Ok, I think I’m grasping how .999… =1, but now I’m really stumped. On this web-page is the following riddle:
How could one perform an infinite amount of actions in two minutes? Shouldn’t it be possible to count how many actions were performed, if they were performed in a finite amount of time?
As far as the answers for the two questions, I’m assuming the answer to the first question is it’s neither on or off, and to the second, no, it wouldn’t matter.
Actually the answer to number one is that it will be both on and off.
Changed my mind, I think you’re and right neither an on or off value can be assigned to the lamp at T=2mins.
Ok, I’ve got it: the button will be both depressed and not depressed and a value of on or off cannot be assigned.
Good old Xeno’s paradox turned on its ear. Well you couldn’t perform a physical action within an infintesemal time slice because of quantum limits. Actually the limit will happen sooner because even if you could control a physical switch of some kind it takes a finite amount of time for a single electron to cross the contacts.
The question is only meaningful as a pure thought experiment once you remove the realities of mass, momentum, electrons and quantum limits. The answer, hell if I know. Any discreet answer is invalid becuase the switch never stops.
OK this time I have absolutely postively go the answer: it will be both on and off.
The problem I was having is that I was modelling whther the button was depressed or not, this way you can’t assign an on or off value at t=2, but if you take a simplier model of whther the lamp is on or off you find at t=2 the lamp is on and off.
This is a classic problem, and I’ve read discussions of it without fully understanding the nuances. But I think the answer is, we have no mathematics to describe the completion of infinite tasks. So not only is the situation described not physically possible (as Padeye says), it’s not mathematically possible either.
I would quibble with that a little bit. Some infinite sequences of tasks are perfectly describable mathematically. For a trivial example, imagine in the first minute the gremlin (or whatever) turns the lamp on, and then at every other step just checks to see if the lamp is on. (I said it was a trivial example.)
But in this particular case I agree with your conclusion: the state of the lamp is not well-defined.
The problem here is that the on and off values are discrete and mutually exclusive.
I suppose any answer (including the ones I gave above) is essentially meaningless.
The trick is that each action is done in a geometrically smaller unit of time (1 minute, 1/2 minute, 1/4 minute, 1/8 minute, …). The sum of this series is 2. Ok, technically the limit of the sum as the number of terms approaches infinity is 2. If I sound overly cautious, it’s just from seeing some of the train wrecks in the 0.9999… =/!= 1 thread.
Whether this actually means you can do an infinite number of actions in finite time by doubling your speed each iteration is something I’ll leave for the GD forum. That takes care of the infinite actions in finite time. The two questions on whether the lamp is on or off are basically restatements of:
inf
[symbol] S[/symbol] (-1)[sup]n[/sup]
n=0
(that’s supposed to be a Sigma if the symbol doesn’t come out right). Since you may not be a math geek, that means that take n from 0 to infinity, plug each into the equation (-1)[sup]n[/sup], and add them all up. So you’re looking at:
1 + (-1) + 1 + (-1) + …
A 1 means turning switch to on, a -1 to turning the switch to off. After adding up the terms, if the sum is 1, the light is on, if it’s 0 the light is off.
This series doesn’t converge to any single value since it flips between 1 and 0. I know this series has a name, but I can’t for the life of me remember it, so I’m having a hard time finding whether or not this has an agreed-upon value. And whether or not starting n at 1 (so that after the first term it turns off) would change the value. After some poking around on Mathworld, it looks like you can equally argue for an value of 1, 0, or 1/2. Take that as you will for what happens to the lamp!
(on preview, I’m wondering whether MCMofC is trying to determine this experimentally with his mental state 
)
OK I’ll tackle this one. Like Padeye said, there are various physical limitations to actually trying this out, so we’ll keep it as a thought experiment.
Imagine a line with the minutes marked off and you move halfway along it, then another quarter way, etc etc: you will always occupy a space in between 0 and 2, but never actually get to 2. The reason you can do this is that between any two distinct real numbers there MUST be another distinct real number; this is the argument used to show that 0.999…=1 (there is no number between 0.999… and 1 so they must be the same).
Point is that every time you move closer to two, there must be a new number between your present position and 2 which you can jump to next. I suspect the problem you’re having is the concept of ‘doing an infinite amount of things in a finite amount of time’ - this is directly analagous to moving along a number line as described above and is easier to think about.
Anyway, that should hopefully clear up the infinite operations thing; what about the actual answer? Thing to imagine here is a graph with a line which gets closer and closer to one of the axes but never actually touches it (http://mathworld.wolfram.com/Asymptote.html); you can see that although you get veeeeeery close, you never actually reach your ‘two minute’ goal. The guy switching the lamp never actually reaches two minutes, he just get’s very very close, and so here’s why it’s a trick question: there is nothing in the question to indicate what happens at two minutes.
What we DO know is that as he gets closer and closer to two minutes, the guy switches the lamp more and more rapidly.
On re-reading my post I’m not sure I was all that clear, so here’s my point:
There is no information in the question allowing you to deduce what happens at two minutes; it would effectively be something which happened AFTER yer man was done playing with it, as everything he does happens before the 2 minute mark, for reasons outlined in my main post.
How can the guy switching the lamp never reach two minutes? Of course he does, if two minutes goes by for us, it goes by for him also.
Infact you can argue that the value is any number between 0 and 1 inclusive.
I’ll explain how I modelled it, which will also explain my schizophrenic answers.
I modelled, in my mind, as an arbitary wave function with the maxima at ‘on’ or 1 and the minima at ‘off’ or 0 (not strictly correct I suppose as the values are discrete) on the y axis and the z axis representing t. The problem is at t=2 the wave function is a straight line parallel to the y axis thus encompassing 0,1 and all the values inbetween.
The light would be off because the switch would melt from the air friction.
Isn’t this the same as asking what the sum is of the sequence:
1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 …
Depending on how you arrange the parens, you get different answers:
1 - (1 + 1) - (1 + 1) - (1 + 1) - (1 + 1) - (1 + 1) - (1 + 1) - (1 + 1) - (1 + 1) …
or
(1 - 1) + (1 - 1) + (1 - 1) + (1 - 1) + (1 - 1) + (1 - 1) + (1 - 1) + (1 - 1) + 1 …
etc.
Although the quantities halve each time in the map problem, I don’t see a conceptual difference. I think this means that the sum in indeterminate.
Infinity is a dangerous mistress.
Ok that’s my fault for not being clear. Here goes:
Consider the information we have about the guy’s actions: we know what he’s doing at time t=1 minute, at time t=1 minute 30 seconds, t=1 minute 45 seconds etc.
We are not given any information as to his actions at 2 minutes, as although the time gets extremely close to 2 minutes, he never actually get’s there (see the link about asymptotes in previous post). So, yes, two minutes do pass and we know exactly what happens inside of the two minute interval, but not what happens at two minutes…at two minutes he has stopped switching the switch in the way the question describes.
All the times we have information about are in fact, less than two minutes; that’s why it’s a trick question.
Well, first off, the statement “, it is easy to see that all these infinitely many time intervals add up to exactly two minutes.” is intentionally misleading. It does not.
There is such a thing as a time quanta. An indivisible unit of time, the shortest possible duration. I don’t know what this is offhand. It is incredibly small.
Therefore, one cannot divide beyond this point. When the genie reaches a single chronon (??) or fraction thereof, that’s it. He’s done. It doesn’t continue indefinitely.
So therefore, all ya gotta do is the math… 1 second, plus 1/2, plus 1/4, plus 1/8… plus one chronon, and it’s over. One presumes he waits the tiny fraction of a second that remains, to make two full seconds, at this point.
Then just count the clicks. q;}
Phnord We’ve already established that this is a theoretical exercise, and things like the Planck time etc are being ignored, and in fact aren’t necessary to reaching a definite answer anyway.
Green_Dragon has it essentially right. It is not that the demon won’t reach the two minute mark–it wil–but that the statement of the problem specifies only what he it does before the two minute mark is reached. At two minutes, the demon will do what it wants, whatever that is.
There is a well-known function whose value at positive or negative values of x is sin(1/x). If you ask what is its value at 0, well, I haven’t told you. Suppose I tell you it is 42. Now its value at 0 is 42. Of course, it is discontinuous, but where is it written that every function is continuous. So it is not that the button is both on and off; it will be one or the other, but there is nothing in the statement of the problem that tells me which it is.
It is not different in principle from the demon that keeps the switch off for all time, except at exactly the two minute mark. That is not realistically possible, you say? No argument here, but then neither is the original question.
And to answer one post, mathematicians talk about completed infinities all the time. The was contoversial in the 19th century but by the beginning virtually all mathematicians were comfortable with the idea (ultra-finitists such as Yesnin-Volpin excepted, but they are vanishingly rare).