Don’t know about that, but mostly I was just getting at the fact that the system described isn’t closed, so we’d need to know the values relevant to the stuff that’s entering and exiting the system.
Simplify the example to this: You have a bicycle with wheels of diameter one unit. It’s pedaled at such a rate that the wheels make one revolution per unit of time. After one unit of time, the bike has moved forward pi units.
Well… I’m not trying to nitpick, I’m just trying to ask the question I’m asking. At the end of the revolution, it’s not clear that there are two objects in this scenario that are pi units apart. There’s just one object that has traveled pi units, which is different from what I was asking.
It might be just as good though. But I’m not sure. What’s the distance traveled through spacetime? I ask because to say that the object moved pi units in a single unit of time seems to rely on an idea that units of time are independent of units of space (or else why be confident we can stipulate that the time measured exactly one unit?). I think, but do not know, that time can be measured in the same units as space when dealing, as physics does, with space and time as dimensions of a single structure.
There are two bikes, one is just stationary.
Are we good now?
Maybe there could be a pully system where when the pully wheel turns one revolution, the rope will have moved an object on the end of the rope exactly the same distance, i.e., the diameter of the pully wheel.
Maybe–though the thickness of the rope probably throws this off.
Anyway, your own idea doesn’t work as an example of what I was after because now the two objects end up right next to each other, not pi units away from each other.
I’m mainly concerned about the question whether we can just stipulate that the time taken is a single unit of time, or whether we’d have to express time measurements in terms of the same unit we’re using to measure space.
If one bicycle moves while the other doesn’t, will they not end up pi units apart?
Yes, sorry, I misunderstood the example.
I think that whatever the answer is, it is going to involve a circular object.
The bicycle is problematic to me because of the constantly changing force vectors that the rider is applying to the petals during revolutions. Anyone want to do the calculus?
Perhaps you could have a wheel on an axle that has a gizmo that seizes the wheel (perhaps via a spoke) after a half revolution? Make the wheel have a radius of one length unit, making the circumference 2pi, set the wheel to one half revolution before the catcher, then apply a gentle, rational force to the extent that when the wheel hits the catch, it stays being pushed and held gently against the catch without sufficient force to break/overcome it. When you hit a rational time interval, declare the experiment over.
Anyone want to come up with an idea using a ball?
Related question:
If I understand correctly, quantum physics always predicts only probabilities that things will be found in particular positions. Do these probabilities ever contain transcendental numbers in their numerators or denominators? Is there any way it makes more sense to stipulate purely rationally measurable initial conditions in QP than in the kind of large scale physics I seemed to be talking about in the OP?
You’re missing the point of Chonos’ response, which is not just about the ability to measure a position to a specified precision, but the ability to even define the position of a fundamental particle to beyond a certain precision per the indeterminacy prinicple. This isn’t just a problem of measurement–that your ruler doesn’t have fine enough graduations to make a measurement of such precision–but rather that the fundamental physics of matter doesn’t allow for such localization of a real energy field such as a particle to exist in a conceptual manner.
If the topography of space is truly continuous then yes, this is true, but we have every reason to believe that his assumption breaks down at the quantum level; certainly, we’re fundamentally unable to determine the particle’s location to an infinite degree of precision. A particle doesn’t pass through every point on a geodesic line through space, but instead jumps from cell to cell like a frog jumping across a series of lily pads. One cannot then just take the limit from each side as it approaches pi and call it a day, but instead you have to acknowledge that below a certain limit there is no longer a real distance between two divisions.
Of course, it would be possible to divide a standard unit, like 1 meter, by pi, and call that a unit like [sub]pi[/sub]meters, and then move a particle by that unit. The issue still remains in the conversion and precision to which the particle can be defined or measured, i.e. if you move it by 1.00000 meters it is only located to 3.14159 [sub]pi[/sub]meters +/- 0.0000017 [sub]pi[/sub]meters.
And so the answer to the o.p. is that, no, you can’t move any real particle by exactly pi, because pi isn’t a rational number. Nor could you move a real particle to a precise distance specified by a rational number smaller than that allowed by the indeterminacy principle. Sorry if that violates the scholastic result that comes from dialectic reasoning, but in the real world insofar as our ability to describe our observations, reality is granular and imprecise at very fine resolutions.
Stranger
Here’s one going the opposite way, using a ball.
Two people stand on a true spherical (non squashed, like Earth) planet together. The planet’s surface is smooth, hard rock. One of them travels via some convenient rationally measured transportation method to the other side, traveling in a straight line along the surface around half the circumference. Now, measure the distance between the two people, disregarding the planet. Voila, it’s the diameter.
It’s late at night here - anyone want to play around with the algebra here to get an answer for the OP?
Maybe you should offer the janitor a nice barometer if he can tell you the answer?
Right, I figured this would come up. I tried to ask the questioni n the OP in a way that made it easy to imagine what I was talking about–for my own sake more than for anyone else’s of course–but I knew I’d have to take this seriously eventually.
Here’s an abstract version of the question.
Forgetting about locations in space or anything like that, I just want to askt whether, if you start with a set of non-transcendental values in the specification of initial conditions, you can end up with transcendental values in a prediction of what will result from that set of initial conditions.
Values of what? Whatever it is that physics assigns values to when it says anything of the form “When initial conditions have values x, y, z… the result will have values of a, b, c…”
As a question separate from that of my previous post, I also wanted to ask:
What the hell was this about?
I don’t even know what “dialectic reasoning” is supposed to mean here.
How is this any different from the examples given above? If you have a circle of diameter one and mark a point on its circumstance, then move it from coordinate 0,0,0 one revolution, then by definition the endpoint is pi,0,0. (And what does time matter?) If you continue to insist that only one point is involved, start two circles moving in opposite directions, in which case the endpoints will be 2pi units apart. (And the amount of time for the two can be any two finite times.)
To my ears, you keep asking the same question and rejecting the obvious answers to that question. I’m afraid I don’t understand what question you’re trying to ask.
Stranger is saying that no macroscopic example will yield a “yes” answer to the OP due to things like the uncertainty principle.
But even ignoring that, the reason I wasn’t sure about the examples you’re talking about is that the open nature of those systems invites the question whether the measurements involving objects and forces entering and exiting those systems would also be rational.
I didn’t specify I was thinking about a closed system in the OP, and I should have. But I thought this would be clear enough anyway–if it’s an open system, then of course you can start with any initial conditions you like, and engineer a transcendental measurement in the results by having a force with transcendental measurement act on the system in the right way. The question is trivially answered in this case.
So to answer your question, here’s the question:
If you start with a set of non-transcendental values in the specification of initial conditions of a closed system, can you end up with transcendental values in a prediction of what will result from that set of initial conditions?
(Values of what? Whatever it is that physics assigns values to when it says anything of the form “When initial conditions have values x, y, z… the result will have values of a, b, c…”)
Secondary question: Is it possible for there even to be a closed system which can be specified entirely using only non-transcendental values?
Consider a piece of farm machinery then. There’s a roller that travels across the ground. For every revolution, It pokes a hole in the ground and deposits a seed. Measure the diameter of the roller. The seeds end up pi * d distance apart.
i want second that. i’ve rarely been interested in pure math in the way i have been recently from reading these threads. i don’t think ever for something entirely hypothetical. i will engage in a little hyperbole and state that SD performs an invaluable service for in the quest for knowledge.
This really isn’t an answer to the OP, but I think it is closely related enough to be of interest.
If you have a one inch long needle and drop randomly it on a floor made of two inch wide strips of wood the probability that it will land so that it crosses one of the seams between the floorboards is 1/pi.
For more info see: Buffon’s Needle.
Well, it depends on what you consider the values, surely. I mean, the numbers don’t exist out there; they’re just part of how you might choose to describe the situation. You could say “A and B are 3 meters apart”, and what you mean by this is that if you took three meter sticks and stuck them together they would reach from A to B. Or you could say “A and B are 3π curved-meters apart”, and what you mean by this is that if you took a circle whose circumference was 3π times as long as a meterstick (and this in turn means something we could spell out in lower-level terms…), it would just reach from A to B. Or you could say “A and B are ln(3)/ln(2) doubling-meters apart”, and what you mean by this is that if you were to start one meterstick away from A in the direction of B, increasing this distance exponentially over time, doubling it each second, then you would reach B after ln(3)/ln(2) seconds. And so on and so on…
So what’s the value which is actually physically present in the situation? Is it 3? Is it 3π? Is it ln(3)? Is it 8? Is it 2.7? Well, the physical situation itself doesn’t present the numbers, any more than the physical situation cares whether you use meters or yards; it’s just the statements you choose to make which incorporate them. And you can always make various statements involving whatever numbers or other abstract concepts you like in some way or another; they just might not be the kinds of statements you find interesting or useful. But what’s interesting or useful is up to you; you’re just as free to speak of the “curved-meter” and “doubling-meter” type measurements as you are to speak of the more familiar ones. They’re all equally present in the situation.
Fine. Then, presumably, to say something more concrete and coherent, we should really ask something specifically like “Can we construct a situation where the distance between point-like objects A and B is 2π (or some other transcendental multiple) times the distance between C and D without having to take advantage of a pre-given such transcendental distance ratio?” [which fundamentally translates easily enough into whatever statement about distance-measuring-devices (lining up of sticks or what have you)]. But though this question is suitably concrete, its answer is trivial: of course you can, in high school-style physics; take any method of constructing an odometer which displays its result as the distance between two objects, and run it around the circumference of a circle. Bam; a distance ratio of 2π between the odometer result and the distance between some object on the circle and some object at its center. Pretty much by definition. Like I said, trivial.
Even more straightforwardly than such an odometer based setup, though, pretty much any nontrivial differential equation ends up spitting out transcendental quantities, and physics (at least in a continuous picture) is chock-a-block with differential equations. For example, take Coulomb’s law (or any of the myriad similar inverse square laws): the repulsive force between two particles is proportional to the product of their charges divided by the square of the distance between them. Consider two particles of equal charge at some distance from another, initially at rest with respect to each other; picking units such that the Coulomb constant, this charge, and this distance are all 1, we obtain that the second derivative of d with respect to time is equal to 1/d[sup]2[/sup], with initial conditions of d being 1 and its first derivative being 0, where d is the distance between the two particles. After one unit of time, this distance will be just slightly above 1.4377. What will it be exactly? Well, I have no better way of describing it than to reiterate the differential equation defining it. I don’t have a proof handy, but I’d be shocked if this value wasn’t transcendental.