I guess many of us at many a times have acted illogically, and probably have made statements that are logically incompatible to each other. If at one point in time you say something ‘is’ and your consequent statements directly imply something ‘is not’, then your ‘is’ and ‘is not’ statements attack each other and together form a contradiction. Contradictions can be quite interesting, so think about it; how often have you contradicted yourself? If your answer is ‘not much’ well good for you else others might call you a hypocrite. If you are someone who likes to take part in sporty debates with your friends or colleagues, once in a while you will find yourself making contradictory statements. Debates are helpful in making new ideas and to accelerate thought processes since you are pushing your brain to come up with new ideas to prove your point but sometimes you loose track of your previous idea and comes up with a new one that basically pulls your leg! Another related or equivalent term is `paradox`. A set of logical statements that concludes in a contradiction is a paradox. Paradoxes seems to defy logic and certain good ones will give a bang on your head and make you smile. Wikipedia just showed me that there are two kinds of paradoxes. Ones that are seemingly logical but are in fact absurd, called Falsidical paradoxes and ones that seem absurd but in real are logical, Veridical paradoxes.
In the reminder of this post I am gonna talk about a Falsidical paradox. And how the proof for the falsity of its logic made me arrive at an idea, which now becomes the title of this post. In the next post I shall give you a very interesting Veridical paradox along with its philosophical impact.
Two friends of mine and me were walking the roads of our college campus at night talking this,that and technology. When suddenly one of them asked the other to tell me about a conversation they had the day before, about a paradox. The other, Sreekumar, said its puzzling but flawed and that he already proved that by a piece of code. Probably you must have heard about proof by contradiction and alike but never ‘proof by code’? but sigh, that’s what programmers do. Nijil, the one who started this talk, quickly jumped in to defend that its not flawed and that its been puzzling people for ages. They then put aside their fight and finally settled to tell me about this strange thing.
Assume two guys X and Y decided to run a race but one, Y, starts 10 seconds ahead of the other. At some point in time Y crosses a checkpoint,p1, and some seconds later X also moves past it. By the time X crossed the checkpoint p1, Y has advanced some distance ahead of p1. Lets mark the distance advanced by Y as p2. By the time X manages to move past p2, Y has again moved a little more further. And thus by the virtue of the fact that X allowed an advantage to Y of starting early, X can never win the race however fast he runs because each time X crosses Y’s current position Y would have moved further, however small it be.
So, did you feel the bang, did you smile? This interesting paradox belongs to a set paradoxes called Zeno’s paradoxes, and is due to a Greek philosopher Zeno of Elea (a place in Italy). Told through the ages, this paradox replaces our X with Achilles (the great warrior of Homer’s Iliad, or remember Brad Pitt’s Troy(2004)? ) and our Y with a Tortoise (that wise Tortoise of Aesop which outwitted the hare in another race!). The clever Tortoise succeeds in convincing Achilles that he would fail, given the tortoise starts first, and Achilles accepts defeat before the race! Even though this paradox seems to prove by some mathematical reasoning that Achilles can never defeat the Tortoise in that race, from our real world experience we know it never happens that way. If someone gets a head start all you need to do is to run faster than you would if the race was fair. So how did Zeno manage to fool us? Hold on,let me continue from where I left off in the conversation with my friends. Feeling the bang I started to think, smiling. But before I came to something solid, Sreekumar jumped up and started describing his idea of what went wrong. The first few words that came from his mouth brought a flurry of thoughts into my mind, and there I was seconding Sreekumar even before he spoke a line! I am not sure if I am exaggerating this, but that’s what I remember of that night. Before I get to what we shoveled down Nijil’s ears that night, let me give you a similar picture. Normal motion, I guess, is usually captured in 24 frames per second (fps) in a camera (i.e. the camera takes 24 photographs in one second and plays it to you one after the other). Once captured by the camera if you watch this video using the same frame rate you get to see the motion as you would see it live, i.e. 24 fps recording and 24 fps playback. What happens when the frames per second is increased from 24 to say 240 for recording? Assuming that the playback fps is never changed (24 fps). Of course you get a slow motion video. The reason for this is, in one second the new camera setting caused the camera to produce 240 frames per every second instead of 24. And in playback 240 frames take much more time than 24 frames. A 10 times increase in playback time causes a slowdown of 10 times in the video. This new video though is much slower than the actual motion, is much more precise in its description of the motion. You get to see exactly how the motion happened. But then you might still be missing some detail. So we crank up the camera settings to a higher fps, resulting in much more precise motion. Increase the fps even more (tending to infinity?) and the resulting video would be so slow that there will be no perceivable motion, or the object that was in motion is now in rest, not moving at all. Even though in reality the object is moving the video shows us that the object is at absolute rest. So, did you find the similarity of how in reality Achilles would have easily won (well he is mighty warrior!) but Zeno showed us he could never win that race? Yes, this is exactly how Zeno fooled us.
Consider the Tortoise has lead of 1 feet at some point in the race. Every one knows between two points infinite new points can be defined. Zeno roots his paradox in the fact that its impossible to cover infinite number of points in finite amount of time (yes, P!=NP). Well that’s true but here the infinite number of points is within the one feet of distance on the road, and all those infinite number of points could be covered by a single footstep! Achilles really don’t have to step on all these points separately. This is where the title of this post comes in. What if you have a ruler that shows you say 30 cm in total, along with markings for each millimeter, but this is no ordinary ruler. Look closer between two consecutive mm markings and the ruler expands the distance between those two markings giving you markings for a smaller unit, allowing you more precision. This expansion continues infinitely giving you infinite precision. Zeno considered the race track as a scale (ruler) of infinite precision that produces the same results as the camera with infinite fps. Just like our camera that showed that moving object was at absolute rest, Zeno when he measured the race track with the scale of infinite precision effectively froze Achilles on the track. How can Achilles win the race now? he is frozen on the track, paralyzed and helpless.
So now you have new tool in your tool-chest, other than pausing time what else can you do with this scale-of-infinite-precision? I don’t know, may be you could use its idea to prove or disprove some other twisted ideas. I myself have tried to use it in a discussion with a good friend of mine on philosophical grounds, not sure I was successful though. But I am certain of one thing you can never use it for the same purpose as you would use any ordinary ruler, to measure that is. Coming back to my friends at college, I wish I could include here Sreekumar’s code which will help convince Achilles that he will win the race but this post is already too long. And I am glad Nijil didn’t google on this paradox before telling about it that night!