The peculiar results of an infinite universe

[Telepath]

Don't worry, not everyone missed your serious point....

Science, has room for speculation... for our reasoning, until we have solid, and unquestionable evidence, is, essentially, speculation built upon or connected to some learned item.

I mean... surely the following is a common and valid scientific methodology:
1. Present a hypothesis. (ie. Speculation)
2. Try to prove/disprove it with mathematics. (Theoretical proof)
and/or
3. Find physical or experimental evidence.

The order and occurence of 2 and 3 may vary, but surely the first step in scientific inquiry is "speculation". If there was no speculation then there wouldn't be a lot of science going on.

JM2CW

[scaddenp]

[quote="fsgregs"]
Assuming the observable universe is a sphere centered on Earth, how long in years would it take Apollo 11 to depart from Earth and reach one "edge" of the known universe? I gave them (a) an estimate of the [u]diameter [/u]of the Hubble Bubble (known universe) (~156 billion LY), (b) the distance in a LY (5.87xe12 miles), and (c) the speed of Apollo 11 (25,000 mph). I told them to ignore inflation and changing dimensions of space. Simply assume we are in the center of a sphere 156 GLY in diameter and solve the math problem. We kept it in English units instead of metric, cause American students have no conceptual image at all of metric units.
Frank
[/quote]

This I found very interesting. I am pretty critical of maths teaching in NZ classrooms where I think good work was overturned by old politicians who perceived (accurately I think) that my generation couldnt do arithmetic as fast as theirs. I tried your problem (in metric) on my 10 and 14 year old as a dinner table problem. Now I wouldnt draw too much from a sample of 2 siblings and bright to boot, but both could do it. What was interesting was how. The 14yr who has maths as worst subject immediately grabbed pencil and drew a picture. It was easy from there. The 10 yr who is good at maths immediately stabbed by intuition as much as anything an didnt spot the radius/diameter trick till he saw his sisters picture. Maths teaching I think vastly understates the value of visualisation. The discussion on a finite universe is vastly illuminated by the 2D analogies. I just wish more emphasis would go onto diagrams in solving word problems. My daughter does it because I taught her to, so as help in struggles she had with word problems. Our childrens teachers so far have seem blind it unless its geometry.

[Startyger]

Well, unless we start to study the universe from more than one point at a time (for instance four separtate satellites working in unison from great enough distances), our observation methods are basically, spherical. At any one point a sattelite has to rotate to get a full perspective based on its bearings. (of course it can do that.. than move to anouther point, but the resulting data will not be from a synchronized effort). So, its observations are 'naturally' spherical. Now.... If we used 6 points with a 7th being Earth, the Sun, the Center of the Milky Way, or the center of the Local Group, we might gain a different vantage of our cosmic enironment.

[selden]

Don't forget that the procedure that you describe (looking at the stars from a satellite at different places in its orbit) is exactly the procedure used by the Hipparcos satellite to measure the distances to the stars shown in Celestia. It managed to measure parallaxes with an accuracy of about 2-4 milliarcseconds. GAIA's standard error is expected to be about 10 microarcseconds: more than 100 times more accurate.

[speedfreek]

Well, this has been an, errr... interesting discussion so far

The original poster claimed that many people believe the universe is infinite without knowing what that actually means. He wanted a method to illustrate to his students the implications of an infinite universe.

The universe may indeed be infinite, or it may not - we have no way of knowing this as things stand at the moment. Without violating Einsteins theory that states that nothing can travel faster than light, we will never know the truth. There is no way to look at the universe as a whole entity. The further we look, the more "out of date" the information is. When we look 13.7 billion light years in any direction, all we see is the cosmic background radiation that was left behind after the big bang.

Yet someone 13.7 billion light years away from us right now would, if looking towards our sun, see cosmic background radiation too. The implication of this is that however far away the observer was, they would see cosmic background radiation 13.7 billion light years away in any direction.

So the universe may be infinite, but in order for us to confirm this as fact, the universe would have to be infinitely old?

[Startyger]

i know this... and believe me, it's still logical, however...

i have a question....


ever look at some friends with flashlights in a valley while you were in a mountain?

the light that comes from the flashlight is vastly dispersed by air, dust etc. Is its speed reduced at all? The originating point, the flashlight is so much brighter than the dispersed light (why we see points of light). If the speed is constant.. why is that light so much brighter... shouldn't it be dispersed by the time it reaches my eye? The 'beam' ahead of the flashlight is diminished in luminosity because of the dispersal .... but... the point of origin?

does this work for stars as well?


yes, this is a newbie question, though i've read things.. maybe i'm missing a simple point... pardon the pun.

[selden]

The broadening of the beam includes photons that get to you later than the ones that get to you first. This isn't so much caused by them travelling more slowly, but rather by the photons being reflected several times in various directions before they finally get to where your eye detects them. To first approximation, the speed of the photons is unchanged along each of the reflected paths. (To second approximation, you do have to take into account that light travels at different slower speeds through different densities of gasses than it does through a vacuum.)

[Startyger]

Thanks.. i was wondering about that.. though, i've never heard or read it...(just more the disspersal of light) The beam of photons being more like a river, or cloud of energy releaset in any particular direction. This would fit the idea of it, as a form of energy, moving in waves (not that i'm saying light is a liquid... but sharing the property of things which can carry energy via waves. Though light is energy. hmmm... [i wonder if there's a carrier we haven't discovered? ie tachyons, strings, are photons nnot particulatet but something else?...... too much thinking!!!])
i guess for the stars.. gravity wells and nebulae etc.. would affect the light aswell as the vast distances. Ambient light is, then, full of vast histories of starlight.

[selden]

There are situations where light acts like waves (e.g. diffraction, radio waves) and there are situations where light acts like discrete particles (e.g. the photo-electric effect used in solar power cells). This dichotomy is one of the foundations of quantum mechanics.

Photons (another term for light) are themselves the carriers of the electric and magnetic forces. They aren't carried by something else.

One of the first experiments to show that there is no underlying "ether" acting as a carrier for light was the Michleson-Morley experiment. There are many descriptions of it on the Web.

[Crowley]

You didn't convince me that there would be infinite copies of everything in an infinite universe. Your basic assumption was that in an infinite amount of objects, every object would be contained unlimited times. (Or at least more often than once.)

That's wrong. Take the natural numbers for example. I could say: "Since there is an infinite number of natural numbers, the amount of natural numbers that have the property to be equal to the number three must also be infinite." - This would be trivially wrong, and it is a proof that not every object that is once contained in an infinite number must also be contained more often.

Then came the "proof" using probability. Sorry to say that, but it was wrong.

What went wrong? - You tried to estimate the probability for an object to exist. But an actual object does not have a probability of existence. (Don't try Heisenberg here, it would be pointless...) If the object exists, the "probability" of its existence is 100%. If the object does not exist, its "probability" of existence is 0%. These probability equations are only for estimating how great the chance is to find such an object if we search for it (and if we don't know if it exists or not). They don't have anything to do with the actual existence of the object. Likewise, that estimated distance to the next earth twin in an infinite universe out of the Scientific American does not give any chance for such a twin to exist. It's just a probability estimation, nothing more and nothing less.

Why does the equation seem to work out so well? - Because You make a mistake by dividing by infinity.

I don't know how to write the infinity symbol here, so I use "i" instead. Let's say I have a number "a" which is neither zero nor infinity. Your assumption is that:

a / i = 0 (for all (a != 0 and a != i))

This is wrong. It is just as wrong as

a / 0 = i

would be. Tempting, but wrong. There is a trivial proof to it. Take the equation

a / i = 0

and multiply it with i:

a = 0 * i

0 and i are constants, but a is a variable. Remember? a stands for every number that is neither 0 nor i! So a could be 1, 2, 3 or -2.9873 . How could two constants mulitplied with each other result in a variable? So now: What's the result of zero mutliplied with infinity?

You just cannot compute it, that's all. This is why something can exist only once in an infinite universe.

(If You don't believe me, ponder about this question: How great is the probability of the number "3" to exist in the set of natural numbers if You take the following equation for probability:

(probability of existence) = (number of items that exist) / (total number of items)

?)

[Cham]

I agree with Crowley.

That probability argument in an infinite universe doesn't mean there REALLY are an infinite number (or any number) of copies of myself "out there" (or Earths, or wathever else). I was never convinced by those articles that appeared in some vulgarisation magazines. Probabilities doesn't tell anything about reality, they are only telling something about our knowledge of reality.

When we say that an event has probability 1 to occur (100% likely event), does that mean that it should absolutely occurs ? I don't think so. Likewise, if an event has a null probability (0% likely event), does that mean that is should NEVER occur ? No. It could occur even if its probability is exactly 0. Take for example the coin tossing experiment. You are asking what is the probability to get the A face. Then you drop the coin a very large number of times (infinite limit process) and counts all the successes (face A observed). The probability is then

P(A) = (number of A successes) / (total number of trials).

Suppose now that the coin is weighted, so the A face is largely favored. You drop the coin N times (N very large), and get N - m successes (m is a fixed, finite number of "exceptional" failures). Your probability now counts as

P(A) = (N - m)/N == 1 - m/N

When N is very large (N -> infinity), you get P(A) = 1 while there was m failures (face A not observed) in the process. Then we must accept the interpretation that a 100% likely event doesn't necessarily mean the event should absolutely occur.

[scaddenp]

Thank you Crowley for that lucid explanation. I've argued about this in idle moments but never really put my finger on the flaw. Your example about
the recurrance of 3 is something for me to tuck away.

[selden]

It seems to me that trying to equate a random collection of molecules with a unique element of a number system isn't appropriate. Or am I misunderstanding what you're saying?

In principle, there's nothing stopping one collection of molecules from coming arbitrarily close to matching the organization of some other collection of molecules. That is, after all, what happens in chemical reactions. The discussion here just seems to be about somewhat larger collections than usual.

[Crowley]

It seems to me that trying to equate a random collection of molecules with a unique element of a number system isn't appropriate. Or am I misunderstanding what you're saying?

You are right. This is an imprecision in my explanation. Numbers are unique whithin their set by definition. The same is not true for persons, objects or whatever things in the universe - because the universe is (probably) not a set of numbers. But: We can count the objects, thereby assigning numbers to them!

In principle, there's nothing stopping one collection of molecules from coming arbitrarily close to matching the organization of some other collection of molecules. That is, after all, what happens in chemical reactions. The discussion here just seems to be about somewhat larger collections than usual.

Of course there is nothing from stopping molecule "configurations" to be similar to one another. Let's just make it more precise:

It is hard to find a computable basis for identifying - let's say for instance - a person. What we want is a function like that:

is_selden(c)

Were "c" is a molecule configuration. The result of the function is either "true" or "false", telling us if "c" stands for the person that we search. Of course, we can't put a molecule configuration into a formula. So what we have to do is to find a system for assigning a unique number to every possible molecule configuration, and that's what "c" is.

Let's say we have a set of all molecule configurations that appear inside our infinite universe. I call it "U". Of course, it is infinitely large. Now comes the question:

If "U" is infinitely large, does that mean that there an infinite subset "T" of "U", such that for all elements "t" of "T", the expression "is_selden(t)" is true?

I tried to figure out in mathematical terms what was meant by the natural language question: "In an infinite universe, is there an infinite number of copies of myself?"

Now a simple example for the above question being answered with "no":

Let's imagine that "U" is the set of prime numbers. (Of course, the natural numbers would not be sufficient, we would need a larger set of numbers. But what I will write will be true for any infinite set.) Let's further define is_selden(c) such that it is only true if "c" is an even number. In that case, the set "T" (Your twins, Selden!) would be:

T = { 2 }

Which is obviously not an infinitely large set. Even though the universe is infinitely large!

This was, told with more words, what I meant. The idea of identifying molecule configurations with numbers is a little strange, but I guess it works out. It does not mean that molecule configurations are numbers, it just means that they are countable (even though infinite in amount).

[fsgregs]

Some of the discussion here points out how unlikely it would be to find an exact duplicate of us somewhere else in an infinite universe. Selden put it in molecular/atomic terms. What are the odds that there exists another body in the universe that has the same number, and same arrangement of atoms as us?

There are approximately 7 x 10^27 atoms in the human body. That is a big number but obviously quite finite.

There are close to 90 natural elements in our body, arranged in particular patterns. I am not a mathematician and would shutter to calculate the # of ways that 90 atoms could be arranged to form a whole that contained 10^27 such atoms. The arrangement of these atoms is a finite possibility but the probability that another "object" in the universe has the identical arrangement of atoms is ... well ... very SMALL! Yet, it is not infinitely small. It may be 10^50000 or more, but it is finite. In my understanding of infinity, that means that somewhere in the universe, there MAY BE another human identical to you and me. IF you so chose to find two of you, you could theoretically do so ... and three ... and 30 ... and ....

Obviously, this does not mean there are an infinite number of humans exactly like us ... just that there could be ... IF the universe were truly infinite in size and IF the # of atoms in the universe was infinitely large. Since the universe does not appear to be infinite, the point may be moot, but for those who choose to believe the universe is infinite ... this still seems to be one consequence of that belief!



Frank

[Cham]

fsgregs,

your argument is of probabilistic nature, and that's where the weakness stands. Having a probability for some event doesn't mean that event will/should occur, even if that probability is 1. Statistics are just a way to formulate our state of knowledge, and doesn't talk about reality (which is just a particular unknown configuration among many possibilities).

[Crowley]

Some of the discussion here points out how unlikely it would be to find an exact duplicate of us somewhere else in an infinite universe. Selden put it in molecular/atomic terms. What are the odds that there exists another body in the universe that has the same number, and same arrangement of atoms as us?

Every basis that You have for this is pure speculation. In reality, there are no odds for something to exist. It either exists, or it does not exist. And this is where You misinterpret statistics.

What are the odds for anything to exist? Is the chance for the existence of the keyboard I am typing on right now greater or lower than the chance for the existence of the bottle of Fanta out of which I am drinking? Which of both is more probable to exist?

Do You notice that these questions are basically senseless?

If everything in the universe would be truly random, then computing such chances would really make sense. But even in an infinite universe, You can never-ever know if something exists infinitely often, only twice or even only once. You can only evaluate a probability when You know in which way the configuration of matter in the universe is structured. But You don't know it, therefore You assume it is random. At least You do so for the rest of these thoughts:

There are approximately 7 x 10^27 atoms in the human body. That is a big number but obviously quite finite.

There are close to 90 natural elements in our body, arranged in particular patterns. I am not a mathematician and would shutter to calculate the # of ways that 90 atoms could be arranged to form a whole that contained 10^27 such atoms.

Did You think of the following: How do You know that every of these possibilities even can exist in our universe? Maybe the configuration of matter in our universe follows such rules that a large deal of those "thinkable" configurations would not have been possible at all!

Your basic assumption is that every configuration has roughly the same chance of existence. And I don't see at all why this should be so.

The arrangement of these atoms is a finite possibility but the probability that another "object" in the universe has the identical arrangement of atoms is ... well ... very SMALL! Yet, it is not infinitely small. It may be 10^50000 or more, but it is finite. In my understanding of infinity, that means that somewhere in the universe, there MAY BE another human identical to you and me.

There may be, or there may not be. Quite right: No matter how the theoretical chance is - it may even be that the object in question is unique whithin an infinite universe! That's why the next sentence is wrong:

IF you so chose to find two of you, you could theoretically do so ... and three ... and 30 ... and ....

No, you would not find two, or 30, if there is only ONE of You in the universe.

Obviously, this does not mean there are an infinite number of humans exactly like us ... just that there could be ... IF the universe were truly infinite in size and IF the # of atoms in the universe was infinitely large. Since the universe does not appear to be infinite, the point may be moot, but for those who choose to believe the universe is infinite ... this still seems to be one consequence of that belief!

Still, I don't see why it counts as proven that the universe is finite in size. Of course, we would get into problems with gravity if we think of an infinite amount of matter in the universe. But still, we only observed a very small and limited part of the universe, for only a few thousand years. Who could really expect that we had found all the laws dealing with it? - It's obvious that there are still more discoveries ahead than lie behind us. At the current point of knowledge, it can neither be taken for granted that the universe is finite nor that it is infinite.

[fsgregs]

Crowley:

Your postings are well thought out and I can agree with them ... up to a point. All argue that if an object or thing is random, it MAY only exist once and not be reproduced, anywhere ... at any time. You repeatedly point out that by favoring the strict definition of infinity, I refuse to recognize that a random event may occur only once ... even through all of infinity.

OK, I am biased. I base my bias on scale. As I argued, the "odds" of me occurring again somewhere else in this universe are vanishingly small. If we follow strict evolutionary events, that means every event on Earth from the beginning of the planet to now ... every birth ... every death ... the Permian extinction ... the K-T extinction ... the invasion of Europe by Rome ... my great great great gandparents meeting each other at that exact moment ... their mating ... my own birth and life experiences ... all would have to be duplicated somewhere else, to lead to another me. If someone tried to calculate the odds of such an enormous population of random events re-occuring, those odds would clearly approach the limits of infinity! Good GRIEF! How could anyone (me) argue that our physical being could happen not only once again, but an infinite number of times again somewhere out there.

I clearly understand your argument. I clearly understand your refusal to consider my postulate. Yet ... in an infinite universe, we COULD exist an infinite number of times. It all depends upon how big is the timeline (infinite), how many atoms can combine in a particular set of ways, how big is your search grid (infinite), etc. . A duplicate of me may not exist more than once, but then again ... it may. There is no law that states a random event cannot be duplicated. Isn't that correct? Well ... if we have infinite space to search and infinite time to do it with, my particular assumption is that I may one day, run across an exact duplicate of myself (however unlikely), and that if I keep searching ... for all of eternity ... I MAY discover thousands ... millions ... billions of my twins.

Even a skeptic has to accept that this MAY be true!

Frank

[Crowley]

fsgregs:

I think now we found a point that we can both agree to. It MAY be that You find infinite copies of Yourself in an infinite universe. But it does not follow as a coerciveness out of the universe's infinity that You find infinite copies of Yourself.

When You fly through the infinite universe in a spaceship for, let's say, 10 billion years and don't find a copy of Yourself, then I will jump out of a corner and shout: "Har har! I told You this COULD happen!" - On the other hand, if You find one or more twins (or something like an infinite number of twins, all probably partying), I will be among the first to congratulate You. (And, by the way, I would ask You kindly if I could borrow Your spaceship for a few million years to find twins of myself.)

[BlindedByTheLight]

fsgregs:

I think now we found a point that we can both agree to. It MAY be that You find infinite copies of Yourself in an infinite universe. But it does not follow as a coerciveness out of the universe's infinity that You find infinite copies of Yourself.

So then what are the odds that one of fsgregs copies has ALREADY found a copy of themselves?