Have I found a paradox, or is the universe digital? Or am I just plain wrong?

Question

If the universe is analog, there must exist an infinite number of positions. This raises an interesting question.

Let me boil it down to something familiar: a table and an ashtray. I'll let the ashtray be square, just for convenience.

In an analog universe, the table alone contains an infinite number of locations on its surface. No matter which two points you select, you'll always be able to point out a location between them. Like the way there are infinite rational numbers between any two rational numbers, unequal to each other.

We imagine the table is 50 x 50 cm, and we say the origin (0,0) is at the lower left. If I put the ashtray on the table, at a random position, what is the probability that it ends up with its lower left corner at the exact coordinate (10,10)?

According to other answers regarding infinity and probability I've found on the web, the chances of hitting a certain position out of a quantity containing an infinite number of positions, is zero.

The problem is that all positions on the table have the same probability. That means the probability for the ashtray to hit any position on the table is zero.

Nevertheless, obviously it is possible to place an ashtray on a table. So, in spite of a zero possiblity, one position is chosen anyway. This cannot be!

If the universe is digital (so a minimum distance exists no matter how small), there is no problem at all: the table contains a finite number of positions, all have the same probability greater than zero of being chosen at random, and one of them is selected when I set down the ashtray.

I remember Max Planck concludes something about the radiation of energy, because an object would radiate equal amounts of energy on all wavelengths, and since there are an infinite number of wavelengths, all objects would radiate infinite energy. This could not be true, so Planck concluded that the energy of atoms could only have discrete values.

So, does my example prove the universe to be digital? Or, is the universe analog, and this is a paradox? Or does my logic fail somewhere?

Answer 1

You have re-found a well-known paradox -- Zeno's.

This is just a compact version of Zeno's paradox. Instead of time and space, what you are dividing against one another are two spaces. The corner has zero probability of ending up anywhere, the same way that the point where Achilles overtakes the tortoise has zero breadth.

In fact, in modern statistical theory the probability of any given point sample from any distribution is almost always zero. But that works because modern probability theory already presumes a solution to Zeno's paradox.

The normal way out of Zeno's model nowadays is Newton's integral calculus. From that point of view, any infinitesimal sliver of anything still has zero area, but these zero contributions do add up to nonzero results when accumulated over continuous ranges.

From modern analysis point of view, this is just the nature of space. Space is continuous so we can always extrapolate values at points from whatever attributes are shared by every neighborhood around them, provided something that consistent exists. We can compute the sums of areas made up of idealized slivers from approximations that converge to the right shapes.

From the point of view of Nonstandard Analysis, this works because a continuous range involves infinitely many points, and a zero times an infinity can be finite.

More functionally, for any proper measure there is a nonzero uncertainty built into the means of measurement. In order for Newtonian mechanics to really work for you, you need to give a tolerance for any measure, a zero tolerance often just automatically creates a misleading zero or infinite answer because nothing real is absolutely specific.

Answer 2

We run into essentially the same problem almost any time we try to combine the real numbers as described by mathematics with probability theory.

When applying probability theory to something like a coin flip, a die, or a deck of cards, we use what's known as a Probability Mass Function to assign a probability value to each possible result. What's the probability rolling this fair six-sided die will give 3? 1/6.

But as you noticed, this type of probability model doesn't work well when trying to work with real numbers, because there are just too many of them. Since there are more real numbers even within a min-to-max interval than there are natural numbers, if you try to assign a single non-negative probability to each number in the domain and require they add up to 1, almost all of them will need to be zero.

So instead, when dealing with a domain of real numbers, probability theory normally switches to using a Probability Density Function. This function assigns a real value to every possible outcome in the domain, but these values are not probabilities. Instead, to get a probability from a probability density function, you need to take the integral of the function from a start point to an end point, and this represents the probability the result will be between those points.

For a simple example, a process that generates a random real number x chosen out of the interval from 0 to 1 such that the likelihood is the same over the entire interval could be represented by the function p(x)=1. The probability of "x is between 0 and 1/4" is the integral from 0 to 1/4 of 1, which is 1/4. The probability of "x is exactly 1/4", if meaningful at all, would need to be the integral from 1/4 to 1/4 of 1, which is 0. So even though the probability of any particular exact value is zero, you still have a probability of 1 that the value is somewhere in the interval, and meaningful probabilities for any smaller pieces of the interval you care to calculate.

Real numbers tend to be useful in physics because generally speaking you can divide ordinary quantities of distance, time, mass, and energy into arbitrary intervals (and sometimes non-algebraic numbers like pi and e pop up in the useful models). Of course as modern science ran into observations now attributed to atoms and photons, and developed the Standard Model of particle physics, we needed new theories that no longer assume mass and energy are indivisible on the smallest (non-ordinary) scales. Could the same happen for distance and time? Who knows?

By the way, probability density functions turned out to be rather useful in physics too: basic quantum mechanics describes position and momentum of an object in the form of complex-valued "wave functions" which obey Schrödinger's Equation. The squares of the absolute values of these wave functions are actually probability density functions for the variable being described. These wave functions do still assume real-valued position, time, and momentum.

Finally, some of the promising looking ideas about quantum gravity (including string theory) suggest there may in fact be a minimum or fundamental unit of space-time, somewhere around the scale of the Planck Length. But as far as I've heard, none of these are quite on the level of an established useful theory yet.

Now, all this goes to show that there are ways around the paradox, and ways to still describe our observed universe fairly well using both real numbers and probability. But that doesn't necessarily imply that's the true fundamental nature of the universe, or anything like that. It's also possible to construct working physical laws using just rational numbers, or numbers which are all multiples of some single chosen very small number. (But those systems may not be as easy to calculate with!) All any physical theory or law can claim is that it does well at explaining and predicting observations, and possibly covers more observations than other theories, and/or is simpler and easier to understand and use than other theories.

So when it comes to questions like: Does the universe truly work this way or that? or Does a "better" set of physical theories get us closer to understanding a true underlying nature of the universe? or Why can we predict that things will happen at all? ... we're into the interesting realm of metaphysics. But that's another story (and one I'm not so qualified to comment on).

Answer 3

Zeno's Paradox is not a paradox. It is an attack on loose thinking. By emphasising the infinite nature of one thing, and not mentioning the infinite nature of another, it confuses people into thinking something is impossible.

The emphasis, in Zeno's Paradox, is upon the infinite number of times a distance can be subdivided, giving the impression that it will take an infinite amount of time to cross those infinites number of subdivisions. It carefully ignores the fact that every time you halve the distance, you also halve the time it takes to cross that distance (at a given speed). It doesn't matter how finely you subdivide the distance if it always takes the same amount of time to cross all of the subdivisions.

So to the original question - how accurately can you measure the position of the ashtray? The chance of its corner being measured as being at any specified location is dependent on how accurately you can measure it. If you could only measure it to the nearest 10cm, and the table is 50cm square, then there are 25 possible locations, and (assuming random distribution), the probability is 1/25 (0.04). If you can measure to the nearest 1cm, then there are 2500 possible locations, and the probability is 0.0004. And so on. Do bear in mind that Heisenberg's principle puts a lower bound on how accurately you can measure its location (assuming a relatively stationary ashtray!).

In other words, the more accurately you can measure position, the more positions there are, and thus the tinier the probability that your object is occupying a specific one (again, assuming random distribution). The probability is never zero, but it gets very small (but so does the target area).

Another answer said pretty much the same thing by saying that you need to specify a tolerance for position.

In other words, your "paradox" is no more a paradox than Zeno's.

Answer 4

The problem is that probability 0 does not mean 'impossible'.

If you have someone flip coins forever, what is the probability that he will never encounter a head? Well, it's zero. But it's possible! In fact, every specific infinite sequence of heads and tails is infinitely improbable; that is, its probability is zero. Still, none is impossible: one of them indeed will happen.

In the same way, the ashtray will land on a point where it had zero probability to land, but that doesn't mean that it was impossible to land there. Indeed it does.

I think this is unrelated to quantum physics telling us distance is quantized. There is a minimum distance you can measure, but that's only a limit on how accurate our instruments can be. It's not like the world is really disposed on a square grid of 1 Planck length, at least to my understanding.

Answer 5

What you describe is a fundamental aspect of probability distributions (in Mathematics), i.e. the probability of each point tends to zero, which is why probabilities are calculated on intervals or areas (in case here is an introduction). The smaller the interval the smaller the probability. At the level of a point, it is considered zero. Nevertheless, you can sum it over an interval. Note that this is a mathematical construct.

The more general question is often posed as to whether the "real" universe is discrete (I believe it is what you meant as digital) or continous.

You will find here, a treatment of the exact argument you are making.

Briefly, the idea has long been that matter is composed of atoms i.e. indivisible particles. Physics adopted that idea, save that what it thought of atoms were in fact divisible in particles, which were then subdivisible in subparticles, etc.. We are now into a hypothesis of "strings", but to say whether are discrete or continuous, only students of those things would be able to argue about it (but since quantum mechanics relies on probability distributions, we might be back to square one).

There is also the question of whether space is itself actually discrete or continuous. Space and time are considered interconnected according to Einstein mechanics.

An important thing to remember is that mathematical constructs and physical models tell us a lot about the interrelations among objects, space and time in the physical universe, and about how larger objects can be decomposed into smaller objects. But it doesn't tell us about what these things are. For all we know, we could have a completely 'wrong' idea of the essence of things, but as long as the theory cannot be faulted and it is applicable, it is good enough.

In the end, it might be a philosophical question on which our science has no definitive answer (in that sense we might call it metaphysical).

Answer 6

The problem is that all positions on the table have the same probability. That means the probability for the ashtray to hit any position on the table is zero.

Not quite. The probability for the ashtray to hit any given position on the table is zero. But as you have observed, you can most certainly lay the arrow on the table and get a new position.

Nevertheless, obviously it is possible to place an ashtray on a table. So, in spite of a zero possibility, one position is chosen anyway. This cannot be!

Since you are not targetting a given position, there is no paradox. There is zero probability that you will pick up the ashtray and put it back down at the exact same given location, but you can certainly put down the ashtray and get a new position, just like you did the first time.

Answer 7

According to other answers regarding infinity and probability I've found on the web, the chances to hit a certain position out of a quantity containing an infinite number of positions, is zero.

I think it's sloppy speech about the word "infinity". Your statement is a "divide by zero" error, undefined, meaningless.

I learned in Maths that the correct way to use "infinity" is that it isn't a number, it's a limit ... or more specifically it's no limit, it's unbounded. The definition of "infinity" is "bigger than any number": for example if you pick any number (e.g. "100") I can find another number bigger than your number (e.g. "101"), therefore the number you picked isn't "infinity" ... no number that you can pick is "infinite".

The correct way to phrase your ashtray problem is something like,

• Start with a table
• Put a grid of size "n" on the table
• The number of grid cells is "n squared"
• Therefore the probability of randomly selecting a grid square is "one divided by n squared"
• This formula for probability is true "for all n, even as n tends towards infinity"
• But you can't talk about the probability "when n equals infinity", because infinity is't a number ... infinity is a synonym for "unbounded" or "without an upper bound".

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The logic of the paradox in the OP depends on "zero multiplied by infinity equals zero" (i.e. zero-probability-per-point multiplied by infinite-points equals a paradox).

You get that equation by doing a division-by-zero, "one divided by zero equals infinity" (i.e. finite-area-of-the-table divided by the size-of-a-zero-area point equals infinitely many points).

But "division by zero" is an undefined operation. Using it results in well-known, trivial (grade-school level), easily recogizable algebraic fallacies, for example as shown here: Division by zero -- Fallacies

It doesn't prove anything about the universe, it's pushing the numbering system until it stalls. You can divide one by n, where n is any non-zero number.

Answer 8

If we assume that space actually is analogue (and I'm not sure that that is what contemporary physics tells us), it follows that the probability of any point getting hit is indeed zero.

This, however, is as things should be: No matter how fine-grained your methods of analysis are, if space (or area) really consists of infinitely many infinitely small points, any method of determining the location of the point that got hit must have some sort of uncertainty to it: It will give you the location up to a certain number of decimals, and up to that number of decimals, points in space are discrete: If you measure precisely up to .01 mm, every cm² consists of exactly 100 different discrete locations.

If you do, for the sake of argument, invent a machine that gives you the location with all its (infinite, if space isn't digital) decimal places, nothing is gained, because you will die before you can read off that infinite amount of information, and humanity will go extinct, so there is no meaningful way to say that it has been measured to more than any arbitrary but finite amount of decimal places.

What that means is that we in fact only assume that any point got hit, because we can never find out which point it was. Therefore, to all intents and purposes, it is as if no single infinitely exact point got hit, but only an area of which we can say that a point inside it got hit. So even if space isn't digital, nothing follows from it.

Answer 9

The paradox you mention and Zeno's paradox rely on the irrationality of numbers, which humans invented. I argue, that this irrationality occurs as a result of our current irrationality. My counter example of this is that irrational numbers do not exist in computers. And according to some of the bright minds of today, they say the universe can be computed. So it just may be digital.

Answer 10

In complement to other answers, I'm wondering what conception of probabilities you have to assume for your paradox to go through.

The paradox seems applicable not only to discreteness, but to the idea that the world would have an infinite number of possible configurations more generally. So even if the world is discrete rather than continuous, it is enough that it would be infinite in size (and so could have an infinite number of possible state) for its actual state to have probability zero. But in such case one could be tempted to say that there's no contradiction in the idea that the universe would have a specific state among an infinity of possibilities.

If you assume ignorance probabilities, that would mean that the actual state of the world, or the table in your example, is "unbelievable". Now we only form beliefs on the basis of our observations and concepts so perhaps conditional probabilities would be more appropriate, but you could obtain the same paradox with conditional probabilities (given that our knowledge is finite). But one could object that what we are disposed to or have reasons to believe is irrelevant to the actual state of the world and that your reasoning is spurious. So you have no reasons to believe that the table is at any specific position, yet it is, but there's no contradiction here.

Assuming nomological probabilities instead of ignorance, the conclusion of the paradox would be that it's nomologically impossible that the universe or the table is in the state it is. But in the case of nomological probabilities, why assume an uniform distribution on possibilities? For example, if the universe is deterministic, there's probability one that it is in its current state given its past state and no paradox ensues, except for the initial conditions of the universe, but why assume that all initial conditions are equally probable? This seems quite metaphysical at this point.

Finally, note that in the case of quantum mechanics, nothing has a perfectly determinate position and probabilities are generally defined over quantified (discrete) possibilities so the paradox doesn't occur, even though space itself is continuous: this is because measurements would be discrete events and would never be of infinite precision (at least in some interpretations).

In any case you raise an interesting paradox having to do with the relation between mathematical formalism and reality. It seems that physics needs continuous space (real and complex numbers, differential calculus) to accurately describe reality, even though our measurements are always discrete, and even though infinities and continuity leads to paradoxes.

Answer 11

Right, I think I have an answer to this, but I can't find out how to write math in this, so bear with me. (And someone please tell me how to format math, can't find it in the formatting help.)

Anyway to the point, I don't necessarily think the chance is zero. Lets start with if you have 10 points then you have a 1 in 10 chance of hitting any individual point. The chance is the reciprocal fraction of the number of points, so 10/1 becomes 1/10. Now infinite points wouldn't technically make 0 chance, it would make the chance be 1/infinity. Now obviously measuring this chance is impossible, but it seems to be a non-zero chance.

If you have N points or N / 1 the chance of hitting any individual points is 1 / N

For any value N that is greater than 0 you will have a chance greater than zero. You can raise the value of N infinitely while still having a chance to hit a specific point.

That or I really need to work on my math skills.