Infinities
4 Feb 2006 5:42:23 UTC
Topic 190713
(moderation:
I have a real issue with infinities. If you wanted to find the area of a wierd shape you would have to use a limit. As far as i understand, using calculus in this way is the same as placing an infinite number of squares inside the wierd shape and then find the area of all the squares. The problem i have with this, is how can an infinite number of shapes fit in a finite space. The other problem i have with this is how can you find the area of an infinite number of shapes. Area should equal infinity.
Typing this, i think i just answered my questions. An infinite number of squares could fit because the lines making up the squares are dimensionless. Just like a point has no width or length. I still dont see how its possible to find the area of an infinite number of squares.
You might not think my stupid calc questions have any business here, but my dilemna with infinites comes up again every time i listen to stephen hawkings audio book. Before the big bang, all mass was at one point. This point had infinite mass. How is this possible. Points dont have mass, and mass cant occupy the same space. Another question slightly off topic, what would have caused the big bang in the first place? For one moment, all the matter is in one place, the next it was being blown millions of miles away. Also, how could this point be infintely dense? There certainly isnt an infinite amount of mass in the universe. And if the point were infinitly dense, then even today the earth would be surrounded by debris. In other words, there would be no empty space. And if there were an infinite density, how is it possible that all mass is moving away from us?
I know there is already a theory for the whole black hole equals worm hole thing. Ive been wondering if perhaps the big bang was the "other end" of a black hole. Im bringing this up because im curious if there is already a theory like this.
Well, thats a lot to swallow. I hope i didnt sqeeze to much on one thread. Thnx
Infinities
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I cant help much with the infinite number of shapes in a finite space. It seems like a trick question or I don’t fully understand the question itself. However, with regard to infinite mass at one point at the time of the big bang I can answer very simply: there could not have been infinite mass at the time of the big bang. Cosmologists use the number OMEGA to measure the amount of material in our universe. Now this number is not known but plays a very important role in the formation of galaxies, diffuse gas, and dark matter. OMEGA tells us the relative importance of gravity and expansion energy in the universe. If this ratio were too high (say infinite as you questioned) relative to a particular critical value, the universe would have collapsed long ago. Had it been too low, no galaxies or stars would have formed. The initial expansion speed seems to have been finely tuned.
Secondly, you say that all mass is moving away from us. Its true that it seems the universe is expanding. We see this in what cosmologists call the Redshift. Galaxies are the building blocks of our universe and by studying the light from them we can infer how they are moving. If it is moving toward us its light moves to the blue end of the spectrum and if it is moving away from us its light shift toward the red end of the spectrum. The light from all distant galaxies is shifted towards the red (all except for a few nearby galaxies in the same cluster as our own.)
With regard to black holes I think your understanding of black holes may be slightly flawed. If your theory were accurate then there must be as many big bangs occurring even now as there are black holes in the universe. Our current understanding of black holes is simply the terminal state of massive stars or perhaps the outcome of collisions between stars. Their gravitational force is a million million times fiercer than earths trapping everything in its vicinity. According to einsteins theory of relativity there would not be an exit within its gravitational pull but simply a slow down of time itself. Maybe even a standstill of time, but either way there would still be a surface to the collapsed star, not an exit.
RE: I have a real issue
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Just about all sensible uses of 'infinity' resolve to 'bigger than anything you can think of', and it's soulmate 'infinitesimal' which resolves to 'tinier than anything you can think of but non-zero'. I know that mathematicians have some fancier concepts but this is the original and best version in my opinion. What makes this workable for any practical calculation is the idea of limit demonstrated by this sequence:
1 , 1/2, 1/3, 1/4, 1/5 .....1/n
an ordered set of numbers, which in this case has a simple formula a(n) = 1/n for the nth element ( n = 1, 2, 3, 4, 5...... ). This sequence clearly converges to zero, meaning zero is the limit. To make this more precise, suppose I choose any number x 'near' but not equal to zero, then if I chose N as the first integer such that N > 1/x, then a(N) = 1/N N. That is for any x that is chosen, I can go far enough along in the sequence so that all remaining elements will be within a distance x of zero. In particular x can be as close as I like to zero and I will still have an infinite number of elements of the sequence closer again. To contrast, the number 2 would not be a limit of this sequence, as I could go within a distance of 1/2 of 2 ( ie. between 3/2 and 5/2 ) and have no elements closer to 2 than my choice of 1/2.
All limiting arguments devolve to a process somewhat like the above, whether the number sequence represents areas, volumes, vector lengths or whatever.
For an infinite slicing of areas you could start with a standard square of sidelength 1, hence of area 1 * 1 = 1. Cut each side in half by placing a point midway, connect such points on opposite sides and you have cut it into four smaller squares. Area of each is 1/2 * 1/2 = 1/4, but as you have four little squares the total area is 4 * 1/4 = 1 as before. Generalize now by cutting each side into n equal parts, connect in correspondence, now you have n * n little squares each of area 1/ ( n * n ), so the total area of the lot is 1 again.
For any general two dimensional shape just imagine any finite number of squares which approximates your surface of interest, find the total finite area of this set of squares, express this total area ( call it A(n) ) as a function of the number of squares ( n ) you have. Now imagine that same surface cut up into 'n + 1' squares - these squares may not correspond to any of the squares that you used for the 'n' case - find the total finite area of this set of squares, express that total area ( call it A(n + 1) ) as a function of the number of squares ( n + 1 ) you have. Repeat for n + 2, n + 3, n + 4 .... ( or at least imagine that you could ). So now you have a sequence of numbers A(n), A(n+1), A(n+2), A(n+3)........ which represent consecutive approximations to the area of the original surface. If you've constructed these sets well then each set in the sequence has a total area which is a better approximation to the 'true' surface area. If you can deduce the limit of the sequence of A's then that's your result. In the simple square of sidelength 1 above, this sequence is 1, 1, 1, 1....that is A(n) = 1 for all n. Pretty boring and obviously converges to 1.
In fact you can use this process to define what is meant by the surface area. For this to work there are some restrictions on the mathematical 'behaviour' of the surface - you have to beware of holes, discontinuities, jumps, and fractal crap - and requires a sensible definition of distance ( 'metric' ), which thus assumes some properties of spacetime that you are ultimately modelling here.
You can go down to a one dimensional curve, just add up lots of little line segments that approximate that curve and go for the limit. Go up to a 3 dimensional volume, add up lots of little blocks which approximate that volume, and go for the limit. All you need is 'reasonably' behaved objects, a good metric, a method for constructing your sequences and hopefully a limit!
( NB. I refer to the number of dimensions of the object, not the dimension of the space it is embedded in )
As regards infinite density, this is simply saying the density is bigger than anything we can think of because: the mass is bigger than anything we can think of and/or the volume this mass is in is tinier than any thing we can think of. In physicist speak: 'our theory is in trouble'. This is why they want a 'quantum theory of gravity' to remove said infinities at small scales and humungous masses, and thus find out what is really happening. As you point out, accepting infinities willy nilly can lead to absurdities. :-)
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
RE: I have a real issue
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Mike Hewson did a pretty good job with the infinities, so let me address the big bang.
In the standard big bang model, the big bang is not an explosion of matter into space, but it is more like an explosion of space. This is actually a pretty tough thing to get your head around, becuase it relates directly to the counterintuitive idea that space itself can expand. In this context, the most general statement about the big bang theory is that space is expanding, carrying all the matter in it with it, and that the big bang is the start of this process.
Now, I've been talking about space expanding, so let's see if we can understand what that means and how it's different from matter moving through space.
Imagine that you take a balloon and draw a bunch of dots on it with a marker. Now, if you inflate the balloon, the dots will be farther apart than when you drew them, but they haven't moved on the balloon. We can identify the surface of the balloon with space and the dots with galaxies (or galaxy clusters). By staying in the same place in space, the galaxies end up progressively farther apart. I should probably add that the galaxies themselves do not expand because they're bound together by gravity.
There is one other thing we can take from this balloon model. With the balloon we only need to track one parameter - the balloon's radius - to know how far apart two dots will be if we know how far apart they are a some particular time. Similarly, in the big bang model, there is only one parameter we need to track the evolution of. This parameter is called the scale factor. The big bang itself is that point at which the scale factor was 0, which amounts to all of space being at a single point. The thing is, if we track the evolution of the universe backward from the present, we reach a point where all of the physics that we know breaks down. And, this happens before the scale factor gets all the way back to zero. Beyond this, we need, at minimum, a quatum theory of gravity to go any farther; and we don't have that yet. So, as of the present, the question of infinite density isn't really relevant because we don't actually know whether that's what is predicted. There have been ideas suggested which avoid the infinite density problem, but they're unverified as yet.
As for black holes and wormholes, these are two geometrically distinct objects, and there's not really any reason to expect an identification between them. For that matter, we have no reason to think that wormholes should actually exist, since they usually require there to be matter with negative mass.
And, even if black holes did somehow dump matter somewhere else, it would not be equivalent to a big bang, as the geometry involved would not take the same form. In particular, it would not admit expanding space.
Mike Hewson did a pretty good
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Mike Hewson did a pretty good job on the infinities, but there is another angle that is worth considering.
I fall out of a tree and accelerate downwards at 32 ft /sec /sec. This means that after 1 sec I am travelling at 32 ft /sec.
How far do I fall in that first second?
Well, at the start I was not moving. I must therefore fall more than 0 ft.
At the end I am moving at 32 ft/sec, so must have fallen less than 32 ft.
Can we get any beter than that? Well lets split it into two halves.
In the first half second I went from 0 to 16, so must have travelled between 0 and 8 ft, in the second half second I went from 16 to 32, so must have travelled between 8 and 16 ft. Adding them up, I must have travelled more than 8ft and less than 24 ft.
Try four stages:
1: 0-2 ft travelled
2: 2-4
3: 4-6
4: 6-8
Total - I fell at least 12ft and at most 20ft.
Spot the pattern - the upper and lower bounds get closer each time.
If I want to estimate how far I fell to an accuracy of 1 ft, I can split the fall into 32 stages of 1/32nd second each. You may like to try this - you find that I fell more than 15.5 ft and less than 16.5 ft.
In fact, I can prove with some algebra that for N steps the estimates will be (16/N) feet apart, and I can also prove that for any number of steps the midpoint of the estimates will be 16ft.
Now, all I am asking you is to think, if we did have an infinite number of stages in my fall, how far apart do you think the estimates would be?
And for an infinite number of points, what do you think the midpint of the estimates would be?
And just how, if at all, would that be diffeent from the sum?
Not quite.
The mathematical way is more prolonged than that.
1: find an approximate area when a finite number of squares fit inside the wierd shape with some gaps as needed, and another approximate area when the same number of slightly larger squares just cover the shape, extending over the edges as necessary. Call these the lower and upper bound on the area.
2: do it again with more squares, and then with even more
3: find a formula for doing it with N squares. Find a formula for the difference between the lower and upper bounds.
4: if and only if this difference goes to zero as N gets arbritarily large then the upper and lower bounds converge on a single area. Call this the total area of the wierd shape. If the difference does not go to zero, look for another system for arranginf the multiple finite squares.
5: go to pub and have philosophical argument about whether this really is the area of the wierd shape or just a mathematical fudge
6: go on to curry house and have philosophical argument about whether this really is the same as adding up an infinite number of zeroes and getting something that is neither zero nor infinity
Steps 5 and 6 seem important to all the maths or physics students I know, but for some reason are not documented in any book on calculus. In my personal experience steps 1-4 are only useful for passing exams, but steps 5 & 6 are the important ones for true understanding.
To summarise: the approach is to start from the finite, which we understand intuitively, and use that as a platform to reach for the infinite which we don't understand at first. And then with the addition of discussion, alcohol and curry, in time the infinite becomes intuitive also.
~~gravywavy
RE: the approach is to
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well said. infinite is very hard to understand.
so as far as i can tell your telling me were not going all the way to infinty, but going very close to see what area we get. this makes sense considering the formula is
lim
delta x ->0
as the change in x APPROACHES 0. Okay, im getting it now.
Heres another example. If you continuously compound money in a bank account for 1 year, you get some number. This doesnt make sense because if you compound something an infinte number of times, how can you get a number besides infinite?
Also, what caused this "explosion of space"? And (we prob dont know this) but what created matter?
RE: Also, what caused this
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We don't know. In fact, there's a pretty good possibility that this is unknowable, at least from a scientific perspective. The problem here is that everything that we know about the universe come from our observations of it. We can't observe anything about the process that cause our universe to come into being because that process must have occurred outside the spacetime of our universe. We have know way of knowing whether the laws of physics that we know would even apply in such a situation.
Strangely enough, this is an easier question than the previous one. I'm sure you've seen Einstein's equation: E = mc^2. The most important content of this equation is that mass (and hence matter itself) is a form of energy. This means that if there is enough energy present it should be able to spontaneously become matter. For example, if two photons, each with an energy of ~.5 MeV, collide head-on, they may annihilate and create an electron and a positron at rest. If the photons had higher energy, they might create a moving electron/positron pair, or perhaps some higher mass particle/anti-particle pair. So, all we need is for some form of energy to have been present, and we should expect to end up with matter.
A subtler question is why there's any matter now. Everything I just discussed would have ended up created an equal amount of matter and anti-matter. While in the early universe there everything was dense enough that we'd expect particle/anti-particle pair to be created faster than they could annihilate, this wouldn't hold true after the universe had some time to cool off.
So, the presence of matter today means that either more matter than anti-matter was created in the early universe, or that somehow some anti-matter turned into matter. This is actually a question still being researched, but the current thinking is that there is an asymmetry between matter and anti-matter at high energies, which is related to (I believe) the unification of the electromagnetic and weak forces.
RE: Heres another example.
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Look up Euler's constant e, the base of the natural logarithms. One way of defining this number is
[pre]e = lim_n->0_ (1 + n)^(1/n)[/pre]
--can you see the similarity to a compound-interest formula?
On the one hand, since the exponent tends to infinity you could indeed argue that the result should be infinite, as expected for raising anything greater than 1 to an infinite power. On the other hand, though, I could argue that the result ought to be 1, because (1 + n) tends to 1 and raising 1 to any power, no matter how large, gives you 1.
In fact the answer turns out to be 2.71828..., a transcendental number, like Archimedes' constant pi. Try the above formula on a calculator, using smaller and smaller values of n to see how it converges on e. I like to think of the number as the halfway point between 1 and infinity.
First, I'm not a scientist,
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First, I'm not a scientist, mathematician, etc. but just another number cruncher, but it's easy to visualize the problem of requiring an infinite number of squares to equal the volume of a circle. First draw a circle, then draw the largest square you can inside of it. The largest square would be one with all four corners touching the circle. This would take up *nearly* the whole area of the circle, but since the border of the circle is not straight, but instead curved, there is a little space left over outside of the box. Now draw four more squares, one on each side of the first square. (I mentioned "box" by accident, but this example would work the same for cubes inside of a sphere). Make these four smaller squares large enough that two of the corners lie on the edge of the first square and the other two corners just touch the circle. You have taken up even more of the area, but now you have twelve more areas that lie between the circle and the boxes. Granted the area of these spaces is much much less in total than the original area, there is still areas that need to be filled. Now make squares as large as possible in each of the twelve spaces you have left over. Four of these squares would fit exactly as the four squares you originally drew, meaning they would share two corners with the line formed by the square it is sitting on, and the other two corners would touch the circle. The other eight squares would share two sides with a previously drawn square and one corner would touch the circle. However, since again the squares do not fill this remaining space perfectly, there's still room left. You could repeat this using smaller and smaller squares (or cubes inside a sphere) to infinity and still not be able to completely fill the area since no matter how small you make the squares, the curve of the circle (or sphere) would still leave an area that was not filled. In the end you could say you used an infinite number of infinitely small squares (or cubes) and there would still be an infinitely small bit of area (or volume) left to fill! Hope this made it a little easier for the "non-theoretical mathematicians" among us (like me!) to understand the concept.
When asked a question and you are not sure of the right answer, I've found that the best answer is always "I don't know for sure, but I'll find out!"
RE: find an approximate
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I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
RE: I have a real issue
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Well, i used to have headaches too when trying to imagine infinity in our normal 3D world.
I can help you about the big bang thingy : cartesian space, euclidian or riemann does not exist at big bang "threshold". This threshold is currently defined by the Planck unit of length (roughly 10 ^ (-34) meter). Below that, current theories let infinites appears in the formulas, "thats all folks!"... it means current (~quantum) theories does not apply here.
When scientists said how huge is the universe and insignificant the Earth is, I really wonder why I bother shave myself... :)