Exploring Infinity: Is it a Math Concept or Real?

[CRGreathouse]

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.

I don't find the argument "I don't understand __, therefore no one can" very convincing. Suffice it to say that most topics in math are harder than this to understand, and most are less well-understood.

 

[Skaffen]

I don't find the argument "I don't understand __, therefore no one can" very convincing.

Me neither, which is why you cannot find me saying it. I'm suggesting that without experiencing something it is hard for anyone to relate to or understand it. If your experience leads you to play with Mathematical infinities then I'm sure many can relate to those infinities. However the OP seeks illustration of an Infinity beyond circles and parallel lines or many other such abstractions. 2 dimensional infinities don't burn the eyes out of your head...well mibbe hmmmmm,

I merely point to the folly of pursuing non-mathematical Infinity, it cannot exist in our Universe ...and leave any room for anything else

 

[jambaugh]

Thanks, jambaugh, for your kind and generous reply.
...

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

I thought my first reply addressed this. We invoke "infinity" as a place-holder for cases where we do not wish to or cannot specify a boundary.

Being explicit, if you so desire I'll say straight out:

No. Reality (or a better word Actuality) is finite. Infinities arise only in counting the possible, not in counting(/measuring) the actual.

 

[CRGreathouse]

Me neither, which is why you cannot find me saying it.

Please explain what you mean by

Knowing is not understanding - I know the configuration analogy (Infinite Hotel) but I am no closer to understanding the Infinite

then.

I merely point to the folly of pursuing non-mathematical Infinity, it cannot exist in our Universe ...and leave any room for anything else

This thread was posted on a math forum, so we're explicitly addressing mathematical infinities. If you want to talk about samsara, the omnipotence paradox, the unboundedness of space, Olbers' paradox, or the like you should probably look at some of the other forums here.

 

[Robinsmermaid]

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

Energy is infinite

 

[zetafunction]

for us the PHYSICIST is just a 'misunderstood' of nature, in NATURE there is no 0 or INFINITY it something is 0 is just because approximations or because we have neglected something

if something is INFINITE we must 'substract' some finite part from it in order to get physical answers

the main example of how INFINITY is just an 'illusion' is renormalization theory, in renormalization mathematical calculations tell you a mass or charge is infinite however when testing you get finite masses.

 

[M Grandin]

But still infinite is not more than that you can see every point of a for instance infinite
straight bar. You even see the end of it - as a distinct point on the hemisphere. Who said infinity was endless?

 

[DaveC426913]

for us the PHYSICIST is just a 'misunderstood' of nature

No. It is a hint that we may have misunderstood nature.

I doubt any physicist would agree with you that there are no zeros or infinities.

 

[Robinsmermaid]

for us the PHYSICIST is just a 'misunderstood' of nature, in NATURE there is no 0 or INFINITY it something is 0 is just because approximations or because we have neglected something

if something is INFINITE we must 'substract' some finite part from it in order to get physical answers

the main example of how INFINITY is just an 'illusion' is renormalization theory, in renormalization mathematical calculations tell you a mass or charge is infinite however when testing you get finite masses.

It all seems like a grand illusion doesn't it.

 

[Robinsmermaid]

No. It is a hint that we may have misunderstood nature.

I doubt any physicist would agree with you that there are no zeros or infinities.

Well if there are no zeros then what does zero even represent except a concept? If I have 5 apples and take all of them away, I have zero of course...but it's simply the concept of that b/c the apples are "Still" in existence. Correct? And if they become eaten they still exists but in another form. So numbering is the math, the numbers really just the concept? Or is the math an expression of what is?

 

[Boy@n]

Existence must be infinite, or say eternal, and proof of that is our own existence.

If that wouldn't be true, it would mean we (or Universe) came into being out of pure nothing.

(Quantum fluctuations are not "pure nothing" IMO, that which is capable of creating, even if not detectable by us, is still something.)

 

[Robinsmermaid]

Existence must be infinite, or say eternal, and proof of that is our own existence.

If that wouldn't be true, it would mean we (or Universe) came into being out of pure nothing.

(Quantum fluctuations are not "pure nothing" IMO, that which is capable of creating, even if not detectable by us, is still something.)

well let me ask you this...if mass functions one way in one form and completely differently in another form, does this mean the previous form of functioning has in and of itself come to a complete end? Such as human beings? It reminds me of Humpty Dumpty.

 

[M Grandin]

But still infinite is not more than that you can see every point of a for instance infinite
straight bar. You even see the end of it - as a distinct point on the hemisphere. Who said infinity was endless?

My statement here may appear somewhat strange or even nonsense. But from a perspective point of view you may say that the infinite straight bar is limited and seen in its full length. The whole length is mapped to a finite line - from close end to "vanishing point". Or between two "vanishing points" if both ends are att infinity, where one of them is behind you. Maybe somewhat philosophical and by "seeing" all points of bar I of course don´t mean you are able watching farthest points in practice.

 

[yuiop]

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.

A google search on this subject shows that defining a circle as a polygon with an infinite number of sides is slightly contentious. It might be slightly better to say that in the limit that the number of sides of a regular polygon goes to infinity, the perimeter approaches 2*pi*r and the area approaches pi*r^2. Here is my little proof for the limit of the circumference for a regular polygon as the number of sides goes to infinity:

Divide the circumference of a circle into n points. Join each point to its closest two neighbours and to the centre of the circle. You now have n isosceles triangles each with an angle of 2*pi/n at the centre. Divide each isosceles triangle into two again at the corner nearest the centre so that each isoscelese triangle is now two right angle triangles. The length of the side of the right angled triangle opposite the centre is simply r*sin(pi/n). (See attached diagram). The total perimeter length C of the polygon is then 2*n*r*sin(pi/n). This can be expressed as:

$$C = \frac{2 r \sin(\pi/n)}{n^{-1}}$$

In this form the limit can be found using L'Hopital's theorem and differentiating the top and bottom of the fraction to obtain:

$$\frac{-2 \pi r \cos(\pi/n) n^{-2}}{-n^{-2}} = 2 \pi r \cos(\pi/n)$$

Since cos(0)=1 it is easy to see from the above, that in the limit that n goes to infinity, the perimeter length of the regular polygon goes to 2*pi*r.

 

[yuiop]

An interesting example of infinite is found in the logarithmic spiral. http://www.mathpages.com/home/kmath492/kmath492.htm When following the path of the spiral into the centre, an infinite number of turns is required, but the length of the path itself is finite. That is just downright weird to me :tongue:

In physics, the coordinate time for a object to fall to the event horizon of a Schwarzschild black hole is infinite, but the proper time recorded by a clock attached to the object is finite. In this case infinite can be transformed to finite. This points to the "many kinds of infinite" that George Cantor introduced us to. Some infinites are bigger than others. Some can be counted and others can not be counted even in principle. Some can be transformed to finite values as mentioned above. Time *might* be infinite and space *might* be infinite depending on what the geometry of the universe turns out to be. Wikipedia says "As to date, analysis of the radiation patterns recorded by the WMAP spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe". I certainly find it hard to imagine a boundary to the edge of the universe with a sign saying "no more space beyond this point - turn back!".

 

[slider142]

I certainly find it hard to imagine a boundary to the edge of the universe with a sign saying "no more space beyond this point - turn back!".

There is also the possibility of a finite large-scale topology with no boundary; ie., in a 2-dimensional universe, the surface of a sphere. A sphere with vanishingly small curvature would be difficult to distinguish from a flat large-scale topology.

 

[tomwilliam]

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.

Yrreg

I'm not sure the physicists would agree with this. Isn't the big bang thought of as the point at which time and space began? I think time and space are essentially one concept, and they are not necessarily infinite.

 

[vish_al210]

Well I removed the post myself... sorry for not being able to define infinite as all of you perceive, as well nothing is infinite...

 

[SpeedOfDark]

For a long time infinity was used to prove things such as why 1 + 1= 2 but infinity was a very tricky subject and yes many things behave infinitely. Numbers are infinite, there infinitely number of infinities. Picture this every irrational number is an infinite of decimal places, infinity is a very useful concept and it is one we need in mathematics. Although it doesn't help us build cars, planes, or nuclear bombs it's an awesome concept.

Georg Cantor is the mathematician who really started all the work on infinity, and he is one of the greatest mathematicians of all time, and he many suppose he went crazy because of the ideas of the infinite.

 

[Hurkyl]

For a long time infinity was used to prove things such as why 1 + 1= 2 but infinity was a very tricky subject and yes many things behave infinitely. Numbers are infinite, there infinitely number of infinities. Picture this every irrational number is an infinite of decimal places, infinity is a very useful concept and it is one we need in mathematics. Although it doesn't help us build cars, planes, or nuclear bombs it's an awesome concept.

Please don't confuse people (or propagate your confusion if you are confused yourself). The notions of an ordered set of numbers being infinite, a particular number being infinite, and a particular representation of a number being infinite by some measure are all very different things.

While it is true that the set of real numbers is infinite¹, and that most real numbers have only infinite² decimal representations, it is also true that every real number is finite³.

1: Meaning that the cardinality of the set is not a natural number
2: Meaning that the cardinality of the set of places where the decimal has a nonzero digit is not a natural number. (Specifically, it is $\aleph_0$)
3: "x is finite" means that there exists a natural number n such that |x| < n

 

[SpeedOfDark]

Please don't confuse people (or propagate your confusion if you are confused yourself). The notions of an ordered set of numbers being infinite, a particular number being infinite, and a particular representation of a number being infinite by some measure are all very different things.

While it is true that the set of real numbers is infinite¹, and that most real numbers have only infinite² decimal representations, it is also true that every real number is finite³.

1: Meaning that the cardinality of the set is not a natural number
2: Meaning that the cardinality of the set of places where the decimal has a nonzero digit is not a natural number. (Specifically, it is $\aleph_0$)
3: "x is finite" means that there exists a natural number n such that |x| < n

While I mean be entirely confused on somethings I'm certain that there is an infinite number of numbers, and an infinite number of numbers between any to numbers.

For this next part I hope someone better mathematically will come alone, and I pose a question.

Isn't a boundless limit infinite and can't this be described by some exponential increase that is infinitely expanding?

 

[Char. Limit]

Human stupidity can be infinite sometimes.

 

[vish_al210]

I have a question folks..
Even a random number becomes finite once we have generated it. though the possibility of what the number will be is infinite, the number by itself is finite as the bounds are finite.
Even in case of an ADC read of an AC analogous value, independently changing with time, the value measured at any static reference of time is finite, where as the possibility of what may be read is infinite.
So as the question posed in this thread, Is anything infinte? (In a static time reference frame, I guess not.) I mean once it occurs or is generated or penned down, the value the system takes is finite (could have been anything but once taken is finite)
Please clarify.

 

[SpeedOfDark]

I don't think anything is infinite... I think it's only a concept and not a physical reality.

 

[bcrowell]

I'm not sure the physicists would agree with this. Isn't the big bang thought of as the point at which time and space began?

The way general relativity is typically formulated is in terms of a metric on a manifold. In that type of formulation, the big bang isn't actually a point on the manifold. On the other hand, you can do tricks like constructing conformal infinities, like in Penrose diagrams.

 

[HallsofIvy]

I have a question folks..
Even a random number becomes finite once we have generated it. though the possibility of what the number will be is infinite, the number by itself is finite as the bounds are finite.

I think you mean that there are an infinite number of possible numbers, not that a specific random number was "infinite" before it was generated.

Even in case of an ADC read of an AC analogous value, independently changing with time, the value measured at any static reference of time is finite, where as the possibility of what may be read is infinite.
So as the question posed in this thread, Is anything infinte? (In a static time reference frame, I guess not.) I mean once it occurs or is generated or penned down, the value the system takes is finite (could have been anything but once taken is finite)
Please clarify.

Hey, it was your idea! Only you can clarify it.

 

[Hurkyl]

I have a question folks..
Even a random number becomes finite once we have generated it. though the possibility of what the number will be is infinite

...

You appear to be quite confused -- you are treating "the cardinality of the sample space^* of a random variable" and "the magnitude of an outcome^**" as if they were talking about the same thing.

As an analogy, it would be like holding a deck of cards and saying "this is 52", then withdrawing the five of diamonds and saying "52 changed to 5".



*: the sample space is, loosely speaking, the set of 'possibilities' of a random variable
**: loosely speaking, an outcome is one of your 'possibilities', assuming I understand you're meaning.

 

[robert Ihnot]

If I remember my Calculus right: Suppose we have bored a hole with radius beginning at 1, and equal to 1/x the depth of the hole at that point. Then the total area of this hole is: $$\int_1^\infty\frac{2\pi dx}{x}$$, which is infinite.

But when it comes to the volume, we have $$\int_1^\infty\frac{\pi dx}{x^2} =\pi$$

So that this hole can not be painted, but it can be completely filled with paint!

 

[disregardthat]

In physics, the coordinate time for a object to fall to the event horizon of a Schwarzschild black hole is infinite, but the proper time recorded by a clock attached to the object is finite. In this case infinite can be transformed to finite. This points to the "many kinds of infinite" that George Cantor introduced us to. Some infinites are bigger than others. Some can be counted and others can not be counted even in principle.

This theoretical phenomenon has nothing to do with different infinite cardinalities!

 

[M Grandin]

If I remember my Calculus right: Suppose we have bored a hole with radius beginning at 1, and equal to 1/x the depth of the hole at that point. Then the total area of this hole is: $$\int_1^\infty\frac{2\pi dx}{x}$$, which is infinite.

But when it comes to the volume, we have $$\int_1^\infty\frac{\pi dx}{x^2} =\pi$$

So that this hole can not be painted, but it can be completely filled with paint!

Interesting case. The same result also holds for a cylinder of radius r and length
1/ r ^ 2 if r approaches 0.

 

[Hurkyl]

Interesting case. The same result also holds for a cylinder of radius r and length
1/ r ^ 2 if r approaches 0.

The radius of a cylinder is a number, it cannot "approach 0".

While you are imagining an (infinite) family of different three-dimensional shapes, all having the same volume but different surface areas, the post you are responding to is referring to a single three-dimensional shape that has infinite area and finite volume.

 

[elabed haidar]

infinte doesn't always mean a number it also means that a number does not exist that's why if we talk about infinite -infinite it doesn't exist

 

[vaibhav1803]

infinity..?..a professor of mine defined it by this statement which is pretty easy to understand.
"pick a number, infinity is a number always greater than any number so chosen."
so ultimately infinity is a concept number...our way to imagine the end of a number line, or increasing sequence of numbers.
Speaking from a mathematical P.O.V. ur not really allowed to do algebra with infi.
but its like the are "types of infinities", so to speak some may be identical so may not be so.
for a better insight i'd suggest u pick up a book on limits & indeterminates to you know get a better look at algebra in limiting situations.
infi. (+/-) infi. may or may not exist, depending on what function produces the infinity. like
Exp(x)/x -1/x ---> 1 as x--->0 (note that i have an infi.-infi. form on the left.)

 

[chwala]

infinte doesn't always mean a number it also means that a number does not exist that's why if we talk about infinite -infinite it doesn't exist

infinity is like a tour that starts and no destination arrived at...it really cannot be defined...beyond any human thinking to be conceivable...and uncountable in the language of real analysis.

 

[M Grandin]

The radius of a cylinder is a number, it cannot "approach 0".

While you are imagining an (infinite) family of different three-dimensional shapes, all having the same volume but different surface areas, the post you are responding to is referring to a single three-dimensional shape that has infinite area and finite volume.

Thanks for comments. But how is it possible not understanding what I say here?
Maybe it is easier understand if you imagine the cylinder as elastic - and the radius resp length as temporary measurements expressed by numbers. Also a shape may change into a kind of limes when a variable is approaching a certain value.

I am not less confused by what you say after that: Where did I say this was the same
thing as what "OP" mentioned? I just mentioned you got the same values for area and volume as in "OP" example. But the kernel for this "interesting" result is the same. You can say my very simple example is a kind of "average" of OP example along total length approaching (excuse that word) infinity.

My example shows the "mechanism" behind "OP" example. How infinite area and limited volume may match.