Can We Really Grasp the Concept of Infinity in the Universe?

[Temporarily Blah]

I gave roughly an hour's thought to infinity, and came up with these statements and questions;

If infinity exists, where would it go? The only infinity that would make sense is the size of the universe, because you couldn't have an infinite amount of any matter without it taking up literally every available itty bitty bit of space within the universe. Time: It has a start, and an ending, but what's outside of that? If there is no motion without time, how can we have come into existence, from nothing to something? (Impossible question at this time alert)


Anyway, I forgot what else I was going to post, so i'll post that later. Cya!

 

[amcavoy]

It seems impossible to imagine these things. It's like trying to visualize... a hypercube.

 

[selfAdjoint]

It seems impossible to imagine these things. It's like trying to visualize... a hypercube.

Exactly; it is as hard to imagine a simple geometric figure in four-dimensional Euclidean space as it is to imagine infinity. Therfore our imaginations are not to be depended on when we study these things, and we resort to mathematics with its axioms and theorems and proofs.

 

[Johann]

The only infinity that would make sense is the size of the universe

I would argue that space and time are infinite.

Time: It has a start, and an ending, but what's outside of that?

It can't be known if time will ever end, although I would agree what we call "motion" must have had a starting point.

If there is no motion without time, how can we have come into existence, from nothing to something?

Things can only exist if their opposites also exist. Light can't exist without darkness, matter cannot exist without emptiness, life cannot exist without death, and so on. As such, "nothing" can only exist if "something" also exists, so the universe exists out of logical necessity, and it must have come from nothing also out of logical necessity.

As to how the universe can be created from a motionless nothing, the answer goes along the same lines. Movement is what allows motionlessness to exist. The "nothing" couldn't be motionless in the absence of a universe to serve as a point of reference, and thus logic forces a moving universe to exist due to the fact that a motionless nothing also exists. In fact, notice that if the universe is finite and space infinite, the universe is just an infinitesimal point in a sea of nothing. The nothing which preceded creation is still out there.

It's all a play with words, I know, but the basic logical structure is there. Which particular words we choose is irrelevant. If you've ever written a computer program, you know how you call your functions and variables is just a matter of personal preference, what really matters is the logical structure.

 

[NewScientist]

there is a theory that we cannot imagine any object or idea that had more than dimensions than the 3 dimensional world we live in, and indeed an entity in any dimension cannot see into dimension which is more 'complex' than its own.

For example a particle that is constrained to move along a line has only 1 dimension, this is all it can experience, it cannot imagine a 2-d world.

And the particle constrained to a page cannot see beyond its 2-D world and so on and so forth.

Thus imagining 4D Euclidean Spaces is perhaps impossible?

I am not sure about this theorty however i heard it somewhere...once...

 

[Gokul43201]

Anyway, I forgot what else I was going to post, so i'll post that later. Cya!

How about your definition of this "infinity" ?

How can we discuss something effectively unless we know what you are talking about ?

 

[Temporarily Blah]

Infinity is unlimitedness, or a never-ending amount of something.


By the way, I heard this definition of a line: A line is an infinite amount of points connected, no matter how long or short.
How can this be? Wouldn't Planck length be the minimum amount of distance between each dot, creating limitedness?

That, and it makes no sense for a 1 inch line to be made of the same amount of 0 dimensions as a 2 inch line.

That, and it doesn't make sense for a 2 foot cube to be made of the same number of planes as a 1 foot high cube.

 

[Pi_314B]

The only infinity that would make sense is the size of the universe, because you couldn't have an infinite amount of any matter without it taking up literally every available itty bitty bit of space within the universe.

I think you are misunderstanding what space is, or matter for that matter.

 

[rocketboy]

Time: It has a start, and an ending, but what's outside of that? If there is no motion without time, how can we have come into existence, from nothing to something? (Impossible question at this time alert)

What is time? No really, I'm serious. I've always (ok not always but recently) considered time as the idea representing...wow, how can you put something like this into words? I've been typing and then holding down <backspace> and then typing again for the past 5 minutes trying to think of how to word this.

Somebody want to help out?

 

[Hurkyl]

That, and it makes no sense for a 1 inch line to be made of the same amount of [points] as a 2 inch line.

It does make perfect sense: it's very easy to show how to "count" the points of one line segment using the points of the other line segment.

The thing you're forgetting is that there's another concept at play here: that of a measure. The measure is the gadget that tells you how long your line is.

 

[Pi_314B]

What is time?

In our universe there are only ones, one at a time, where time is the nothing ones are composed of.

 

[Temporarily Blah]

hmm, maybe I'm just confused about the "dots into line" part.

Anyway, time is the expression of motion, I believe.

If there is no time, there can be no motion... can there?

 

[Gokul43201]

Infinity is unlimitedness, or a never-ending amount of something.


By the way, I heard this definition of a line: A line is an infinite amount of points connected, no matter how long or short.
How can this be? Wouldn't Planck length be the minimum amount of distance between each dot, creating limitedness?

That, and it makes no sense for a 1 inch line to be made of the same amount of 0 dimensions as a 2 inch line.

That, and it doesn't make sense for a 2 foot cube to be made of the same number of planes as a 1 foot high cube.

This quandary is easily resolved. The "infinity" that appears in the definition of a line is different from your concept of infinity (this was why I asked you for its definition). The first has a methematical definition, the second does not.

 

[RAD4921]

The illusion of infinity

Infinities are created from linear thinking which in itself is an illusion.

 

[Hurkyl]

Infinities are created from linear thinking which in itself is an illusion.

Would you care to:

(1) Define what you mean by linear thinking
(2) Explain why you think linear thinking is an illusion
(3) Explain what it means for an infinity to be created
(4) Explain why you think infinities are created from linear thinking
(5) Explain why you think there is no other way to create an infinity

?

And for good measure

(6) Your definition of infinity

would be good to have as well.

 

[Lifter0569]

Ok there are a few things I believe are incorrect here. For instance, "Things can only exist if their opposites also exist." While this seems to a person as a logical conclusion, "darkness" and "emptiness" themselves do not exist but infact are defined to be the absense of something that we define exists. Light could not be DEFINED if darkness was not also defined. Darkness is just absense of the perception that we get when we "experience" light.

Also, "How can this be? Wouldn't Planck length be the minimum amount of distance between each dot, creating limitedness?" is false logic because you apply an infinite mathematical concept to our limited universe. From what I have gathered from any science that studies our universe at the smallest of scales, state that the universe does not go infinitely small. Therefore, you can not say that you can "zoom" in, in our universe infinitely.

Further, "If there is no time, there can be no motion... can there?" This is correct. Motion is completely dependant on spacetime. Without one or the other, motion itself can not be reasoned to be apart in this universe. If there is no time, there is no space. If there is no time or space, there is no matter to exist in the stage of spacetime, our universe. Therefore, there is not an existing universe at all. These concepts are fundamentally defined to be dependant on one another.

Finally, "Infinities are created from linear thinking which in itself is an illusion" is a vague statement which nobody here can definitively specify what he meant, exactly. From what I can guess, he is defining the concept of infinity to be false reasoning because of the way it is perceived, through linear thinking. What he is trying to get of this I really don't know (except to disprove infinity). If anybody can shed light on the subject or to even correct myself, it would be appreciated.

 

[Dayle Record]

Infinity is not a one size fits all kind of thing. There are several versions of infinity and we can't really know if we have properly conceptualized anyone of them. I have a simplistic view, and say you are a line in one dimension that somehow can consciously inspect every point from here to there, there is a lot to see, and for infinity to be inspected, our species runs up against a problem with time. I think that going farther and farther into subatomic space, and then outward from there to macro phenomenon, in a straight line, in either direction from the small to the large, or large to small, there is an infinity there, in any direction you choose to travel.

If we are to do with infinity what we have done with everything else, then infinity will look like a huge human being (with a gun *snicker*).

 

[Smurf]

I think by linear thinking he was referring to duality or something.

 

[bola]

As for the physical nature of infinity.. Obviously there is a limited amount of atoms in the universe, or energy, if not a constant value of energy, therefore spacetime is not infinite.
However people start to ask themselves, "what is outside the universe?"
And that's when it gets complicated because we have no idea what lies outside spacetime, unless you want to get into string theory and the higher dimension.

Regardless, the question is infinitely recursive, and to me pretty pointless, since we can never imagine what infinity is.
About time, here's my little theory on it;

Like every rule in the universe, time is also a rule, or property, of the way particles interact.
The particles themselves make up certain rules, like gravity's force, the speed of light, electromagnetisms force, et cetera.
So when these particles bind, you cannot for example travel faster than light.
Now what if time is an emergent property of the elementary particles?
Time itself doesn't control the motion of the particles, rather time emerges from the particles as they themselves create more and more rules the more they bind and gain complexity.
That's maybe why time and space was warped during the big bang, because the universe was at its simplest.
As the particles started to bind and cool down, it gained complexity and certain possibilities were erased from the plane of reality.
Time may be one possibility that emerged.

 

[Hurkyl]

Obviously there is a limited amount of atoms in the universe, or energy, if not a constant value of energy, therefore spacetime is not infinite.

No, it is not obvious.

And that's when it gets complicated because we have no idea what lies outside spacetime, unless you want to get into string theory and the higher dimension.

String theory doesn't go outside of space-time -- it says that space-time has more dimensions than we think it does.

since we can never imagine what infinity is.

A rather bold statement, in face of the many people who think they can imagine what "infinity" is, not to mention give precise definitions and make fully rigorous arguments involving it.


As for the rest of your post, it appears you're trying to hijack the thread -- what does your pet theory on time have to do with infinity?

 

[bola]

-- No, it is not obvious.

Oh isn't it?

-- String theory doesn't go outside of space-time -- it says that space-time has more dimensions than we think it does.

I meant the higher dimension, our universe is one membrane.

--A rather bold statement, in face of the many people who think they can imagine what "infinity" is, not to mention give precise definitions and make fully rigorous arguments involving it.

They are wrong. They can't imagine infinity nor give explanations of it.

--As for the rest of your post, it appears you're trying to hijack the thread -- what does your pet theory on time have to do with infinity?

If you read the other posts you will see someone asked what time was, and I gave my opinion on it. And by the way I don't have pet theories I only try to look at it the way I see it.

 

[Hurkyl]

So are you going to back up anything you say, or just continue to preach?

 

[bola]

The only thing I had to back up was the statement about other people who said they could describe infinity.
However I didn't because you never came with any arguments against mine.

 

[Hurkyl]

Could you point out to me where you argued that

(1) There is a limited amount of atoms in the universe, or energy, if not a constant value of energy
(2) We can never imagine what infinity is.
(3) They can't imagine infinity nor give explanations of it.

please? I can't seem to find it.

(I'm not bothering with your theory on time)

 

[Lifter0569]

I think I can somewhat help with his statement that ----

"(1) There is a limited amount of atoms in the universe, or energy, if not a constant value of energy"

I would agree, is the universe we live in not generally accepted fact that it has a finite amount of atoms? Generally, I would say it is fact that it is finite. Just through logic and reasoning, I do not think anything in this universe is truly infinite in any respect, but perhaps I have not thought long enough on the subject.

---------------------------------

"(2) We can never imagine what infinity is."

When someone truly understands something, they inherently limit the concept that they understand. This is then applied to understanding infinity, and it being an infinite concept, one inherently limits infinity which goes against its very definition. I think that is what he means, and I agree. A human can "get" the concept but can not truly understand it in it's fullest.

---------------------------------

"(3) They can't imagine infinity nor give explanations of it."

We can always try and imagine infinity but whether we are correct or not, is the true question. We can give explanations to the concept of infinity, but we really can not truly grasp it's meaning.

---------------------------------

 

[bola]

We can imagine infinity as something that never stops, but to me at least, this is something that is always moving out and out in all directions.
However infinity can't move, because it is infinite, if something moves it has an edge, so basically an infinite universe must be everywhere.
The word everywhere is not even the correct use of the word, there is no english word to describe the location of infinity, because it has no location.

As you can see it's impossible to describe or talk about infinity in any purposeful way.
This was an addendum to Lifter's post.

Timespace must be finite, because the mere concept of infinity is such that everything that can possibly ever happen, has already happened, and will happen infinitely into the future.
Once again, the words "future" and "everything" are meaningless when talking about infinity, but it's all I got with the english language.

 

[Hurkyl]

I would agree, is the universe we live in not generally accepted fact that it has a finite amount of atoms?

No, it is not generally accepted, which is why I objected! There are two basic possibilities for the universe:

(1) You can keep going in one direction, and eventually "wrap around" and come back where you started (more or less). Think of Pac-Man or Asteroids, if you remember way back then. This is a closed universe.

(2) You can keep going in any direction, and never return. (so it is infinite in extent) This is either flat or open.

There can be mixes of these -- open in one direction, but closed in another. I get the impression that current theories of gravity rule out these mixed possibilities.

According to the WMAP Mission page, the data seems to indicate that the universe is flat, and thus infinite in extent. (Flat means that spatially, it is very similar to Euclidean space that we learned in geometry)


But, it's still possible for the universe to be flat, but with a finite amount of matter in it -- but as I understand, the evidence seems to indicate a homogeneous universe, which means on the largest scales, it looks more or less uniform. In particular, it is not an island universe where we have a finite "island" of matter, and then nothingness.


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Okay, now I've gotten the physics out of the way, I can speak about the infinite!

There are lots of nebulous thoughts you can toss around, but "infinite" does has a simple, easy to understand definition: not finite.

So, the question is what is finite? Well, the natural numbers seem to be the prototypical finite things -- when we say something is finite, we mean that there is some sense in which we can compare that thing to a natural number, and find a natural number that is bigger.

For example, we might say the number of people on Earth is finite, because we can find a natural number bigger than the number of people. We might say that a line segment is finite, because we can find a natural number bigger than its length.

However, we would not say that a line (from good ol' Euclidean geometry) is finite. One would often say that the line is unlimited in extent, and I posit this sort of thing is from where the notion that the infinite is something "unlimited" stems.

Of course, in another sense, the line is clearly limited in its extent -- it cannot go outside of itself! Also, it can't go outside of the Euclidean plane. I assert that this sort of "shifting" of the meaning of the word "limited" is one of the major sources of confusion people have when trying to understand the term "infinite".


Now, the word infinity. It is a rather unfortunate word, because it seems to get people to identify all things infinite into a mangled mess. But, let's look at some of the phrases in which the word "infinity" is used:

A Euclidean line just keeps going in both directions off to infinity.
The size of the Euclidean plane is infinity.

In both of these cases, and many other similar cases, the word "infinity" is simply referring to some "place" further away than any finite distance, or some "value" larger than any finite value.

These notions are very easy to capture in a rigorous way. I will define something called the extended real numbers, as follows:

The extended real numbers consists of all the ordinary real numbers, and two additional things which I will write as +&infin; and -&infin;. I will will then extend the definition of "<=" as follows:

-&infin; <= a <= +&infin;

for any extended real number a. I will also define |±&infin;| = +&infin;.

For a more thorough exposition (including the arithmetic operations) see:
http://en.wikipedia.org/wiki/Extended_real_line
http://mathworld.wolfram.com/AffinelyExtendedRealNumbers.html

And *poof*, we now have a line that contains the old real number line, but has added two "endpoints", whose magnitude is greater than any natural number -- in other words, infinite.

(In some sense, this is directly analogous to starting with the interval (0, 1) and adding its two endpoints to get the interval [0, 1])

So, here we have a precisely defined notion of something that is bigger than any natural number. It is also a useful notion, as evidenced by the fact that the extended real number line is what is used in calculus.


This isn't some modern abstract nonsense either: it's been around since at least the early 1800s. One of the great successes of analytic geometry was the projective plane which adds a whole line "at infinity".



Does any of this capture the notion of "infinite"? I have to say yes:

First off, it satisfies the definition of the infinite in the fact that it involves things "beyond" the finite.

Secondly, it accurately captures the things people are trying to express when using the words "infinite" or "infinity". e.g.

If we extend the Euclidean plane to get the Projective plane, then the every Euclidean line does, in fact, contine to infinity, and contains a single point of the line at infinity.

If we use the extended real numbers for measuring things, then the area of the Euclidean plane really is +&infin;.


There are, of course, other places where "infinity" or "infinite" are used, but I think I've written enough, so I will not go into detail on things like:

Cardinals and ordinals -- mathematical things generalizing the notions of quantity and counting to infinite sets.
Hyperreals -- a number system consisting of infinite numbers and infinitessimal numbers, which is "internally" indistingushable from the ordinary real numbers.

 

[bola]

I'm not into the whole math thing, more the philosophy, so if what I'm about to say is incorrect please correct me so I can learn something.

Regardless, if you have a universe that you can map with cartesian coordinates, each coordinate is unique.
The problem is, does the universe contain an unlimited amount of coordinates, or is there a highest number?
And if so how can you prove that it does from earth?
How can we trust that the mathematical model is portraying 100% of the universe?
Is that even possible?

I dunno, enlighten me.

 

[Hurkyl]

Well, the modern philosophical viewpoint on infinity (at least according to Wikipedia) is that it's a topic to be addressed mathematically, so .

if you have a universe that you can map with cartesian coordinates, each coordinate is unique.
The problem is, does the universe contain an unlimited amount of coordinates, or is there a highest number?

Based on the hypothesis, the answer is that there are infinitely many coordinate values, because there are infinitely many cartesian coordinates.

(Though, I suspect that you meant to say that you're mapping the universe to some subset of cartesian coordinates)

In the mathematics currently used to model the universe, a coordinate chart consists of all possible combinations real number values that can be assigned to the components -- this is true whether you have a coordinate chart that covers the entire universe (if such a thing is possible), or a little tiny portion of space-time.

How can we trust that the mathematical model is portraying 100% of the universe? Is that even possible?

I'll have to leave that question to the scientists, but I would presume the answer is that it's not pragmatic to assume that the universe hoodwinking us into thinking its infinite and more or less homogeneous and isotropic.