Actual Infinity in Reality?

[Neo-Scholastic]

Are the laws of logic considered descriptive?  If so, how could they exist outside of space-time?  Without space-time, there couldn’t be an A to be either A or not A.  Right?

IMHO logic is prescriptive since what is being described could not be otherwise. It transcends any particular circumstance and applies universally regardless of the type or degree of any particular thing.

[The Grand Nudger]

In a different universe...things would be different....and logic would describe the state of things in that different universe.

[SteveII]

Therefore the the abstraction of actual infinity (from the first quote) is based on an axiom that there exists at lease one infinite set. Appropriately, an axiom in mathematics is defined as: a statement or proposition on which an abstractly defined structure is based. In case we are still unclear, the third quote defines an Abstraction. I highlighted the key theme all the way through this. Abstract. 

You have NOT made an argument (in this or the previous thread) where you show how this abstract concept in mathematics applies to the real world. It should be simple to propose some thought experiments or examples for us all to consider. I understand your point that the concept in mathematics exists--now you need to provide some evidence that it applies to real objects. Re-iterating infinite set theory from mathematics will not further this discussion.

Yes, I get that. I will happily admit (as I have done repeatedly) that we do not *know* if there is an actual infinity in the real world. None has been given, I agree.

But that is distinct from the question of whether it is *logically contradictory*. The fact that it works in the abstract is enough to show there is no *logical* problem with the concept.

Also, all claimed 'dis-proofs' are based on a confusion between 'size' in terms of subsets, 'size' in terms of cardinality, and attempting to 'subtract' when it isn't well-defined. In other words, the claimed 'dis-proofs' simply don't manage to disprove the concept.

You are answering a metaphysical question with a mathematical axiomatically-driven abstraction. Mathematical axiomatically-driven abstractions are not part of the pool where logic goes for information. By the definition of the terms, you have not *shown* anything to be logical.

[RoadRunner79]

[polymath257]

You are answering a metaphysical question with a mathematical axiomatically-driven abstraction. Mathematical axiomatically-driven abstractions are not part of the pool where logic goes for information. By the definition of the terms, you have not *shown* anything to be logical.

You really can't mean that. Mathematics (and certainly NOT metaphysics) is the prototypical logical subject.

Here is an argument:
1. It is possible that space is flat (in fact, the observed evidence suggests this).
2. Spatially flat manifolds can be infinite in extent. (R^4 is an appropriate example)
3. Hence it is possible that space is infinite in extent.
4. As far as we can see, space is homogeneous (the same at all locations and in all directions)
5. Hence, it is possible that there are an infinite number of stars.

Now, what is you 'metaphysical' issue with either mathematical 'abstractions' or physical possibilities?

[SteveII]

I don't think the B Theory of time solves the underlying problem of having a series of cause/effect relationships. It seems to me that even if all points of time are equally real, they are still ordered by a structure we call cause/effect--a tangible series of objects we can use in thought experiments. [NOTE: I say this to start because there are some here who deny even this].

I don't see how you get that this is a problem unless you assume that any series must have a first member.  In that case, you would be assuming what you need to prove.  If time is infinite, like the idea of spatial infinity, then the 4-space manifold that is time+space simply has no boundary in either direction, temporally.  I don't see how the idea that there is a cause/effect relationship between every successive part of that manifold undermines the possibility of it being infinite.

That is the question: does a series of events need a first member? Since this thread is on infinity and not something like the PSR, I will continue to limit it to just the infinity question. I brought up cause/effect because some deny there is as a way to avoid the question. I wanted to make sure we were not talking past each other. 

Perhaps a variation of Hilbert's Hotel:

We can conceive of a possible world (much like the one you are proposing) with a beginningless series of discrete successive events of equal duration leading up to the present (real or perceived present). 

[ ...en, ... e5, e4, e3, e2, e1, e0]

We can conceive of another possible world with exactly the same events in the same order, but in between each of those events, another event occurs.

[ ...en, En, ... e5, E5, e4, E4, e3, E3, e2, E2, e1,E1, e0]

In this series, an infinite number of additional events have been added to an already infinite series of events. Are there more events? No. Infinity + infinity = infinity. We can also do the subtraction example from Hilbert, and imagine all the events prior to e3 could have been left out of the chain. 

[e3, e2, e1, e0]

In this series, we have subtracted an infinite number of events from an infinite number of events. Infinity - Infinity = 4.  Alternately, every other event could have been left out:

[ ...en, ... e4, e2, e0]

In this series, we have left out an infinite number of events from an infinite number of events. Infinity - Infinity = Infinity. 

This is not just "counter-intuitive". Actual infinities of real objects leads to absurdities (metaphysical impossibilities). Therefore an actual infinite is not logically possible.

I don't see that Hilbert's hotel demonstrates that an actual infinity is a metaphysical impossibility.  What metaphysical truth is it contradicting?  You seem to be arguing on the surface here, claiming that it is a metaphysical impossibility without showing any actual metaphysics.  It seems to me that you've simply argued in a circle.  Your metaphysics doesn't admit of an actual infinity, so to you an actual infinity is impossible (metaphysically).  In doing so you seem to have simply assumed what you need to demonstrate.  I don't offhand see how Hilbert's hotel advances your argument any.  To me, it's just a distraction.  The hotel produces results that seem absurd.  It's not clear that Hilbert's hotel demonstrates impossibilities.  You need to show the latter, not the former. 

Possible worlds semantics helps us identify metaphysical possibilities/impossibilities. I set up four such thought experiments above. We clearly have contradictions that arise when comparing these possible worlds. But the real problem is that your position is that all of them are true despite the obvious contradictions. You have not shown why we should accept the contradictions other than to wonder if that's just the way it is. It seems to me that you have some burden of proof to shoulder if you are proposing ignoring obvious contradictions. 

If anything, Hilbert's hotel demonstrates that our understanding of the meaning of reference is undermined by an actual infinity, and that seems true. We can't form a sensible relationship between referents and the things they reference under operations involving infinity. Is that a metaphysical problem, I don't think so. 

Why isn't that question begging? The proposition is that an actual infinity does not exist. To dismiss thought experiments on the basis they don't work with actual infinities needs a little more support.

You need to go further than simply recounting Hilbert's hotel to show that any essential metaphysical assumption has been violated.  When you do, I think you'll find that you've simply assumed your conclusion.

It seems to me you are accepting an actual infinity as a brute fact. Can accepting a brute fact really be considered "logical". 

[Example language from a paper from Wes Morrison - http://spot.colorado.edu/~morristo/EndlessFuture.pdf]

Thanks.  I'll look at this in more detail at a later date.  According to Morriston, "It is controversial, of course, whether there is genuine absurdity in either case."  I don't see that you've eliminated the controversy so much as arbitrarily championed one side of it.

The other side seems to be to claim there are no absurdities. Hard to pick that argument apart when you just showed there is.

[LadyForCamus]

Yes, I get that. I will happily admit (as I have done repeatedly) that we do not *know* if there is an actual infinity in the real world. None has been given, I agree.

But that is distinct from the question of whether it is *logically contradictory*. The fact that it works in the abstract is enough to show there is no *logical* problem with the concept.

Also, all claimed 'dis-proofs' are based on a confusion between 'size' in terms of subsets, 'size' in terms of cardinality, and attempting to 'subtract' when it isn't well-defined. In other words, the claimed 'dis-proofs' simply don't manage to disprove the concept.

You are answering a metaphysical question with a mathematical axiomatically-driven abstraction. Mathematical axiomatically-driven abstractions are not part of the pool where logic goes for information.

Heh?!  Maybe I’m misunderstanding you, but the first time I ever saw a truth table was is in math class, lol.

[The Grand Nudger]

Logic -is- an axiomatically driven abstraction.  Math and logic both find themselves forced to draw from the same "pool" when attempting to describe some x. Working from axioms, logic itself only describes the relationships between propositional statements and their material implications. That;s why it;s so useful. We can apply it to a broad range of statements and implications once we define ur variables. The same is true of math. In the absract, it signifies nothing. Variables defined, it describes the universe.

Where they differ is in the added difficulty of natural language as opposed to math.  Were it not for that..we'd never have to wonder about truth..we could plug any statement into a simple calculator and determine it. Ulimately, that may be what we're doing when we work shit out with our headmeat, tho, so caveat emptor.

Neither math nor propositional logic are crystal balls that force the cosmos into accord with their axomatic assumptions. It's the reverse, factually. Their utility might explain why some of us elevate both beyond their station...but, that;s just a quirck of humanity. We invest incredible value in useful things, and bestow those things with fanciful properties as a function of our admiration. I talk to my axe, too.

[Edwardo Piet]

[SteveII]

You are answering a metaphysical question with a mathematical axiomatically-driven abstraction. Mathematical axiomatically-driven abstractions are not part of the pool where logic goes for information. By the definition of the terms, you have not *shown* anything to be logical.

You really can't mean that. Mathematics (and certainly NOT metaphysics) is the prototypical logical subject.

I clearly said "Mathematical axiomatically-driven abstractions" are not sources of logic. The noun in that phrase is abstractions not mathematics. 

Here is an argument:
1. It is possible that space is flat (in fact, the observed evidence suggests this).
2. Spatially flat manifolds can be infinite in extent. (R^4 is an appropriate example)
3. Hence it is possible that space is infinite in extent.
4. As far as we can see, space is homogeneous (the same at all locations and in all directions)
5. Hence, it is possible that there are an infinite number of stars.

Now, what is you 'metaphysical' issue with either mathematical 'abstractions' or physical possibilities?

That's a mess in addition to not being a valid argument. 5 does not have anything to do with the other 4. At most, 1-4 is trying to argue for infinite space. Stars came out of left field. 

I don't think 3 follows from 1 and 2. In other words, 1 and 2 could be true but that does not give any indication that 3 is true.