I’d thought of that actually! It would be exactly the kind of thing I’m looking for, except it’s my understanding that it’s physically impossibl to drop a needle “randomly” in the sense called for in that theorem.
Incidentally, there is a very beautiful and shockingly simple proof of this fact which, for some reason, I rarely see used, despite actually establishing a much stronger result. To wit, pick units such that the width of the floorboards is 1. Then the result is that for any particular shape and size of needle, whether straight, curved, or what have you, if the needle has length L, then when thrown with orientation and position chosen at uniform random, the expected number of seam-crossings is 2L/π.
Why is that? Well, we can imagine any particular needle as composed of lots and lots of tiny straight segments, all size epsilon; by linearity of expectations, the expected number of seam-crossings for the whole needle is just the sum of the expected number for each segment. As all these size-epsilon straight segments are essentially the same as each other (they differ only in position and orientation within the needle, which is irrelevant to their behavior under our random throwing), it follows that the expected number of seam-crossings for a needle is simply proportional to its length. I.e., it is kL for some k.
All that’s left to be done is to find the particular constant of proportion. Well, consider a circular needle of diameter 1. It always has exactly 2 seam-crossings. Furthermore, its length is π. Therefore, k = 2/π, completing the proof.
In the particular case of a straight needle of length 1/2, this tells us that the expected number of seam-crossings is 1/π. And since such a needle is so short it can only cross at most once (well, except in the probability zero case that it lines up along a seam), the expected number of seam-crossings is the same as the probability of having any seam crossings at all. Thus establishing the result as it is usually phrased.
I’ve already noted all of this above, in the posts where I was dealing with initial observations that units of measurement of arbitrary. The idea is to ask whether, whatever units you are using, you can start with rational multiples of that unit and end up with transcendent multiples of it.
Not at all. As Stranger brought up above, none of these macroscopic constructions will work as an example of the kind of thing I’m looking for, due to things like the uncertainty principle. That’s why I’ve finally worded the questions o abstractly. As I took Stranger to be emphasizing, physics doesn’t actually strictly speaking make statements of the form “given initial conditions x, y, z the resulting condition will be a, b, c” about macroscopic objects concerning their positions in space, if x, y, z and a, b, c are to be taken as locations and forces. They might be probabilities or something (I don’t know, that’s why I worded the question abstractly) but strictly speaking, they’re not locations and forces, apparently, so no example using locations and forces will work. I originally asked for examples in terms of locations and forces, but I now think that’s a no go–the answer isn’t a trivial “yes” but a trivial “no” for cases like that.
ETA: Post 29 may ask a better question:
Setting aside the locations/forces problem mentioned just previously, this looks a lot closer to what I was hoping for than any of the odometer style examples.
Can we stipulate units of time measurement independently of units of space measuremnt? In other words, do units of space measurement determine what units we’re using to measure time as well, if we’re treating spacetime as a single four-dimensional structure?
Ah. It was unclear to me whether you really were exclusively concerned with uncertainty-drenched physics or were interested in, e.g, the Newtonian picture as well.
Yes, again if the circumference of a circle is pi*diameter, then draw a circle with a diameter of 1 meter. What’s the circumference? It is pi meters.
So. Wrap a length of string around this circle. It is pi meters long. Put any two objects at either end of the string and they are pi meters apart. I know pi is an irrational number, but maybe if you use base 12 it works out better.
(I mean, I guess we could specify different length and width units and engineer an example that way. But that would cover over the way that the three dimensions of space can be measured all using the same unit, and physics typically measures them using a single unit. Do I understand correctly that when dealing with space and time together as a four dimensional structure, there’s a single unit of length that gets used in all four of those dimensions?)
I think perhaps I’ve failed to clearly express what I meant to express with that passage. The point is, it’s not just a choice between meters vs. yards and that sort of thing. That’s one kind of ambiguity over what the numbers are which are present in the physical state, sure. But the ambiguity runs far deeper than that, which is what I tried to illustrate with, for example, the business about measuring in “double-meters”, which is to say, measuring in a scale which is related to meters not by a multiplicative constant, but rather logarithmically.
That is, the way I see it, a number is present in a physical state only in the sense that there is some measurement which produces that number. Which is to say, there is some way of performing experiment such-and-such, occasionally get readings on various devices and manipulating them according to algorithms so-and-so, which eventually produces that number from that state. But, of course, there will be other experimental algorithms one can run which will produce whatever number one like; which is to say, any number can be considered some kind of measurement of some kind of quantity in that state. It doesn’t do to simply say “Fix the units, and then there will only be so many numbers which are present in the state”, as though, up to a multiplicative constant, there were only so many different kinds of experimental-results-into-number procedures one could perform. One has to also limit oneself to only the outputs of some particular measurement procedures of interest (e.g., only the results of measurements of distance, not the results of measurements of logarithmic distance, or other more esoteric number-producing-instructions).
This was all only aimed at the secondary question, “Is it possible for there even to be a closed system which can be specified entirely using only non-transcendental values?”. My point was that I don’t think this is a particularly coherent question, because there’s no distinguished choice of how specifications are to be given, and thus of what it amounts to to be a value within a specification. The information in knowing all the distances is the same as the information in knowing all the logarithmic distances, and the same information can be represented in rather more esoteric ways as well, for example; but which of these should count as the values in the specification? This is ambiguity far worse than mere choice of units.
Let me put it another way: supposing there were a series of Yes/No experiments/measurements/observations I could carry out upon the universe, E1, E2, and so on, each of which could be taken as giving some information about the physical state of the universe. And from these experiments I could therefore choose to define a number Omega concisely capturing the total information contained in all of them; the sum of 1/2[sup]k[/sup] over all k for which the k-th experiment produces a Yes result, for example. Should Omega therefore be considered a value in the specification of the universe? But just as well, I could reorder the experiments, flip what counts as Yes and No, change the rules to calculate Omega in arbitrarily many other ways, producing arbitrarily many other numbers (and we needn’t only talk about real numbers; we could talk about any other kind of abstract entity we cared to). No criterion has been given to single some of these out as values in the specification of the universe and others as merely distorted ways of describing those values. And, indeed, I would say there are no distinguished such criteria; it’s just a question of what one is interested in.
Something like this is what I was trying to get at in response to your secondary question, though I fear I still haven’t put it well. But discussion of thoughts like “Are there states where all physical quantities are algebraic as opposed to transcendental?” runs close to common but ill-supported assertions like “All physical quantities are real as opposed to complex”, in a way which sets my buzzers off. There’s no question of whether all physical quantities are like this or like that. It’s just about how you choose to use the abstract entities to model the behavior of physically observable events.
Sorry, those last lines should be “There’s no question of whether all physical quantities are these kinds of abstract mathematical entities or those kinds of abstract mathematical entities. It’s just about how you choose to use abstract mathematical entities to model the behavior of physically observable events.” Got caught up in editing and frozen out before I could fix everything.
But I think I’m flailing around muttering to myself incomprehensibly, so I’m going to go to sleep for now. Generally a good idea around this time, I think. 
(In fact, I’d been recently thinking of taking a break from the board for a while, but this is the second time a thread of yours disrupted those plans by catching my attention and enticing me to post within it)
I think I get what you’re saying, and I think I don’t get what you’re saying.
I think I get it because… I read it and it don’t have any feeling that I’m failing to understand. If I understand correctly, you’re pointing out that even after units have been settled on, it’s not been decided what the values are in the specification of the initial conditions since it hasn’t been decided how to convey the information that this measurement has value X and that measurement has value Y and so on. One way of conveying this might give all algebraic values, another way of conveying it might give transcendental values, and so on.
I think I don’t get it because… it seems like since I just asked for a situation in which the initial conditions can be described using only non-transcendental values, all of the above is moot. Yes, there’s an arbitrariness involved in deciding how to convey information about the initial conditions. But can’t we just ask whether there is some way to decide how to convey that information which yields only algebraic values for some particular situation? (And then, using the same method of conveyance of information, we can look at the final conditions and check and see whether we get any transcendental values. What’s the “same method of conveyance of information?” Asking this seems to get close to Grue territory, and my impression of your philosophy of science is (both that you may not like having a philosophy of science ascribed to you! and) that you’d definitely not want to worry about Grue-someness.)
Well, what I’m wondering is in what way (if any) the practice of physics commits the practitioner to to realism concerning the real number line. So I guess I’m at least close to the territory you’re worried about.
If physics predicts transcendental situations resulting from algebraic situations, that’d be evidence that physics commits practitioners to a strongly physicalistic style of realism concerning the reals. (I.e., not only are they real in some abstract platonic sense, and their not just real in the sense of being necessary elements of some social construct or other given that social grouping’s interests, but rather they’re really really real in the physicalist sense that some physical situations simply can’t be described without reference to them.* That’s not necessary for realism, but it’s sufficient and would make things easy… which is why I suspect the answer to the question I’ve been asking in this thread is “no”.)
*ETA: Ah shoot; just because a situation can be described in a way that satisfies what I’ve been asking after, doesn’t mean it must be so described as I imply here. No time to re-think this right now, sorry…
Then my evil scheme has worked.
Yes, and it’s continuing to work momentarily, but I really will nod off after this post:
Something along the lines of grue-someness (and, specifically, “What’s better about blue than grue? Nothing, intrinsically, except that I’m more interested in the former”*) is precisely the concern I have regarding trying to just say “Is there some way to specify this state using only algebraic values?”. I might say “Sure, this state is Jangbat(3, 4, sqrt(2)). That only uses algebraic values. Specifically, the krong-distance between particles A and B is 3, and the krong-distance between particles A and C is 4, and …”. What’s the Jangbat function? Well, it’s a particular crazy function that happens to return this state on the input (3, 4, sqrt(2)). What are krong-distances? They’re a particular crazy way of reporting the information which we could otherwise report as distances in meters or logarithms of distances in meters or whatever, but in which the distance between particles A and B counts as 3, and that between A and C counts as 4, and…
I could do this for any state, with some Jangbat function or some krong-measurements. Making the answer to the question “Is there some way to specify this state using only algebraic values?” always trivially “Yes”. The only nontrivial questions would be “Is there some nice way to specify this state?”, where one gives particular criteria for what it takes to be nice. But there aren’t any intrinsically distinguished niceness criteria; it’s just a matter of what one’s interested in.
Something like that.
*: This isn’t necessarily my actual position on the grue problem. I’m just making an analogy which may be helpful for illuminating the concern I’ve been struggling to express here.
In practice, we never use gruesome predicates; we always use nice ones. That we can’t specify what the nice ones are doesn’t mean that the nice ones aren’t there for us to use, and doesn’t mean that we don’t in fact use them. They’re there for the using, and we use them. (Sounds a little dogmatic but I’m just being brief… 
)
By making use of a concept like “blue” rather than “grue” I commit myself (in my actions if not by my thinking, since in my thinking I know about possibilities like grue) to the reality of blueness and the non-reality of grueness.
Similarly, as far as I know, in the practice of science we use nice functions and not gruesome ones. Since my concern is to know what practitioners of science are committed to (if anything) by their practice,* and how committed they are to it, I don’t need to worry about gruesome possibilities here. I’m not wondering about the nature of reality, I’m wondering about the nature of scientific practice. Scientific practice doesn’t worry about gruesome possibilities, so I’m not worrying about them either.
(Link on Grue for anyone who’s following this conversation can be found here, though I hate that it’s in an article on relativism. Traditionally, the issue is discussed as a type of problem of induction.)
*By the way, “nice” here might just mean “used in practice” making my claim that science uses nice functions and not gruesome ones tautological. I’m okay with that, I think a tautology at this place in the conversation works.
But remember, only the INITIAL values are determined by the OP