of mathematical intuition is the ability to apprehend and even visualize this realm.
Since the ever increasing modernization of technology makes our lives more complex, an attempt is made to conceptually express the relationship between the mathematical complex numbers and the ‘real world’, and then to progress from the complexity to reality. In mathematics, a complex number is a number of the form ‘a + ib’, where a and b are real numbers and ‘I’ is the imaginary unit with the unique property, i2 = -1 (scientific philosophers call it as mystic-i). When the imaginary part is ‘zero’, the complex number is just a real number ‘a’. This statement has a direct relevance to our complex lives, where-in one should try to live as close as reality. In the imaginary term ‘ib’, the ’I’ cannot be zero, which means always there will be an imaginary term, which gets nullified only when its multiplication term ‘b’ becomes zero.
The above conditions, resembles the Hindu philosophy where the Universe as a ‘whole’ is a combination of ‘sat’ and ‘asat’, where ‘sat’ is real (truth) and ‘asat’ is unreal. ‘Sat’ may be related to the ‘Absolute truth’, where as ‘Asat’ is a time-depended reality. Then the Universe as we see, feel, know and perceive, may be always a ‘Sadasat’; because the change is imminent. Then it may strike to us that why should we bother about these imaginary and unreal things, when one has to live in reality. In this regard, the following Hindu philosophical verse is quoted:
“Asato ma Sadgamaya, Tamaso Ma Jyotir gamaya,
Mrityorma Amritam gamaya, Om Shanti shanti Shantihi”
Which means, Lead me (by giving knowledge) from the ‘unreal’ to the ‘real’; from darkness (of ignorance) to the light (of knowledge); from death (sense of limitation) to immortality (limitless liberation).
Swamy Krishnanada (of Divine Life Society) says that the knowledge is identified with learning, the academic acquisition of information regarding the various objects of the world. But the spiritual wisdom is the same as insight, known as ‘intuition’, whereby the object of knowledge is possessed in completeness and does not anymore remain as extraneous (namely, irrelevant to the subject or unrelated) something. This is in relevance to the detailed meaning of complexity, where the ‘parts’ of a ‘whole’ remain connected.
Further to the similarity in meaning, the properties of the complex numbers are also seemed to be relevant to our spiritual wisdom. Some of them are:
a) The terms ‘a’ and ‘b’ cannot be separated as parts, they should be treated as ‘whole’,
b) They represent a vector (it means a value with a direction),
c) If the complex number (a+ib) is multiplied by its complex conjugate number (a-ib), the resultant is (a2 + b2),
d) It is best used for representing planar structures and polynomials with negative axis,
e) The value of i2 is -1.
The relevance of the first property is already discussed earlier in this article. The second property says it has a value, means significance or standards of behavior with a direction. This implies that one should have a directional behavior, which should be aimed at the ‘absolute truth’. In the third property, it implies that the complexity is the additive term of real and imaginary parts. If one can try to subtract the imaginary part from the real part, it is known as the complex conjugate and their multiplication (to be productive for achieving a great deal), then the resultant is the summation of the squares of the real terms, which means certainly we are in ‘reality’.
Regarding the fourth property, the complex numbers represent circles or polynomials with negative axis which reminds about the ‘higher’ levels on the spectrum of consciousness (circular, point and radial). According to Bell’s theorem, human consciousness and the physical world cannot be regarded as distinct and separate entities. We are intimately associated not only with the earth we inhabit, but with the farthest reaches of Cosmos also. Coming to the negative axis it encompasses, it is none other than our negative experiences in our lives. Swami Sukhabodananda, while explaining the role of experience in our lives, reiterated that there are things which can be understood only by experience and it is inexpressible. So, while accepting our negative experiences as it is, we should also realize that our lives will have positive and negative experiences, so as to minimize the complexity (by getting the additive (a + ib) and difference (a-ib) terms) and to enjoy the ‘absolute reality (a2 + b2).
Finally, no square of any number (positive or negative) should be negative that is the peculiarity of the ‘ˇ’ here. William Rowan Hamilton (Irish Mathematician) who in 1835 first suggested treating complex numbers as pairs of real numbers so as to reduce the complexity. Mathematicians argue that since ˇ is the solution to x2 = -1, thus it is ‘real’ as any other number. However, stalwarts like Leonhard Euler, David Hilbert, Asimov, Kurt Godel & Peter (2006), all tried to explain the reality of complex numbers as well as that of mysterious √-1, which is i. This mysterious ‘ˇ’ is synonymous with spiritual ’ˇ’, which remain illusive (a thing that seems to be something it is not) for a common man.
Discussion and Conclusion:
In this age of nanotechnology, the intrinsic nature of complex ‘ˇ’ and the spiritual ‘ˇ’ still remain intrigue. It is a fact that, without the applications of imaginary numerical system, no engineering sciences would have existed as our every day lives depend on them. As everything in our lives becomes more complex, so is our spirituality. We live in an increasingly complex technological world where nothing works like it is supposed to, and at the end of the day makes all of us hunger for simplicity to some extent. Simplicity is not the opposite of complexity; rather it is the complementary key, the formula that permits us to access and utilize multi levels of virtual reality and global knowledge networks.
While simplifying the complex number to a real number, the first step is to make the real number ‘b’ equal to zero, which is attached to the imaginary-i. That means, reduce the imaginary thoughts to its minimum to come close to the reality. Spiritually, lessen the attachment to the time-dependent ‘Asat’ and try to be as close as to ‘sat’ real. Ramana Maharshi teaches that the world and reality are effectively negations of each other. One can never be described in terms of the other.
The verse 2.16 of Gita states that ‘there is no non-existence of Sat (or Atma) and no existence of the Asat. The reality of these two is indeed certainly seen by the seers of the truth. Using the mathematical derivable of reality from the complex, one has to live in reality. If the condition forces us to face the complexity, use the same mathematical treatment to convert the complexity to reality, by using your intuition, knowledge and experience. As per my intuition, if some of the philosophical terms cannot be digested by a layman, it is appropriate to use our academic knowledge also to get a better meaning.