Is Infinity only a mathematical concept or is there anything infinite in reality? (1 Viewer)

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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

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Re: Is "Infinity" only a mathematical concept or is there anything infinite in realit

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!
 
Re: Is "Infinity" only a mathematical concept or is there anything infinite in realit

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.
 
Re: Is "Infinity" only a mathematical concept or is there anything infinite in realit

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.
 
Re: Is "Infinity" only a mathematical concept or is there anything infinite in realit

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...
 
Re: Is "Infinity" only a mathematical concept or is there anything infinite in realit

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.
 
Re: Is "Infinity" only a mathematical concept or is there anything infinite in realit

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

DaveC426913

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Re: Is "Infinity" only a mathematical concept or is there anything infinite in realit

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.
 
Re: Is "Infinity" only a mathematical concept or is there anything infinite in realit

A straight line is also a circle with infinite diameter. Gotta love mathematics. :biggrin:
 
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Re: Is "Infinity" only a mathematical concept or is there anything infinite in realit

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

DaveC426913

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Re: Is "Infinity" only a mathematical concept or is there anything infinite in realit

A straight line is also a circle with infinite diameter. Gotta love mathematics. :biggrin:
I think though the OP is wondering if there are any infinities that really exist in nature, not just conceptually.
 
Re: Is "Infinity" only a mathematical concept or is there anything infinite in realit

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?
 
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Re: Is "Infinity" only a mathematical concept or is there anything infinite in realit

In optics, when the rays of light are parallel, the object is at infinity.
 
Re: Is "Infinity" only a mathematical concept or is there anything infinite in realit

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
 
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Re: Is "Infinity" only a mathematical concept or is there anything infinite in realit

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

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Re: Is "Infinity" only a mathematical concept or is there anything infinite in realit

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" [Broken], 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" [Broken])

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).
 
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CRGreathouse

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Re: Is "Infinity" only a mathematical concept or is there anything infinite in realit

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

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Re: Is "Infinity" only a mathematical concept or is there anything infinite in realit

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 [tex]\infty[/tex] with less than [tex]\infty[/tex] steps?
 
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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

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Re: Is "Infinity" only a mathematical concept or is there anything infinite in realit

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 [itex]2^{\aleph_0}[/itex]. 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 [itex]+\infty[/itex].

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 [itex]\infty[/itex].
 

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