Actual Infinity in Reality?

[LadyForCamus]

@steve,

I was a bit confused about this ‘no start’ as well, but I think the relevant distinction here is in how we are viewing time. If time actually flows from moment to moment the way we experience it, then I agree with you that if we assume infinite time, it seems like a logical impossibility to not ever start anywhere, and somehow still get somewhere. But, the point that Poly and Grand are making is that it could very well be true that time does not flow. That there is no, ‘first this one, then the next, and then the next, ect.) but rather, time is a static collection of an infinity of points in existence; past, present, and future. All points exist simultaneously, and every time slice represents a section, or point, in the ‘set’ of infinite time. If that is the case, then infinite time is very much like an infinite set of numbers in mathematics, and completely logically possible. You don’t have to start at a number and actually count one after the other in order to have a completed infinite set of numbers in mathematics. You couldn’t; isn’t that the point? Similarly, you don’t have to start at ‘event one’ in time and wait for the next one to happen in order to have a completed set of infinite time. All the points are already there.

My understanding is that B theory of time is an active competing hypothesis under serious consideration by the scientific community, and not without evidential support. If that is the case, then I have to agree with Poly and Grand that there is no logical contradiction here.

(I’m a dunce in the corner compared to the rest of you, so if I got something wrong above, please jump in and course-correct. Thanks!)

[SteveII]

[polymath257]

At least as far as I understand it, Einstein's views were not *exactly* in line with the B theory of time. But they certainly were not at all in line with the A theory. Let me explain.

The B theory seems to suggest that time is a universal, objective thing in a way that contradicts general relativity.  For example, if I measure the time between two events to be 5 seconds, so will everyone else. But that is NOT the case in relativity! It is quite possible in relativity for one observer to measure 5 seconds between events and another to measure 3 seconds between those same two events. From what I can see, the B theory isn't very comfortable with this aspect of reality.

General relativity, on the other hand, considers time and space together to be a single entity: spacetime, which has an objective reality. So, while distances in space and durations between events are not separately well-defined (they depends on the motion and position of the observers), a combination of them *is* objective and is agreed upon by all observers. In this system, time does NOT 'flow'. ALL of space and ALL of time are, together, the geometry of spacetime. Time, like space, simply exists. Neither is well-defined in and of itself. BOTH are required for a consistent system.

One of the issues here is that it is possible that one observer will see event A as happening before event B and another may see it the other way around. So the idea of causality needs to be somewhat modified: instead of causes only having to be in the past (which is observer dependent), they have to be in the *past light cone* (which all observers agree to). The past light cone is the collection of events in spacetime from which light could reach the event in question. Nothing outside of this light cone can be a cause for an event.

With this modification, however, causality *is* a central aspect of general relativity. But it isn't in the sense that time flows from one set of events to another. It is that the particular distribution of particles in spacetime has to obey certain rules that reflect this causality. The spacetime is *still* one complete entity, but the 'pictures' on that manifold are not completely random: they reflect constraints on the solutions of the basic dynamical equations.

---

OK, we have a good question here:

Is {1,2,3,4,5....} an actual infinity or a potential infinity?

Steve claims it is a potential infinity. I claim it is an actual infinity. The difference is that Steve seems to think of it as a process of creating that set and I see the set as simply there and our attempts to *list* it are doomed to failure.

But I can tell what is in and what is not in that set with perfect precision. For example, the number 59749873 is in that set and the number 82.389587 is not. That is *all* that is required to completely define the set: to know what is in and what is not. You don't have to list the elements, just know how to tell when something *is* an element.

So the set {1,2,3,4,5....} is an actually infinite set. It is the *complete* collection of counting numbers. Nothing is being added or taken away from it.

And we are NOT taking the succession of finite sets
{1}, {1,2}, {1,2,3}, {1,2,3,4},....
NONE of those is the set above. THAT I certainly agree with. At no finite stage in this progression do you get the set above. I agree with that also.

But that isn't the same question. The question is whether the completed set
{1,2,3,4,.....}
is an actually infinite set or not. Since it is well defined and since it is none of the finite sets above, it is.

[SteveII]

@steve,

I was a bit confused about this ‘no start’ as well, but I think the relevant distinction here is in how we are viewing time.  If time actually flows from moment to moment the way we experience it, then  I agree with you that if we assume infinite time, it seems like a logical impossibility to not ever start anywhere, and somehow still get somewhere.  But, the point that Poly and Grand are making is that it could very well be true that time does not flow.  That there is no, ‘first this one, then the next, and then the next, ect.) but rather, time is a static collection of an infinity of points in existence;  past, present, and future.  All points exist simultaneously, and every time slice represents a section, or point, in the ‘set’ of infinite time.  If that is the case, then infinite time is very much like an infinite set of numbers in mathematics, and completely logically possible.  You don’t have to start at a number and actually count one after the other in order to have a completed infinite set of numbers in mathematics.  You couldn’t; isn’t that the point?  Similarly, you don’t have to start at ‘event one’ in time and wait for the next one to happen in order to have a completed set of infinite time.  All the points are already there.

My understanding is that B theory of time is an active competing hypothesis under serious consideration by the scientific community, and not without evidential support.  If that is the case, then I have to agree with Poly and Grand that there is no logical contradiction here.

(I’m a dunce in the corner compared to the rest of you, so if I got something wrong above, please jump in and course-correct.  Thanks!)

You are not a dunce. I observed not too long ago that I didn't know anything about this subject and read quite a few articles. Even the process of this discussion fills in some things for me. 

First, you don't need time to "flow". You still have causal connections from one time slice to another that are countable. A rotting apple has very predictable steps that are easy to plot. You still need a baby horse in a prior time slice to get an adult horse in another. These causal connections are what the whole enterprise of science is about. To say that the theory dismisses this fundamental aspect of reality is to not understand the theory. 

Second, I think it is quite a leap to say that every and all universes ever exist simultaneously in a giant spacetime manifold like the one that we experience in our universe. That just seems to be a convenient assertion. However, as we have been discussing, the B theory of time does not eliminate the aspect of reality that each event is inextricably linked in prior-to and after than relationships/connections. 

Third, the idea that all events that will ever happen are equally real simultaneously is NOT the same as infinite set theory in mathematics. The first is an actual infinite and the second is a potential infinite. To be clear, mathematicians are talking about potential infinities when the talk about sets. This is because one side is bounded and only the open side is potentially infinite. These two terms have very different definitions and cannot be used interchangeably or use one to prove the other.

[polymath257]

Third, the idea that all events that will ever happen are equally real simultaneously is NOT the same as infinite set theory in mathematics. The first is an actual infinite and the second is a potential infinite. To be clear, mathematicians are talking about potential infinities when the talk about sets. This is because one side is bounded and only the open side is potentially infinite. These two terms have very different definitions and cannot be used interchangeably or use one to prove the other.

This is just false. Mathematicians are talking about actual infinities. They even talk about different sizes of actual infinities (for example the difference between countably and uncountably infinite sets).

There is no requirement of having an 'open end' in the description of a set:

N={x: x is a natural number}

is a perfectly well defined, infinite set. No 'open side' and no 'bounded side'. The description in terms of a list,

N={1,2,3,4,...}

is more a convenience for those who cannot read mathematics than anything else.

What you seem to think is that the second is some sort of process. It isn't. It *does* appeal to your understanding to know what things are in the set and what are not. But the list itself is just a notational convention and nothing else. In the same way, we can define an uncountbaly infinite set

R={x: x is a real number}

or

[0,1]={x: x is a real number and 0<=x<=1}.

Both of these are uncountably infinite sets.

There is no such thing as a 'potentially infinite set' in math. Sets are either finite or infinite. In the latter case, they are actually infinite. They can be countably infinite or uncountably infinite. For the uncountbaly infinite sets, there are infinitely many different possible cardinalities (although very few are used in practice)

Look in *any* math book and you will find NO distinction made between potential and actual infinity. The reason? Those notions have been replaced. They are no longer used because they are confused and ill defined.

So, I will make a challenge. Look in *any* advanced level math book produced in the last half century. Find *any* reference that discusses *at all* the notion of 'potential infinity'. I challenge you to find a single source *in math* for your claims from the last 50 years (I'll even go 75 years).

[Jenny A]

I am not asking if there is a concept in mathematics that deals with infinity or if there exists sets with an infinite number of members (although you might use the concept in a larger argument). I am not asking if we can theoretically divide something an infinite amount of times (although you might use the concept in a larger argument). I am not asking about a potential infinite. 

I am asking about an actual infinite of something concrete (not abstract). Can it logically exist? Why or why not? 

No mention of God either. Philosophy subforum--let's stick with pure metaphysics.

Anything may exist as a posibility unless there is an inherent logical contradiction built into the possibility of it's existence. For example, invisible bachelors might exist (though not probably) without defying logic, as might 1000 year old bachlors. But married bachlors may not logically exist. What exactly is it about an infinity that makes you think it is logically impossible? And why do you think an infinite number of objects or time is illogical but an infinite division of an object or period of time is not?

[GrandizerII]

And in fact, Einstein himself was a B-theorist with regards to time, calling time (or rather, the flow of it) an illusion.

I'm not so sure. Perhaps he was, but relativity/time dilation is not B Theory.

He may not have called himself explicitly a B-theorist, but the things he did say about time clearly indicate that he was. The B-theory of time, by the way, is consistent with relativity and time dilation and such (see the video). It doesn't contradict it, as it doesn't strictly assert that every observer must observe the same moment in time, or anything like that.

From your same article:

According to Einstein, then, time is relative to the observer, and more specifically to the motion of that observer. This is not to say that time is in some way capricious or random in nature – it is still governed by the laws of physics and entirely predictable in its manifestations, it is just not absolute and universal as Newton thought (see the section on Absolute Time), and things are not quite as simple and straightforward as he had believed.

Yes, and? The B-theory of time goes a bit beyond the science, but it is the one philosophy of time that is consistent with modern science (unlike the A-theory of time, with which you would have to come up with heaps of unwarranted assumptions to get it to even remotely account for relativity and such).

Again, see the video that links relativity and time dilation and length contraction to the B-theory of time (and eternalism, which is virtually equivalent to the B-theory of time).

I understand B Theory of Time just fine. I have actually seen the video and read at least a dozen articles on it. My problem with you is that you think that the theory does away with causation. It does not. Even if you believe that all time slices (past, present, future) are equally real, they are all ordered by a causal connections in one direction. Fruit rots in one direction. Baby horses always preceded adult horses. These connections can be drawn with exact precision--that is literally what science does. There is nothing in B Theory that gets around this. You are adding a layer on top and calling it B Theory.

Not adding any layer on top of what is clearly an implication of the B-theory of time. Maybe it's the word "illusion" that puts you off, but if you want "causality" to still be implied by the B-theory of time, you do have to redefine the word to mean something that doesn't indicate the flow of time or actual change or actual motion. Fresh fruit does not cause rotten fruit in the sense that it morphs into it; the "antecedent" fresh fruit is still there and the "consequent" rotten fruit is there as well (just in different time moments). Baby horses do not actually eventually morph into adult horses; the baby horse still exists along with the adult version of it. And the key reason we have this one perceived direction happening in this one particular local universe is because of the increase of entropy from one state of this universe to the next, with entropy being at its lowest in the moment closest to the supposed "Big Bang singularity". But otherwise, as far as the laws of physics themselves are concerned, there is no direction of time. And even then, direction does not mean temporal causation anyway; both "cause" and "effect" simultaneously and eternally exist.

I see one of your problems. You said "causality is an illusion". You will not find that phrase anywhere in an article of B Theory. The typical description is the "flow of time is an illusion".

You can say "illusion" or you can say "derived/emergent", same thing, different words. Let's not get into an argument over semantics. Physicists don't geenrally agree with you that causality is a fundamental part of reality.

That is a metaphysical claim with very thin reasoning. We only know about the laws of physics in our universe. Extrapolating what appears to be a finite universe into a spacetime manifold for all universes ever is a stretch.

Then why are you acting like you know what would happen if there were multiple universes as opposed to just this universe? The view I hold to is nothing more but a logical extension of what is very well-accepted in both scientific and philosophical fields. Rather than denigrate me for having a "bold" view that, AFAIK, isn't contradicted by modern cosmology, why not just stick to challenging my view? After all, you're one to talk, considering your insistence that God exists against all odds.

HOWEVER, that doesn't really answer the question. Regardless of your insistence, there would still be measurable causal connections all the way back through the time slices and you are back up against my point above.

Except they have always been there (under the view I hold to), so no successive addition occurring at all. Successive implies a flow of time. There is no flow of time at all happening, whether in this local universe or beyond.

Not so at all. If the universe had a beginning, the spacetime manifold had an absolute beginning. Has nothing to do with observer position. The fact that all timeslices were created at once does not mean there was a first time slice in the sequence.

It's clear you don't get it. Let's go with the mathematical analogy again. The infinite set of all positive integers has what we may consider to be a "starting number" (namely, the number '1'), but it doesn't mean that '1' causes or leads to '2', that '2' causes or leads to '3', and so on. If they are already there in the set, then they don't need to begin to exist in order to exist.

And the timeslices weren't created. They're there, and that's that.

Your example is a potential infinite set -- not an actual infinite. Two very different things. This distinction is important for this whole discussion.

No, it isn't. The elements have always been in such a set. How many times do you need to be told this?

[pocaracas]

You guys are doing great. I've learned quite a few things with this thread.

But, to thwart the philosophers, one need only add the possibility that time is not what we intuitively think it is.

Because, if time does not really flow from event to event, then temporal causation may break under certain conditions... which are those? don't know.
What I do know is that Special Relativity narrowed the scope of temporal causation to the speed of light. Prior to that, philosophers would be happy to claim instantaneous causation everywhere, as the speed of light is too fast for our human senses to grasp it as anything else than instantaneous... who knows what lies behind the next stone in the way?
Better not make any definite pronouncement.

[LadyForCamus]

I am not asking if there is a concept in mathematics that deals with infinity or if there exists sets with an infinite number of members (although you might use the concept in a larger argument). I am not asking if we can theoretically divide something an infinite amount of times (although you might use the concept in a larger argument). I am not asking about a potential infinite. 

I am asking about an actual infinite of something concrete (not abstract). Can it logically exist? Why or why not? 

No mention of God either. Philosophy subforum--let's stick with pure metaphysics.

Anything may exist as a posibility unless there is an inherent logical contradiction built into the possibility of it's existence.  For example, invisible bachelors might exist (though not probably) without defying logic, as might 1000 year old bachlors.  But married bachlors may not logically exist. What exactly is it about an infinity that makes you think it is logically impossible? And why do you think an infinite number of objects or time is illogical but an infinite division of an object or period of time is not?

@Jenny, 

The strangest thing; earlier today you randomly popped into my mind and I wondered what ever happened to you (I remembered you being an awesome contributor), and less than eight hours later, here you are!!!

ESP?  ?

[polymath257]