Exploring an Infinite Ocean: Questions & Answers

[LightningInAJar]

Summary:: Ideas concerning an ocean without limit.

I asked this question in a philosophy forum and got quite a bit of feedback but it didn't quite answer my initial question.

If there was an infinite ocean and I scooped a cup of water out of it, was anything actually taken/lost from the ocean and/or gained? I am assume the ocean is a finite depth but infinite in all other directions so there is a place for that cup of water to go.

And also when the water is taken (if we can agree on that much) and ripples are created from the point of removal will there be entropy from this event? Won't the ripples shrink down below a Planck length before it reaches the edges which don't exist?

 

[hilbert2]

This is not a scientific question. How would an infinite ocean not collapse gravitationally due to its own mass?

 

[anuttarasammyak]

If your point is infinity not only in physics but in logic or mathematics you may be able to find some "paradox" discussions.
e.g. {1,2,3,4,... } and its subset {2,4,6,8,...} have the same size.

 

[hmmm27]

Or, if the philosopher's forum's answers sucked too much...

re: Infinite ocean : there's not much in the way of infinity metrics regarding salt water, so you get to choose. Personally, I'd tend towards keeping it at "infinity minus a cupful" until inconvenient.

But, there's many different forms of "infinity". Currently, another thread is arguing math using at least three different kinds... some in combination (I think).

If you've a planet completely covered in water, then the ocean is infinite because no matter how far in which direction at what speed you row, you'll never reach a boundary. If you remove a cupful from the planet, then the radius will decrease insignificantly, as will the distance to the horizon. But, until you remove many many cups of water, the ocean will still be "infinite". [edit: I stand corrected]

If salt water made up every nook and cranny of the universe, spacetime, multiverse, etc. etc. etc. pulling out all the stops, then - black holes aside - your question is moot, because you couldn't remove a cupful : nowhere to remove it to.

A (capital P) Planck length is the (current) smallest base unit of measurement of length. Since trying to find a smaller base unit using (advanced forms of) current technology would create black holes (gotta love Wikipedia, sometimes), it might be awhile before we get smaller.

Depending on who you ask a wave would either approach zero while approaching infinity(math), decompose into heat long before "infinity"(physics), and whatever QM has to say about it.

The forum has a Search function.

 

[jbriggs444]

If you don't idealize the situation, then it's an engineering problem, not a philosophical one.
If you do idealize the situation, then it's a mathematical problem, not a philosophical one.
Physics does not figure in much other than introducing practical impossibility issues.

The question of "whether anything was taken/lost" seems to be primarily a linguistic one, not a philosophical one. Can you state clearly what you mean by the question?

With my mathematician's hat on, being familiar with Hilbert's hotel and taking "whether anything was taken/lost" to be asking whether what was left behind is equivalent to what was there originally I would answer "no, nothing was lost".

 

[Grasshopper]

If you remove a liter of water, the liter next to it moves to fill it up, then the liter next to that fills that hole up, and after the nth liter of water moves to fill the n-1 hole, the n+1 hole moves to fill the nth hole, and then the problem becomes “is there a largest integer,” does it not? In which case the answer is no, and the water is still infinite.

But is that the same thing as saying nothing has changed? Everything has shifted one spot. But then again, it’s all the same anyway.

 

[Hornbein]

Summary:: Ideas concerning an ocean without limit.

I asked this question in a philosophy forum and got quite a bit of feedback but it didn't quite answer my initial question.

If there was an infinite ocean and I scooped a cup of water out of it, was anything actually taken/lost from the ocean and/or gained? I am assume the ocean is a finite depth but infinite in all other directions so there is a place for that cup of water to go.

Sure. A cupful of water has been taken from the ocean. The remainder remains infinite. Unless you remove an infinite amount of water, the depth remains unchanged. Even if you somehow remove an infinite amount, the depth might remain unchanged. It depends.

And also when the water is taken (if we can agree on that much) and ripples are created from the point of removal will there be entropy from this event? Won't the ripples shrink down below a Planck length before it reaches the edges which don't exist?

Eventually the ripples will be indistinguishable from heat. The ocean is then out of thermal equilibrium to some minuscule degree.

Now to respond to other posts. Water is made of molecules. The molecules are countable. So the ocean has a countable infinity of atoms. (Real numbers don't really exist, so I doubt that uncountable infinities physically exist.)

It is believed that our Universe is infinite and homogenous. So it contains both an infinite amount of matter and an infinite amount of space. Nevertheless we are able to measure the density of said matter thusly determining the ratio between these infinities. If an infinite ocean existed then it could very well be a fraction of the total volume, even a very small fraction. As long as this fraction is greater than zero then the ocean is infinite.

Let's say that the ocean is a huge tube of water molecules. I can go to one end of that tube but the tube goes off endlessly into infinity with constant radius. In this case the fraction of the volume of the Universe is zero. By playing games with the distribution in space, making a non-homogenous universe, you could have infinitely many such tubes but their fraction of the volume of space is still zero.

Let's say that the ocean starts out having an arbitrary shape and begins to collapse gravitationally. It will never become a sphere, because the collapse would take an infinitely long time. If it separates into an infinite number of pieces then it is possible that that an infinite number of those pieces are also infinite.

Let's say the ocean started out as a sphere. It must have an infinite radius. (Definitely a degenerate sphere. We could equally well call it an infinite cube.) Nevertheless this water does not necessarily fill all space. It's surface would be an infinite plane. In this case we could never say what percentage of the Universe it occupies. A submarine would never reach the other side unless it travels infinitely fast. But that would unphysical :-).

 

[jbriggs444]

so I doubt that uncountable infinities physically exist

My opinion is that by the time you've defined what it means to "physically exist" sufficiently that the question is answerable, the answer will turn out to be physically irrelevant.

 

[DaveC426913]

If you've a planet completely covered in water, then the ocean is infinite because no matter how far in which direction at what speed you row, you'll never reach a boundary.

No. This state is known as 'finite yet unbounded'.

 

[DaveC426913]

If you remove a liter of water, the liter next to it moves to fill it up, then the liter next to that fills that hole up, and after the nth liter of water moves to fill the n-1 hole, the n+1 hole moves to fill the nth hole, and then the problem becomes “is there a largest integer,” does it not? In which case the answer is no, and the water is still infinite.

But is that the same thing as saying nothing has changed? Everything has shifted one spot. But then again, it’s all the same anyway.

Indeed. With a few tweaks, The Infinite Ocean Paradox can be reduced to Hilbert's Grand Hotel Paradox, which has already been well-documented.

And that's the beauty of mathematics, is it not? Take a problem without a solution and reduce it to a problem that does have a solution.

 

[jbriggs444]

Indeed. With a few tweaks, The Infinite Ocean Paradox can be reduced to Hilbert's Grand Hotel Paradox, which has already been well-documented.

And that's the beauty of mathematics, is it not? Take a problem without a solution and reduce it to a problem that does have a solution.

Old but good. Mangled a bit to keep the length down...

An engineer's bed is on fire and he extinguishes it with his half full pitcher of water. The bed of his mathematician room-mate is also on fire. The mathematician drinks half of his full pitcher of water and, satisfied that a solution exists, goes to sleep.

 

[DaveC426913]

(Thank you BillTre for giving me my 3,000th 'Like', awarding me the coveted PF Einstein Trophy.)

 

[DaveC426913]

Thank you BillTre for ruining my perfect 3,000th 'Like' score.

 

[Grasshopper]

Old but good. Mangled a bit to keep the length down...

An engineer's bed is on fire and he extinguishes it with his half full pitcher of water. The bed of his mathematician room-mate is also on fire. The mathematician drinks half of his full pitcher of water and, satisfied that a solution exists, goes to sleep.

Doesn’t the physicist approximate the fire as a spherical cow and calculate the approximate amount of water needed to put out a fire in the shape of a spherical cow, or something to that effect?

 

[LightningInAJar]

Wouldn't any type of "magic" need to exist to accommodate an infinite ocean? For example the idea of two unmovable objects heading at one another would apparently need to phaze through one another in order for that situation to logically work. Even though we know of nothing that does that. So maybe if an infinite ocean existed and we took a cup of water it wouldn't be like Hilbert's hotel. The water could maybe double itself both wear it was taken from and into the cup? So no ripples at all occurred? Almost like an event occurring without forward or backward passage of time.

 

[jbriggs444]

Wouldn't any type of "magic" need to exist to accommodate an infinite ocean?

You can imagine taking a cup of water from a bathtub, right?
And a cup of water from a swimming pool?
And a cup of water from a small lake?
Or a cup of water from an ocean?

In each case we have a transient local disturbance that eventually dissipates to become small enough to ignore. I believe that it is this vision that we are invited to share. Nothing terribly magical. No two things occupying the same place at the same time. Just a cup of water drawn from an unbounded ocean and enough time for the ripples to become ignorable.

Certainly, one can quibble about it taking more than any finite time for the ocean to perfectly re-equilibriate (the speed of a water wave being finite).

 

[russ_watters]

It annoys me that philosophers would ask this question.

 

[LightningInAJar]

It annoys me that philosophers would ask this question.

Why? Don't some in physics think the universe may be infinite? Isn't it important to know what we’re talking about?

 

[anuttarasammyak]

The universe may well be infinite but only finite part of it would interact with us.
ref. https://en.wikipedia.org/wiki/Cosmological_horizon
I hope I said is not a digression.

 

[Hornbein]

Why? Don't some in physics think the universe may be infinite? Isn't it important to know what we’re talking about?

Yeah. The consensus is that our universe is infinite, so the discussion has real-world pedogogical value.

 

[Jarvis323]

Water is made of molecules. The molecules are countable. So the ocean has a countable infinity of atoms. (Real numbers don't really exist, so I doubt that uncountable infinities physically exist.)

The following is just a bunch of thoughts I've had about it. I hope it doesn't sound too out there.

I think it's important to be clear in how we assign existence to numbers in general. For example, does the number 9 exist, and if so, in what form? Well, it exists as an abstract concept. And any instance where that concept exists physically is presumably through some physically embedded information that encodes it. But encodings are observer/system dependent also. There are no atomic 9's floating around in space-time. What we know of and see is that some physical system acts on our instruments and or our senses and then our minds through some chain of events, and the final result is a mental cognition of something classified a 9 with some context attached. This is the case even when simply doing pure mathematics I think. In all cases, 9 is not what it seems like to us (at least me), and existence, even of abstract mathematical objects might not be what it seems at the metaphysical level.

Even though 9's aren't floating around as raw atomic things, if you hypothetically had some kind of system for unambiguously identifying some underlying configuration of something, through the chain of events that eventually reaches your mind, as a one to one representation of a natural number, then maybe you could say some microscopic physical representation of the number 9 exists, with respect to that system.

When it comes to any computable number, including numbers like $\pi, \sqrt{2}$, those numbers could be represented by a finite discrete configuration interpreted through some such system. As an example, we can take the set of all Turing machines, and definitely one of them prints the digits of $\pi$, and there is one that prints every other computable number too (although there will be multiple different ones that print the same numbers), hmm?

But it would seem that such a system to identify an uncomputable number would not be accessible to beings like us. At least that is what the Church-Turing thesis seems to imply (although that is not proven) depending on how we define computable and what consciousness is. Even our models of quantum and analog computing are no exception. But we cannot rule out some form of hyper-computation made possible through something we don't know about. We can say that if the universe is doing hyper-computing, nobody seems to know how that might be exactly. If the universe isn't doing hyper-computing then I suppose we can say that it is not computing uncomputable numbers.

But what would that mean for the universe to compute an uncomputable number anyways? All such numbers are infinite. So whatever sense you can think of it being computed, maybe it is through some kind of 1d signal through some kind of temporal fluctuation, or maybe it's somehow producing some infinite configuration that exists all at once and represents the number. In the temporal case, it's hard to fathom. The number surely has a beginning. So do you consider infinite extension of time in both dimensions? Or maybe just infinite extension of time in one direction. If that is the case, you could say some finite time has passed at any moment, if you haven't produced infinitely many fluctuations in that finite amount of time, then the number isn't finished, and the number that has been computed thus far is actually a prefix also of a computable number.

If there were infinitely many fluctuations over finite time or space, then you've basically got infinitely deep physical systems in terms of time and length scales occurring.

Now suppose we think of spacetime as a grid that is infinitely divisible, and fluctuations occur at the smallest scales, and all of the way up. We are now defining some fluctuation of an infinitely small point, and imagining whether those fluctuations across time or space somehow are printing uncomputable numbers. Now we want to figure out how they could print all real numbers. Well, first we want some hypothetical system to be able for an oracle to break down these local processes in ways that uniquely represent all real numbers at once. So what are the rules? Can you have a combination of spatial and temporal encoders? Can they overlap (e.g. share fluctuations)? If you allowed non-local patterns and overlap, then you could trivial say all real numbers are covered (like a crossword puzzle where the letters spelling words can be shared and don't need to be connected, all you need is all the letters and every word is on there). So maybe that's not allowed. Maybe overlap is allowed, but only locally (like a crossword puzzle). I don't know.

Maybe we can start with a model of something basic and study that. So we could suppose a scalar field on a grid with infinitely many points (like $\mathbb{R}^2$). Then imagine each point in that space has a scalar value that can fluctuate infinitely many times in a finite amount of time. But somehow those fluctuations are all part of a single system, with local causality, like a cellular automata. And now we can ask, could each of those points (which there are uncountabley many of) be a generator for a unique real number? If not, how about $\mathbb{R}^n$ for some $n$? Or do we need $\mathbb{R}^{\infty}$? Anyways, I don't know the answer, but maybe someone does?

Or maybe in such a space, without temporal evolution, could we have a crossword puzzle including all real numbers?

Or maybe the space can be a countably infinite grid with infinite dimensions instead and the numbers can be in some crossword puzzle in spatial form?

Anyways, maybe instead of thinking of the physical embedding of the numbers themselves, we want to consider a number conceptually as a potential property. For example, just having the points in some space like $\mathbb{R}^n$, so that we have an uncountable number of points. But an uncountable infinity of nothing isn't very satisfying.

That a single uncomputable number could exist in some physical form less trivial than that is hard to imagine, let alone an uncountable infinity of them.

I guess all of these questions are really just questions about math (some of them probably have answers I don't know about), since I've just tried to figure out some formal concepts that could be used to map our notion of existence to our concept of numbers. But I'm not sure how it could be any other way that we talk about anything and be precise without it ending up being an exercise of some sort of logic and mathematics.

 

[russ_watters]

Why? Don't some in physics think the universe may be infinite? Isn't it important to know what we’re talking about?

Because it isn't really a philosophy question, just a bad science question. It's as if they think that's what philosophy is!