Zeno’s Paradox

Zeno’s paradox is my favourite conversation filler. The beauty of the paradox is how universally simple it is to explain it to people from different ages and backgrounds. Here’s how it was originally told:

The Tortoise challenged Achilles to a race, claiming that he would win as long as Achilles gave him a small head start. Achilles laughed at this, for of course he was a mighty warrior and swift of foot, whereas the Tortoise was heavy and slow.

“How big a head start do you need?” he asked the Tortoise with a smile.

“Ten meters,” the latter replied.

Achilles laughed louder than ever. “You will surely lose, my friend, in that case,” he told the Tortoise, “but let us race, if you wish it.”

“On the contrary,” said the Tortoise, “I will win, and I can prove it to you by a simple argument.”

“Go on then,” Achilles replied, with less confidence than he felt before. He knew he was the superior athlete, but he also knew the Tortoise had the sharper wits, and he had lost many a bewildering argument with him before this.

“Suppose,” began the Tortoise, “that you give me a 10-meter head start. Would you say that you could cover that 10 meters between us very quickly?”

“Very quickly,” Achilles affirmed.

“And in that time, how far should I have gone, do you think?”

“Perhaps a meter—no more,” said Achilles after a moment’s thought.

“Very well,” replied the Tortoise, “so now there is a meter between us. And you would catch up that distance very quickly?”

“Very quickly indeed!”

“And yet, in that time I shall have gone a little way farther, so that now you must catch that distance up, yes?”

“Ye-es,” said Achilles slowly.

“And while you are doing so, I shall have gone a little way farther, so that you must then catch up the new distance,” the Tortoise continued smoothly.

Achilles said nothing.

“And so you see, in each moment you must be catching up the distance between us, and yet I—at the same time—will be adding a new distance, however small, for you to catch up again.”

“Indeed, it must be so,” said Achilles wearily.

“And so you can never catch up,” the Tortoise concluded sympathetically.

“You are right, as always,” said Achilles sadly—and conceded the race.

zeno_achilles_paradox

[ from  https://upload.wikimedia.org/wikipedia/commons/6/66/Zeno_Achilles_Paradox.png ]

The Tortoise claims that if he is given a head start Achilles will  never be able to catch up with him. Why? By the time Achilles reaches the point where the Tortoise started from, some time would have elapsed and in that time the Tortoise would have moved a tiny bit ahead. Now the Tortoise is again some distance ahead of Achilles and the same argument can be repeated. If this argument was true, Achilles would never be able to catch up with the Tortoise. But we know from experience that Achilles would definitely overtake the Tortoise as he is faster than the tortoise. Therein lies the paradox that has puzzled people for centuries.

I like to annoy my friends by bringing up this paradox for discussion. Most of them are stunned by how correct the argument sounds in spite of knowing there must be something wrong in the argument. Some of them have heard this before and say that the resolution of the paradox has something to do with discretization of space-time. However, in the form the above proof is presented it doesn’t really depend on discretization explicitly.

There are two informal ways how I think we can resolve the paradox (which are not necessarily new or unique):

  1. We claim we know the speed of the Tortoise and Achilles with the certainty that it is not zero. If their speeds are not zero it becomes difficult to pinpoint where each of them is in space. If we can say for sure a given object is at a given point in space, it must be at rest. If it is moving, it is impossible to say that the object is at a particular point in space. This is related to the Heisenberg’s Uncertainty Principle. If we know the speed of the Tortoise or Achilles with uncertainty that tends to zero, the uncertainty in determining the position of either one of them increases to infinity. That is where the fallacy creeps in when trying to analyze the problem by looking at points in space to mark the position of the Tortoise and Achilles.
  2. Suppose our measurements of where both Achilles and the Tortoise take some infinitesimally small but non-zero time. If you were taking a video this measurement time is determined by the frame rate. At a point in time when the positions of the Tortoise and Achilles are almost the same. The distance which Achilles would have moved by the time we finish making our measurement would be greater than the distance between Achilles and the Tortoise because the Tortoise is slower than Achilles. This makes me wonder about how we try to judge which horse won the race by looking at the video. May be if we had seen the video a fraction of a second later, the horse that came second would have won. Recent advances in camera technology using which you can capture 20 billion frames per second should be helpful.

It’s amazing how such a simple thought experiment tugs at our very naive understanding of space and time.

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Afterthoughts on a Thesis

Most of last year I worked on my thesis which attempted to answer one simple question: can an autonomous agent learn the rules of a game by observing people play it?

As we started working on the problem, we realized how difficult the first part of the problem was. Effectively, we had to solve the computer vision problem where you want to  extract information about the world from visual input just like humans. To build a model of the world, you need to separate the players in the video from the background and the players from the pieces and the pieces from the board, essentially separate entities need to be identified separately (image segmentation). That is you need to group pixels together to form ‘discrete’ entities or objects. The next step would be tracking the objects you have identified(video tracking). Lighting changes, occluding objects, cluttered environment and shape changing objects are only some of the issues you would face while tracking objects in a scene. To be crude, all the above steps are still low-level vision.

The next step would be to keep track of how the relationship between objects is changing. That is high-level vision. Simply segmenting pixels and tracking them is not enough. This part is extremely tough to do given that the very concept of ‘relationship’ between objects is not easily described to a machine. For instance, say you have a glass. A person pours water into it and drinks it. You might teach a toddler or better yet, he might learn from observation that a glass is something that can hold a liquid and it can be then used for transferring liquids. But then one fine day, the toddler observes that an upside-down glass can’t be used for storing liquids. The water just spills. But now that glass is not totally useless. It can be used for keeping something on it i.e providing support. So the relationship between objects is not a bland static thing – it changes depending on their relative orientation among other factors(the theory of affordances). High-level vision is where a lot of ‘intelligence’ should come in. The problem is how do you formalize this problem for an autonomous agent to comprehend. In fact, the problem of linking pixel-level processing to object and event level processing is a topic of active research in both neuroscience and computer vision(mid-level vision).

Some assumptions and lines of code later we had a system that looks at a Kinect stream of a game being played and generalizes from the visual observation the rules of simple games like Towers of Hanoi and Peg Solitaire. We also learnt the spatial structure of the game played using some heuristics. Couple of thoughts on the game rule learning system:

  1. The rules we learnt are severely limited by the logical framework of the world provided to it.  Some simple concepts need to be fed into the system like what a board is, what up and down are, what backward and forward are etc. We used an inductive logic programming framework to learn the rules. The rules can be learnt only in terms of old concepts present already in the system. In fact learning new concepts about the world is an aspect of intelligence. Think about how the language you think in limits the thoughts you have. To be more specific, the way one represents the world in his own head decides how he is going to think and act. For instance, a C++ programmer would probably be thinking in terms of for loops to solve a problem while someone in MATLAB would always look for ways how he can avoid for loops to solve the same problem. It is a two-fold problem: representation of learnt concepts and addition of new concepts. Heck, imagine an autonomous agent learning a new concept which changes the way it represents its old concepts!
  2. The problem that we attempted to solve was to make sense of the world from visual observation. It is in essence similar to what humans/scientists are doing all the time. Physicists are attempting to find the rules our world and the objects in it follow. Newton coming up with the Laws of Motion from his experiments is a perfect example. He made observations of the world around him and made a rational guess of what the rules governing the world(physics) might be. Thankfully for the rest of us, he realized how inefficient our representation system(maths) was and gave us calculus. A lot of hard-work(see this page from his notebook for example) went in to develop calculus. He used his old concepts to learn something new and one of the new concepts he learnt was how his(and the rest of the world’s) old representation system sucked. Newton is what a really intelligent agent should aspire to be. (Yeah, I get the impending joke. A guy who calculates so rigorously is already a robot.)
    Newton’s notebooks where he calculates the area under the curve (from http://zetatrek.tumblr.com/post/91642082140/clockwork-to-chaos-an-online-workshop)

    An agent that can play all games( General Game Playing system) has applications in real world like carrying out search operations and strategizing in military operations and electronic commerce. But the system needs to know the rules before it can start strategizing. This system needs its complement: an  agent that can learn the rules of any game presented to it. Many real world problems can be represented in form of games(nudge nudge Game Theory). A truly intelligent system would be able to learn all the games presented to it i.e. become a General Game Learner. And to do that an agent should be able to learn new representations, assimilate new concepts and come up with ‘legibly’ elegant solutions.

    How do we get there is a tough question to answer. Are we stuck with the wrong hardware to actually come up with true intelligence? How does a computer realize motivation? Can we define motivation in any way for a being which has no life? Is vision necessary for intelligence or can we come up with better sensors for agents to learn about the world? This list never ends.

House Hunting Stories from Bangalore

The search for a house can be frustrating but that doesn’t mean the series of interactions with brokers and landlords is without its fair share of humour. Of course, all real-estate comedy is only funny in hindsight.

1. Meeting the broker

After hours of scrolling on rental listings on websites, we managed to find the ‘perfect’ house and contact its broker. Copy, paste, dial, exchange hellos. A call is made to set up a rendezvous point, just as smugglers would do in old movies. And that’s how easy we would expect the deal to go down. Here’s what happens instead:

Me: We talked over the phone about a 3BHK in HSR …

Broker: Yes. 

Me: So let’s go and see that shall we?

Broker: Woh ghar(pause for dramatic effect)ab nahin raha. (That house is (available) no more.) 

Me: So why is is still listed on the website?

Broker: We will tell our guy to bring it down. We have other 3BHK flats. Hum dikhaata hai na aapko badhiya ghar.

That house is no more now. But don’t fret, I’ll show you a great house!

That’s like their motto. They tell you that and the next second you are off on the pillion of the motorcycle with a random stranger who is tripling, without a helmet and talking on the mobile phone with only one hand on the handle. But they can’t miss a call because that is worth a month’s rent in brokerage(somewhere between 20K to 30K). With the number of brokers in this city on the rise, risking lives is just an occupational hazard.

2. An actually complicated relationship

The broker-househunter relationship is complicated. It’s like an open marriage. You know pretty well your broker is going to show your bookmarked house to other prospective buyers. Your broker knows pretty well too that you will be consulting the services of rival brokers and seeing other houses. But you can’t let the above trivialities sour your relationship. He might just strike a great deal for you with the owner. 

Unhappy with the houses being shown by a broker, we contacted another broker in the same area. You can imagine the awkwardness of the entire situation when our new broker hitched a ride with our old broker to meet us the first time. Apparently they were friends. 

Old Broker: Bhaiya, yeh kya hai?(What’s this?) 

New Broker: You know him?

Old Broker: Yes. I’ve been showing him houses since a week.

New Broker: You didn’t like any of the houses he showed? That isn’t possible.

Me: But I didn’t.

The next few seconds were punctuated with an awkward silence. Extremely awkward.

Old Broker: I have to get back to a customer. 

Phew. 

3. On having friends come over

There is a lot of euphemism going on in the renting business on the topic of bringing your friends over. And by friends they mean lovers and house owners have taken offense at the prospect of friends coming over. One particular owner referred to us as pious-looking people who wouldn’t bring the ladies home. An usual request from middle-aged people renting homes to bachelors phrased in a slightly creepy way. There was another broker who put the above policy in plain broken English:”You bring friends. You take them inside. You bring girlfriends. You kick them out.” That seems slightly harsh. Especially the bringing people home to kick them out part.

You said we can’t bring the ladies home. Challenge accepte.. oh wait.. we already play that game and it’s called life.

4. Dealing with Car-beds

We browsed through a house on display which was described as fully furnished with two beds. The only thing they forgot to conveniently leave out was that one of the beds was a car-bed. The following is a car-bed:

“I tricked you when I said,’Let’s go for a long drive tonight!’ “

That sleazy caption is the only reason a person in his twenties would like to sleep in a car-bed. The house was good in more than one way. So we talked with the broker and asked him to ask the owner to do something about the car-bed. Imagine our disbelief when the owner came back with the epic reply, “You must deal with the car-bed.” We confirmed what deal with it meant? Can we throw it from the balcony and pretend it never existed? Can we keep in locked in the cabinets and pretend it never existed? No, apparently deal with it meant that one of us needed to man up and deal with a car-bed being in our house and start sleeping in it. The car-bed had to stay. We had marked the place for later but never got back to it.

There’s plenty more tidbits on how we lost out a house to female bachelors, how we were told to show up as a family to occupy family-only houses(set of three brothers one of whom is already married), how we must learn to keep our religions off the walls and in our hearts, how we could hangout on the balcony in lungis to enjoy the Bangalore wind… uhmm.. I think we saw too many houses. 

On Bollywood’s Ageless Coupling

The Vulture did a brilliant analysis on how Leading Men Age, But Their Love Interests Don’t in Hollywood. They go on to show that the age of the female lead doesn’t scale linearly with the age of the male lead. For example:

Denzel Washington’s Age plotted with the Ages of his Love Interests in different movies

In this particular criteria, Bollywood is no different from Hollywood. The age of female love-interests lies well restricted in the 20-30 years range. Let me make my case with some evidence:

  1. Shah Rukh Khan
    Shah Rukh Khan's Age plotted with the Ages of his Love Interests in different movies
    Shah Rukh Khan’s Age plotted with the Ages of his Love Interests in different movies

    Some observations:

    1. SRK had female love interests in 2 movies who were half his age: Anushka Sharma(Rab Ne Bana Di Jodi) and Deepika Padukone(Om Shanti Om).
  2. Salman Khan
    Salman Khan's Age plotted with the Ages of his Love Interests in different movies
    Salman Khan’s Age plotted with the Ages of his Love Interests in different movies

    Some observations:

    1. The number of data points i.e. distinct actresses he has worked with.
  3. Aamir Khan
    Aamir Khan's Age plotted with the Ages of his Love Interests in different movies
    Aamir Khan’s Age plotted with the Ages of his Love Interests in different movies

So our favourite actors have no qualms about having love-interests half their age. So that makes them creepy, right? Well, not exactly! Many actresses have marked discontinuities in their careers after they hit 30.  But the above problem doesn’t justify Bollywood’s age-agnostic casting. The root of this difference between ages is that many a time the above actors have been cast to play characters much younger than themselves like Aamir Khan playing a college student in 3 Idiots, Salman Khan playing characters whose age doesn’t matter to the story in… well in any movie post-Wanted etc. But that leaves us with a more dangerous question: why aren’t female leads being cast as characters whose age differs significantly from their own? Not-so-subtle chauvinism.

But the times they are a changing. This year we have had female leads playing central characters in movies like Queen. Hopefully, directors and producers get better at casting and ten years down the line we should find multiple intersections between the two lines in the above graphs.

The Tao Lives On

This is a true story of one simple idea making its way across space and time.

The 1970s, USA

Aah the 70s. It was a good time to live in. In those good times, an Austrian-born American physicist called Fritjof Capra “struggled to reconcile theoretical physics and Eastern mysticism.” He adds “… was at first helped on my way by ‘power plants’ or psychedelics, with the first experience so overwhelming that I burst into tears, at the same time, not unlike Castaneda, pouring out my impressions to a piece of paper.” [1]

I am not exactly sure why you would want to reconcile differences between these two supposedly diverse fields, but then again I wasn’t alive in the 70s. Anyway, that resulted in a book called The Tao of Physics subtitled “An Exploration of the Parallels Between Modern Physics and Eastern Mysticism.”  The book is about the parallels of Eastern philosophies derived from Hinduism and Buddhism, and experimental results in quantum physics. It became a best-seller.

Image of Lord Shiva Overlaid on particle tracks in the bubble chamber

I quote from the book:

“For the modern physicists, then, Shiva’s dance is the dance of subatomic matter. As in Hindu mythology, it is a continual dance of creation and destruction involving the whole cosmos; the basis of all existence and of all natural phenomena. Hundreds of years ago, Indian artists created visual images of dancing Shivas in a beautiful series of bronzes. In our time, physicists have used the most advanced technology to portray the patterns of the cosmic dance. The bubble-chamber photographs of interacting particles, which bear testimony to the continual rhythm of creation and destruction in the universe, are visual images of the dance of Shiva equalling those of the Indian artists in beauty and profound significance. The metaphor of the cosmic dance thus unifies ancient mythology, religious art, and modern physics. It is indeed, as Coomaraswamy has said, ‘poetry, but none the less science’.”

The metaphor of Shiva’s dance symbolizing the creation and destruction of subatomic particles is very tempting to anyone with the slightest of religious inclinations. It does help one “reconcile” the differences between mysticism and quantum physics. But from a scientific point of view isn’t all this poetry still just poetry? You are not going to get those “patterns of cosmic dance” from just that image of Nataraja. It requires advances in maths, physics and engineering to get those patterns. The end result might help one feel more in sync with nature or closer to God but let us not for a moment forget that it was still philosophy.  This Capra understood well. He said “Science does not need mysticism and mysticism does not need science, but man needs both.” That might also explain why his book was wildly popular.

But academics did warn the average reader to not get carried away. Leon M. Lederman said,“Starting with reasonable descriptions of quantum physics, he constructs elaborate extensions, totally bereft of the understanding of how carefully experiment and theory are woven together and how much blood, sweat, and tears go into each painful advance.”  More reviews[1].

The 2000s, Switzerland

On June 18, 2004, an unusual new landmark was unveiled at CERN, the European Center for Research in Particle Physics in Geneva — a 2m tall statue of the Indian deity Shiva Nataraja, the Lord of Dance. The statue, symbolizing Shiva’s cosmic dance of creation and destruction, was given to CERN by the Indian government to celebrate the research center’s long association with India. The Indian government acknowledged the significance of the metaphor of Shiva’s dance for the cosmic dance of subatomic particles, observed and analysed by CERN’s physicists. [2]

Statue of Lord Shiva at CERN

In fact, a proud moment for a number of Hindu Indians to see their God in front of the building where the greatest scientific minds are at work unraveling the mysteries of the universe. And what an ominous shadow it casts on the building behind! But the thing to keep in mind is that it was still a symbolic gesture, by no means a scientific evidence of a fact that the dance actually causes creation and destruction of sub-atomic particles.

The 2010s, India

The Internet age has reached India. The Tao has found its way back to where it all started. And people keep citing this book as a must-read if you are into “quantum physics and Hinduism” on Quora. Popular Facebook pages claim how the statue depicts the acceptance of quantum physicists of Lord Shiva’s role in the creation and destruction of sub-atomic particles. But as it happens in 30-40 years a lot has changed in quantum physics.

Joseph Wang(Ph.D. Astrophysics) answers on Quora,”The sad part is a lot of change has happened in quantum physics. It’s only use is as a historical guide to see how people in the 1970’s were trying to merge Eastern mysticism with particle physics.  The trouble is that the physics in the book is now known to be largely wrong.  This isn’t a problem with the author, it’s just we just know a lot more about QM(Quantum Mechanics) now than we did in the 1970’s.”  Peter Woit, a mathematical physicist at Columbia University, criticized Capra for continuing to build his case for physics-mysticism parallels on the bootstrap model of strong-force interactions, long after the Standard Model had become thoroughly accepted by physicists as a better mode.[1]

Interestingly, last year, Rajnath Singh, President of BJP, touched upon some ideas of Capra.

“Heisenberg learnt the Uncertainty Principle from the philosophy of Veda of this country. Heisenberg came to India in 1929 and met Rabindra Nath Tagore. In this meeting he discussed with Tagore different topics related to Vedic philosophy and theoretical physics. Assistant of Heisenberg and Austrian scientist Fritjof Capra has himself written in his book ‘Uncommon Wisdom’ on page no 42-43, “In 1929 Heisenberg spent sometime in India as the guest of celebrated Indian poet Rabindranath Tagore, with whom he had long conversations about science and Indian philosophy. This introduction of Indian science brought Heisenberg great vision, he told me. He began to see that the recognition of relativity, interconnectedness, and impermanence as fundamental aspects of physical reality, which had been so difficult for himself and his fellow physicists, was the very basis of the Indian spiritual traditions. ‘After these conversations with Tagore’, he said, ‘some of the ideas that had seemed so crazy suddenly made more sense. That was a great help for me.”[3]

The purpose of this reference might have been to illustrate how the traditional knowledge contained in the Vedas (might) have helped Heisenberg get to the Uncertainty Principle. While that might be true but the Indian connect is on rather flimsy grounds. It definitely took a lot more than just Eastern philosophy for Quantum Mechanics to get where it is today. We can be proud of our scientists and be inspired by them. But what one shouldn’t take pride in is in the nationalistic ownership of scientific facts. There is a dearth of stories about current Indians in science and an abundance of stories of Indian (pseudo)science. That needs to change. May be one day we will be citing not the Vedas but modern Indian scientists in the context of nation building.

And as far as science is concerned, it is way cooler to play God than just be amazed. People are onto some really cool stuff. Like moving single atoms. Imagine moving an atom which is so small, there are about 5 million atoms in the period at the end of this sentence.[4]

It is interesting how the same idea came back to where it originated: India. Only now it’s remixed for the modern age. Isn’t this the same reason why old Hindi songs are remixed too? Time indeed is a flat circle.


References:

[1] http://en.wikipedia.org/wiki/The_Tao_of_Physics

[2] http://www.fritjofcapra.net/shiva.html

[3] http://www.bjp.org/documents/national-executive-documents/2013/speech-sh-rajnath-singh-in-bjp-national-council-meeting-at-talkatora-stadium

[4] More on the science behind moving atoms: http://www.research.ibm.com/articles/madewithatoms.shtml#fbid=U1zTqCv33Oo

Recency Effect and the Oscars

Call this a passing observation, if you will.

Serial position effect is the tendency of a person to recall the first and last items in a series best, and the middle items worst. People tend to begin recall with the end of the list, recalling those items best (the recency effect). Among earlier list items, the first few items are recalled more frequently than the middle items (the primacy effect)[from http://en.wikipedia.org/wiki/Serial_position_effect]

The term recency effect  is relevant in short term memory recall tests. But for lack of a better term, let’s ask ourselves the question “Do the Academy Award Nomination for Best Motion Picture  suffer from a recency effect?” In other words, are the nominations for Best Motion Picture biased in favour movies which are released closer to the deadline for Oscar nominations?

Some quick data-scraping from IMDB  for last 10 Oscar nominations(2005-2014) for the Best Motion Picture  yielded this dataset. The results for the 72 movies are as follows:

Number of Best Motion Picture Nominations v/s Month of Release Date of Movie
Number of Best Motion Picture Nominations v/s Month of Release Date of Movie

The graph clearly shows a recency bias for the nominations. But that doesn’t necessarily imply causality. Most probably the studios and producers are well aware of this and schedule releases of big movies which they think are good enough for an Oscar nomination in the final four months leading up to the Oscars.

UPDATE:

I was pointed out on Reddit that it might be the case that the number of movies released in those months might be more which can be a reason why the number of nominations are more.

 

Number of movies released v/s Month of release(in 2012) [from http://hollyquant.files.wordpress.com/2013/05/average-monthly-grosses-and-release-count1.jpg ]

It so happens that the number of movies is more but not proportionate to their nominations. What is more interesting is how even though the number of movies released during summer is less but they are able to gross more may be because the viewership is more during those months because of the vacations.