Tuesday, April 07, 2009

Achilles and The Tortoise

Long long ago in Athens, there was held the man-tortoise run off. Achilles made a nice (arguably so) gesture of giving Tarquin the tortoise a 100-yard head start. Tarquin's tactic was to keep moving constantly, however slow it might be. If Achilles wants to overtake Tarquin, logic states that he must first get to where Tarquin is, when the race starts. This will take a few seconds. In these few seconds, Tarquin who is focussed on constantly moving, is sure to have moved a short distance away from Achilles. Achilles needs to now get to this distance, by which time Tarquin would have moved a teeny distance further. And so on...you get the drift? It seems as if it's "logically and mathematically impossible" for Achilles to beat the tortoise. But as we all know, Achilles did win the race, overtaking poor Tarquin. (Source: "The Ancient Paradox of Achilles and the Tortoise", attributed to Zeno (born c.488 BCE))

The feeling one has when one knows the implicitness of a situation/outcome and yet cannot verbalize in as many intelligent words and logical statements to prove it, is indeed very pitiable. It's always crippling when I can no longer use mere words to churn out a logical proof and need to resort to mathematical means to prove a point, however implicit it may seem. I'm never good with abstract mathematical proofs (as you will or have evidenced before), but let's see how far I can get with mere words alongside some logic.

The basic logic holds true - in order for Achilles to overtake the tortoise, he needs to first get to the current state of the tortoise by which time the tortoise may have moved a little farther away. But will this turn out to be an endless loop that can't be won over? Intuitive sense argues no. Time and space if viewed as being continuous chunks, it is reasonable to envisage dividing space into 'n' points... is n is close to infinity? I don't know... we can get down to an atom and a molecule but beyond that I haven't kept track of Physics to know more :). But if "n" could be infinity, does it make sense? In a physical sense, where do we draw the boundary on "space"? My room, my home, my street, country, continent, world... universe? I think scientists have conceded that "space" does tend to extend to infinity. But in this particular example, space would be with respect to distance between Achilles and the tortoise; this is obviously finite by itself, but seems infinite to the extent to which it can be divided. Anyway, having chunked up space into so many such divisions, we move onto time. We start dividing time into infinitesimal chunks. Then it follows that, due to this phenomenon called "speed", the rate of movement of Achilles is such that he traveled at a dimension of space and time, wherein the tortoise's motion was almost non-existent for a certain extent of space and time, thus causing him to overtake the tortoise.

I guess Achilles and tortoise makes this reasoning sound more plausible, for the two are such stark contrasts. It's quite conceivable that if we were to freeze the time frames of the race, the tortoise would seem to appear immobile for several frames. But what about the blindly whizzing cars in formula-one races (or other such races)? To what extent should space and time be divided to fathom a stage where there was no motion?

Does this make sense? As always, please help me out with your views :)


oorjas said...

hmm.. it is something i have never thought about.

mathematically this seems impossible, but when i first read it i said , what, it is so simple one was travelling at a speed a bit more than the other so he reached the point where the previous body was before it could leave. hence overtaking the moving body.

but after reading the chunks of time and space theory i was confused.

in F1 races all the bodies are traveling at almost similar speeds and in matter of fractions they overtake one another. how small the fraction of time would be? that need more mathematics than i know.. :-)

well i think they have cameras which photograph at such rates which shows the photo finish. that can calculate all that.

does my comment make sense..?

Neeraja said...

Oorjas - Thanks for thinking about this problem :). It's not an easy one I agree. Apparently it has it's roots in calculus. (dx) denoting intervals of time/space and along with it is introduced the concept of limits (Lt x-> infinity). It's a mathematical fallacy for it may theoretically be hard/impossible to prove but practically provable. But I wish someone could explain why it's so hard to mathematically prove the concept of speed? Seems simple enough :)

oorjas said...

hey the new theme looks great.. :-)

Neeraja said...

Thanks Oorja, I decided to give my blog a makeover after 3 years of boring templates :)