Wednesday, September 9, 2009

Mathematical Induction Disproved

A long time ago, in a land far, far away, a Java programmer had way too much time on his hands...

A truly momentous event. As I was having lunch today, I happened to think of maths. Now, this, in itself, is astounding - maths being something I usually reserve for my worst nightmares. I'll tell you how it all began. Pay attention, folks. Innocuous, the setting may be. But that is true also for Newton and his apple; Goodyear and his rubber...

I was having lunch inside the building, facing the elevators. It was way too hot outside, and there were other reasons, too - I am not at liberty to divulge those. I was watching the elevator lights, indicating them going up and down. I had nothing better to do because I was sitting next to Ashutosh. Most of you probably don't know Ashutosh, and so wouldn't have a clue why sitting next to him makes you watch elevator lights. But that's a story for another day.

It was then that an interesting observation came to me. Only the people who had more than a floor to walk took the elevators. That is, those who were on the 5th floor (I was on the 6th) were quite content to walk up and down. My mind chose this moment to wander onto the topic of mathematical induction. If I remember my maths correctly (my maths teacher would probably reckon that it's debatable whether I ever knew any to forget - but we all have our critics), then what it states is as follows.

To prove something, you first prove that the thingy is true for X = 1 (or, was it 2? - anyway, it's probably not very important). Then, you go on to prove that if it is true for X = N, then, it is also true for X = N + 1. Sounds convoluted? Well, this is why millions of kids the world over choose to go into the Arts once schooling is over.

Applying all that crap to the elevator problem, you get the following results. Please follow it carefully. This is really important.

To prove : WE DON'T NEED ELEVATORS. (I think the caps are mandated by the MI committee.)

Let us take the "X = 1" case. That is, you're on the first floor. There are no "why"s here, by the way. So... first floor. You don't need to take the elevator to go to any other floor (say N) - unless, as everyone knows, there's a fire. This is because going from the 1st floor to the Nth floor can be broken up as - from the 1st floor to the second floor; from the second floor to the third floor ... from the (N - 1)th floor to the Nth floor - that is, a sequence of movements of only one floor each. And, since, we've already proved, through direct observational analysis, that people don't use elevators for movements of only one floor... You get my drift?

Now, the "X = N" case. Let us assume that you're up on the Nth floor. Now, even here, to go to any other floor (say M - we've already used up N), you don't need the elevator. Just substitute "N" for "1", and "M" for "N" - the original "N" - and you're good to go.

If you've followed the argument so far, and if you're still here, you'll realize that if it holds for "X = N," then, it should also hold for "X = N + 1" - unless, of course, the (N + 1)th floor has very high ceilings. That is, to repeat, any movement from the (N + 1)th floor to any other floor can also... blah blah blah.

And that's it. I think we're done and dusted here. But I need to wrap it up all formally with a sentence that begins with "therefore".

Therefore, we can safely assume that if the hypothesis is true for X = N, it is also true for X = N + 1.

The long and short of it is that, according to mathematical induction, we've just proved that MOST PEOPLE DON'T NEED ELEVATORS.

But we all know how ridiculous that statement is. Of course, we need elevators. Because if we didn't, why do all these buildings have elevator shafts?

The inescapable conclusion, therefore, is that MATHEMATICAL INDUCTION IS FALSE. Which leads us, sadly, to conclude that a lot of theorems being accepted as true by mathematicians are in fact without valid proof.

So... any chance of a retrospective correction of my score of 8% in the second-term Plus Two maths exam?