Request for Help re the Notion of Potential Infinity

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Re: Request for Help re the Notion of Potential Infinity
TH I don't have a strong opinion about what the nature of reality is.
 ThomasHobbes
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Re: Request for Help re the Notion of Potential Infinity
You said space isn't what I think it is.

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Re: Request for Help re the Notion of Potential Infinity
The completed set of integers ( ....1, 2, 3, 0, 1, 2, 3 ...) is an example of the concept of "Actual Infinity". My post specifically concerns the concept of "Potential Infinity". "Potential Infinity" is not something static or completed in any sense, rather it is a PROCESS.chewybrian wrote: ↑June 3rd, 2018, 6:53 amby chewybrian » Today, 6:53 am
ThomasHobbes wrote: ↑Yesterday, 8:39 pm
The set of integers does not exist. It is just an idea...A piece of string of infinite length cannot be expressed by an integer. What have integers got to do with my exampkle.
The notion of "Actual Infinity" ( or "completed infinity" ) is routinely used by mathematicians. Anyone who has been taught the differential "Calculus" in senior mathematics classes at high school will be familiar with using (Actual) mathematical Infinity ( which is given the symbol of a number "8" lying horizontally on its side : 00) when performing certain mathematical operations like integration and so on.
Could we please keep this thread focussed on the notion of "Potential Infinity" ( which is, BTW, a perfectly legitimate concept in philosophy and absolutely not an "absurdity" or a "nonsense") as opposed to "Actual Infinity" (?), and ,in particular, on any interesting metaphysical ( and to be clear, I define "metaphysics" as that branch of philosophy which is concerned with enquiring into the nature of what is real as opposed to that which is not real, or only apparently real) religious implications ( e.g. implications for proponents of theism or non atheism, etc.) that forum members believe the conception of "Potential Infinity" may directly or indirectly entail ?
Finally, to be clear, I define "metaphysics" as the that branch of philosophy which is concerned with enquiring into the nature of that which is real (as opposed to that which is not real, for example, that which is only apparently real, and so forth.)
Regards
John

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Re: Request for Help re the Notion of Potential Infinity
Please delete the last paragraph of the post above which I forgot to erase before submitting.
Dachshund
Dachshund
 Consul
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Re: Request for Help re the Notion of Potential Infinity
"It is common to distinguish between two notions of the infinite: the potential and the actual (or completed). Loosely, the potential infinite is some indefinitely extensible operation; the actual infinite is the state produced when the operation has been performed more than a finite number of times. Aristotle, as he says, is describing the potential infinite, whilst what I have called the infinite is the actual infinite. The 'actual/potential' terminology is not terribly happy. The potential infinite is neither potential nor infinite. It is not potential, at least in the way that most things are potential in Aristotle; for it (as opposed to parts of it) can never come into being. And it is not infinite, since at any stage it is finite. (The sense of actuality in which the actual infinite is actual, might also be thought to be moot.) However, the terminology is now too well entrenched to change easily; so I will use it."
(Priest, Graham. Beyond The Limits of Thought. 2nd ed. Oxford: Oxford University Press, 2002. p. 27)
(Priest, Graham. Beyond The Limits of Thought. 2nd ed. Oxford: Oxford University Press, 2002. p. 27)
"We may philosophize well or ill, but we must philosophize." – Wilfrid Sellars
 Consul
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Re: Request for Help re the Notion of Potential Infinity
"If you were to head out into the cosmos, traveling ever farther, would you find that space goes on indefinitely, or that it abruptly ends? Or, perhaps, would you ultimately circle back to your starting point, like Sir Francis Drake when he circumnavigated the earth? Both possibilities—a cosmos that stretches infinitely far, and one that is huge but finite—are compatible with all our observations, and over the past few decades leading researchers have vigorously studied each."
(p. 10)
"The big bang model of cosmology includes a detail that will prove essential. The model provides not one but a handful of different cosmological scenarios; all of them involve an expanding universe, but they differ with respect to the overall shape of space—and, in particular, they differ on the question of whether the full extent of space is finite or infinite."
(p. 20)
"At the beginning of this chapter, I noted that we don't know whether the universe is finite or infinite."
(p. 25)
(Greene, Brian. The Hidden Reality: Parallel Universes and the Deep Laws of the Cosmos. New York: Alfred A. Knopf, 2011.)
(p. 10)
"The big bang model of cosmology includes a detail that will prove essential. The model provides not one but a handful of different cosmological scenarios; all of them involve an expanding universe, but they differ with respect to the overall shape of space—and, in particular, they differ on the question of whether the full extent of space is finite or infinite."
(p. 20)
"At the beginning of this chapter, I noted that we don't know whether the universe is finite or infinite."
(p. 25)
(Greene, Brian. The Hidden Reality: Parallel Universes and the Deep Laws of the Cosmos. New York: Alfred A. Knopf, 2011.)
"We may philosophize well or ill, but we must philosophize." – Wilfrid Sellars
 Consul
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Re: Request for Help re the Notion of Potential Infinity
"8.7 The domain principle
…
If the sequence in question is not to be understood as an already completed infinity (a set), then it must be understood as something in a state of generation, i.e., a potential infinite. And to speak of the set containing the members of the progression is exactly to speak of the (actual) infinity thus generated. What this principle comes to, therefore, is this:
For every potential infinity there is a corresponding actual infinity.
Following Hallett ((1984), p. 7) let us call this the Domain Principle. I take it to be a formulation of the Kantian insight that totalisation is conceptually unavoidable (6.1), though stated in a much more satisfactory way than Kant ever managed to achieve.
But what, exactly, is a potential infinity? The idea is, intuitively, clear enough. (We have been working with it since chapter 2.) Moreover, traditionally, it has been thought unproblematic; it is the other member of the pair, the actual infinite, that has been thought problematic. It is therefore somewhat ironic that in modern mathematics the notion of the actual infinite is relatively well understood (any infinite set is an actual infinity); it is a precise analysis of the nature of potential infinity that still awaits us (as Hart (1976) observes). I shall not try to offer one here; it will suffice, for our purpose, to leave the notion at an intuitive level .
Whatever else it is, a potential infinity is some kind of variable quantity, whose variation can go beyond any preassigned bound of a certain kind. As Cantor puts it (Hallett (1984), p. 12):
—
The potentialinfinite is mostly witnessed where one has an undetermined, variable finite quantity which either increases beyond all limit . . . or which decreases beneath any finite small limit . . . More generally, I speak of a PotentialInfinite whenever it is a question of an undetermined quantity which is capable of innumerably many determinations.
—
The corresponding actual infinite is, by contrast (ibid):
—
to be understood [as] a quantum which on the one hand is not variable, but rather is fixed and determined in all its parts – a genuine constant – but which at the same time surpasses in magnitude every finite quantity [i.e., preassigned bound] of the same kind.
—
8.8 . . . and its defence
So much for the Domain Principle itself. Let us now turn to Cantor's defence of it. In his own words, this goes as follow (Hallett ( 1 984), p. 25):
—
There is no doubt that we cannot do without variable quantities in the sense of the potential infinite; and from this can be demonstrated the necessity of the actualinfinite . In order for there to be a variable quantity in some mathematical study, the 'domain' of its variability must strictly speaking be known beforehand through definition. However, this domain cannot itself be something variable, since otherwise each fixed support for the study would collapse. Thus, this 'domain' is a definite, actually infinite series of values. Thus, each potential infinite, if it is rigorously applicable mathematically, presupposes an actual infinite.
—
Cantor's argument is a simple and ingenious one. It is based on the equally simple observation that for a statement about some variable quantity to have determinate sense, the domain of its variability must be determinate. (Cantor says 'known beforehand by definition'; but this is too strong; simple determinacy will do.) For example, consider the claim 'Let z be a root of the equation ax^2 + bx + c = 0. Then z has at least one value.' This is true if z may be complex; false if z must be real.
Now consider some statement about a potentially infinite variable; by the observation, if this has determinate sense, its domain of variability must be determinate. Can this domain not itself be a potential infinity (variable)? Perhaps it can; but then the sense of the original claim is not determinate unless the domain of this variable is itself determinate. Could that, itself, be a variable? Perhaps. We are obviously off on an infinite regress. Moreover, this regress is vicious. If it went on forever, the sense of the original statement would never become determinate. Given that the statement does have determinate sense, the regress must bottomout somewhere. And thus the original variable must find its ultimate ground in an actual infinite domain."
(Priest, Graham. Beyond the Limits of Thought. 2nd ed. Oxford: Oxford University Press, 2002. pp. 1235)
…
If the sequence in question is not to be understood as an already completed infinity (a set), then it must be understood as something in a state of generation, i.e., a potential infinite. And to speak of the set containing the members of the progression is exactly to speak of the (actual) infinity thus generated. What this principle comes to, therefore, is this:
For every potential infinity there is a corresponding actual infinity.
Following Hallett ((1984), p. 7) let us call this the Domain Principle. I take it to be a formulation of the Kantian insight that totalisation is conceptually unavoidable (6.1), though stated in a much more satisfactory way than Kant ever managed to achieve.
But what, exactly, is a potential infinity? The idea is, intuitively, clear enough. (We have been working with it since chapter 2.) Moreover, traditionally, it has been thought unproblematic; it is the other member of the pair, the actual infinite, that has been thought problematic. It is therefore somewhat ironic that in modern mathematics the notion of the actual infinite is relatively well understood (any infinite set is an actual infinity); it is a precise analysis of the nature of potential infinity that still awaits us (as Hart (1976) observes). I shall not try to offer one here; it will suffice, for our purpose, to leave the notion at an intuitive level .
Whatever else it is, a potential infinity is some kind of variable quantity, whose variation can go beyond any preassigned bound of a certain kind. As Cantor puts it (Hallett (1984), p. 12):
—
The potentialinfinite is mostly witnessed where one has an undetermined, variable finite quantity which either increases beyond all limit . . . or which decreases beneath any finite small limit . . . More generally, I speak of a PotentialInfinite whenever it is a question of an undetermined quantity which is capable of innumerably many determinations.
—
The corresponding actual infinite is, by contrast (ibid):
—
to be understood [as] a quantum which on the one hand is not variable, but rather is fixed and determined in all its parts – a genuine constant – but which at the same time surpasses in magnitude every finite quantity [i.e., preassigned bound] of the same kind.
—
8.8 . . . and its defence
So much for the Domain Principle itself. Let us now turn to Cantor's defence of it. In his own words, this goes as follow (Hallett ( 1 984), p. 25):
—
There is no doubt that we cannot do without variable quantities in the sense of the potential infinite; and from this can be demonstrated the necessity of the actualinfinite . In order for there to be a variable quantity in some mathematical study, the 'domain' of its variability must strictly speaking be known beforehand through definition. However, this domain cannot itself be something variable, since otherwise each fixed support for the study would collapse. Thus, this 'domain' is a definite, actually infinite series of values. Thus, each potential infinite, if it is rigorously applicable mathematically, presupposes an actual infinite.
—
Cantor's argument is a simple and ingenious one. It is based on the equally simple observation that for a statement about some variable quantity to have determinate sense, the domain of its variability must be determinate. (Cantor says 'known beforehand by definition'; but this is too strong; simple determinacy will do.) For example, consider the claim 'Let z be a root of the equation ax^2 + bx + c = 0. Then z has at least one value.' This is true if z may be complex; false if z must be real.
Now consider some statement about a potentially infinite variable; by the observation, if this has determinate sense, its domain of variability must be determinate. Can this domain not itself be a potential infinity (variable)? Perhaps it can; but then the sense of the original claim is not determinate unless the domain of this variable is itself determinate. Could that, itself, be a variable? Perhaps. We are obviously off on an infinite regress. Moreover, this regress is vicious. If it went on forever, the sense of the original statement would never become determinate. Given that the statement does have determinate sense, the regress must bottomout somewhere. And thus the original variable must find its ultimate ground in an actual infinite domain."
(Priest, Graham. Beyond the Limits of Thought. 2nd ed. Oxford: Oxford University Press, 2002. pp. 1235)
"We may philosophize well or ill, but we must philosophize." – Wilfrid Sellars

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Re: Request for Help re the Notion of Potential Infinity
Thank you for the references above, Consul; they are helpful.
Regards
D
Regards
D
 chewybrian
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Re: Request for Help re the Notion of Potential Infinity
Thanks for taking the time to explain the options. I'm inclined to see the universe as an object in time. I can see how our ability to see time and/or distance might be restricted to the universe, but I still think something could go on before or after, or outside the universe, that we will never be able to validate. Space without matter, time without people, a tree falling in the forest, something... Maybe I'm like a primitive man, assuming that the world is flat because it appears so, and I have no evidence to contradict appearances.Halc wrote: ↑June 3rd, 2018, 2:02 pm'Real' has special meaning in metaphysics. OK, this topic was not posted in the metaphysics section, but the opinions being asserted by ThomasHobbes are metaphysical opinions.chewybrian wrote: ↑June 3rd, 2018, 10:16 amPerhaps 'real' was a poor choice of wording, or maybe it has some special meaning to a physicist? I suspect we might agree ideas are not material, but they exist in some sense, at least impacting material things through us.
'Big crunch' happens when expansion stops, turns around, and the whole universe collapses into a sigularity again, ending time. This seems unlikely given that the expansion is accelerating, not slowing.If you have an idea that time or space or finite, maybe you could point to a good source for the uninitiated to check out this idea.
"Big rip" is runaway expansion where space expands faster than even the nuclear strong force can hold the simplest things together. There is no action after that, no more entropy to create, so an abrupt end to time. You can google any of these terms.
Barrinig those two fates, "heat death" just means the universe dissolves into a haze of noninteracting radiation, and is the end of time like the other two, just not so abruptly.
Time going forever is more of a mathematical sense, but seems to have no equivalent in physics. At some point, there will not be another hour in front of you regardless of your fate of choice.
There is a cyclic theory that the universe bangs over and over, and thus infinite time. This violates thermodynamic law, but that law might not apply over multiple bangs. So in that sense, time would be infinite in both directions. I think this model was proposed only as a solution to a naive view of time (that the universe is an object that exists in time, not that time is defined by the nonobject that is the universe).
As for a limit to space, I have no reference for that. Perhaps it wraps like a balloon, and if you go far enough, you get back to the start. No edge that way, but a finite size nonetheless. If we're shaped like that, the wrap is more distant than the expansion rate lets us see. It is impossible to actually return to Earth by going in a straight line like Dachshund describes. I know of no viable model that says there is an edge, and if you were there, you would see matter all on one side and none in the other direction.
But, of course, there would have been no practical advantage to the primitive man to know otherwise, and I'm not sure we have much to gain in seeing the universe as some kind of Klein bottle, or by having a really, really slowly dropping ball in Times Square. It brings me right back to ethics. We are much more likely to threaten our own existence than to bump up against the limits of time and space, so working to become better people seems more urgent.

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Re: Request for Help re the Notion of Potential Infinity
Halc,Halc wrote: ↑June 3rd, 2018, 2:02 pmAs for a limit to space, I have no reference for that. Perhaps it wraps like a balloon, and if you go far enough, you get back to the start. No edge that way, but a finite size nonetheless. If we're shaped like that, the wrap is more distant than the expansion rate lets us see. It is impossible to actually return to Earth by going in a straight line like Dachshund describes. I know of no viable model that says there is an edge, and if you were there, you would see matter all on one side and none in the other direction.
The only possibility that makes any kind of intuitive / "logical"sense to me is that maybe it is the case that our 3D universe is finite and it is in the process of expanding into a 4th dimension ( which we cannot comprehend in any way with our human minds) and that this 4th dimension perhaps provides the "stuff" for our universe to be expanding /stretching out into.The mathematics could support the possibility of such a 4th dimension for our 3D universe to be expanding into, and if so, then we (in our current 3D universe) would definitely have an "edge" in this extra dimension in the same way one can point to the edge of a 2D beach ball surface, right? But it would then seem plausible to propose that (maybe) this 4th dimension would, in turn, also have an "edge" itself, beyond which there existed a hypothetical 5th dimension with its own "5D stuff" and so on and so on, right (?) so that, in short, a potential infinity of additional dimensions 6th, 7th...102nd....5,567th ... 666,678,901st....is what is "happening"?
Does what I am suggesting make any sense ?
Regards
Dachshund
 Halc
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Re: Request for Help re the Notion of Potential Infinity
Well, if the universe is 3D and expanding into something not 3D, then it would immediately not be 3D. If we cannot comprehend this 4th dimension with our human minds, then I'm not sure how this idea is making sense to you. Not saying any of it is wrong, and not saying I understand what you are envisioning, but the whole bit above seems to just contradict itself.Dachshund wrote: ↑June 4th, 2018, 2:45 pmThe only possibility that makes any kind of intuitive / "logical"sense to me is that maybe it is the case that our 3D universe is finite and it is in the process of expanding into a 4th dimension ( which we cannot comprehend in any way with our human minds) and that this 4th dimension perhaps provides the "stuff" for our universe to be expanding /stretching out into.
A regular explosion of say a bomb happens at a location somewhere, and the bits fly off and expand into 3D space, each bit receding from its nearest neighbors at actually a fairly sedate pace, but receding from its far neighbors at a rapid pace. Space isn't expanding in this example, its just that the stuff is expanding into space that is already there from a bomb that went off at a moment in time, time that was also already there. That model matches the big bang pretty well until you get into the really distant stuff that is so far away that it is receding from us at greater than lightspeed. The model sort of falls apart there, but only sort of. There doesn't have to be stuff that far away since it hasn't had time to get there anyway.
Anyway, it doesn't posit any magical dimension that is beyond comprehension because nothing has ever been explained by something incomprehensible. Best off to just say there are parts you don't know. There are plenty of parts in my view written off to "I don't know that answer". Gaps are areas for exploration, places still in need of filling in.
Yes, I think the universe has 4 major dimensions, not 3. Plenty more minor ones if string theory ever starts predicting things. But the 4th dimension is hardly beyond comprehension, and yes, there is a point where the big bang takes places, and you need to point in that 4th direction to point at it. Not so hard to do it.
There, you envisioned an extra dimension by getting rid of one of the spatial ones. That's how I do it as well. Not so incomprehensible. Yes, the surface (space) of the beach ball has no edge in 2D, but in 3D it is nonexistent except at one fixed radius where the ball surface exists. That's not really an edge, since there is no space on either side of it, just at that one radius. An 'edge' has traditionally has at least some stuff on one side but not the other.The mathematics could support the possibility of such a 4th dimension for our 3D universe to be expanding into, and if so, then we (in our current 3D universe) would definitely have an "edge" in this extra dimension in the same way one can point to the edge of a 2D beach ball surface, right?
That's a model of a 2D thing in 3D, and of expanding space if the ball is inflating. Print a bunch of stars and galaxies on it, and notice that they move apart as it inflates, just like we observe. The big bang takes place at the center of the ball, but on no particular place where the surface is now. To point at it, the inhabitants of the surface need to point in a nonspatial direction at the center of the beach ball where there is no surface anymore.
My beach ball is not empty. It is solid, and the ball surface is just a mathematical curved surface buried somewhere deep in the solid with stuff on all sides, not just in the spatial dimension.
I think only if it helps explain something. Doesn't mean it is wrong, since there is probably things about reality that are so removed from our ability to ever observe that we cannot possibly ever even guess at them. Such things make zero difference to us, and so are beyond meaning to us.But it would then seem plausible to propose that (maybe) this 4th dimension would, in turn, also have an "edge" itself, beyond which there existed a hypothetical 5th dimension with its own "5D stuff" and so on and so on, right (?) so that, in short, a potential infinity of additional dimensions 6th, 7th...102nd....5,567th ... 666,678,901st....is what is "happening"?
Does what I am suggesting make any sense ?
Quite an outofbox thinker. Progress would never be made without people thinking up new ways of looking at things. I have done my share, but not sure if I've actually contributed to the base of ideas that are considered valid interpretations of the world. Somebody would have to agree with me first on something, like that's ever likely to happen...Regards
Dachshund
What I wrote in this post is me agreeing with others. It isn't necessarily truth, but it least is stuff you can find online.
 Consul
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Re: Request for Help re the Notion of Potential Infinity
If space is infinite, it can still expand; but in this case it's not an expansion into anything (else). Infinite space has no boundary, so there cannot be any other space beyond it into which it can expand. The apparent paradox is that when an infinite space grows (or shrinks), it does so without getting larger (or smaller): ∞ x 2 = ∞, ∞ x 1/2 = ∞Dachshund wrote: ↑June 4th, 2018, 2:45 pmThe only possibility that makes any kind of intuitive / "logical"sense to me is that maybe it is the case that our 3D universe is finite and it is in the process of expanding into a 4th dimension ( which we cannot comprehend in any way with our human minds) and that this 4th dimension perhaps provides the "stuff" for our universe to be expanding /stretching out into.
"If space is truly infinite in size, then it always has been and always will be. When it shrinks, its contents are squeezed ever closer together, making the density of matter ever larger. But its overall extent remains infinite. After all, shrink an infinite tabletop by a factor of 2 and what do you get? Half of infinity, which is still infinite. Shrink by a factor of 1 million and what do you get? Infinity still. The closer to time zero you consider an infinite universe, the denser it becomes at every location, but its spatial extent remains unending."
(Greene, Brian. The Hidden Reality: Parallel Universes and the Deep Laws of the Cosmos. New York: Alfred A. Knopf, 2011. p. 26)
Correspondingly, what we observe in an expanding infinite universe is a decrease in the density of matter, a dilution of matter; but there's no increase in the volume of space.
"We may philosophize well or ill, but we must philosophize." – Wilfrid Sellars
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Re: Request for Help re the Notion of Potential Infinity
Space, is that which is not there. Because it is not there, it cannot be measured. Something which cannot be measured is infinite. The non existence of space, is what makes space useful. It’s what makes my coffee cup useful. Existence and nonexistence are two sides of the same coin. Try to imagine a coin with just one side. When you measure an object, you are measuring the finite material of that which is there, and at he same time, the infinite space it is contained within.
Even though you can see me, I might not be here.

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Re: Request for Help re the Notion of Potential Infinity
How do you know?