Paradoxes and self-deception

I met an octogenarian orchardist from an ancient land, who said – “There is only one me and there is only one life to live.”  Don’t worry! I am not going to plagiarize Percy Bysshe Shelley all the way but I did meet a very interesting lady a few weeks ago in Japan and she made such a strong impression on me that I felt compelled to document my learnings. But please allow me to ramble a bit.

It took me a few decades to internalize the fact that time is the most precious commodity we have and another few years to realize how and whom we spend that time with is even more important. Then I got thoroughly confused when I came across Sholem Aleichem’s astute observation on life- “Life is a dream for the wise, a game for the fool, a comedy for the rich and a tragedy for the poor.”  Obvious inquiries such as “who am I” and “what type of game am I playing with my life” ensued. The more I thought about it, the more paradoxes I started to notice in life. At times, I could not explain my self-deceptive behavior and wondered if my approach to life is lacking in some ways or the paradoxes are messing with my mind. The whole process made me very uncomfortable but then it occurred to me, mind itself is a perfect example of a paradox. It is capable of generating both disturbing and comforting thoughts at the same time. May Sarton’s quote came to mind and calmed my nerves somewhat: “We have to dare to be ourselves, however frightening or strange that self may prove to be.” A strong urge came over me to meet the kind of people May Sarton talked about.

Let us explore a few paradoxes before dealing with “one me and one life” quote. Since math is my passion, let us start with a few math riddles.

  1. What is the sum of 1-1+1-1+1-1+1-1+1….to infinity? This is a divergent series, also known as Grandi’s series, named after an Italian Mathematician (circa 1700s).

Let us think of a few ways to solve this series. One way is to group the 1s into pairs of (1-1) for example.

The whole series then can be written as (1-1)+(1-1)+(1-1)+(1-1)+….. = 0

So is zero your final answer? What if we pair the 1s slightly differently, say (-1+1).

We can write the above series as 1+(-1+1)+(-1+1)+(-1+1)…… = 1

Is it 0 or +1? We can argue forever whether the series ends with a +1 or -1 to make our case. However, nobody has seen the edge of infinity (except for Buzz Lightyear – he has gone beyond infinity) and my best guess is we will never know. A lot of issues in life are that way. But let us not get philosophical yet. Let us mess with the series a little more. Let us math the heck out of it.

Let us assume N= 1-1+1-1+1-1+1-1……

Now let us subtract N from 1 on both sides.

1-N = 1-(1-1+1-1+1-1+1-1……), let us open the parenthesis

1-N = 1-1+1-1+1-1+1-1+1-1…. (The right hand side looks pretty much like N)

Therefore 1-N = N

2N = 1 → N = 1-1+1-1+1-1+1-1+…. = 1/2

How about that! Now we have three possible answers: 0, 1/2, 1. Which is it?

Let me try to make a case for 1/2. That does not mean it is right but here is one way to think about it. Imagine turning on and off a light switch extremely fast and measuring the brightness of the room. What we will notice is that the brightness of the room will be approximately half as bright as a fully lit room on average. By the way, what we discussed here is an extremely simplified explanation of Cesaro summation (please look it up if you are interested.)

I don’t know about you but I have a problem with this problem. Sum of integers gets us to a fraction. How can that be? That is why it is a famous paradox.

Let us look at a couple more.

  1. What is the sum of 1-2+3-4+5-6+7-8…..to infinity?

We can approach it three different ways (you know the drill from Grandi’s series).

Let us group 2 consecutive integers at a time as we have done before

=> (1-2)+(3-4)+(5-6)+(7-8)+…..

= (-1)+(-1)+(-1)+(-1)+(-1)…. = -1-1-1-1-1-1-1-1….

We could say the answer is “negative infinity.”

How about if we group the 2 integers slightly differently

=> 1+(-2+3)+(-4+5)+(-6+7)+….

= 1+1+1+1+1+1+…. We could now say it is “positive infinity”

By the way, behavioral psychology has a very good explanation for the logic behind above groupings. Prospect theory is a must read if you want to figure out a way to engineer happiness in your life!

And of course, our third approach…

Assume M = 1-2+3-4+5-6+7-8+….

Let us add M to itself.

M+M = (1-2+3-4+5-6+7-8+…)

+ (1-2+3-4+5-6+7-8+…)

Let us play a small trick here (but a valid operation). Let us group the second number “-2” in the first M series with the first number “1” in the second M series.

M+M = 2M = 1+(-2+1)+(3-2)+(-4+3)+(5-4)+(-6+5)+(7-6)+(-8+7)+…..

2M = 1-1+1-1+1-1+1-1+…..

But we know from Grandi’s series that 1-1+1-1+1-1+1… = 1/2

Therefore, 2M = 1/2 => M = 1/4

M = 1-2+3-4+5-6+7-8+9-10…. = 1/4

There you have it. If you are thinking what’s with all this +1, -1 stuff, fine, let us deal with something real simple.

  1. What is the sum of natural numbers from 1 to infinity (1+2+3+4+5+6…)?

Let us use our math wizardry.

Assume L= 1+2+3+4+5+6+…..

We know from example 2 above that

M= 1-2+3-4+5-6+7-8…. = 1/4

Let us subtract M from L

L-M = (1+2+3+4+5+6+…) – (1-2+3-4+5-6+…)

L-M = (1-1)+(2-(-2))+(3-3)+(4-(-4))+(5-5)+(6-(-6))+….)

L-M = 0+4+0+8+0+12+…. = 4(1+2+3+4+…)= 4L (since L = 1+2+3+4+5+….)

Therefore L-M = 4L => 3L = -M (but M =1-2+3-4+… = 1/4)

3L = -1/4

L = 1+2+3+4+5+6+…. = -1/12

How about that! I bet you did not dream of a negative fraction as a solution for the summation of infinite series of positive integers.

There is another and much simpler way to do the above summation and here it is:

Let us group three numbers starting with integer 2 as shown below:

L = 1+(2+3+4)+(5+6+7)+(8+9+10)+(11+12+13)+……

Then L = 1+9+18+27+36+45+… (All multiples of 9, no kidding!)

L = 1+9*(1+2+3+4+5+…) = 1+9*L

Therefore 8L = -1

Or L = 1+2+3+4+5+6+… = -1/8

So what is the final answer? How can this be?

Above answers do not make much sense and yet math shows otherwise. How do we square this paradox?

Of course, I can sit in a lotus pose and tell you that science and spirituality are one and the same. And that the sum of infinite positive integers tells us that the more we accumulate the less we will have at the end!

If you are still wondering what the deal is with all these silly math paradoxes, please don’t get irritated. Let us first resolve the major issue hiding behind these so called math paradoxes. The problem is with the way we manipulated infinity.

Does it mean anything to say let us add something to or subtract something from infinity? Also, how we group things has a big impact on the final outcome. We simply can’t use our standard finite operations on infinity. Blind application of these operations leads to disastrous results. If you are still not clear let me summarize it in a very simple way. Let us assume Bill Gates’ total net worth as of today is about $100 billion (it’s like infinity for most.) We all can agree that Bill Gates’ net worth does not change much even if he loses a dollar to his best pal Warren Buffett in a game of Bridge. That must mean then a dollar is worth nothing. Crudely that is what we did in our infinite series summation calculations.

We know that 1-1+1-1+1-1+… series is divergent but if the number of 1s in the series is even then the total sum has to be 0 otherwise +1. There is no possibility of 1/2. But infinity complicates our comprehension.

Similarly, 1+2+3+4+5+… series has a simple formula for finite series. Assuming there are “n” numbers, the sum of n sequential positive integers starting from 1 is n*(n+1)/2.

It is easy to see this intuitively. Let us say we want to calculate the sum of first 100 positive integers (1, 2…100). Here is an interesting way to solve it. We can create 50 pairs of numbers from 1 to 100 numbers such that they add up to 101.

For example:

Let us group the first integer 1 and the last integer 100, (1, 100) => 1+100 = 101

Similarly, the second integer 2 and the last but one integer 99, (2, 99) => 2+99 = 101 and so on.

We can create 50 such pairs (add up to 101) as shown below:

{(1,100), (2, 99), (3, 98),…,(50,51)}

There you have it. The sum of the series then must be (the sum of each group multiplied by the number of such groups) 101*50 = 5050

Apparently, the famous mathematician, Gauss did this calculation in a few seconds in his primary school and astounded his teachers and fellow students.

The purpose of above math wizardry is to highlight the issue of paradox. They are paradoxes because our interpretation of a certain term or terms, infinity (in this case), is faulty. Otherwise, there is nothing paradoxical about these series.

Let us move on to the real world. Here is a sophism.

  1. What is better – Eternal bliss or a slice of bread?

What is better than eternal bliss? Nothing!

But a slice of bread is better than nothing. So, slice of bread must be better than eternal bliss.

You are probably up in arms over the usage of the word nothing and wondering why is this even a paradox. Well, nothing is better than eternal bliss but eternal bliss is better than nothing. There lies the paradox. I know it is silly but we do a lot of this in our day to day activities though.

Alright then, let us get philosophical now.

Most of us face three paradoxes in life and these paradoxes mess with us pretty badly. Misunderstanding these paradoxes causes a lot of harm to our well-being.

Paradox -1: Certainty-Uncertainty Paradox: We seek Certainty in life as it gives us peace of mind but unknowingly or knowingly we also yearn for growth, which is an outcome of Uncertainty. For example, most like the idea of a steady income with fixed hours and yet they look for career growth and more excitement at workplace. Deep down they want to work for a dynamic Tech start-up and have an option to make a billion bucks. Certainty does not provide that upside but Uncertainty does. Unfortunately, the downside of Uncertainty is that it may screw up their work hours and may even threaten their livelihood. This causes a lot of heartburn.

In summary, too much of Certainty creates boredom and too much of Uncertainty creates fear and unrest. And yet most of us swing between Certainty and Uncertainty like a pendulum and constantly complain about our work-life dissatisfaction.

Paradox-2: Identity Paradox: Each of us is very unique and yet we all feel connected somehow and seek conformity. This is a hard one to deal with and we spend the least amount of time thinking about it. Genetically we all are unique and seek different things in life. Yet we create various social systems to get along and fit everyone under these arbitrary systems. This process creates a lot of friction among and within us. We get agitated when others disagree with us and try real hard to convince them of our viewpoints.

In short, we are unique and yet spend most of our time trying to fit in and impress/influence others. How bizarre!

Paradox-3: Ethical Paradox: We get a lot of pleasure in gaining/acquiring stuff but we also feel fulfilled by giving. An interesting paradox! Some of us don’t seem to know when/how to take and when/how to give. At times, we justify ill-gotten gains by donating some of it to charity thinking that the giving neutralizes the toxicity of the gaining.  On the other extreme, some of us condone Robin Hood type behavior.

A lot of us spend most of our time gaining/acquiring in the hopes of giving it all or some of it away at the end (at least that is what we keep telling ourselves.) Unfortunately it rarely ever works that way.

These paradoxes are pain in the rear for sure. So, how do we deal with them?

I read a lot of theories on these topics including some esoteric religious literature but none satisfied my curiosity. Theory is one thing but practice is an entirely different ball game.  I don’t know about you but I always knew right from wrong and yet made a few wrong choices along the way using some twisted logic (self-deception.) I figured there must be some wise men and women out there who could shed some light on these paradoxes through their own life experiences.

So, how to find these people! I came up with an interesting strategy after a lot of mind bending contemplation: Process of elimination, i.e., eliminate the type of people I don’t want to meet from the sample size and whoever is left should be worth meeting. This plays into my strengths too. It has always been a struggle for me to know my likes but never been an issue to know my dislikes. I have got my shit-list and it is quite long.

Ok, I used the following criteria for the process of elimination:

  1. No religious practitioners – They accept everything in the name of faith. Not that there is anything wrong with it but it does not work for me.
  2. No ascetics – This group does not have any paradoxes and I don’t want to be one of them. At least not yet.
  3. No poor people and no socialists – The concept of giving up something that I never had is hard to digest.
  4. No sick people – A different mindset.

That pretty much eliminated most of the human population 🙂 But seriously though, the above criteria helped me narrow down the type of people I wanted to meet dramatically. I finally synthesized the outcome as below:

A developed capitalist society with a tradition of self-awareness and civic mindedness but that has also been on both sides of compassion and cruelty. In short, a group of not so perfect people striving for perfection! Japan met my criteria (with usual disclaimers) and there you have it. Now you know my reasons and frame of mind behind the Far East visit.

Sometimes process of elimination works wonders. Mind bending, eh! Well, I have a confession to make. I have been to Japan multiple times before and have a pretty good idea of Japanese culture and history. So, not sure if I identified Japan through my process of elimination or my interest in Japan helped me concoct the screening process. I like to believe it is the former.

I met a lot of interesting and inspiring people in Japan and some of them were more welcoming than the others but I did sense a trace of dissatisfaction/unrest in most of them. I learned a lot about human behavior and my own biases from every one of those interactions.

My love for nature took me to the countryside and Japanese have a very unique way of preserving their natural habitats. They make an effort to blend in instead of blending nature at their will. Ok, I am romanticizing it a bit too much but my goal is to look for the best.  This positive outlook led me to the orchardist (mentioned at the beginning of the article) and she turned out to be one of the best human beings I have ever come across.  I only interacted directly with her for a few minutes but I absorbed a lot by watching her go about her daily routine. I learned the importance of enjoying the process more than the outcome from her. I have met people like that before but I was not mentally ready to understand it then. She confirmed through her actions that happiness is a path and not a destination. Although what she was doing in her orchard might seem trivial to some, she was doing it with complete focus and dedication. Her advanced age has definitely slowed her down but the satisfaction on her face, I bet, has not diminished over time.

I finally got a chance to meet her and ask her for her opinion on Certainty-Uncertainty paradox. She initially shied away from talking about philosophical stuff in a foreign language (there was an interpreter) with a complete stranger but she finally obliged. Her view is that there is no such thing as Certainty. “It is our false arrogance that makes us believe in such absurd things”, she said. She only believes in encountering wonderful things and not so wonderful things in life. Then she asked me, “How can you get bored when you are passionate about something and having so much fun?”

Her answer to the first paradox: Find your passion! Obviously this is not new and not the first time I heard but hearing it directly from a truly passionate person made a big difference and wiped out my constant cynisim. It took all but 20 seconds for her to help me realize my lack of understanding of Certainty.

She then said the most impactful words, “There is only one me and there is only one life to live. I respect others but I can only do what my heart tells me to do. I learned it the hard way after raising three kids and fifty years of marriage.” And she laughed and went back to doing her chores. I found her a few hours later taking a break near an orange tree and asked her regarding social pressures. Her response was even simpler, “Stay away from people that make you unhappy.”

I got my answer to the second paradox: Live your life on your own terms, respect others but stay away from people that make you unhappy.

Finally, I asked her about ethical paradox and she very quickly shut me down by saying, “My responsibility is to be the best person I can be. What I get or give is not important to me. I just don’t want to be a burden to society.” That pretty much answered my third paradox. I thanked her for her time and she gave me an orange and wished me a pleasant stay in Japan. Although our conversation lasted only for a few minutes, she left me with weeks’ worth of ideas to contemplate. I cut short my Japan trip and came back home two weeks earlier than I originally planned. She made me realize that I already have everything at home.

Over the following weeks, I ruminated over my fallacies in those paradoxes and realized my fault. I have been acting as if my life is a summation of series of infinite events. Unfortunately, as we discovered in our math riddles/paradoxes our finite operations don’t work well on infinite series. Please allow me to explain further. We act as if we have a lot of time and not focus on current events. A lot of us waste away our precious “present”.

The three infinite math series we discussed earlier helped me immensely in figuring out the philosophical meaning of it all.

Here is my interpretation: Some of us see life as a sum of infinite events of 1-1+1-1+1-1+1-1+…∞.  In other words, for every good (+1) we face a bad (-1). A pessimist could group his every positive with a negative and say life is a zero sum game (1-1) and come to the conclusion that there is no point in living. However, an optimist could group her every negative experience with a positive one (1+(-1+1)+(-1+1)….) and come to an exact opposite conclusion. Or we can be smart Alec 😉 and get only a fraction of the full happiness that we can achieve. How we group our experiences is key to our long-term happiness.

The second math series injects some volatility into the equation. Magnitude of each subsequent event in this series keeps growing, i.e., 1-2+3-4+5-6+7-8+…..

In other words, we experience a positive event of magnitude +1 followed by a larger negative event of -2 and so on. Again a pessimist could very quickly group the smaller positive event with a larger negative event (1-2) and feel depressed. But an optimist could do exactly the opposite and group a larger positive event with a smaller negative event (-2+3) and feel excited. The second math paradox tells us that we can get a negative infinity by grouping the numbers the way a pessimist does and a positive infinity by grouping the numbers the way an optimist does. Same life events and entirely different outlook on life!

Finally, the third math paradox is even more revealing.  We could feel miserable even with a series of positive events in our life if we don’t focus on the finite present.

We take a lot of abuse in the present in the hopes of better future and that makes us very unhappy. This unhappiness creates various paradoxes. These paradoxes lead us to self-deception. And the whole thing becomes a vicious cycle.

Most of us make our present forgettable hoping for a memorable future. Unfortunately, only memorable present makes for a memorable future. The orchardist reminded me of that. Our life is not a summation of a series of infinite events, it is continuation of the present into the unknown. And the unknown is finite. Why focus so much on the unknown when all that matters is the present. Of course, the present could be either positive or negative but that is entirely within our control.  How and whom we spend the time determines the present.