Zeno's Paradox of Motion:Thompson's Lamp Scenario

[BlindBeauty]

This question is probably so easy that I'm making it too complicated but here it is:

"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?"

Conceptually, without mathematical background, I would say ON since it was ON when it began. But why? Why not? Mathematically?

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

Again, conceptually, I would say yes, it would be ON if it started ON. But what's the answer? Mathematically?

Thanks.
David

 

[TD]

I don't think it will matter. The question of being on or off is related to the number of switches, i.e. if it's an even or odd number of switches.
When you take the limit for the number of switches to infinity (and hence get the result that it will take 2 minutes), the information of odd or even is lost, it can be both.

 

[Doc Al]

Since this is a math question, not really a physics question, I'll be moving this to a more appropriate forum. This issue is: Does the inifinite series (1 - 1 + 1 - 1 ...) converge? I'd say no.

 

[TD]

That's another why of looking at it, but don't you mean the sequence 1,-1,1,-1,... (where 1 represents on and -1 represents off, for example)?
Either way, the sequence doesn't have a limit and the (alternating) series doesn't converge in this case.

 

[matt grime]

It isn't a maths question, at least it isn't a non-vacuous maths question. Parity is a property of finite sets not infinite ones.

Just because someone thinks they have a well formed question doesn't mean they do. I think you'd've been better off deleting the question entirely, not moving it to maths. I doubt anybody here wants it.

 

[Hurkyl]

I agree with matt that this doesn't belong in math -- I believe the main point of this paradox is to deal with the issues of modelling reality. I guess philosophy of math & science is the best fit, so I'll toss it over there.

Making some implicit assumptions explicit, the paradox goes like this:

Assumption: it is really possible for a physical situation to behave as stated in the problem.
Fact: knowing everything about a system before a certain time allows you to determine the state of the system at that time.
Paradox: the state after two minutes cannot be determined, despite the complete information of the previous times!

 

[BlindBeauty]

Thanks. I WAS making it too complicated!

 

[nameless]

This sounds tangential to me.
As there is a definite limited block of time to work within, the switching would have to get progressively faster and faster the closer it drew to the two min mark, in a seemingly tangential fashion. Always closer but never reaching, achieving greater than light speed, double light speed, quad, etc. Given the parameters of the problem, it sounds like a fallacious problem on the face. By the 'end' of two min, the light would appear, for all intents and purposes, continuous.
Like 'motion' and 'time'.