The Control Experiment

I was sipping my morning chai (at noon) and gingerly punching my own thigh. A recent football injury, where an opponent's knee had smashed into the meaty front portion of my thigh  just above the knee had left me sensitive in this region.

Immediately after the incident, I was able to carry on playing. It was after I got back home that the intensity of the pain searing through my legs became apparent. The next 2 days I could hardly walk and the region was swollen. Over time though, the pain became less intense and a visit to the doctor revealed that though the injury was of a painful nature, it was local and not serious - I just had to wait it out and it would be alright.

Now, ever since the injury, I have been eager to get back to playing football or at least jogging, but the past few days have seen a different kind of confusion - am I still injured? Sometimes the pain returns, sometimes I feel I am completely healed. This morning, while punching my own thigh, I noted the slight pain where I had received the blow. At this moment a thought struck me - I am a scientist. I have to truly assess whether I am experiencing injury pain. Think about it - have you ever punched your own thigh softly to know how much it hurts with how hard a punch? So unless there is a really sharp pain for a very soft touch, it is hard to say whether it is a normal pain or an injury-pain. So there was only one way to find out. I punched my other thigh softly in the same place with as near the same intensity as I could manage and felt a similar sting. Still I couldn't discern whether it was of lesser intensity and so I repeated the exercise until I felt ridiculous - here I was, tea in hand, punching my own thighs repeatedly with different forces in different spots.

My conclusion right now is that the lingering effects of the injury are still present and it would be unwise for me to get physically active again soon. And my other conclusion is that the human body (or at least mine) is really really sensitive to even light blows. I don't know how I withstood the blow from the original football incident without fainting.

I have always been an over-enthusiastic sportsperson with not too much talent for any sport nor a very resilient body. It has made for an unfortunate combination and I am thus the veteran of several injuries. They say you live and learn. And I have lived (thankfully) and learnt. My assorted wisdom from my most serious injury (a ligament tear in the knee that kept me limping and in pain for nearly 6 months) is that we have no clue how we do things. I started walking slightly differently to avoid the pain and forgot how I used to walk. I have no idea how my knee used to feel before injury, so now I still don't know whether it feels right. I just go with it in the changed circumstances and there is no way of ever returning truly to the state of my body pre-injury. And this happens with every injury.

When one part of one leg is injured, you will avoid putting your weight on it, so another part of your hip or other knee feels more strain. Then as you slowly heal, you adjust your walking to avoid that other pain and very quickly the "feel" of your own body becomes unrecognisable. You cannot simply say "Ok, the injury is gone, let me go back to how I was doing stuff before the injury". That is simply not possible. Through some indirect, cascading way, your body has a stored memory of each thing that has happened to it. Some incidents are remembered very strongly, others are swept away with time. That one day you had to pretend to limp because you told your class teacher than you missed the previous day due to an ankle-injury. Well now you have lost a bit of your "normal" walk. In fact, there is no normal, I think. It keeps changing but we feel it is constant. I feel I am more aware of this solely because of my numerous injuries. (Once after I sprained my shoulder playing Table Tennis, a great joke I heard at my expense was that my next injury would be from playing chess!)

"A game of two halves" is a common cliche in football, where one team plays really poorly for the first half and then plays really well after the 15 minute break at half-time. In English commentator parlance, this is known as "taking the game by the scruff of the neck". But why does this happen? There are many explanations. You see, the manager yelled at them at half-time so they came back to the field with renewed determination. But also the opposition put in a lot of effort playing so well in the first half that now they're more tired. But also there was a small but crucial tactical change by the manager. And also this one player who was having a really bad game suddenly started playing at his usual level. So what is it? A change in mentality? A change in tactics? The one crucial player? There is no control experiment again. A lot of it is to do with the mind.

One last football analogy - when a team takes the lead, they automatically seem to be more focused on defending only. No matter the coach says "Play as if the score is still 0-0". There is the instinctive desire to defend more. That's why there is a football saying that "You are most vulnerable as soon as you score". It's impossible to play as if you don't know the score when you actually know the score, no matter how hard you try. The act of scoring fundamentally changes how you think and hence who you are on the football field for the rest of the game.

When I was mid-way through my 5 year undergraduate course, my grades started falling. Well, actually they didn't "start" falling. They were alright and then suddenly they fell and also stayed down for a while. I began to think how I can rectify this situation. First step to rectify this is to find out why it is happening. Excellent question!

Well, college is harder than school. But the first few semesters were a breeze. Well, first few semesters are designed to be easy. But also I've lost a bit of interest in the subject and in studying in general. Also, have my initial good grades made me little complacent? The fear factor of exams is completely absent. But it was the same even during the initial semesters, I still studied without any fear or anxiety. What made me study then? Or am I studying the same amount, but that is not enough? Or my mind is getting less sharp? Maybe the first instance of a bad score demoralised me and I'm in a vicious cycle. Ok, let us take a step back. Grades dipped in school also. How did I get out of that cycle?

Of course, at that point of time I didn't think with so much clarity. And I didn't give this so much precise thought, just random questions in the shower or while sleeping. But now I understand something else - no solution from school would ever have helped me. I had changed so much as a person. From a structured school life staying at home to the complete freedom of hostel life. From learning really random things (like how some p-block elements react differently with cold sulphuric acid and hot sulphuric acid) to having courses in Humanities on "rational enquiry" and "scientific communication". These fundamentally altered who I am, how I think and how I do things, so I couldn't fall back to what I was doing before to solve the same problems. Priorities changed, opinions changed.

But maybe I noticed this in this phase because I was thinking about how to tackle one particular challenge after a drastic change in a short period. Maybe we are always changing. Every step alters fundamentally how we walk, every thought alters fundamentally every next thought we will ever get, and before we know it, we are constantly becoming a new person while being in the illusion that our personality is very constant. It is like the ship of thesus situation. Old solutions will not work anymore. Or maybe old attempts at solutions will suddenly become fruitful. Maybe changing the formation is counter-productive at 0-0, but at 1-0, it completely changes the game. But we can never know what works precisely. Because there is no control experiment. There never will be. No football game where only exactly one thing changed at one time. The change in formation changed your attitude along with it. And the opponents' attitude. In football and in life.

And now I will return to trying to re-create and remember how I used to walk when I have never paid attention to how I used to walk. I will return to facing new challenges everyday that I am clueless about how to solve, but 5 years from now I will write a grand blog post on how to think about it and what to learn from it and throw some more gyaan at you!

That is the Whole

There's something beautiful about log-log plots. Let me explain. Let us say you have 3 numbers - 0.001, 0.1 and 10. Now you draw a line of length 10 in your favourite unit with your pencil, mark one end as 0 (the left end, unless you write in Arabic or you are a psychopath) and the other end as 10. Now I ask you to make a point at these 3 values. You've already marked one end as 10, so you put one point there. But how do you make these two points 0.001 and 0.1 without them smudging each other? They're too close.

Not necessarily. If your favourite chosen unit was kilometre, then it's easy to make these marks distinct and far apart. But let me know when you find a kilometre of paper or blackboard for this excellent demonstration. The distance between 0.001 and 0.1 is 0.099. The distance between 0.1 and 10 is 9.9. The distance between 0.001 to 10 is 9.999. The 0.099 extra is nothing from the perspective of 10, but from the perspective of 0.1 it is big and for 0.001 it is massive. For the elephant, the ant and the rat are practically the same tiny size. For the rat, the elephant is a giant and the ant is tiny. For the ant, even the rat is a giant. So what's the solution?

But 0.1 and 0.001 aren't "practically the same" are they? The first is 100 times larger than the second. And 100 times larger than 0.1 is 10, which we all agree is large. So in a way, the "separation" between 0.1 and 0.001 is tiny, but it is also really large when seen in another perspective. So what's the way forward here for our original problem - to make a pencil mark at 0.001 and 0.1?

This is where the logarithm (log for short) comes in. You draw a line of length 4. You mark one end as (-3) and one end as 1. (-3) corresponds to 0.001 and 1 corresponds to 10. Because 10 raised to the minus third power is 0.001. And 10 raised to the first power is 10 itself. What you've done here is you have made "scale" more important than the absolute separation. This somehow recognises the fact 0.1 is equally separated from 0.001 and 10. It's like Gulliver peering close to the Lilliput world and finding that is all exactly identical to his world, but just at a lower scale.

Let me ask another question (rhetorically). How many numbers are there between 0 and 10? Our first instinct would be to say 9 numbers. But these are 9 whole numbers. If we draw a line of length 10, how many numbers can we mark with our pencil? These are called "Real numbers", and there are infinite real numbers between 0 and 10. Take any number from 0 to 9, put a decimal point after it and then cook up any sequence you want of how many ever digits - that's a real number between 0 and 10, from 0.78346873647823 to 9.2 to 3.000000000001. Best of luck marking them with a pencil though.

But what does this "infinite" mean? How many numbers lie between 0 and 100? Obviously, infinite again. But surely, this infinite is different in some way. A line of length 10 has infinite numbers. Now, a line of length 100 has 10 lines of length 10, each of which has infinite numbers. So this new infinite should be 10 times more than the old infinite.

There was something that I learnt in my first year maths course that amazes me - the number of numbers (real) between 0 and 100 is the same as the number of numbers between 0 and 10. Not 10 times greater, as one would expect. This is a mathematically exact fact. Of course, it is also a very basic set theory fact, but maybe I am still so stuck-up on this because I'm a physicist.

But let's go further. What about between 0 and 1000? The same. Between 0 and 1000000000000000000000000000? Exactly the same. The same as between 0 and 0.00000000000000000000001. It doesn't matter what (positive) number you stick there, the number of numbers you will find in between them are exactly the same.

Is this related to the log thingy we discussed earlier? I think so - if not directly, then at least as a way of visualising and understanding this. You saw how we first took the interval from 0.001 to 10, worked our log magic and made it in such a way 0.001 and 10 are equidistant from 0.1. So we made the intervals from 0.001 to 0.1 exactly equal to 0.1 to 10. For any such interval, we can take combinations of logs and additions and multiplications and to make them equal. This is an implicit acceptance of the fact that between any 2 numbers, there exist the same number of numbers, so we can always scale it in such a way that the distance between them can be a number of our choosing. We did exactly this by taking a log and making the distance between 0.001 and 0.1 the same as the distance between 0.1 and 10.

Ok, so this is for finite numbers. What about if we call in the big guy, the guy who changes the rules when he walks in? Infinity. How many numbers are there between 0 and infinity? The answer is again the same. The same as the number of numbers between 0 and 1. And 0 and 0.00001. And between 0 and 1000000.

What is infinity? Infinity is that number which trumps all other numbers. You give me any number and ask me which number is greater, I can say infinity and I would be right. In fact, it was basically invented to do this job. It exists only in our heads.

Now let us go back to the numbers between 0 and 1. ALL the numbers.Can you give me a number greater than all these numbers? Yes, of course. 1.1 is greater than all numbers that are between 0 and 1. So is 1.000001. So is 100000. and so is infinity. In this regard, for the all the numbers between 0 and 1, the number 1.1 or 2 play the same role that infinity plays for the entire number line. Pick any number and infinity is greater. Pick any number between 0 and 1 and 1.1 is greater. We can extend this to say 100000000+1 is like the infinity for the numbers between 0 and 100000000. And maybe, we can extend this is ALL numbers and infinity itself. Usually, just invoking infinity changes the rules, but here it doesn't. And with my slight mathematical training, I am confident that if I sat down and read about it for a few days I could get it, but this kind of abstraction has never been my strong suit, so here I will believe the mathematicians blindly. (nothing new for a physicist to do)

Each little section of the real line, no matter what the length, in a way, contains the whole real number line. Every chunk has this property, as does, quite obviously, the whole real line, which stretches to infinity. So the next time someone at a party asks you to draw a line all the way to infinity, just draw a straight line and label the 2 ends with your favourite real numbers and tell them it's practically the same thing. You'll surely be invited to the next one.

My grandfather has studied Vedanta and I love to pick his brain whenever I get the chance. I remember him quoting a mantra during a discussion about something -

पूर्णमदः पूर्णमिदम्

पूर्णात् पूर्णमुदच्यते |

पूर्णस्य पूर्णमादाय

​पूर्णमेवावशिष्यते ||

This is the "Shanti Mantra" of the Ishavasya Upanishad. Poornam could be translated as the whole, or the complete, or everything. The gist of the translation my grandfather gave me is this (with help from the first couple of results when you google this) - 

That is the whole. This is the whole. From the whole comes the whole. Taking the whole from the whole, what remains is still whole. 

This is a view on the cosmos (or maybe Brahman, I'm not sure, I'm not even close to being an expert). And I found it really cool that the imagination of the cosmos here is very similar to the number line, which when you consider the infinity at the end of it, is also an abstraction, a figment of our collective imagination. 

This little section of the real number line is the whole real number line. That also is the whole real number line. If you take a section out of it (which is whole), the remaining section is still the whole real number line. It's a beautiful idea, that there is the entire cosmos in every small bit of matter. It reminds me of a (probably fake) conversation where someone asked Michelangelo how he made such a beautiful sculpture of David out of a shapeless chunk of rock and Michelangelo said that the David has always been in the rock, his job was merely to scrape out the rock that was covering it. Every rock is a potential David, every stone, made of stardust, contains within it as many secrets as the entire cosmos itself. And by knowing this stone perfectly, you can understand the entire cosmos. By knowing your own self completely, you can know the higher self.

But there's something deeper in this. Every bit of the real number line can be identified with the whole real number line, but it isn't identical. They have some properties which make them identical in some scale. But each number, or each collection of numbers has its own identity and properties. It is why Kanaka Dasa could compose thousands of Krithis in dedicated specifically to Kaginele Adikeshava or Basavanna could write his vachanas addressing Kudala Sangama Deva. These are really local deities, perhaps the temples and the forms of the deities here not known to people outside their district, not far outside Karnataka for sure. Yet, these learned men spoke of these deities as the masters of the universe. Not because they are one and the same as the entire cosmos, but simple because they are a part of it. It does not diminish the deity in the temple two streets away, who is also local, but also the master of the universe.