Exploring Infinity: Is it a Math Concept or Real?

[kaleidoscope]

Is "Infinity" only a mathematical concept or is there anything infinite in reality?

I mean ∞ is indeterminate in a way such that any of the following expressions can be constructed:

∞ - ∞ = 1,
∞ - ∞ = 0
∞ - ∞ = ∞

Is there anything in reality that can actually behave like that?

 

[CRGreathouse]

Mathematically, your statements aren't well-defined, so to understand them we must translate them in some way. If we're allowed the same latitude in 'translating' into reality, it's very possible we'll find analogues. But the answer will depend very much on who's doing the translation and what choices they make!

 

[Tac-Tics]

What is reality?

Actually, don't try to answer that. There's a thread in the philosophy forum for that. It's 100 pages long and doesn't get any closer to any useful answers.

Not only is infinity not an actual number, it's not even a single concept. It would have been better if infinity were called "boundless" or "limitless", because then you introduce a new, very flexible idea of a "boundary" or a "limit", which can be defined in a number of totally different ways depending on the context of your problem.

Consider a few distinct uses of infinity:

Consider a car race. We use a timer to measure how long it takes for each car to make it to the finish line. But suppose one car explodes during the race. We could say it took an "infinite" amount of time to finish, because it will never finish.

More abstractly, here are two kinds of infinity that only involve sets of real numbers.

The cardinality of a set is a way to give a "size" to a set. If a set is finite, its cardinality is also finite. But there are also many, many kinds of infinite cardinal numbers. For instance, we often say there are "more" real numbers than there are integers, but there are JUST as many integers as there are rationals.

But cardinality is not the ONLY way to talk about the "size" of a set. There is a totally different concept called a measure. The measure of a set is based on the idea of the "width" of an interval. Where as [0, 1] has an infinite cardinality, it has a measure of 1.

Then, there are dozen subtly different kinds of infinities used in calculus and analysis.

Statements like "∞ - ∞ = 1" are totally meaningless. While we all agree on what "1+1" means, "∞+∞" depends very much on context. As it turns out, after algebra class, there are very few standard definitions for things, and symbols get recycled all the time to describe totally different ideas.

 

[trambolin]

Depending on what you mean by infinity, the rules differ. If you use the infinity as a compactification of, say the real line, as tac-tics used in the exploding race car, then, it is not the same infinity as in "x--> ∞" typically used in the limiting arguments. I know you will not believe me as I put it in one sentence. But:

As I have written a few times on this forum, please read the book "Where mathematics come from" by Lakoff, Nuñez.

This issue is treated completely in their book together with other issues that are seemingly paradoxical or definition dependent. It is a tough book to read (sometimes boring and quite repetitive) but pays off if you can bear with it.

Try to get an idea of the context in which infinity is definedand avoid using them in the places where they don't belong.

 

[kaleidoscope]

I mean ∞ is indeterminate in a way such that any of the following expressions can be constructed:

∞ - ∞ = 1,
∞ - ∞ = 0
∞ - ∞ = ∞

Is there anything in reality that can actually behave like that?

Thanks for the replies, this is what I had in mind:

If i have infinite coins and give away all but 1, then ∞ - ∞ = 1
If i have infinite coins and give away all of them, then ∞ - ∞ = 0
If i have infinite coins and give away every other coin, then ∞ - ∞ = ∞

So, I was wondering if we can really have an "infinite" amount of anything in the real/concrete world?

Keep the replies coming...

 

[planck42]

Infinity is much more of a mathematical concept than anything real, but periodicity is a good way to disguise it. For instance, closed paths are essentially periodic, and it's possible to go an infinite distance along them with a finite amount of space.

 

[trambolin]

If you have infinitely many coins how do you give away all of them? All at once, one by one?

 

[DaveC426913]

A physical circle is a polygon with an infinite number of sides... you could use a ruler to measure its circumference by adding the lengths of its infinite sides.

 

[TheAlkemist]

A straight line is also a circle with infinite diameter. Gotta love mathematics.

 

[Dadface]

A straight line is also a circle with infinite diameter. Gotta love mathematics.

Doesn't a straight line have two ends?Perhaps we could have a straight line of infinite length whose two ends meet to make a circle of infinite diameter.The thoughts making my brain ache with the severest of aches.Time for a cuppa.:yuck:

 

[DaveC426913]

A straight line is also a circle with infinite diameter. Gotta love mathematics.

I think though the OP is wondering if there are any infinities that really exist in nature, not just conceptually.

 

[TheAlkemist]

I think though the OP is wondering if there are any infinities that really exist in nature, not just conceptually.

Other than the infinite stupidity that characterize some of our politicians, I for one don't know about any. Are there any cases where mathematics has used infinities to model real world phenomena?

 

[skeptic2]

In optics, when the rays of light are parallel, the object is at infinity.

 

[mxbob468]

A physical circle is a polygon with an infinite number of sides... you could use a ruler to measure its circumference by adding the lengths of its infinite sides.

except there are no perfect circles so every physical circle has a finite number of edges

 

[yrreg]

Originally Posted by DaveC426913
I think though the OP is wondering if there are any infinities that really exist in nature, not just conceptually.​

Other than the infinite stupidity that characterize some of our politicians, I for one don't know about any. Are there any cases where mathematics has used infinities to model real world phenomena?

Before anything else, I am not a mathematician except that I know the operations of + - * and / and also operation of squaring a number like 9, and I can square any number that does not take so long to complete as to tire me out before I can finish it.

That said, what I know about the concept of infinity is that a thing that has the attribute of infinity has no beginning and no ending.

In regard to time it is always existing, so that you cannot come to a point in time before which it was not existing nor a point in time after which it does not exist anymore.

In regard to space it is everywhere, so that you cannot come to a point in space beyond which it is not existing.

And in regard to power to produce something, it can produce anything you can think of or it itself can think of, and it can think of anything that is thinkable.

What about infinite repetition of a question like if God made everything who made God and who made the God who made everything and who made the God who made the God who made everything and on and on, the socalled infinite regression that is supposed to be a argument that God does not exist.

Of course that kind of an infinite is not anything but a repetition of the same sentence, and I will just say that you cannot repeat that sentence infinitely because you don't live forever to repeat it, besides there was a time you were not existing.

Now that I have brought in God, I would say that the only entity that is infinite as I understand infinite in regard to time and space and power, God is the only infinite entity.


What about in mathematics, is the concept of infinity a mere concept without any objective existence?

The way I think I would say yes, it is only a mere concept unless you apply it to God as I have described God as being infinite.

And when mathematicians use the concept of infinity in their computations or calculations, the infinite is no longer infinite, not anymore in the sense of not bounded by any beginning or any ending, but is already understood as being finite or bounded by borders.

Asking mathematicians here, is that correct? so that in computations or calculations the infinite means something so huge but still limited or so small but still not endlessly small for there is a smallest beyond which it cannot be any smaller.

Otherwise, if mathematicians in using the infinity in their computations or calculations as really something that is without any limits, then it is impossible for them to ever finish their computation or calculation (are they not the same, computation and calculation?) of anything that needs computation or calculation to arrive at an answer.

Why? Because mathematicians are humans who at one time did not exist and eventually will stop existing upon death.


What do the mathematicians here say?




Yrreg

 

[jambaugh]

Infinite = in-finite = not finite = not bounded.
Typically you see an "infinity" (in application) as a place holder for a boundary which does not exist or which we wish to leave ambiguous.

For example, in considering intervals (a,b) = {x: a<x<b}
Then "infinite intervals" are intervals lacking a boundary e.g.
{x: x<b} = {x: -infinity < x < b} = (-infinity, b).

Similarly when measuring we in actuality measure to finite resolution and with practical upper and lower limits so our set of measurable values is necessarily finite (e.g. the set of marks on your measuring tape or meter scale or the number of values your digital meter can resolve). But we wish to work with different resolutions and bounds in the same context so we invent the concept of all the rational numbers or all the real numbers to express the lack of a boundary to scale or resolution. (We also need it to consider averages of arbitrarily large numbers of measurements.)

Mathematically of course we can define anything we like as long as we are consistent and rigorous. Note there is a distinction between cardinal and ordinal infinities. (Cardinality = count, ordinality = position in a sequence)

It is instructive to see the sequence of definitions leading to the extended real numbers (the reals plus + and - infinity).

(Note what follows is one of many variations.)

First we define the whole numbers as finite cardinal numbers (counts of elements of constructed sets). The entire set of whole numbers is defined by defining a set upon which one may iterate to yield successors and postulating that there is no largest element. (0 is in the set, given n is in the set then so is n++ = n+1).

In essence the whole numbers as a set is the container of all the nested containers of the form:
{}=0 subset of{0}=1 subset of {0,1}=2 subset of {0,1,2}=3 ... subset of N.

Next we define the negatives to get the whole set of integers. Typically you can define the integers as acts of translation on the whole numbers, i.e. as discrete vectors. With this we get the definition of addition as the composition of acts of translation.

Next we define multiplication (iterated addition) and quotients (equivalence classes of ratios) to define the rational numbers. We can order these rationals, larger to smaller and so draw a "rational number line".

To get the real numbers the classic method is to define http://en.wikipedia.org/wiki/Dedekind_cut" , pairs of sets containing together all the rationals, and segregating them so all the elements of one set is greater than each element of the other. Picture a cut in the ordered sequence of rational numbers. These cuts define the real numbers. (A Dedekind cut is "rational" if the set of larger elements has a smallest element or if the set of smaller elements has a largest element.)

Finally, noting that one can map the points on the real number line to the points on an open line segment one defines the "end points" as + and - infinity to yield the extended real number line. We however loose the normal arithmetic properties we had with the real number line. This is a nice way to express the calculus concept of a limit.

One may also map the real line to a circle so that instead of +/- infinity one has a single point at infinity (the point opposite 0 on the circle). This carries nicely into higher dimensions (see http://en.wikipedia.org/wiki/Stereographic_projection" )

In nearly all cases the infinite objects emerge as boundaries tacked on where no boundary previously existed. (biggest container of natural numbers, boundary between rationals, endpoints for the reals).

 

[CRGreathouse]

That said, what I know about the concept of infinity is that a thing that has the attribute of infinity has no beginning and no ending.

Philosophically, perhaps. Not mathematically.

[0, 1] has a beginning (0) and an end (1) but is infinite -- in fact, uncountably infinite. omega has a least element, {}, but no greatest element. It's not clear how you'd define "beginning" or "end" for beth_2.

You see, there are many mathematical objects described as infinite (that's why mathematicians don't use the term "infinity"), but they all have different properties, and these properties are mostly at odds with your understanding.

 

[CRGreathouse]

What do the mathematicians here say?

Almost everything you wrote was wrong.

In regard to time it is always existing, so that you cannot come to a point in time before which it was not existing nor a point in time after which it does not exist anymore.

This is cosmology, not math. The principle you refer to is being eternal, not infinite.

It's not at all clear that this is true. Current research suggests this is false.

In regard to space it is everywhere, so that you cannot come to a point in space beyond which it is not existing.

This is cosmology, not math. The concept you refer to is unboundedness, not infinity.

It's also not clear that this is true. Current research suggests that this is partially true and partially false: that the universe is finite but unbounded.

And in regard to power to produce something, it can produce anything you can think of or it itself can think of, and it can think of anything that is thinkable.

This is theology (and ontology), not math. That property is called omnipotence, not infinity. I don't believe it's well-defined in either field, though!

What about infinite repetition of a question like if God made everything who made God and who made the God who made everything and who made the God who made the God who made everything and on and on, the socalled infinite regression that is supposed to be a argument that God does not exist.

Of course that kind of an infinite is not anything but a repetition of the same sentence, and I will just say that you cannot repeat that sentence infinitely because you don't live forever to repeat it, besides there was a time you were not existing.

This argument belongs to metaphysics and theology, not math. I will point out that it can be phrased in finite form, and that there's no need to pronounce an argument to make it valid. (Philosophers and theologians debating this point typically have stronger arguments on both sides; Google for some.)

Now that I have brought in God, I would say that the only entity that is infinite as I understand infinite in regard to time and space and power, God is the only infinite entity.

Honestly, I don't think you can even define the key terms here ("God", "infinite") so I tend to dismiss your assertion.

What about in mathematics, is the concept of infinity a mere concept without any objective existence?

The way I think I would say yes, it is only a mere concept unless you apply it to God as I have described God as being infinite.

I don't understand how you can claim that (1) time, (2) space, (3) omnipotence, and (4) God are infinite, and tacitly assume that (1) - (4) exist, while still apparently denying that things can be infinite.

And when mathematicians use the concept of infinity in their computations or calculations, the infinite is no longer infinite, not anymore in the sense of not bounded by any beginning or any ending, but is already understood as being finite or bounded by borders.

Asking mathematicians here, is that correct?

No.

so that in computations or calculations the infinite means something so huge but still limited or so small but still not endlessly small for there is a smallest beyond which it cannot be any smaller.

This is absolutely not what mathematicians mean by the infinite. It may be similar to what physicists refer to as infinite, though; you'd have to ask them.

Otherwise, if mathematicians in using the infinity in their computations or calculations as really something that is without any limits, then it is impossible for them to ever finish their computation or calculation (are they not the same, computation and calculation?) of anything that needs computation or calculation to arrive at an answer.

Why? Because mathematicians are humans who at one time did not exist and eventually will stop existing upon death.

You wrongly assume that calculations involving "infinity" must be infinite in length. I can manipulate the number 1,000,000 mathematically in less than a million steps, so why would you assume I can't manipulate $$\infty$$ with less than $$\infty$$ steps?

 

[yrreg]

Is "Infinity" only a mathematical concept or is there anything infinite in reality?

Thanks, jambaugh, for your kind and generous reply.


I just want to ask you now to focus your attention on the title of this thread -- no I am not the author, but I just happen to have the same interest as the author -- the title namely:

• Is "Infinity" only a mathematical concept or is there anything infinite in reality?

Please give a concise and precise and clear answer to the question in the title of this thread.


Just first choose one concept any of infinity in mathematics and answer the question posed in the title of this thread; or you can give several concepts and answer the question in regard to each concept of infinity you bring up from your knowledge of mathematics.


Yrreg

 

[Hurkyl]

I just want to ask you now to focus your attention on the title of this thread -- no I am not the author, but I just happen to have the same interest as the author -- the title namely:

• Is "Infinity" only a mathematical concept or is there anything infinite in reality?

Please give a concise and precise and clear answer to the question in the title of this thread.

There were already two examples. Modifying slightly, all of our successful theories of physics assert that there are "points" in the universe, and while there are limitations in our ability to identify and distinguish them, the cardinality of the set of all points is clearly $2^{\aleph_0}$. Several theories assert there is something called "volume" one may be interested in about spatial regions, and have distinguished spatial slices whose volume is the extended real number $+\infty$.

It's not too hard to take our latitude/longitude coordinate system and tweak it so that we name places on the Earth's surface with points of the projective complex numbers. Under one natural scheme, the North geographic pole is the number $\infty$.

 

[CRGreathouse]

See how Hurkyl uses meaningful mathematical terms

$2^{\aleph_0}$

$+\infty$

$\infty$

instead of the philosophical term "infinity"?

I just thought I should point that out. It's one easy and fairly successful way to distinguish those with and without mathematical training in these sorts of discussions.

 

[disregardthat]

"Infinity" in mathematics is only a semantic term we attribute whenever it is intuitively appropriate. We can say that a set if "infinite", but formally we may mean that there is an injection from the natural numbers to the set. We can say that a straight line in the real plane is infinitely long, but formally mean that it has a parametrization as such: x = t, y=at+b for real t. "Infinite" is just a label; the mathematical equivalent in each situation has nothing to do with infinity per se.

We can extend the real numbers with $$\infty$$ by incorporating the symbol with applicable formal rules which allow us to say e.g. that $$\infty > r$$ for all real numbers r - and appropriately call the symbol infinity - but $$\infty > r$$ is only a syntactical statement and has no "deeper" philosophical meaning beyond what ordinary real numbers have.

 

[themaestro]

Dont think the equations are vaild as infinity is not a number so can not have e.g A=infinity. You can only say A tends to infinity. (and here A can not be a number because a number is fixed and can not tend. A must be an expression, e.g the sum of a infinite series, if for example y=1/x, the you would say y tends to infinity as x tends to 0. y has no value at x=0, it does not equal infinity at x=0)

now if A tends to infinity and B tends to infinity then you can say A+B tends to infinity, but not A+B= infinity.
what A-B tends to depends on exactly what A and B represent. If for example A is the sum of all integers and B is the sum of the squares of All integers, then A-B tends to -infinity (As B tends to infinity faster than A). If B=A-1 then A-B=1 (as you would expect!) However remember that neither A or B equal infinity here. So You can't say what does Infinity- infinity equal as infinity is not a number, just as you can't ask what does BOB - Alice equal, Doesn't mean Bob and Alice don't exist

 

[HallsofIvy]

Thanks, jambaugh, for your kind and generous reply.


I just want to ask you now to focus your attention on the title of this thread -- no I am not the author, but I just happen to have the same interest as the author -- the title namely:

• Is "Infinity" only a mathematical concept or is there anything infinite in reality?

Please give a concise and precise and clear answer to the question in the title of this thread.

Then you have posted this in the wrong section. People have been discussing "infinity" in the mathematical sense because this is the "mathematics" section. If you want to find out what, if anything, "infinity" has to do with reality you will have to post in the Physics section.

Just first choose one concept any of infinity in mathematics and answer the question posed in the title of this thread; or you can give several concepts and answer the question in regard to each concept of infinity you bring up from your knowledge of mathematics.

As I just said, you can't. Mathematics does not concern itself with "reality". Mathematics can be applied to reality but you will have to ask this question of those who apply mathematics.

Yrreg

 

[mxbob468]

There were already two examples. Modifying slightly, all of our successful theories of physics assert that there are "points" in the universe, and while there are limitations in our ability to identify and distinguish them, the cardinality of the set of all points is clearly $2^{\aleph_0}$. Several theories assert there is something called "volume" one may be interested in about spatial regions, and have distinguished spatial slices whose volume is the extended real number $+\infty$.

It's not too hard to take our latitude/longitude coordinate system and tweak it so that we name places on the Earth's surface with points of the projective complex numbers. Under one natural scheme, the North geographic pole is the number $\infty$.

the assumption of these theories is that space is continuous (versus discrete, maybe contiguous is better). this is not known for certain.

 

[disregardthat]

Dont think the equations are vaild as infinity is not a number so can not have e.g A=infinity. You can only say A tends to infinity. (and here A can not be a number because a number is fixed and can not tend. A must be an expression, e.g the sum of a infinite series, if for example y=1/x, the you would say y tends to infinity as x tends to 0. y has no value at x=0, it does not equal infinity at x=0)

now if A tends to infinity and B tends to infinity then you can say A+B tends to infinity, but not A+B= infinity.
what A-B tends to depends on exactly what A and B represent. If for example A is the sum of all integers and B is the sum of the squares of All integers, then A-B tends to -infinity (As B tends to infinity faster than A). If B=A-1 then A-B=1 (as you would expect!) However remember that neither A or B equal infinity here. So You can't say what does Infinity- infinity equal as infinity is not a number, just as you can't ask what does BOB - Alice equal, Doesn't mean Bob and Alice don't exist

This is how classical analysis treats the notion of infinity, but it does not mean however that infinity as a symbol cannot be axiomatized in the same way as numbers and manipulated in a similar fashion. We are perfectly capable of treating infinity as a number (in a general sense). Physical intuition does not stop the imagination of mathematics.

 

[Hurkyl]

the assumption of these theories is that space is continuous (versus discrete, maybe contiguous is better). this is not known for certain.

Of course it's not known "for certain". But unless you have a better way to study reality than science, it's the state of art of knowledge of reality.

 

[mxbob468]

Of course it's not known "for certain". But unless you have a better way to study reality than science, it's the state of art of knowledge of reality.

yes i wasn't controverting that.

but OP asked for known examples of infinity.

 

[Feldoh]

In physics you come across a lot of problems where the units in one part of the problem might be a few orders of magnitude larger or smaller than another part of a problem. All of these problems can be approximated as an infinite difference.

Infinities come up a lot in approximation schemes, which typically are needed when analytic solutions cannot be found.

 

[cuallito]

For a physicist infinity just means "as the numbers get really large." It only makes since as a limit. There have been attempts for formalize the idea of infinity, like hyperreal numbers: http://en.wikipedia.org/wiki/Hyperreal_number

Georg Cantor went insane pondering infinity.

So to answer your question, your original formulae don't make much sense but we can modify them to:

lim x->∞, x - x = ? which is of course, zero.

However, a quantity can "go" to infinity at different speeds: lim x->∞, 2*x - x = x = ∞

Infinity isn't a number.

 

[Hurkyl]

For a physicist infinity just means "as the numbers get really large."

Well, yes and no. Many applications are like that (and in math and computer science too). However, physics also has a great need to work with notions of the infinite infinite in exact and precise forms as well. Consider, for example, how often delta functions are used.

Infinity isn't a number.

*sigh*

There is not a real number, an integer, or a complex number named infinity. There is a projective complex number named $\infty$. There are two extended real numbers named $+\infty$ and $-\infty$. There are many cardinal numbers, ordinal numbers, and hyperreal numbers that, while they don't have "infinity" in their names, are infinite.



It's like nobody ever wants to actually learn about the subject, and just want to continue spouting all the half-baked notions that circulate around.

 

[Skaffen]

An Infinite hotel filled with infinite people can still hold another infinite amount of people if they shuffle up through the rooms (i.e. people in room 1 move to 3, 2 to 4 etc).

The finite mind cannot grasp the Infinite. Then again Maths is where you go for answers that the mind barely grasps the questions to, so...

 

[CRGreathouse]

The finite mind cannot grasp the Infinite.

Sure it can. Most common conceptions of the infinite (e.g., omega, aleph null, beth_1, the projective $$\infty$$, the extended real $$\pm\infty$$, the $$\infty$$ of the Riemann sphere) are easy to understand, certainly much easier than, say, Hardy & Littlewood's circle method.

 

[CRGreathouse]

Then you have posted this in the wrong section. People have been discussing "infinity" in the mathematical sense because this is the "mathematics" section. If you want to find out what, if anything, "infinity" has to do with reality you will have to post in the Physics section.

I agree on the "wrong section" critique. But it's broader than just physics -- I suggest also philosophy, at the least -- metaphysics, ontology, and theology at least.

 

[Skaffen]

Sure it can. Most common conceptions of the infinite (e.g., omega, aleph null, beth_1, the projective $$\infty$$, the extended real $$\pm\infty$$, the $$\infty$$ of the Riemann sphere) are easy to understand, certainly much easier than, say, Hardy & Littlewood's circle method.

Mathematical infinities are not Infinities when they are bounded by the context they were framed in. A circle can be considered a local infinity, so in a well understood regulated context you are right...not difficult.

Knowing is not understanding - I know the configuration analogy (Infinite Hotel) but I am no closer to understanding the Infinite because it can never be demonstrated (wouldn't be any room left for anything else), like thinking of Nothing (Including space)...just not built that way.

Easy in Maths cos Maths works in 2 dimensions, plenty room for a few infinities :)