What does infinity mean to a physicist

[curiouschris]

Infinity has always been a problem for me, apart from the concept I do not believe it exists. numbers go on to infinity and even negative infinity but they don't really exist so they don't count (hehe). some people say the universe is infinite in size. well apart from infinite in the same way as a spheres curved surface is infinite then no the universe cannot be infinite.

I raised this with a colleague, he said from a maths point of view (and presumably physics) that infinite just means immeasurably large but finite.

Is that true?

If so please call it something else other than infinite may I suggest $$\infty$$-1 (lol)

If its not true how can infinite be used within equations to describe 'real' things because its impossible.

 

[Pythagorean]

I don't think we've ever discovered infinity in nature. I tend to think it doesn't exist either. In physics, whenever we say infinity, we really mean "a huge number". So big, as it were, that it causes terms to go to zero in the approximation.

This simplifies mathematics and allows for series approximations. I don't think anything is really going to infinity though.

 

[zeromodz]

We do use infinity in nature. For example, a gravitational singularity which is found at the center of every black hole. We know that there is no degenerate matter that can maintain equilibrium with the force of gravity because it is too strong. There is no known force in the universe that can stop this incredible warping of space.

The result is an astronomically large quantity of matter being crushed into a point in space with zero volume and infinite density.

P = M / V
∞ = M / 0

 

[Stonebridge]

Infinity doesn't exist in the real, physical world. However it does exist in mathematics, a world we have invented.
We use mathematics to model the real world, and it works just fine until we get an infinity. Then we know that our mathematical model of the real world has broken down at that point.

 

[Superstring]

Infinite usually means that something is wrong with the current equations. In a singularity there seems to be infinite acceleration of gravity as well (which doesn't make any sense):

$$g=\frac{Gm_1m_2}{r^2}$$

Plug in the 0 for r:
$$g=\frac{Gm_1m_2}{0}=\infty$$

 

[zeromodz]

Infinite usually means that something is wrong with the current equations. In a singularity there seems to be infinite acceleration of gravity as well (which doesn't make any sense):

$$g=\frac{Gm_1m_2}{r^2}$$

Plug in the 0 for r:
$$g=\frac{Gm_1m_2}{0}=\infty$$

Actually, that makes perfect sense. It only has an infinite acceleration at zero volume. There is no such thing in space where there is zero volume, so I don't see the problem.

 

[curiouschris]

The problem is there can be no infinite. so there can be no infinite acceleration of gravity. if the singularity is the sum of all things that exist today (laymans interpretation) then the sum of all things today must equal the singularity.

So if there is no infinite gravity today then there wasn't in the singularity.

So in my most humble opinion the singularity cannot be modeled by the math or there was no singularity because it is an impossible state.

CC

 

[Superstring]

There is no such thing in space where there is zero volume, so I don't see the problem.

Well the problem is that GR predicts all matter in a black hole to be crushed into a singularity with zero volume. So yes, infinite acceleration due to gravity is a problem.

 

[Hurkyl]

Well the problem is that GR predicts all matter in a black hole to be crushed into a singularity with zero volume.

Actually, GR insists on a hole; filling that hole with a point of space-time is a violation of GR.

(And trying to use Newton's formula for gravitational "force" is absolute nonsense in this context)

 

[zeromodz]

Well the problem is that GR predicts all matter in a black hole to be crushed into a singularity with zero volume. So yes, infinite acceleration due to gravity is a problem.

Well, its only infinite at the very singularity. Even a 10^-99 meters apart from the singularity, the acceleration becomes finite.

 

[Bigbio2002]

Actually, that makes perfect sense. It only has an infinite acceleration at zero volume. There is no such thing in space where there is zero volume, so I don't see the problem.

Even the "singularity" at the center of a black hole has a finite, albiet small, volume. A singularity would imply that it has an exact position, which violates the uncertainty principle.

Infinity can be thought of as a tool, but I don't think it really exists in the conventional sense (except maybe in an infinite multiverse thingy, the universe itself is presumably finite. I say this with a B.S. degree in Layman's Physics).

 

[zeromodz]

Even the "singularity" at the center of a black hole has a finite, albiet small, volume. A singularity would imply that it has an exact position, which violates the uncertainty principle.

Infinity can be thought of as a tool, but I don't think it really exists in the conventional sense (except maybe in an infinite multiverse thingy, the universe itself is presumably finite. I say this with a B.S. degree in Layman's Physics).

What type of degenerate matter can stop gravity from crushing it into zero volume?

 

[collinsmark]

Infinity has always been a problem for me, apart from the concept I do not believe it exists. numbers go on to infinity and even negative infinity but they don't really exist so they don't count (hehe). some people say the universe is infinite in size. well apart from infinite in the same way as a spheres curved surface is infinite then no the universe cannot be infinite.

I raised this with a colleague, he said from a maths point of view (and presumably physics) that infinite just means immeasurably large but finite.

Is that true?

If so please call it something else other than infinite may I suggest $$\infty$$-1 (lol)

If its not true how can infinite be used within equations to describe 'real' things because its impossible.

Mathematically, infinity must exist because there must exist the infinitesimal.

For example, there are an infinite number of real numbers between 1 and 2 (technically the more appropriate term might be uncountable, but by its definition, still infinite).

One might argue that any practical number, such as the $$\sqrt{2}$$ which represents the diagonal of a square, might trail off with all zeros eventually, if one were to calculate out the decimal places far enough. But if that were true, it would be possible to represent $$\sqrt{2}$$ as a ratio of integers, albeit very, very large integers.

But here is rough proof that this is not possible. Suppose $$\sqrt{2}$$ can be represented by a ratio of integers, M/N (where M and N might be very large, yet still finite).

Suppose that we reduce M/N such that M and N are the smallest units possible. Now there are 3 possibilities regarding M and N.
(a) M and N are both odd.
(b) M is odd and N is even.
(c) M is even and N is odd.
Note that M and N cannot both be even, because then we could simply divide both by 2, and retain the same ratio.

Given the above, we could say,

$$\sqrt{2} = \frac{M}{N}.$$

Squaring both sides gives us

$$2 = \frac{M^2}{N^2}$$

which is,

$$2N^2 = M^2.$$

Here we can be confident that M is even because 2N is even: The the square of an odd integer gives an odd number. 2 times any integer is even thus 2N is even. Thus if M² is even, M must be even. That also implies that N must be odd (because M and N cannot both be even).

Since M is even, let's make a substitution that 2L = M.

$$2N^2 = {2L}^2,$$

$$2N^2 = 4L^2,$$

$$N^2 = 2L^2.$$

But this means that N must also be even, since multiplying any integer by 2 creates an even number. making both M and N even, which contradicts the original proposition. Thus $$\sqrt{2}$$ cannot be represented by a ratio of integers. $$\sqrt{2}$$ belongs in some different class of numbers, that cannot be represented by a ratio of finite integers (namely irrational numbers).

One way to interpret this is $$\sqrt{2}$$ has a truly infinite number of decimal places to represent it -- they do not trail off to all zeros eventually. Another way to interpret this is that $$\sqrt{2}$$ cannot be represented by a ratio of integers -- at least not finite integers -- at least not unless both M and N are infinite.

So if one believes that squares exist, and if one believes that squares have diagonals (i.e. one believes that isosceles right triangles exist), arguably one must believe that infinity exists (at least mathematically speaking). This logic might be debatable, but it does present a very strong argument.

 

[Bigbio2002]

Despite the fact that QED is a really nice, really accurate theory, there's no way it can be used to predict exactly how degenerate matter would behave at the core of a black hole. With that said however, it may indeed be packed into a small space, but it can't possibly have a zero volume due to the uncertainty principle. Degenerate matter is still matter, and matter takes up space. Just because relativity is flawed and suggests that an infinitely small singularity is possible, doesn't mean that it IS possible, especially in light of the fact that it's incomplete (non-quantum).

 

[Cryptonic]

There's no such thing as infinity. It's a fabrication of the human imagination. The important thing about "infinity" is that it shows up mathematics as an imperfect model of reality (I'm sure mathematicians/physicists hate to hear this, but it's TRUE).

 

[curiouschris]

Mathematically, infinity must exist because there must exist the infinitesimal

Why?

Unimaginably small is still finite. any thing smaller than that doesn't exist. The discussion is about reality not mathematical modeling.

Proof of infinity in math well that's easy, $$\frac{1}{0}$$ = $$\infty$$. it exists in the mathematical model and that is true.

Just like zero, a very important number in maths it does not relate to a real thing. I am holding zero rocks in my hands means there's no rocks there. I could quite easily say I am holding zero universes in my hand and be 100% correct. If I said I am holding an infinitesimally small universe in my hand, I'd be lying but no-one could PROVE IT. If I found it, then it must be finite. so whatever real thing I found at that scale it must be smaller again MUCH smaller, so no-one could prove I am lying.

infinitesimally small is as impossible in the real universe as infinitely big.

CC

BTW: each time you add a decimal point that's not zero it gets bigger not smaller so infinitesimally small must be by definition All Zero's

 

[Billson555]

Hey, this is my first post on the Physics Forums. I just find the idea of infinity interesting, so I though I'd comment. Zeno's Paradox comes to mind. and this might be a good example of how math is an imperfect representation of the world. the distance from one point to another can be separated into pieces, each 1/2 the length of the last (.5D, .25D, .125D, etc...). using summation notation, you could technically add up all the parts to equal the distance, D. but also, theoretically I think (as Zeno thought), one can infinitely divided D into an infinite number of pieces. thus, you would have to add an infinite number of increasingly infinitesimal distances. applying this to the real world, if one crossed one section of the distance at a time, then you could never get to the other point, because you would cross and increasingly infinitesimal distance, thus crossing and infinite number of distances. and infinite number of real numbers exist, both outside of and inside of 1. so if one actually applied this to the real world, because matter has volume and thus a dimension of length, there could theoretically be an infinite number of pieces of matter, increasingly infinitesimal. so you could make the conclusion, perhaps, that infinity exists in matter because the space around (and of) us is infinitely "deep" (is filled with an infinite number of infinitesimal particles of matter). which also brings to mind the oddity that since one does appear to go from one point and reach another, one can traverse infinity, just as one traverses infinity in math by counting from 1 to 2. but since math is really not a perfect model of the world, this likely isn't true. And although I don't really understand a singularity, is supposed to be an infinitely dense/small thing with extreme gravity? because one could make the argument that it has an infinite pull because it is nothing. nothing is technically infinity. I think of infinity as having no point in it because it has no beginning and no end. the only other thing I can think of that has no start or end is 0-nothing. if a singularity is nothing, then technically it is infinitely deep and pulls in matter infinitely. you can't measure it because it is infinite because it is nothing. I don't know. this is all really hypothetical. I'm not a real scientist, I'm just interested in physics, so I joined the forums.

 

[Hurkyl]

Sigh, allow me to correct some errors.


The only infinitessimal real number is zero.

1/0 is nonsensical, when viewed as arithmetic of real numbers.

Whether or not a mathematical structure has an element called "infinity" is entirely up to the mathematical structure.

I don't know of any (non-contrived) mathematical structures that:

• contain a non-zero infinitessimal number
• that number has an (infinite) reciprocal
• the reciprocal is called "infinity"

All math is, more or less, a "fabrication of the human imagination".

 

[collinsmark]

All math is, more or less, a "fabrication of the human imagination".

Yes, I suppose you're right. :tongue2:

Leopold Kronecker has a famous quote saying,

"Die ganzen Zahlen hat der liebe Gatt gemacht, alles andere ist Menschenwerk.
(The dear God has made the integers, all the rest is man's work.)"
-Leopold Kronecker

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

Anyway, slightly off topic (or maybe not!) here is fun game to play. I've actually stumped many mathematically inclined people with this riddle (many for well over 10 or 20 minutes).


Proof that 2 = 1
Find the flaw! (game/riddle)

We start with the postulate that a = x.

$$(1) \ \ \ \ \ a = x$$

Add a to both sides.

$$(2) \ \ \ \ \ a + a = x + a$$

Simplify left side,

$$(3) \ \ \ \ \ 2a = x+a$$

Subtract 2x from both sides,

$$(4) \ \ \ \ \ 2a - 2x = x + a -2x$$

Factor the left side,

$$(5) \ \ \ \ \ 2(a - x) = x + a -2x$$

Simplify the right side

$$(6) \ \ \ \ \ 2(a - x) = a - x$$

Divide both sides by (a - x)

$$(7) \ \ \ \ \ 2\frac{(a - x)}{(a-x)} = \frac{a - x}{a-x}$$

Simplify

$$(8) \ \ \ \ \ 2 = 1$$

Therefore 2 = 1. QED.

Obviously 2 is not equal to 1, so there must be a flaw in the above logic. Specify the step which has the (first) flaw and the reason it is flawed.

If you already know the answer, or if you figure it out, you might wish to keep it to yourself for awhile (on this thread) lest spoil it for the rest. :rofl:

 

[DrGreg]

Although there are some advanced branches of mathematics where an entity called "infinity" exists, in this post I refer to standard real numbers and complex numbers, which are what most physicists would use.

In the standard real number system there is no such number as infinity. Mathematicians do use the words "infinity" or "infinite", but only a shorthand for some other way of expressing it. When being rigorous, you should never write $x = \infty$, but you can write $x \rightarrow \infty$, which has a specific technical meaning.

To give an example, you might say that the energy of a particle tends to infinity as the particle's velocity tends to the speed of light ($E \rightarrow \infty$ as $v \rightarrow c$). But this means the same thing as "no matter how much energy you supply to a particle, it will never reach the speed of light". You specify an energy, as large as you like, and the particle can exceed that energy by going fast enough (but still slower than light). Similarly, any sentence containing the word "infinity" or "infinite" can be rephrased in terms of finite quantities.

If an equation of physics seems to give you an answer of infinity, that means you've misused the formula and it doesn't apply in that circumstance.

 

[curiouschris]

Thanks DrGreg. Thats about as concise an answer as I could have hoped for.

 

[IttyBittyBit]

I raised this with a colleague, he said from a maths point of view (and presumably physics) that infinite just means immeasurably large but finite.

This is simply not true. In mathematics, something that is finite, no matter how big, cannot be infinite. If he really said that then it reflects poor understanding of basic mathematics principles. Also, if you look at most number sets we use for calculations, such as N (natural numbers), Q (rational numbers) or R (real numbers), none of them include infinity. So if we consider the word 'number' to mean 'real number', then saying 'infinity is the largest possible number' makes no sense, because infinity is not a number.

In the end, mathematics could be considered as just manipulation of symbols. We are free to 'define' infinity in any way we want. Cantor, for example, defined many different types of infinity, some of which were infinitely larger than others. Others have created number systems which include infinity (for example, R plus +/- infinity) but these number systems cause a lot of problems and you need to be very careful to avoid making them self-contradictory. For example, if I naively defined infinity as 1/0, then I could also argue that 2/0 = 2*(1/0) = 2*infinity = infinity, so 2/0 = 1/0 ==> 2*x = 1*x, where x=/=0, so I have proved that 2 equals 1. Even if you manage to create a logical, consistent number system that includes infinity, it would wind up losing many of the nice properties that 'normal' number systems enjoy.

Infinity is, admittedly, a hard concept for beginners in mathematics to understand. In mathematics, we (try to) have a solid set of rules which we follow to reach conclusions, and infinity is no different from anything else in this regard. Infinity does not occupy a special or problematic position in math.

 

[curiouschris]

Thinking back on what my colleague said I probably owe him an apology. his quote was more like "immeasurably large" I think I may have added the "but finite"

I come here as a layperson. trained in neither maths nor the other sciences. but I also like to ponder these areas. Thus if something doesn't equate I like to discover why. Usually its just my poor understanding of the subject matter.

When physicists discuss matters with people not in their fields its not uncommon for words to mean different things to different people. thus when a word like infinite is used it conjures different images to me (as the layperson) than to say another physicist or mathematician. Say someone says the universe is infinitely large (why do I hear Carl Sagan whenever I think that) then it most probably means a different thing to me than the speaker.
To me it means never ending, and not in the way walking around the Earth forever is never ending, that is simply repetitious. In the way that traveling at 1 Billion * C in a true straight line (no space time distortion) for trillions of years would result in me being billions of trillions of light years away from the starting point and still the journey has not gone long enough to say its even begun.

To me this is clearly not possible.

The use of the above description is not meant to provoke a discussion.

Infinite acceleration of gravity was another one mentioned in this thread, again clearly impossible when looked at from my understanding of the word infinite.

So I sought a better definition of 'Infinite'

CC

 

[Superstring]

Yes, I suppose you're right. :tongue2:

Leopold Kronecker has a famous quote saying,

"Die ganzen Zahlen hat der liebe Gatt gemacht, alles andere ist Menschenwerk.
(The dear God has made the integers, all the rest is man's work.)"
-Leopold Kronecker

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

Anyway, slightly off topic (or maybe not!) here is fun game to play. I've actually stumped many mathematically inclined people with this riddle (many for well over 10 or 20 minutes).


Proof that 2 = 1
Find the flaw! (game/riddle)

We start with the postulate that a = x.

$$(1) \ \ \ \ \ a = x$$

Add a to both sides.

$$(2) \ \ \ \ \ a + a = x + a$$

Simplify left side,

$$(3) \ \ \ \ \ 2a = x+a$$

Subtract 2x from both sides,

$$(4) \ \ \ \ \ 2a - 2x = x + a -2x$$

Factor the left side,

$$(5) \ \ \ \ \ 2(a - x) = x + a -2x$$

Simplify the right side

$$(6) \ \ \ \ \ 2(a - x) = a - x$$

Divide both sides by (a - x)

$$(7) \ \ \ \ \ 2\frac{(a - x)}{(a-x)} = \frac{a - x}{a-x}$$

Simplify

$$(8) \ \ \ \ \ 2 = 1$$

Therefore 2 = 1. QED.

Obviously 2 is not equal to 1, so there must be a flaw in the above logic. Specify the step which has the (first) flaw and the reason it is flawed.

If you already know the answer, or if you figure it out, you might wish to keep it to yourself for awhile (on this thread) lest spoil it for the rest. :rofl:

And that's why infinite is non-nonsensical. (**If you can't figure it out, try plugging in numbers)

 

[curiouschris]

My answer was 0=0 not 2=1. But I got it wrong! I also would have stopped at step 4 to proceed would have been wasted effort. When I stepped through the whole thing I got 2=1

So its proof that there is an anomaly in a mathematical supposition. The true answer is 0=0 but if mathematical rules are applied as I know them then 2=1

Are there any similar mathematical absurdities that don't involve that particular rule set?

The question this raises to me is, are there any mathematical proofs that rely on this absurdity? If so they should be reworked because even wolfram alpha got it wrong!

Please excuse my phrasing year 10 maths is as far as I got.

I am not sure what infinity has to do with it.

CC

 

[Hurkyl]

even wolfram alpha got it wrong!

I bet it was user error. x/x=1 is a valid identity if x denotes a polynomial. However, if x denotes an indeterminate real number, then x/x=1 is^* equivalent to the assertion x is nonzero.


*: In an appropriate syntax where this is not nonsense.

 

[curiouschris]

No not user error at all.

Why did wolframalpha assert that 0/0 = 1?

Heres my simplistic logic, assuming real numbers. any number divided by itself is 1. This is the rule WA applied but its wrong!

Its also the basis for collinsmark's little teaser.

The exception is where the number is 0, 0/0 = 0

If you have nothing to start with and divide it by nothing you still have nothing. plain and simple child's logic.

Apparently the exception is not taken into account by WA.

It also raises what I consider serious concerns about any algebraic formula that doesn't take into account this exception.

CC

 

[Hurkyl]

Why did wolframalpha assert that 0/0 = 1?

It looks like an honest-to-goodness software bug -- it looks like wolframalpha is supposed to be using Mathematica to compute a response, but Mathematica's response would be Indeterminate, as you can see with other similar queries like 1*0/0.

The bug may be recent -- I found this page demonstrating it was returning the right answer last May.

 

[Billson555]

My answer was 0=0 not 2=1. But I got it wrong! I also would have stopped at step 4 to proceed would have been wasted effort. When I stepped through the whole thing I got 2=1

So its proof that there is an anomaly in a mathematical supposition. The true answer is 0=0 but if mathematical rules are applied as I know them then 2=1


CC

uh, no the answer is 0=0. at step four, using law of substitution, x-a=x-x=0. 2*0 is 0, so you can't factor a 2 out of it. the right side of the equation should be 0 as well by law of substitution.

 

[curiouschris]

Thats what I said Billson555. At step 4 the answer is 0=0 and that's why I wouldn't have bothered proceeding.

Hurkl, The reason I went to WA was in my head 0/0=0 (not indeterminate) but I wanted to confirm that because anything divide by 0 is well let's say infinite and also Superstring intimated that it would have something to do with that. Remember I am not a mathematician by a long shot. A student of maths may well have accepted 0/0=1 and deduced 1=2 therefore the universe is about to implode. which is exactly what collinsmark was relying on. I was also taught x/x=1 no exceptions.

I knew that wasn't acceptable and that's why I said WA was wrong. You assumed I was some sort of dullard and managed to type in something else (normally I do )

Now you say its a software bug. perfectly acceptable explanation and maybe WA should be notified.

You also say it should be indeterminate. Theres probably good reason for that but my mind says 0/0=0.

What you didn't say was sorry :grumpy:

CC

 

[curiouschris]

lol
http://www.starkeith.net/coredump/wp-content/uploads/2007/10/divide-by-zero.gif