Flashing light bulb problem

[Mobius]

True, I am more interested in the maths as it has been establised thta the situation cannot happen.

 

In your example x satisfies x(t)=1/t for all strictly positive t. what is x(0), there is nothing forcing the t to 0. In the case of the flashing light bulb time is forcing the function f(t)=0 to f(2), (unless time stops of course and it never reaches the 2 minute mark, but I wouldn't want that situation to arise... it would however solve our problem ;-).

[Mobius]

If your doing a search for xenos paradox, spell it Zeno's paradox!!!

[YT2095]

If your doing a search for xenos paradox, spell it Zeno's paradox!!!

aha! no bloody wonder, I was spelling it as you guys did LOL

[matt grime]

True' date=' I am more interested in the maths as it has been establised thta the situation cannot happen.

 

In your example x satisfies x(t)=1/t for all strictly positive t. what is x(0), there is nothing forcing the t to 0. In the case of the flashing light bulb time is forcing the function f(t)=0 to f(2), (unless time stops of course and it never reaches the 2 minute mark, but I wouldn't want that situation to arise... it would however solve our problem ;-).[/quote']

 

 

what? there's nothing that forces your time to be 2 either, by your logic (not that it seems logical to me)

 

if you must think of time elapsing (but we're nto oding physics)

 

then try x(t)=1/(2-t) for t in the interval (0,2), what is x(2)?

[Mobius]

ok, I see your point...

 

I also see how maths and physics should have a restraining order...

Nothing in maths forces time to 2 minutes (in this senario anyway).

but in physics there is nothing to stop the 2 minutes approaching.

 

Just as a bizzare twist to this post. If this flashing light bulb was happening at the event horizon of a black hole then the time would slow down and eventually become 0 (according to us) thus satisfying our required condition. Of course according to the lightbulb time would seem normal...!

 

I don't know where this thread is going but it seems that such an unassuming question has many problems....;-)

[YT2095]

entering the "Black hole" would be the equiv of forcing a resolution to the decimal point count I mentioned earlier with the 10/3= scenario.

 

btw, this is getting off topic now.

[matt grime]

any question that has a multitude of unstated assumptions (each of which could be one of many choices, most of which are inconsistent with each other) willl naturally lead in many directions. personally i don't think the question is at all interesting beyond showign that there is a lack of understanding of the word "function" in the physical sciences, yet somehow i keep trying to explain what maths does and doesn't say.

[Mobius]

Well it looks like maths doesn't have a solution to this question ;-)

[matt grime]

Of course not. Nor should it.

[YT2095]

Awww, don`t give up Matt, your point is basicly why I said this a while back.

 

perhaps it`s a mental flaw on my part then' date=' if it cannot occur, The question is irrelevant (to my way of thinking).

 

the question as outlined in the OP is invalid.[/quote']

 

it was a little frustrating for me, and so I can certainly appreciate how much more so it would have been for you!

I had to do a complete "mental changeover" from my default Practical to the "Conceptual" and still found flaws in it! LOL

 

Mobius, can ya ask us something SIMPLE next time? like maybe the meaning of Life or something? LOL

 

all`s not wasted anyway, I know how to spell Zeno!

[matt grime]

As far as I can tell, the question in its purest mathematical terms is, reversing the direction of time as we may, defein

 

f(t) to be n mod 2 (ie 1 for odd n 0 for even n) when t is in the interval (1/2^{n+1},1/2^n], ie start at 1 and count down to 0

 

for n in the natural numbers (including 0)

 

is there any way to "naturally" extend this to a function valid at t=0? The answer is "no" not in the terms given. There is a reasonable, natural extension to negative integer n, that is we can extend to a function on the intervals (1,2], (2,4] and so on.

 

OK, so what if i were to tell you now that we were modelling a theoretical object that can switch on and off? well, since it is theoretical there is no obvious way to say what happens when t approaches zero. we have no intuition to rely on since it is an inherently unintionistic (to invent a new word) situation. is time even infinitely divisble? might it not be discrete? (actually, all models for discrete time, i'm unreliably informed, have problems and would lead to contradictions with known observations at large scales; this doesn't mean that time is continuous jsut that no good discrete model exists).

 

so it is practially impossible and theoretically not enough information is given.

 

there is always danger in extrapolating beyond known data anyway.

[BigMoosie]

Here is a similar idea I heard once, and it is less theoretical than the one being discussed:

 

There are two racers: A (fast) and B (slow). B gets a headstart and when A starts running, B is at the 100m mark. When A gets to 100m he reconsiders his situation and sees that B is at the 150m mark. When A gets there he reconsiders and sees B at 175m, and when he gets there B is at 182.5m, etc etc.

 

As you can see A will never reach B because each time he gets to his next destination B will have inched forward ever so slightly.

[matt grime]

and there goes xeno's (or zeno's) paradox.

[BigMoosie]

Ahh, should have looked that up. Sorry.

[YT2095]

OK' date=' so what if i were to tell you now that we were modelling a theoretical object that can switch on and off? well, since it is theoretical there is no obvious way to say what happens when t approaches zero. we have no intuition to rely on since it is an inherently unintionistic (to invent a new word) situation. is time even infinitely divisble? might it not be discrete? (actually, all models for discrete time, i'm unreliably informed, have problems and would lead to contradictions with known observations at large scales; this doesn't mean that time is continuous jsut that no good discrete model exists).

 

so it is practially impossible and theoretically not enough information is given.

 

there is always danger in extrapolating beyond known data anyway.[/quote']

 

I`m with you 100%, hence I posted:

another point' date=' that needs mentioning.

if we ignore the actual Dynamics of this setup, the On/Off rate will be at a frequency impossible to attain at even a quantum level. it`s on a par with ideas about "what happens when we travel facter than C" or "irresistable force meets an imovable object" and the likes. Simply the situation cannot Occur.

 

Any answers to these, would be pure speculation.[/quote']

 

the question itself is unacceptable

[Mobius]

Ah yes some interesting points there on time. If time was discrete it would solve our problem, as I hinted at earlier. The 10^(-43) seconds is Planck time and the associated length is Planck length (1.6*10^(-35)m). Shorter time intervals are not understood, it doesn't necessarily mean that they don't exist.

 

Sure we could always abandon time altogether and believe (like physicist Julian Barbour does) that time doesn't exist and we live in a world of 'nows'.

 

Matt, I get your point, just not sure how to add information that you say is lacking. I set initial conditions and allowed time to evolve, I don't see where any further information is required. this is probably due to my lack of maths skills (or rather my way of looking at maths). My trusty computer does most of my maths for me;-)

[YT2095]

think of it as an exponential curve, with a divide by 2 condition as the Y axis and time as the X axis, it`ll start off steep and then gradualy taper off towards flat, but it`ll never end even if you ran it for a million years

 

ok sure, eventualy it`ll look like a flat line almost but it WILL increase in amplitude, but it`ll Never reach its end point.

[Mobius]

yes I do understand an infinite sequesnce, what I don't understand is what happens when the limit of an infinte sequence is reached and we are not talking about 1 million years, we are talking about 2 minutes....!

[YT2095]

ever heard of the chess board and the grains of sand? where each square has exactly double the amount of sand as the one next to it?

 

you start with ON for part of a minute then off for half the same part and so on...

 

of course since it`s only /2

it`s pretty easy, if it starts at ON, then after 2 mins it should end as off, and the other way around if it starts as Off.

 

the whole thing is a binary count (like the 8x8 chess board that =64).

 

the same will apply if we stick to the leading edge OR trailing edge of the pulse throughout. the pulse width is imaterial since it opperates at a /2 at the same time constant.

 

THERE is the answer to your question

 

(sorry for the time taken, I DO like time to work things out!)

[Mobius]

I think you have oversimplified the situation. you are thinking of finite counts. This is not a finite count. The time intervals between the flashes become shorter until they reach their limit of 0 at the 2 minute mark. therefore technically being both on and off which is an unacceptable situation.

 

The chess board and binary count you are talking about are based on finite counts.

[matt grime]

yes I do understand an infinite sequesnce, what I don't understand is what happens when the limit of an infinte sequence is reached and we are not talking about 1 million years, we are talking about 2 minutes....!

 

 

but there's the crux: you've told me how it behaves at every point in time prior the the elapse of two minutes but you've

 

1. not told me what happens at 2 minutes

 

2. not told me how i am to use the previous information to conclude what happens at the limit point.

 

i need to know how things evolve through the "barrier" as it were.

 

i have no absolutely canonical theoretical method of passing the behaviour to the limit, nor do i have any "fall back" physical intuition i can use.

 

 

recall the eaxmples i gave about nested sets and so on. there is no reason to suppose that there is any meaningful way to pass to the limit point and have the "bahaviour carry on" that is we cannot infer from the previous information about all the preceding points in time what the behaviour at 2 minutes is.

 

sometimes properties can pass to the limits, some times it doesn't make sense

 

forget the elapsing of time which is a complete red herring anyway, suppose i told you that at time

 

t=1 i have 1 banana, t=2 i have 2 bananas, t=3 i have 4 t=4 i have 8, at time t=5 i have 16 bananas. how many banans do i have at time t=6 or 7?

 

you might suggest 32 and 64, assuming the pattern continues, or you might recognise that it is the infamous trick sequence (of the maximum number of segments you can partition a circle into by joining up dots on the perimter, and suggest that the answers are whatever that sequence is) but truthfully there is nothing compelling you to make either of these deductions.

 

the ideas of limits and so on that you're talking abuot are actually totally irrelvant to the reason why mathematically this makes no sense. the problem is simply that given the information we cannot deduce what is going on at t=2.

[YT2095]

you make it Finite by putting the 2 minute limit on it.

 

I`ve abandoned "Practicals" and "Maths" pure Logic states that what ever the condition is at the outset, in a binary operation involving even numbers.

that when Stopped at an even number the condition will be at the same state, regardless of what occurs in between this duration or however impractical. if all conditions remain constant to the specified algorythm, it will remain True.

 

your problem is INDEED solved!

[matt grime]

I think you have oversimplified the situation. you are thinking of finite counts. This is not a finite count. The time intervals between the flashes become shorter until they reach their limit of 0 at the 2 minute mark. therefore technically being both on and off which is an unacceptable situation.

 

The chess board and binary count you are talking about are based on finite counts.

 

but the problem is that you *have* only specided what happens after a finite number of switches, you have not told us *how* one passes to the limit. passing to a limit is a delicate question that needs context; you've not give us any mathematical context and there is no physical context.

[YT2095]

I know one thing, he`ll get no more Paradigm shifts from me today!

[matt grime]

let em give another exmaple, suppose for the sake of argument that we know something, x(t) is a continuous function of t defined for all t, and that x(1- 1/2^n) = 7+2/n^2, then for n in the natural numbers, then what is x(1)? well, since x varies continuosly and 1-1/2^n tends to 1 as n tends ton infinity, uit must follow that x(t) tends to 7 since 7+2/n^2 tends to 7 as n tends to infinity.

 

see, more information, and i can make a deduction.

 

I can give a funtionf defined for all of t with range {0,1} that exactly models the bulb for the period of time youve defined it for and that is 1 (on) when t=2, and i could give one that is 0 when t=2, indeed i can give infintely many different models for either situation.