A Challenging Question about an Infinite Universe.

[sjastro]

Given the recent thread about the size of the Universe lets assume the Universe is flat and infinite (no flat finite Universes such as 3-torus allowed).

A question that spans both science and philosophy is how is how a Big Bang cosmology which has a finite expansion velocity and occurred in a finite time frame (13.8 billion years ago) leads to an infinite Universe.
In other expansion based models such as the Steady State this is not a problem as an infinite Universe is also infinitely old.

The question arises because of our intuitive association of "infinity" with something "very very big."

Steven

[N1 (Mirko)]

I understand the Universe is infinite in a similar way a circle or spherical surface is infinite. Reduce its diameter, and it's still infinite. Reduce it to a dimensionless point and it's still infinite because it lacks dimension. So at no stage was it finite.

[sjastro]

Originally Posted by N1

I understand the Universe is infinite in a similar way a circle or spherical surface is infinite. Reduce its diameter, and it's still infinite. Reduce it to a dimensionless point and it's still infinite because it lacks dimension. So at no stage was it finite.

The spherical surface is finite but unbounded. It has a finite surface area for any given radius.
Similarly the circumference of a circle, is finite for any given radius.

For an n-dimensional hypersurface to be infinite means it cannot be enclosed in an n+1-dimensional "ball". In 2 dimensional Euclidean space the ball is a circle, in 3 dimensional Euclidean space a sphere etc. We can make the radius of the ball as large as we want, there will always be points on the infinite surface that lie outside the radius of the ball.

Regards

Steven

[xelasnave]

Well Steven I assume you have an answer but I will use the opportunity to comment.
The reason I find steady state comfortable is infinity needs little explanation if any.
I find inflation difficult to accept so avoiding it in the mix appeals to me.

I would think you can not double up finite to ever become infinite.

But I suspect maths can provide a path.

[xelasnave]

Sorry Steven somehow I missed your last post before I posted.

[julianh72 (Julian)]

Originally Posted by xelasnave

The reason I find steady state comfortable is infinity needs little explanation if any.
I find inflation difficult to accept so avoiding it in the mix appeals to me.

There are many concepts in modern cosmology and quantum mechanics which relate to the very big and the very small that my brain simply cannot grasp - yet I still accept them as being (almost certainly) true.

The human brain evolved to deal with human-scale problems like sabre-toothed tigers and rivers and mountains, but there was no evolutionary imperative to develop a brain which can truly understand and visualise fundamental particles, wave-particle duality, quantum entanglement, or finite-but-unbounded n-dimensional space-time.

Analogies such as an inflating balloon or the raisins in an expanding plum pudding help, and I get that "Ah-Hah!" moment when I think I "get it", but that is always followed by a crashing disappointment when my 3-dimensional brain puts the balloon or pudding into the physical context of my larger universe, where I can sit god-like outside the balloon / pudding and see its boundaries, and also what lies outside it. Somehow, my brain can't do the reverse, and put itself inside the pudding, or on the surface of the balloon, and accept that there is no "outside".

I often wonder whether the people who develop these theoretical frameworks can somehow truly "see" in their mind's eye what an n-dimensional hyper-surface looks like, or are they limited by the same limited 3-dimensional visualisation, and a mathematical model of n-dimensions?

[xelasnave]

Yes Juliahn I hold similar thoughts.

I often find myself thinking about the Universe and somehow sitting outside which of course could not be done in any reality.

I find cosmology very interesting but fortunately realise one must embrace difficult concepts often counter intuitive.

Thank goodness for scientific method it gives us supportable theory.

[sjastro]

Alex,

The answer I provided to Mirko is not really what I had in mind in conceptualizing infinity.

I doubt there is a satisfactory answer to the question, only a redefinition of infinity which makes one more comfortable with the concept and doesn't make the question of how a Big Bang with a finite expansion velocity and finite time frame lead to an infinite Universe, a seemingly impossible one to answer.

In the late 19th century transfinite mathematics was developed which revolutionized the notion of infinity. It was so controversial at the time and opposed by mainstream mathematics, it led to its creator Georg Cantor in having a nervous breakdown and being committed to a mental asylum.
Today the subject is textbook pure mathematics.

A simple example is the real number line which is an infinite set. It extends to -infinity and +infinity.
If you take any interval on the real number line such as [1,2], you find it is also contains an infinite number of real numbers.
The intuitive concept of infinity is blown out of the water, you have two infinite sets, one which contains "fewer" numbers and is a subset of the "larger" infinite set.

If you extend the idea of the number line as forming a 1-dimensional space, you have a smaller infinite space "encapsulated" in a larger infinite space.

Hence the intuitive idea that infinity is larger than anything we can imagine is not terribly useful.

Steven

[xelasnave]

Thank you Steven.
I knew about Cantor from a humble TV show.

So unfortunate.

He probably was never able to appreciate his contribution.
I am not sure if he recovered.

Let's try something easier... what is dark energy and how is it causing an accelerated expansion.

[Eratosthenes (Peter)]

Originally Posted by sjastro

......Hence the intuitive idea that infinity is larger than anything we can imagine is not terribly useful.

Steven

Mathematicians work with different conceptual frameworks involving "infinity". Some refer to a "countable" infinity and an "uncountable" infinity.

When infinities appear in derivations or calculations, they are usually a good sign that something is wrong. (Hilbert, the great German Mathematician came up with the Hotel reservation thought experiment). It's useful to get infinities to vanish before they appear in the calculations or derivations, or perhaps cancel each other out. Then they dont need to be dealt with. mathematicians deploy some clever techniques that avoid infinities. On the other hand one can integrate to infinity and arrive or approach a limit which is a valid result. Or Sum up a series that approaches a certain finite number, or perhaps infinity itself.

There are some strange Summation series which can be described as nonsense such as {1+2+3+4+5+6......to infinity} = -1/12

Now does the series {1+2+3+4+5+6......to infinity}, a set of positive integers btw, approach a value -1/12 which is not only less than 1, but also negative (less than zero)???? And yet this result is used in several areas of Physics, including String Theory. (String Theory, Joseph Polchinski, Vol. 1, p. 22). Now one can argue that some tricks were used to arrive at this result which contradict mathematical logic and so result is not a valid result. The result is nevertheless use in other areas of mathematics and Physics. Mathematicians use a method called analytic continuation and when applied to divergent series, its possible to rationalise the -1/12 result. Anyway back to the subject at hand...

Some infinities are "bigger" than other infinities

So what type of infinite Universe are we talking about here Steven?

[xelasnave]

I think of infinity as something without limit on it's dimension.
It has no top, bottom or sides and without a start in time or an end of time.
You can apply math yet it can not grow or shrink
There could be no concept of time.
A set of numbers has a start so by my definition one could not describe a set as infinite.
As cosmology has a starting point for time can we still entertain infinity.
On this basis is it unreasonable to say the Universe must be finite.

[sjastro]

Originally Posted by Eratosthenes

Mathematicians work with different conceptual frameworks involving "infinity". Some refer to a "countable" infinity and an "uncountable" infinity.

The correct term is countable and uncountable infinite sets.

A countable set is equivalent to the set of natural numbers N, if one can define a mapping function which is one-one and onto between the sets.
An infinite set that is not equivalent to N is uncountable.
Given that N is the common denominator in defining countable and uncountable sets you are wrong in asserting that mathematicians are using different conceptual frameworks involving infinity.

When infinities appear in derivations or calculations, they are usually a good sign that something is wrong. (Hilbert, the great German Mathematician came up with the Hotel reservation thought experiment). It's useful to get infinities to vanish before they appear in the calculations or derivations, or perhaps cancel each other out. Then they dont need to be dealt with. On the other hand one can integrate to infinity and arrive or approach a limit which is a valid result. Or Sum up a series that approaches a certain finite number, or perhaps infinity itself.

What has this got to do with the subject?
Do you seriously think that cosmologists have calculated the Universe as being infinite?
I suggest you try looking at http://en.wikipedia.org/wiki/Friedmann_equations.

Whether the Universe is infinite or not depends on the value of the curvature k used in the cosmological model under study.

There are some strange Summation series which can be described as nonsense such as {1+2+3+4+5+6......to infinity} = -1/12

Now does the series {1+2+3+4+5+6......to infinity}, a set of positive integers btw, approach a value -1/12 which is not only less than 1, but also negative (less than zero)???? And yet this result is used in several areas of Physics, including String Theory. (String Theory, Joseph Polchinski, Vol. 1, p. 22). Now one can argue that some tricks were used to arrive at this result which contradict mathematical logic and so result is not a valid result. The result is nevertheless use in other areas of mathematics and Physics. Mathematicians use a method called analytic continuation and when applied to divergent series, its possible to rationalise the -1/12 result. Anyway back to the subject at hand...

What absolute hogwash.
Analytic continuation applies to analytic functions which are functions of the complex variable z.
Sorry to disappoint you but {1+2+3+4+....} are not complex numbers, they don't even form a power series on which analytical functions of both complex and real variables are based on.

As far as {1+2+3+4+.....} converging to -1/12 well I think any reasonably smart high school student will tell you otherwise.

I suspect Mr Polchinski is stating something very different.
Why don't you provide the relevant source.

Some infinities are "bigger" than other infinities

So what type of infinite Universe are we talking about here Steven?

Try reading the thread instead of cherry picking statements.

[N1 (Mirko)]

Steven, how do you know that a spherical surface is finite when your entire existence is inside same surface?

[Eratosthenes (Peter)]

Originally Posted by sjastro

The correct term is countable and uncountable infinite sets.

A countable set is equivalent to the set of natural numbers N, if one can define a mapping function which is one-one and onto between the sets.
An infinite set that is not equivalent to N is uncountable.
Given that N is the common denominator in defining countable and uncountable sets you are wrong in asserting that mathematicians are using different conceptual frameworks involving infinity.



What has this got to do with the subject?
Do you seriously think that cosmologists have calculated the Universe as being infinite?
I suggest you try looking at http://en.wikipedia.org/wiki/Friedmann_equations.

Whether the Universe is infinite or not depends on the value of the curvature k used in the cosmological model under study.




What absolute hogwash.
Analytic continuation applies to analytic functions which are functions of the complex variable z.
Sorry to disappoint you but {1+2+3+4+....} are not complex numbers, they don't even form a power series on which analytical functions of both complex and real variables are based on.

As far as {1+2+3+4+.....} converging to -1/12 well I think any reasonably smart high school student will tell you otherwise.

I suspect Mr Polchinski is stating something very different.
Why don't you provide the relevant source.



Try reading the thread instead of cherry picking statements.

Wow, you're response is very impressive.

Its strange that there are areas of Physics that actually use the result "Summation of positive integers, to infinity equaling -1/12"

If it's hogwash, then so are parts or all of String Theory (which is probably hogwash anyway)

What have you got against Cherries? They contain anthocyanin antioxidants - may well be unique to this fruit. Lets not forget the Vitamin A, C, calcium and iron

love cherries

[tonybarry (Tony)]

Dear Steven,

Eratosthenes may not be presenting hogwash.

{1+2+3+4+... to infinity} = -1/12

http://www.smithsonianmag.com/smart-...949559/?no-ist

Phil Plait does a good rendition of the series according to Ramanujan. He also offers an apology to the conventional mathematicians among us, who take umbrage at the novel use of = .

Would you care to comment ?

Regards,
Tony Barry

[sjastro]

Originally Posted by N1

Steven, how do you know that a spherical surface is finite when your entire existence is inside same surface?

Mirko,

We live on a (near) spherical surface. We can travel around along a great arc on the Earth's surface and end up in the same spot we departed. Clearly this is not possible if the surface is infinite.

Cosmologists have used a similar principle to test if the Universe in finite and spherical. A photon emitted could traverse a spherical Universe and end up near the same spot of emission.
Since the anisotropic regions of the cosmic radiation background can now be well resolved, double images in the background would indicate this phenomena.

Regards

Steven

[sjastro]

Originally Posted by tonybarry

Dear Steven,

Eratosthenes may not be presenting hogwash.

{1+2+3+4+... to infinity} = -1/12

http://www.smithsonianmag.com/smart-...949559/?no-ist

Phil Plait does a good rendition of the series according to Ramanujan. He also offers an apology to the conventional mathematicians among us, who take umbrage at the novel use of = .

Would you care to comment ?

Regards,
Tony Barry

Hello Barry,

Its interesting that Phil Plait ended up back tracking on this.
http://www.slate.com/blogs/bad_astro...ng_result.html

To put this {1+2+3+4+5+......}= -1/12 into perspective, the series is clearly divergent so mathematicians are looking looking for a corresponding convergent geometric series.

Mathematicians start off with an oscillating series
S1={1-1+1-1+1-.......}

And
S2={1-2+3-4+5-6.....}

If you add S2 to itself with terms shifted one place ie -2+1, 3-2, -4+3 etc
you get S2+S2={1-1+1-1+1-1....}= S1
S2=S1/2

If S3={1+2+3+4+5+6...} and you subtract S2 from S3
You get (0+4+0+8+0+...}=4S3
S3=-S2/3

Now S1={1-1+1-1+1-.......}=1/2, hence S2=1/4 and S3=-1/12 the desired result.

You might wonder where S1={1-1+1-1+1-.......}=1/2 comes from.
Its an oscillating series which is neither divergent or convergent.
In order to calculate this value one needs to compare it to a convergent series.

This is where the analytic continuation comes into the picture involving complex numbers which involves a geometrical interpretation.
The oscillating series sits on a boundary of a circle. A geometric series outside the circle is divergent, a geometric series inside the circle is convergent.
When a convergent geometric series approaches the boundary, its summed value is found to be S1=1/2.

Hence S1=1/2 S2=1/4 and S3=-1/12.

This shows we are not evening summing the series, we are summing a convergent series that is mathematically close to it.

Regards

Steven

[sjastro]

Originally Posted by Eratosthenes

Wow, you're response is very impressive.

Its strange that there are areas of Physics that actually use the result "Summation of positive integers, to infinity equaling -1/12"

If it's hogwash, then so are parts or all of String Theory (which is probably hogwash anyway)

What have you got against Cherries? They contain anthocyanin antioxidants - may well be unique to this fruit. Lets not forget the Vitamin A, C, calcium and iron

love cherries

Your whole argument is based on a wonky video where the only reference to String Theory is a guy who points to an equation in a book.
That's all there is?
String Theory has been caught out by an ill directed finger?

Since you didn't bother with my request to provide the source, a simple google search found the appropriate document.

http://stringworld.ru/files/Polchins...nic_string.pdf

As expected cherry picking or in this case finger pointing doesn't tell the whole story.
It doesn't tell you that the author already knows that {1+2+3+4+5+6....} is a divergent series.

The story leading up to the infamous finger pointed equation 1.3.32, involves treating the energy levels of a bosonic string like a quantum mechanical harmonic oscillator. In this case summing over all the harmonics leads to a divergent value.
String Theorists use a mathematical technique from Quantum Field Theory known as renormalization. In this case the higher frequency harmonics are cut out of the sum as they are above the energy threshold level of the system.
Equation 1.3.32 basically indicates the summed energies after renormalization.

Rather than being a mathematical trick, renormalization has a physical significance and is related to the Casimir effect where there has been a shift in the ground energy state of the bosonic string.

All of this is of course totally irrelevant to a thread involving Cosmology.
Once again you are in engaging in Gish Gallop tactics.

[Eratosthenes (Peter)]

Originally Posted by sjastro

Rather than being a mathematical trick, renormalization has a physical significance and is related to the Casimir effect where there has been a shift in the ground energy state of the bosonic string.
.

Can you provide a source for any evidence for the existence of strings, Bosonic or otherwise.

You are swimming in String Theory which isn't even a science, but rather a Mathematical Philosophy as it stands today.

In any case Bosonic String Theory isn't even a viable Physical Model.

You seem to hop from Metaphysics to Physics to Philosophy and back again - which isnt necessarily a negative, but at least inform posters when you do this otherwise the demarcation between solid mainstream Physics (which can be challenged incidentally) and mysticism and metaphysical conjecture becomes shiftingly blurry. Very shiftingly blurry indeed Steven

[sjastro]

Originally Posted by Eratosthenes

Can you provide a source for any evidence for the existence of strings, Bosonic or otherwise.

You are swimming in String Theory which isn't even a science, but rather a Mathematical Philosophy as it stands today.

In any case Bosonic String Theory isn't even a viable Physical Model.

I have made it clear in the past that I don't accept String Theory as is not falsifiable. My comments relate to how a String Theorist sees the physical significance of renormalization.
What you don't seem to understand is your song and dance routine over a set of numbers that supposedly adds up to -1/12, relates to the mathematics of renormalization, not string theory per se.

You seem to hop from Metaphysics to Physics to Philosophy and back again - which isnt necessarily a negative, but at least inform posters when you do this otherwise the demarcation between solid mainstream Physics (which can be challenged incidentally) and mysticism and metaphysical conjecture becomes shiftingly blurry. Very shiftingly blurry indeed Steven

You have got to be joking.
I made it perfectly clear in the very first post of the parallels between Science and Philosophy in this topic.

In case you missed it.

A question that spans both science and philosophy is how is how a Big Bang cosmology which has a finite expansion velocity and occurred in a finite time frame (13.8 billion years ago) leads to an infinite Universe.
In other expansion based models such as the Steady State this is not a problem as an infinite Universe is also infinitely old.

Where does the metaphysics and mysticism show up in this thread?