Okay, Frank FINALLY explained to me, in terms that I can understand, why it is possible to move from one place to another in a finite amount of time despite the fact that every distance is divided into infinite parts.

However, in his answer, he linked me to this site which gave this question:

"THOMPSON'S LAMP: Consider a lamp, with a switch. Hit the switch once, it turns it on. Hit it again, it turns it off. Let us imagine there is a being with supernatural powers who likes to play with this lamp as follows. First, he turns it on. At the end of one minute, he turns it off. At the end of half a minute, he turns it on again. At the end of a quarter of a minute, he turns it off. In one eighth of a minute, he turns it on again. And so on, hitting the switch each time after waiting exactly one-half the time he waited before hitting it the last time. Applying the above discussion, it is easy to see that all these infinitely many time intervals add up to exactly two minutes.

QUESTION: At the end of two minutes, is the lamp on, or off?

ANOTHER QUESTION: Here the lamp started out being off. Would it have made any difference if it had started out being on?"

The site doesn't give the answers. Can any of you?

I believe the answer is that the light will be dim. If you consider that as the light is switched on and off towards the infinitely fast, the switch will be on half the time and off half the time so half the amount of current will be allowed to pass through.

But, my theory is that your light bulb has burnt out from all the abuse you've put it through.

If you are actually saying, "we stop messing with the lamp after two minutes, when we stop turning it on and off infinitely fast, is it on or off?" That isn't a mathematical question... that is a question for whoever is working the switch.

Realistically, there is no practical application as such for limits. Nobody can turn a light on and off infinitely fast so it seems you are trying to take a principle of the abstract apply to the physical world and that isn't ever going to produce a correct result.

The issue with this type of problem is that the presumption is that you can apply properties of a finite set to an infinite one. More specifically, if the light switch were originally on and we flipped it an even number of times within a certain time span, it would be on by the end of it; likewise, if we flipped it an odd number of times, it would off by the end of the time limit. But to classify infinity as an odd or an even number is inherently incorrect-infinity is not a number (even though when we try to conceptualize it, we do apply numerical properties to it).

At the exact point the timer hits the two minute point, the light will be being flipped on/off every 1/infinity seconds, for lack of a better expression, which means that it stays in its on/off state for exactly 1/infinity seconds. However, 1/infinity is mathematically defined as having the value of 0, so it remains off for 0 seconds, and then it remains on for 0 seconds, ad infinitum.

So, to answer the question, "After the 2 minutes, is the light on or off?" The answer is:

Yes.

You all must be right, that this is an impossible question to answer.

I think this is so mind-boggling, it's driving me crazy! It is impossible for me to really understand how there can be an infinite number of moments between one point in time and the next, and yet the next point in time arrives.

How could this supernatural power ever get to the end of the string of turning on and off his supernatural lamp if there are an infinite number of times that he will turn it on and off in the meantime?

How can an infinite string come to an end?

How can we get to the end of a minute if there are an infinite number of moments that we must live through in the meantime?

*Sigh.* Oh well. I guess I'll just have to chalk this up to yet ANOTHER thing I'll never really get.

Thanks, everyone!

quote:

I think this is so mind-boggling, it's driving me crazy! It is impossible for me to really understand how there can be an infinite number of moments between one point in time and the next, and yet the next point in time arrives.

I think it is easier to grasp if you keep in mind that the infinite number of moments between two points would also necessarily be of infinitely short duration, i.e: 1/infinity = 0. And then, try to stop thinking about it.

A mathematical approach might go something like this:

Let f(t) be a function of time in minutes. Represent the on-off state of the lamp as a 1 (ON) or 0 (OFF).

The lamp starts at t=0 in the ON state, so f(0)=1. With me so far? It goes OFF in one minute: f(1)=0. Then f(1.5)=1, f(1.75)=0, and so on.

The question is: What is the value of f(2)?

The problem is, f is not a well-behaved function. It should sit in the corner If you try to take the limit of f(t) as t approaches 2 you will find it has no limit, and the function is undefined and discontinuous at t=2.

Suppose the lamp's state had a value f(2)=V. Let e (epsilon) be a small number. Modern analysis asks questions such as, what happens to |f(2-e)-V| as e approaches zero? This is usually discussed in pre-calculus or introductory calculus.

So is the lamp on or off at 2 minutes? It's a trick question, because it presumes that there's an answer. Mathematically the question is meaningless because the lamp's state is undefined.

Now, one caveat: We are doing a "thought experiment" with an ideal lamp that functions perfectly according to our specificiations. Were this a real lamp in the real world, it could not faithfully reproduce the required on-off behavior on small time scales, as disucssed above by aminator2002. You could not "do the experiment" of Thompson's Lamp in the real world. Similarly you can't generate a perfect square wave in the real world, no matter how good the equipment, so we are asking a mathematical question, not a physical one.

Another example is the function G(x) = sin(1/x). What happens to G as x approaches 0? Same deal. It has no limit.

This is unlike Zeno's paradoxes (asked about by Sarai here), which seek to analyze real things such as flying arrows and footraces. But we already know the outcomes (as did Zeno, of course): The arrow hits its target, Achilles overtakes the tortoise, etc. The same mathematical methods applied to Zeno's paradoxes, unlike Thompson's Lamp, yield well-behaved functions with certain mathematical properties: they are well-defined, continuous and differentiable, they generate convergent series, etc. What you'd expect of descriptions of the real world.

It's just that Zeno, five centuries BCE, didn't have mathematical methods to deal with infinities, infinitesimals, convergence, continuity, etc. He was asking insightful questions that took more than an additional 2000 years to answer!

This message has been edited. Last edited by: Professor,

Here's another famous problem in the same vein:

Consider the infinite sum,
S = 1 - 1 + 1 - 1 + 1 - 1 + ...
Does S converge? If so what is its value?

That's simple, says Student A: Just group the terms as follows:
S = (1-1) + (1-1) + ... = 0 + 0 + ... = 0.

But wait, says Student B: I grouped the terms differently:
S = 1 + (-1+1) + (-1+1) + ... = 1 + 0 + 0 + ... = 1.

These are different answers. They can't both be right.

In reality, S does not converge and the infinite sum is undefined. Both students are wrong!

Very interesting. Thanks, Professor!

I seem to agree with animator2002 here, as the situation explained seems to form an asymptode. That is, the half of any fraction could never equal to zero but continue to get smaller and smaller.

The first things that popped into my mind after reading this question Sarai were:
Iteration and Geometric Progression.

GP can help us devise an equation.

1 + 1/2 + 1/4 + 1/8 + 1/16 +...
= 1 + 1/2*(n)
where n is the number of times, after the first time switching off the light, the switch is clicked.

S = 1 / (.5)^n

Hmn........now what?

Let us divide the two minutes into segments.
from the two minutes, upon starting time left is
1, 1/2,.........OH, I see some light now:

S=2=1 / (.5)^n
which implies .5^n = .5
OhGod, I have gone wrong somewhere.
Guess it has been a while since I practiced Mathematics.

Pin~Jinx / anarchist ashamed at being baffled