## Do you believe there can be an Actual Infinite

• 2.1k
Following is a restoration of the opening post that set off this thread:
People are in love with the magic of infinity. It’s ideal for research as it generates all sorts of mad ideas. Anything that can happen will happen; an infinite number of times! Endless possibilities! But whilst infinity is fun I can’t help but think it is a source of confusion in many instances and is holding back scientific progress.

Many, many paradoxes and speculative theories disappear at a stroke if we are simply willing to acknowledge the actually infinite cannot exist.

There are strong arguments against the actually infinite. For example, no matter how many times you add one you never reach infinity so actual infinity is impossible to achieve.

On paradoxes, for example, Zeno’s paradoxes, there is a very simple solution if you take the view that the actually infinite is impossible:

- Assume time is continuous
- Examine any system over a fixed time period
- Then the system goes through an actually infinite number of states in a finite period
- Actually infinite is impossible so reductio ad absurdum time is discrete
- Time is discrete so Archilles only has to cover a finite number of steps to reach the tortoise

In the physical sciences we used to be quite strict with infinity:

- used only as approximation of very large/small
- indicate of logic error when occurs elsewhere
- even in maths infinity = divide by zero = logic error

But I guess belief in the actually infinite keeps cosmologists in a job for an actually infinite period of time...
— Devans99
• 659
I voted "no." One of the problems is that "infinity" has slightly different meanings depending on the context. In terms of future time being "infinite", it connotes an unending temporal sequence of events - so it entails incompleteness. This is the nature of the potential infiniite. Another sort of infinity is the number of real numbers between 0 and 1: this is a conceptual infinity, uncountable - and it may not correspond to anything in the real world (is there truly such thing as "length" below a Planck unit?)
• 2.6k
Yes, in the sense of the infinite as answer to (consequence of) certain questions. I understand a no vote as the view that aspects of the infinite are not realizable under certain conditions - which is unarguable. But that clearly doesn't cover all conditions.
• 1.9k
Yes, to clarify I mean actual infinity in the context of the physical sciences. I.e. when it is applied to real world entities.
• 1.4k
I mean actual infinity in the context of the physical sciences. I.e. when it is applied to real world entities.
Then I have to say no. A lot of things can be modelled by using infinity in some way or another, but I don't see it in reality as applied to real world entities. Modeling is one thing, reality another. The number 3 or pi don't physically exist, even they are extremely useful in modeling reality.

As a mathematical entity the question is totally different.
• 1.9k
Good for modelling and approximations I agree. It’s the use of actual infinity to stand real world quantities that disturbs me. Statements like space or time maybe actually infinite... nonsense.

Certainly has lots of uses in maths but it’s interesting to note that Actual Infinity is not constructable geometrically or otherwise mathematically (in the limit is not the same as actual infinity). So even in maths, actual infinity is unrealisable.
• 1.4k
. It’s the use of actual infinity to stand real world quantities that disturbs me. Statements like space or time maybe actually infinite... nonsense.
What disturbs me is when people just mix math with reality in general and forget the model part.

This happens especially in physics, but can happen also in let's say economics. It's when people take the idea "Math is the language of science" a bit too far and literally replace the real World with math.
• 4.5k
Statements like space or time maybe actually infinite... nonsense.

I think “nonsense” is too strong. But there is certainly a real metaphysical question here. Our mathematical models lead to rather glib beliefs about infinity. And our current physics makes it a much more complex and interesting issue.

Principally I’m talking about the discovery that reality is quantum and so individuation is contextual. To arrive at some located number of entities, you have an emergent limit on how many can exist for a given material extent. This is the holographic bound on information or the light cone principle.

So, in practice, space and time are materially constrained. They may be modelled as infinite dimensions, unlimited. Yet once matter and energy are added to the picture, then things look actually quite different. You have an ontology which is about finitude emerging from ambiguity, rather than one which presumes an underlying continuity that can be infinitely divided - at no physical cost - as is the case with the ur-model of the mathematical number line.

So infinity is a mathematically revered notion. Folk like to apply it to metaphysics as if it were true. But modern physics points to a very different ontology of actualisation now. The maths is out of date.
• 1.9k
I don’t have your knowledge of quantum mechanics admittedly but I’m sticking with my claim - infinite space and time is nonsense:

- Empty space has vacuum or dark energy associated with it. Total energy content of universe has to be finite. Space is finite.
- If time reaches back infinitely then it’s impossible to reach today. Time is finite.
• 763
in many instances and is holding back scientific progress.

Name three instances.

Many, many paradoxes and speculative theories disappear at a stroke if we are simply willing to acknowledge the actually infinite cannot exist.

There are strong arguments against the actually infinite. For example, no matter how many times you add one you never reach infinity so actual infinity is impossible to achieve.

On paradoxes, for example, Zeno’s paradoxes, there is a very simple solution if you take the view that the actually infinite is impossible:

Or we can just use calculus, which requires multiple levels of infinity and resolves such apparent paradoxes. I mean sure, dropping infinity is a "simple" solution if you completely ignore how much of mathematics crucially requires infinity. But hey, what do I know... (I know this is snarky but come on, do you really think it's a "very simple solution" to just drop an indispensable concept???)

In the physical sciences we used to be quite strict with infinity:

- used only as approximation of very large/small
- indicate of logic error when occurs elsewhere
- even in maths infinity = divide by zero = logic error

Division by zero is not infinity, it's undefined (in most math systems, some do give it a result). According to Relativity, space is a continuum so it is infinitely divisible. No serious physicist is going around saying Relativity is fundamentally incoherent.

But I guess belief in the actually infinite keeps cosmologists in a job for an actually infinite period of time...

Oh yea, I guess mathematicians and non-cosmologist physicists don't use infinities at all. Nope, the natural numbers are finite, as are the reals and so on. Guess space is actually finitely divisible, sorry Einstein.

And before you drum up a response like "I'm not disputing its use in maths but..." consider this. We take what mathematicians and logicians say seriously when we adopt the formal systems they create. That means that to use such systems we are committing ourselves to a particular kind of metaphysics. If you accept standard mathematics you cannot possibly claim that actual infinities are impossible in virtue of a contradiction. You might say that not every aspect of our particular universe can be infinitized, but there's no argument that the concept itself precludes instantiation in the world. After all, that would mean either:

-Infinite collections are a category mistake: False because collections are the very things which can be infinite
or
-Infinity entails a contradiction: False because we know that standard math systems are consistent (or rather, no contradiction is yet provable in them).

So since neither of those has any merit, there's no argument against infinities on the basis of the concept alone. You either accept the well-studied math systems or you don't, but you can't use them and yet deny the very assumptions they're built on. That's complete crankery.
• 1.9k

Three instances where infinity clouds the issue:

- The big rip
- Black holes and the singularity
- The debate over continuous/discrete

I’m not suggesting dropping infinity from maths. It’s fine with the limit concept and set concept. I’m suggesting dropping actual infinity in physics and metaphysics when used as the value for real world quantities.

Relativity is an approximation of reality not reality - We don’t know for sure if space is continuous. Anyway continuous space is a Potential Infinity whereas I’m talking about Actual Infinity.

There is an argument that the natural numbers are only potentially infinite - we have used finitely many of them so far and that will remain the case.

I’m hardly the only person to have a problem with actual infinity. The great German mathematician Hilbert posed his Infinite Hotel paradox. Just one of many paradoxes that stem from actual infinity.

I know it’s possible to construct consistent mathematical systems around infinity; that is not what I’m objecting to. I’m objecting to the use of actual infinity in physical sciences.
• 12
- Empty space has vacuum or dark energy associated with it. Total energy content of universe has to be finite. Space is finite.
- If time reaches back infinitely then it’s impossible to reach today. Time is finite.

But if one removed all phyiscal mass and energy, both the visible and dark, wouldn't empty space simply be infinite vacuum? Would you then believe that if space is finite, expansion of the universe will hit the "edges" of space one day? Or do you believe that the expansion of the physical universe determines the size of the space it's in, which is finite?

If time reached back infinitely, it would reach the future infinitely as well? I'd assume being infinite would imply a two way relation, past and future? otherwise you'd make the infinite time finite by assuming that it reaches back infinitely, it cannot reach today. Understanding time as a simple line, the line would be clearly infinite pointing left as it would be pointing right. If one marked a point P on the line, then the line going from point P and left would be infinite paralell to "if time reaches back infinitely". However, we already made ourselves a starting point and hence we've already made finite what is infinite. An infinite timeline would have no starting point nor finishing point. you could mark a point P anywhere on the time line, and it would have to be infinitely long on both ends.

However, I do not believe there is anything physical that is quantifiably infinite in actuality/reality.
• 700
It depends on what kind of infinity you are thinking of. Actual numbers are the result of counting operations, which take time and so are necessarily actually finite, but potentially infinite.

On the other hand lines of concurrent explanations necessarily terminate in an self-explaining, and so actually infinite, being. However this infinity is not numerical, but the denial of limitations on the power to act.
• 1.9k

“But if one removed all phyiscal mass and energy, both the visible and dark, wouldn't empty space simply be infinite vacuum?”

Dark energy maybe inherent to space so it’s not possible to remove it.

“Would you then believe that if space is finite, expansion of the universe will hit the "edges" of space one day? Or do you believe that the expansion of the physical universe determines the size of the space it's in, which is finite?”

We live in 4D space time. I’m not sure but maybe as you get closer and closer to the edge of the universe, time slows down and then reaches a stop? So you can never pass the boundaries as time does not exist outside.

‘If time reached back infinitely, it would reach the future infinitely as well?’

Time does not reach back infinity far that would be an actual infinity. Is time future infinite, that is only a potential infinity in a 3D + Time world but it is an actual infinity if you buy Einstein. I buy Einstein so time has an end too for me.
• 1.9k
So you believe God is infinite? The problem with that is you are either infinite or you exist - not both at the same time.

A finite god is much more naturalistic...
• 656
People are is love with the magic of infinity. It’s ideal for research as it generates all sorts of mad ideas. Anything that can happen will happen; an infinite number of times! Endless possibilities! But whilst infinity is fun I can’t help but think it is a source of confusion in many instances and is holding back scientific progress.

Max Tegmark would agree with you.

Not only do we lack evidence for the infinite but we don’t need the infinite to do physics. Our best computer simulations, accurately describing everything from the formation of galaxies to tomorrow’s weather to the masses of elementary particles, use only finite computer resources by treating everything as finite. So if we can do without infinity to figure out what happens next, surely nature can, too—in a way that’s more deep and elegant than the hacks we use for our computer simulations.

Our challenge as physicists is to discover this elegant way and the infinity-free equations describing it—the true laws of physics. To start this search in earnest, we need to question infinity. I’m betting that we also need to let go of it.
• 700
I favor sound logic over naturalist belief.
• 1.9k
• 1.9k
I can sort of believe in an omnipotent god thanks to the wonder of the simulation hypothesis (https://en.wikipedia.org/wiki/Simulation_hypothesis), but in no way can I believe in an actually infinite god and I see nothing in your argument to prove god is infinite.
• 700
Do you agree that every line (regress) of concurrent causality must terminate in a self-explaining cause? If you do, then God has an unlimited capacity to perform any possible act.
• 1.9k

“Do you agree that every line (regress) of concurrent causality must terminate in a self-explaining cause?”

- the prime mover? Yes I mainly buy it.

“If you do, then God has an unlimited capacity to perform any possible act.”
I
- you’ve made a big jump there. God creating the universe is not demonstrative of omnipotence.
• 700
Creating the universe is not the critical point. The critical point is being the end of the line of explanation. Since everything has an adequate explanation, so must the end so must the end of the line; however, because it is the end of the line, its explanation cannot be something else. If it were, we would not be at the end of the line. So, the end of the line must be self explaining. Since beings explain/cause other things in light of what they are, what a self-explaining being is, must entail that it is.

Based on a statement of Plato in the Sophist, let me suggest that if something can act in any way, it exists. In other words, existence is the unspecified capacity to act. Correlatively, what something is (its essence) is convertible with what it can do. If a being can do everything a duck can do, and nothing a duck cannot do, then it is a duck. So, we may think of the essence of a being as the specification of its possible acts.

So, if a being is to entail its own existence, the specification of its possible acts must entail the unspecified ability to act. Clearly, this is impossible if its specification limits the being's ability to act in any way. Thus, to be self-explaining, a being must have an unlimited capacity to act.
• 1.9k
The unspecified ability to act is not synonymous in any way with omnipotence or infinite extent. We have the unspecified ability to act and we are not god.
• 4.5k
We take what mathematicians and logicians say seriously when we adopt the formal systems they create. That means that to use such systems we are committing ourselves to a particular kind of metaphysics. If you accept standard mathematics you cannot possibly claim that actual infinities are impossible in virtue of a contradiction. You might say that not every aspect of our particular universe can be infinitized, but there's no argument that the concept itself precludes instantiation in the world.

There is a big difference adopting the maths because it is a useful model and accepting it as the actual metaphysics. And it should be telling that the central problems of modern physics/cosmology revolve around finding ways to avoid the mathematical infinities, or singularities, that are contained in the current best models.

That is why quantum physics has to be built on kluges like renormalisation that give a semi-arbitrary means of just cancelling away most of the infinite quantum contributions to bare particle properties. The formal maths returns the answer to any question as "the quantum corrections sum to infinity". And then the physicist says we will just introduce a cut off factor that cancels away all that gross excess and leaves us with the exact sums that matches observation.

So the infinity-generating maths can be tamed by introducing heuristic constraints. After that, the maths really works well. But there is then no particular reason why you would think the maths represents a good model of the actual metaphysics.

It is the same everywhere you look in the physics. Particles are explained by symmetry maths. But the maths is too perfect usually. It sums to zero. Some other factor has to be added to the story to explain why there is a faint asymmetry in the mix such that not everything cancels away, leaving nothing. Matter and anti-matter can't be perfectly symmetrical otherwise all of one would annihilate all of the other, leaving no mathematicians or physicists.

A Theory of Everything would aim to offer a completely mathematical description that did away with the various kluges that physics has been forced to develop to get rid of the pesky infinities and zeros. However my view is that this in turns requires a different maths of infinity. The metaphysics of the maths would be what has to give.

Reality is already telling us that now. :)
• 21
Time is nothing more than measure of co-ocurences of events with human–created time measure, which is clock. Time assigned to Universe has different flow, it might as well not exist, because of the absence of evaluator of the co-ocurences. Unverse works as unconscious machine, that removes necessity to count or regard time as important. When time is absent who is to say if Universe is eternal or not? Who is to say that eterniry is not equal to a one second in Universal terms? It is not equal to anything. No humanly constracted system of time apply to Universe. In the same way measure of size is not iportant to Universe, and is not measured and never will be. I see it as neverending supply of life and everything thta life needs. However, human species are deeply self depreciating in thinking that Universe is not generous, and only because of the traiss that humans display and limitation ( may be, not sure) of Earth's resources. We soak in temporary shortage of Earth's abilities to satisfy our never ending demands upon it, and because of it we do not see how generous is Universe, it supplies us with eternal recreation of life, by redesigning of self, it runs on neverending potential.
• 2.1k
Are you aware that denying the actual infinite involves committing to one or the other of the following two propositions?

1. If we travelled far enough through the universe in a straight line we'd end up back where we started

2. The universe has a boundary. In that case, as Aristotle asked, what happens if we go to the boundary and poke a spear through it?

Personally, I find an infinite universe more plausible than either of those.
• 4.5k
But if one removed all phyiscal mass and energy, both the visible and dark, wouldn't empty space simply be infinite vacuum?

No. An empty space is simply a matter field in its lowest possible energy state. This is now a central fact of cosmological thinking. It is what the holographic universe is all about.

So an empty space is still full of the black body quantum radiation that is "generated" by its own event horizons. The universe at its heat death would still radiate internally with a Planck scale jitter - a photon gas. The photons would be as cold as possible - within Planck reach of zero degrees - and so have wavelengths about the size of visible universe. So about 32 billion lightyears in length. Unbelievably weak. Yet spacetime would always have this ineradicable material content there as part of what it is.

Of course, mathematically you could imagine actually empty spaces. Maths does that routinely. In fact it is the basis of how it goes about the job of conceiving of spaces - as devoid of material content.

But physics tells us that spacetime and energy content are connected at the hip. Matter tells spacetime how to curve and spacetime tells matter how to move, as Wheeler famously put it. They are two faces of the one reality.

And so the job for maths is to catch up with reality if it can. At the moment, the existence of this connection is one of the kluges that have to be inserted by hand to make the cosmology work as a scientific model. It would be the big advance to make it emerge as a mathematical prediction.

Why is there this Planckscale cut-off that prevents the universe either being infinitely energy dense (as the quantum corrections to any material particle says it should) or, alternatively, completely empty, as would be the case if the quantum jitter of spacetime itself only had a zero or infinitesimal contribution to make?
• 2.7k
Imagine a spaceship that has an accurate odometer. We set the spaceship in motion and it travels across the universe. The reading on the odometer will always be finite no matter how far the ship goes.

This story denies actual infinity without suggesting that there is some boundary to space.
• 763
I’m not suggesting dropping infinity from maths. It’s fine with the limit concept and set concept. I’m suggesting dropping actual infinity in physics and metaphysics when used as the value for real world quantities.

I know what you're suggesting, I even precluded it. The whole point is you cannot use standard maths and make this argument. Every science uses some math or other. But whatever mathematical formalism is used, they also make recourse to infinity. There's no coherent way of making sense of this if you then drop infinity in your metaphysics because you use the math and treating as true, you're already accepting it. It's ridiculous.

Relativity is an approximation of reality not reality - We don’t know for sure if space is continuous. Anyway continuous space is a Potential Infinity whereas I’m talking about Actual Infinity.

You either think it's true or not. As fundamental assumption of relativity is the use of a Non-Euclidean geometry with an infinitely divisible space (although even Euclidean geometry posits an infinite plane). The theory requires an infinitely divisible spacial structure, this isn't something you can gesture at as an approximation that we need not think about. That's an actual infinity; between any two points of space there's more space. That's not potential at all.

There is an argument that the natural numbers are only potentially infinite - we have used finitely many of them so far and that will remain the case.

That's malarkey. The natural numbers can be put into a one-to-one correspondence with a proper subset of itself. That makes it infinite. The use of the natural numbers has no bearing on the cardinality of the set.

’m hardly the only person to have a problem with actual infinity. The great German mathematician Hilbert posed his Infinite Hotel paradox. Just one of many paradoxes that stem from actual infinity

Hilbert's Hotel is not a paradox in the literal sense. It makes perfect mathematical sense. The only issue is that people import their naive views about infinity when they think about it and it drives them off course. Hilbert's Hotel contains no contradictions, ergo it's only a paradox in the sense of it having a weird conclusion, it is provably the case that infinity results in no contradictions.

I know it’s possible to construct consistent mathematical systems around infinity; that is not what I’m objecting to. I’m objecting to the use of actual infinity in physical sciences.

My whole point has been to ask, On what basis? As there is no contradiction, your only recourse (as I said) was to say there's some category mistake. But collections of things are exactly what we know, mathematically, can be infinite. You haven't given the actual reason to accept what you're saying is anything more than a bias for a view you already held.
• 763
There is a big difference adopting the maths because it is a useful model and accepting it as the actual metaphysics. And it should be telling that the central problems of modern physics/cosmology revolve around finding ways to avoid the mathematical infinities, or singularities, that are contained in the current best models.

If you accept nearly any mathematical system, you're going to assume some infinity or other. If you dont, you're either an ultrafinitist (who nearly all mathematicians see as borderline cranks) or you have some reason why you think that in *particular* instances they ought not be used. The resolution of singularities is in part due to the precedence of them turning out to be the result of mistakes in our models. We still use infinities elsewhere. From the continuous nature of space, to the recourse to infinitesimals (which aren't even allowed in classical math, but physics tends to be braver in pioneering the use non-standard maths), infinities are by no means barred from the metaphysical assumptions made in science, and certainly not as just a useful thing assumed for convenience (no more than other areas of math). It has to be treated carefully, because it doesn't always useful results (hence renormalization).

It's not even that i deny renormalization is used just so I can defend infinity, I just see the wholesale denial of infinity as applied to reality to often require dropping normal mathematics without a clear reason.
• 1.9k
Time is nothing more than measure of co-ocurences of events with human–created time measure, which is clock... Unverse works as unconscious machine, that removes necessity to count or regard time as important.

The universal speed limit - the speed of light - is defined in terms of time (speed=distance/time) and is absolutely fundamental to the universe. Don’t see how you can claim time is unimportant or transitory...
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