Calculating 50 Shades of Grey

The Mathematics of Social Sciences

As I enter my journey at IIMB as a PhD candidate in marketing, I see myself as a social scientist. I see myself as an explorer and investigator of the human mind and behavior. I see myself applying principles of psychology, sociology and anthropology - the beautiful 'soft' sciences of mankind. But suddenly, my vehicle of learning takes a sharp detour into the 'hard' science of mathematics.

Over the last 3 weeks, I have been exposed to a lot of mathematics - a lot and of a different breed. I have been exposed to a lot of concepts whose existence I was oblivious to. After keeping me away from mathematical education after class 12, life decided to finally orchestrate a rendezvous with Linear Algebra. As terms like matrices, vectors, transpose, multivariate calculus, projections, etc. were being 'bombarded' at me, a volley of questions bounced in my mind...

"Why do I need to learn all this? How is this related to my future research work? How will this help me in understanding the human mind? Isn't statistics enough to compute quantitative data? Why do we need matrix algebra here???"

For the first 2 weeks, these questions kept rotating within my mind. But as I entered the third week, consciousness dawned into me. After spending 5 hours with Gilbert Strang (I met him virtually through his book!), epiphany struck its magic wand and I realized that Linear Algebra is one of the coolest things a social scientist can learn. From that moment, I started seeing myself as a potential 'Social Mathematician' as well.

As a social scientist, I would be involved in explaining the human mind and behavior. As we all know, today's world is not a very simple place. There is nothing purely black or white. There are various shades of grey. And as we all know, grey is nothing but a linear combination of black and white.

Let's take a simple generalization of the human mind being a combination of 2 elements namely - Good & Evil. All the people in this world would have different combinations of good and evil in their minds. We can draw parallels with numerous ancient oriental philosophical concepts like Yin & Yang (Combination of 2 opposing elements) and Om (Combination of 3 elements A,U,M).

So, we can start identifying linear combinations in almost everything around us, including the human mind.

Let's go back to our basic example of linear combination of 2 basic colours - Black and White.

Now, imagine that I am mixing these 2 paints on a palette to get a resultant colour Grey. It is possible for me to mix black and white paint in different proportions to arrive at various shades of grey.

When we are given a particular shade of grey, we can find out the specific proportion of black and white that was combined to get this specific shade of grey. Since grey is a linear combination of black and white, we can express them in the form of algebraic equations where w= white, b=black and g=grey

g= xw+yb

Here, x & y are the proportional weights of white & black respectively.
For different values of x & y, I will get different shades of grey.

We can express this particular linear combination in the form on a matrix where the rows equate to the dimension of 'whiteness' and 'blackness' and the columns signify the actual colours:

In a scenario where I am mixing only 2 elementary colors - black and white - that are mutually independent of each other - for every shade of grey I can identify the specific combination of black and white. But there may be situations where we may not be able to arrive at a specific combination.

Situation 1 - No possible combinations!

Imagine you come across a question as absurd as, "What combination of black and white will give you RED?"


No matter what combination of black and white you mix on your palette, you will never get red. Hence, such a situation has no solution.

Situation 2 - Infinite possible combinations!

Imagine you come across a question that says, "What combination of black, white and medium grey will give you dark grey?

Here, we are combining 3 colours to get a fourth colour. But the problem is that the initial 3 colours are not mutually independent of each other. The third colour medium grey is a linear combination of the first 2 colours. In such a situation, we end up getting infinite possible combinations! 

Let's see this with a numeric illustration:

We define medium grey as lg= w+b, and dark grey as dg= w+3b, we can express this as:

As it can be seen above, we can have multiple possible combinations of the 3 colours to arrive at the fourth colour. We can either mix only white and black and none of medium grey, or take a portion of medium grey and add some more black to it to make it dark grey. If we were to include the possibility of having negative values (i.e, if it was possible to remove some colours from a mixture), we would be able to get infinite solutions!

Now, let's calculate the grey areas of human mind

Till now we looked at colours and pigments which are physical objects. But the crux of the exploration of a social scientist is human mind and behavior. So, let's apply the concept of linear combination in a human phenomena.

Let's ask a fundamental question - "What makes me content with my life?"
As a social scientist, I would think of various elements that lead to contentment in life. These elements could be having a job that you love, having a positive family life and earning handsome money.
Linear algebra comes into picture here because the level of contentment in my life would be a linear combination of these 3 elements in some proportion!

For some people, earning good money may be more important than doing something you love. Such people end up doing a job that they hate doing just because it pays them well. For some people, family happiness may be much more important than job satisfaction. Such people may end up doing something they hate just in order to keep their family happy. Instead of just guessing what element could be important in a society, we can use the magical wand of Linear Algebra to precisely find those proportions!

For illustrative purposes, let's assume that we are trying to find these proportions among a homogeneous group of 28 year old, male, graduate, working professionals in Bangalore. In order to find the combinations, we need to measure the individual elements. We can measure the following 4 questions using a 100 point scale where 0=Not at all and 100= A lot:

Q1. How content are you with your life?
Q2. How much do you like the work that you are doing?
Q3. How much higher do you think is your salary as compared to what people get at your age & experience?
Q4. How positive/ pleasant is the overall mood of your family?


We administer these questions to 3 people and their responses are as follows:

Name	Work Likeability	Paycheck Handsomeness	Family Positivity	Life Contentment
Ayush	40	50	10	27
Bharath	20	60	40	38
Chandan	60	30	50	49

As it can be seen, the data that we receive will always be in a tabular or matrix form. Each column stands for a variable captured in a question. Since we are trying to find the importance of each variable on the final variable of Life contentment, we will carry out combination of columns as follows:

On solving this system by linear algebra methods, we arrive at the solution of x=0.3, y=0.2 and z=0.5.
This gives us the solution that family positivity is given more importance, followed by work likeability and paycheck.

With the help of linear algebra, we can find out what is the main driver of contentment. Similarly, this tool can be applied in calculating any other measurable linear combination in this world.

We know that the human brain is made up of grey cells and the human mind is also made up of shades of grey. Mathematics can be used to quantify the human mind and yes, we can calculate more than just 50 shades of grey!

Moral of the story: Embrace mathematics, even if you claim to be a qualitative researcher because beauty lies in integration - the linear combination of qualitative and quantitative research!

Linear Algebra helps us visualize beyond the physical 3 dimensional space through the column-vector concept. Once we are able to visualize multiple dimensions, we may get closer to solving Einstein's dilemma here:

MANJUNATH – Was he an idiot? Or are the rest of us Idiots?

After watching Manjunath – The Movie last night, my mind seems to be stirred with thoughts on what is the real definition of ‘duty’.

Here is the official trailer of the movie to give you the context:

Being a B-School graduate, working in the corporate world, this story is about each and every one of us. This story touches the lives of every young ‘educated’ Indian who has dreams of ‘making it big’ in his/her career.

But the bigger question that emerges in my mind is what is the real definition of ‘making it big’? What is our real ‘duty’?

The movie reaches its inspirational peak when Manjunath sources his courage from the ‘guiding light’ of the Bhagwat Gita. These are some of the shlokhas (Excerpts from Chapter 2, Verse 31-37) that elucidate the consciousness of ‘duty’ in mankind:

sva-dharmam api cāvekṣya

na vikampitum arhasi

dharmyād dhi yuddhāc chreyo 'nyat

kṣatriyasya na vidyate…

…hato vā prāpsyasi svargaḿ

jitvā vā bhokṣyase mahīm

tasmād uttiṣṭha kaunteya

yuddhāya kṛta-niścayaḥ

TRANSLATION: Considering your specific duty as a kṣatriya, you should know that there is no better engagement for you than fighting on religious principles; and so there is no need for hesitation… O son of Kuntī, either you will be killed on the battlefield and attain the heavenly planets, or you will conquer and enjoy the earthly kingdom. Therefore, get up with determination and fight.

These shlokas say that one must never hesitate or fear from doing one’s ‘real duty’. For a middle class young Indian who has ‘slogged’ to get into a premier B-school and worked hard towards landing a high-paying job, what do you think is the ‘real duty’? Before we answer this question, we need to look deeper into the culture that is ingrained in Indians since childhood.

What unites the great Indian middle class is the ‘dream’ – the dream to rise above their ‘mediocre’ lives. Many middle class youngsters in India have grown up seeing some of their ‘richer’ uncles being ‘worshipped’ by their entire family. They have grown up watching extravagant bungalows and lifestyles in Bollywood movies and their definition of an ‘ideal lifestyle’ has been shaped by these factors.

The milestone of success for most Indian youngsters today is to ‘own a house’, interspersed with some touch of luxury in the form of expensive cars and exotic holidays. Although these are physical manifestations of ‘success’, the social benefit of ‘brag value’ designations in ‘brag value’ companies acts as an icing on the cake. This manifestation of ‘success’ is not solely for self-satisfaction. It is for the satisfaction of the Family – because, in India, the Family comes before the Self.

So, now we move back to the question - what do you think is the ‘real duty’ of a young Indian B-school grad?

Is it his duty towards his family’s aspirations of rising above the middle class?

– If yes, then Manjunath was truly an idiot. He failed at his duty of being the ‘ideal son’ to his parents. He gave up his life which wasted a great source of potential monetary and ‘brag’ value to his family.

Is it his duty towards his conscience and personal integrity?

- If yes, then Manjunath was not an idiot. What he did was the right thing to do. We all have one life to live, which is quite short. Can we afford to waste this life in pursuit of just money and compromise on our personal integrity? Or do we choose to just move with the tide and 'hope' that the next generation wakes up?

I would like to quote an interesting classification of mankind shared by my husband. There is a hierarchy of 3 types of people in this world:

1. ·         The lowest level men are those who spend most of their energy in concentrating on the possessions of their neighbours – their desires are always comparative - directed towards beating their neighbor’s possessions.
2. ·         The average level men are those whose desires, although directed towards the self, are confined to material possessions in terms of maintaining/ enhancing their physical body and upgrading to luxury.
3. ·         Then, there are the highest level of men, whose desires are directed towards the self, but beyond material possessions. They seek wisdom, enlightenment and peace. This ‘peace’ can emerge only if our conscience and personal integrity remains satisfied.

Manjunath definitely belonged to the third and the highest level of mankind.

As the shloka says ‘either you will be killed on the battlefield and attain the heavenly planets, or you will conquer and enjoy the earthly kingdom’

This truly depicts where Manjunath is today. He was killed on the battlefield and attained the heavenly state of bliss – the mental state where your conscience is at peace.

So, do you really feel he was an idiot? Or are the rest of us idiots?