How is it possible to win one-in-infinity odds?

[PeterDonis]

The mark is not a selection of atoms on the paper but atoms of ink or graphite that are deposited onto the paper.

That doesn't change my answer: there are a finite number of ink or graphite atoms or molecules and a finite number of ways to deposit them on the paper.

 

[PeterDonis]

It's also a question of being well-defined. It would be simpler to take the position of a single atom at a precise instant. Is that well-defined?

We're talking about actual measurements; you're the one who pointed out that those always have results of finite accuracy. In numerical terms, measurement results are always finite decimals.

 

[Dale]

a finite number of ways to deposit them on the paper.

I don't think that is true. Or at the very least, it is not clear to me that it is true

 

[PeterDonis]

I don't think that is true.

Remember, again, that we're talking about actual measurements. At least, that's my understanding: you are talking about the set of possible results of measuring the centroid of an ink spot on a piece of paper. Since the paper itself is a compact region, and since the result of any possible measurement will have finite accuracy, it follows that there are only a finite number of possible measurement results within the region occupied by the paper. (The basic mathematical fact I am relying on here is that any compact set can be covered by a finite number of open sets. The finite accuracy of measurements means that we can model the possible results of any measurement in a compact region using such a finite cover of open sets.)

 

[Stephen Tashi]

We're talking about actual measurements;

Some people are talking about actual outcomes and some are talking about measurements of actual outcomes - the outcome being regarded as something physically distinct from its measurement.

The assumption that a physical outcome is described by a probability distribution of a continuous random variable is a kind of invariance principle. It says that the behavior of any device sufficient to measure where the outcome falls within a particular a finite set of intervals can be computed from the same continuous distribution.

Does debating the physical existence of continuous probability distributions differ from debating the nature (continuous or non-continuous) of basic physical quantities such as length, time etc ? Or does the probability aspect introduce something new to the debate?

 

[PeterDonis]

Some people are talking about actual outcomes and some are talking about measurements of actual outcomes

In the thread in general, that's true. In the particular posts I was responding to, as I said, my understanding is that the measurement results were what was being discussed.

As far as "actual outcomes" go, I think it is also important to keep in mind the difference between our models and "reality". In our models, spacetime is a continuum, and so are spacetime-related observables like position. But we don't know for sure that that is true in reality. We have no evidence against it at this point, but we have plausible reasons based on quantum gravity to think that, if we ever get to the point of being able to probe Nature at or near the Planck scale, we might find such evidence. (Or we might not--we won't know unless and until we are able to try.)

 

[Dale]

you are talking about the set of possible results of measuring the centroid of an ink spot on a piece of paper

No, I am talking about the ink spot itself as the measured outcome of the experiment. I am not talking about producing a number (or a pair of numbers) from that.

I understand and accept that if you want to store the outcome in a computer or communicate it to a colleague then you will need to use a finite representation of the outcome. But that is the conversion of the physical outcome to a number, not the physical outcome itself.

At this point I am not going to continue on this. I haven't thought it through enough and don't want to continue arguing a point that I am not confident about. It just is not clear to me that the "no such experiment" claim is correct.

 

[PeterDonis]

I am talking about the ink spot itself as the measured outcome of the experiment.

What kind of measured outcome is "the ink spot itself"? I don't understand what actual measurement you are describing.

 

[PeterDonis]

that is the conversion of the physical outcome to a number, not the physical outcome itself.

If you are going to make this fine distinction, then "the physical outcome itself" is unobservable, just like "the actual state in reality" as opposed to measurement results we can actually store and communicate and work with. Which makes me wonder about how useful this fine distinction really is. At the end of the day, the finite representations of outcomes are all we can actually work with.

 

[PeterDonis]

how is it possible to win odds of one-in-infinity in this scenario?

Remember that this scenario cannot actually be realized, so any answer to this question is going to be talking about something that can't ever actually happen.

This means that your basic assumption that some result will always occur--that some number on this infinite roulette wheel will always be picked on every spin--is no longer the obvious truth that it is for a finite roulette wheel. You can't actually make an infinite roulette wheel so you can't actually test what happens when you spin one. And arguing by analogy with finite cases is precisely the kind of thing that can trip you up when trying to reason about infinities; often such analogies are not valid.

So IMO the best answer to your question (although not the only possible one--other alternatives have been suggested in this thread) is that it might not be possible to "win" when spinning an infinite roulette wheel. You can't just assume that it is and ask why. You would have to be much, much more precise in specifying exactly how such a thing would work, in a way that was mathematically consistent, and then try to reason from the mathematical specification. The simple words "an infinite roulette wheel" are not, by themselves, such a specification.

 

[DaveE]

Remember that this scenario cannot actually be realized, so any answer to this question is going to be talking about something that can't ever actually happen.

This means that your basic assumption that some result will always occur--that some number on this infinite roulette wheel will always be picked on every spin--is no longer the obvious truth that it is for a finite roulette wheel. You can't actually make an infinite roulette wheel so you can't actually test what happens when you spin one. And arguing by analogy with finite cases is precisely the kind of thing that can trip you up when trying to reason about infinities; often such analogies are not valid.

So IMO the best answer to your question (although not the only possible one--other alternatives have been suggested in this thread) is that it might not be possible to "win" when spinning an infinite roulette wheel. You can't just assume that it is and ask why. You would have to be much, much more precise in specifying exactly how such a thing would work, in a way that was mathematically consistent, and then try to reason from the mathematical specification. The simple words "an infinite roulette wheel" are not, by themselves, such a specification.

Yes, this!

A practical alternative, used by pretty much everyone except mathematicians and philosophers, is to say that the chance of winning tends to zero as you approach infinity. You can nearly always avoid the "at infinity" case if you want to. You'll learn about this in your first calculus class, or if you look up epsilon-delta proofs and limits.

"At infinity" is really complex, and, IMO, doesn't exist in the real world.

 

[FactChecker]

Remember, again, that we're talking about actual measurements. At least, that's my understanding: you are talking about the set of possible results of measuring the centroid of an ink spot on a piece of paper.

I think that the OP thread may have been hijacked, but maybe this is what the OP was questioning.
There can be calculations after the measurement. An average or other result can be derived from the measurements or a centroid can be calculated, etc. I would still call the derived answers "experimental results".

In terms of the representation of a measurement, it is very dependent on the units involved. IMO, a measurement of a circle's diameter of 1 is the same as a derived circumference of $\pi$.

 

[PeterDonis]

There can be calculations after the measurement.

Yes, but if there are only a finite set of possible measurement results, there can only be a finite set of things you can calculate from them with a finite number of operations.

 

[PeterDonis]

In terms of the representation of a measurement, it is very dependent on the units involved. IMO, a measurement of a circle's diameter of 1 is the same as a derived circumference of $\pi$.

The question is not whether particular numbers in the finite set of possible results are rational or irrational. The question is about the cardinality of the set of possible results. If you are using any real measuring device to measure the circle's diameter, it will have finite accuracy. That's true even if you multiply the result by an irrational number, $\pi$, to get the number you will give as the circle's circumference.

 

[PeroK]

In terms of the representation of a measurement, it is very dependent on the units involved. IMO, a measurement of a circle's diameter of 1 is the same as a derived circumference of $\pi$.

$\pi$ is computable. That's why you are able to communicate it as a specific real number. Only a countable subset of the real numbers are computable. That's why I said in a previous post that unless you know this, there's no discussion, as you are simply claiming what's mathematically impossible.

https://en.wikipedia.org/wiki/Computable_number

This means that (if you want to choose from an uncountable set) you cannot possibly know what number you have chosen. The best you can do is claim that some (unspecified) number meets certain criteria. E.g. in the fixed point mapping theorem. See:

https://mathworld.wolfram.com/NonconstructiveProof.html

The problem is whether non-constructive methods can apply to an "experiment". Or, whether they rely on precise mathematical axioms that cannot be assumed to hold for physical objects (or even for spacetime itself).

 

[PeterDonis]

The problem is whether non-constructive methods can apply to an "experiment". Or, whether they rely on precise mathematical axioms that cannot be assumed to hold for physical objects (or even for spacetime itself).

Some mathematicians don't even consider non-constructive methods to be mathematically valid. Since the Axiom of Choice is logically independent of ZF set theory, their objections cannot just be dismissed even on mathematical grounds (apart from the question of what mathematical objects can validly be used to describe physical measurement results).

 

[FactChecker]

$\pi$ is computable. That's why you are able to communicate it as a specific real number. Only a countable subset of the real numbers are computable. That's why I said in a previous post that unless you know this, there's no discussion, as you are simply claiming what's mathematically impossible.

Whether it is computable is not the issue. In an experiment, if you can not measure or determine whether a position on a line is exactly $3.1415926... (=\pi)$ then you would say, by your earlier posts, that the point is not a possible result of the experiment. On the other hand, if you use a different scale on the line and the position previously identified as $3.1415926...$ is now $1.000000...$, then you would say that the point is a possible result of the experiment.

I would like to distinguish between the result of an experiment versus the recorded result of an experiment. I think that I am talking about the first and you are talking about the second.

 

[Dale]

how is it possible to win odds of one-in-infinity in this scenario? Am I thinking of zero the wrong way? One is infinitely far from infinity, which I think would make it infinitely small.

Ok, leaving aside the question about whether an experiment can produce a real number, the key is that probability 0 doesn’t mean impossible.

Suppose I have a real-valued random variable with a uniform probability in the interval from 0 to 1. The probability that the variable is equal to exactly 0.3 is zero, but 0.3 is possible because it is in the interval from 0 to 1. Its probability density is non-zero. On the other hand, 1.3 is not possible because it is outside that interval. So impossible and probability zero are different concepts.

 

[PeterDonis]

Suppose I have a real-valued random variable with a uniform probability

The OP is actually asking about a countably infinite set (an infinite roulette wheel), which raises a different issue: there is no such thing as a "uniform probability" on a countably infinite set, as @PeroK noted in an early post in this thread. You can find valid probability distributions on such sets (he gave one example), but they will not be uniform: the probabilities for each individual member of the set will not all be the same.

In this respect countably infinite sets are actually more problematic for our intuitions than real-valued variables, for which we know how to define a uniform probability density.

 

[Dale]

The OP is actually asking about a countably infinite set (an infinite roulette wheel),

The OP is asking about a countably infinite set with uniform probability. This is a contradiction. Others have already answered about a countably infinite set with non-uniform probability. I have answered about an uncountably infinite set with uniform probability.

 

[PeroK]

Whether it is computable is not the issue. In an experiment, if you can not measure or determine whether a position on a line is exactly $3.1415926... (=\pi)$ then you would say, by your earlier posts, that the point is not a possible result of the experiment.

Not at all. It's not just about infinite decimal expansions. I can have a scale that reads $0, \frac \pi 4, \frac \pi 2, \frac {3\pi} 4,\pi$.

I would like to distinguish between the result of an experiment versus the recorded result of an experiment. I think that I am talking about the first and you are talking about the second.

This is a different matter. Let's take the position of an electron. The problem is that you have the HUP (Heisenberg Uncertainty Principle), which effectively implies that the electron cannot have an infinitely precise position in the first place. Or, to put it another way, the wave-function that would define a single, infinitely precise position is not a physically realisable state.

This is part of the problem. It only appears at the macroscopic scale that an object has a well-defined precise position. This principle, which you are relying on, does not apply at the microscopic scale.

 

[PeroK]

The OP is asking about a countably infinite set with uniform probability. This is a contradiction. Others have already answered about a countably infinite set with non-uniform probability. I have answered about an uncountably infinite set with uniform probability.

Let's take a closer look at this.

I would say that measure theory has an abstract mathematical definition of probability that does not define a physically realisable process in this case. I would put this in the same category as the Banach-Tarski paradox:

https://en.wikipedia.org/wiki/Banach–Tarski_paradox

You could also look up the proof that there exists an unmeasurable subset of $\mathbb R$. That set, I suggest, does not correspond to any physical object.

There are many examples of an uncountable infinite set or an uncountable process that are well-defined mathematically, but do not correspond to any physical set or process. And, in fact, the Gambler's Ruin is an example of a countable mathematical process that is physically impossible: namely, how to dispose of an infinite number of coins in finite time.

The uniform distribution on $[0,1]$ is probably the simplest and the most contentious. It looks innocent and many people claim (without really thinking about it), that you can choose a real number between $0$ and $1$ from this distribution. However, when they conclude that "something impossible can happen" this ought to make them stop and think more deeply about the relationship between mathematics and experiments: i.e. what it is physically possible to do.

If you assume elementary particles are point particles and that classical mechanics (and not QM) applies, then you can assert that a point particle must be precisely at a specific location, one of uncountable many, and may remain at rest at that point. That is clear. It's still not possible to describe an arbitrary point with infinite precision, but you could claim that your classical particle has chosen for you a point out of uncountably many in an interval.

But, elementary particles are quantum mechanical and cannot be pinned down, even theoretically. It's not possible to appeal to the notion that the particle "must be somewhere".

You can, of course, still claim that spacetime has uncountably many points. If we allow that to be in doubt, and say that there are no uncountable sets in nature, then there is no possibility of physically choosing from an uncountable set. For the purpose of this discussion, therefore, we have to assume that spacetime is uncountable.

The problem is then how to choose an arbitrary point in spacetime, without reference to physical particles, with their inherent uncertainties?

The only idea I can see in this thread is to pick a particle (or large set of particles, such as a dart, or ink spot on paper) and ignore QM. That doesn't cut it for me.

The last try, IMO, is to somehow map some portion of spacetime to another and appeal to the fixed-point mapping theorem to pick out one of the uncountably many points. But, it feels like the transition from pure mathematics to a real physical scenario will remain problematic, even in this case.

 

[Dale]

@PeroK I have already told you that I am not going to further pursue the question about the experimental realization of a real-valued measurement. I don’t find your arguments convincing (especially re finite precision which doesn’t seem relevant to me) nor do I find @PeterDonis arguments convincing. But I also have not thought about this carefully myself and don’t have contrary arguments that I find convincing yet.

Until I have convinced myself all I have is doubts which I am done arguing. Please don’t try to draw me in again. I will simply answer wrt the math without focusing on any experimental realization. As I did above.

 

[FactChecker]

Not at all. It's not just about infinite decimal expansions. I can have a scale that reads $0, \frac \pi 4, \frac \pi 2, \frac {3\pi} 4,\pi$.

This is a different matter. Let's take the position of an electron. The problem is that you have the HUP (Heisenberg Uncertainty Principle), which effectively implies that the electron cannot have an infinitely precise position in the first place. Or, to put it another way, the wave-function that would define a single, infinitely precise position is not a physically realisable state.

This is part of the problem. It only appears at the macroscopic scale that an object has a well-defined precise position. This principle, which you are relying on, does not apply at the microscopic scale.

A distribution (usually) has a mean that is often used as the result.

 

[jbergman]

Yet, some event, $\{ X=r \}$ where $r \in [0,1]$ will result from an experiment.

This isn't quite accurate. There is no real physical experiment where this will happen. In any real experiment there will be some measurement error which necessarily discretizes the space.

 

[jbergman]

@Dale can you suggest an experiment that would produce any real number (in an interval, say) and prove (or at least justify) why any real number could result.

I assume you accept that most real numbers are indescribable (uncomputable), and the set of computable numbers is countable with measure zero. From that point of view, your experiment could at best claim to have chosen a real number, but could not specify which one. And, in particular, if two such experiments were carried out there would be no algorithmic way to test whether the numbers are equal.

This is a key paradox of the real numbers. We can test mathematically that $x = y$, where $x, y \in \mathbb R$. But, there is no terminating algorithm to check whether two real numbers are equal. Unless you restrict things to the computable subset. IMO, that is a good example of where a simple piece of mathematics (If $x = y \dots$), is not actually physically/algorithmically possible.

Exactly. I don't think people understand how nasty real numbers and that most are uncomputable.

 

[.Scott]

Are you sure about that? I think you are maybe claiming that we cannot measure an element of a continuum. But I am not sure that is true.

Forget about measuring a point on a continuum - we'll never get to spin the wheel.

As soon as I ask people to place their bets, I will need enough paper (or other material) to record their selections.
If I number the bins in decimal and the players record their selections on pieces of paper, how many digits does each piece of paper need to hold?

 

[Dale]

If there is a such thing as a real-valued measurement then bets can be placed the same way. E.g. if marking a piece of paper is real-valued then bets can be submitted as marked pieces of paper.

Anyway, you are now the third person that I have had to tell that I don’t find their arguments convincing. Again, I also have not thought about this carefully myself and don’t have contrary arguments that I find convincing yet. Until I have convinced myself, all I have is doubts which I am done arguing. Please don’t try to draw me in again. I will open a new thread when I am ready.

 

[PeterDonis]

Thread closed for moderation.

 

[PeterDonis]

After moderator review, the thread will remain closed.