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The '0topic' turned out quite interesting. Now I'd like to have your views on 'infinity'. What is your concept of it? Is it a 'place', some kind of 'value', or just '...as n increases without bounds'? (I have my own view :) Just recently I saw the following: n/inf. = 0, which leads to n = 0 * inf. (Maybe one should treat 'inf.' like 0: don't divide by it!)
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Not sure if this is what you are after, but I would recommend a book called "Everything and More  A Compact History of Infinity*" by David Foster Wallace. Deals mostly with the work of Georg Cantor. I read the book, but don't claim to have fully understood it.
^{*  The actual book title uses the symbol for infinity which I am unable to reproduce here.}
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Books on this subject isn't my problem. (I can also recommend Peter Rozsa's 'Playing with Infinity'.) I just wanted to hear your views. There are some who disagree with Cantor, for instance.
Oh yes, I can use the 'sleepy eight', but it doesn't come out right on some computers...
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OK, just making sure.
I'll leave the discussion to those who are qualified.
I was going to try using "oo" for the 'sleepy eight' symbol, but that didn't look very good.
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Try this one for a copy and paste: ꝏ
Edit: never mind the forum software destroyed it.
Edited: 1 Oct 2010, 5:14 p.m.
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Well, to my simple mind infinity is a concept; the concept that numbers (or series) go on forever, without end. It's certainly not like a "number" that can be the object of an arithmetic operation.
Middle school kids know what infinity is: numbers go on forever.
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Quote:
Just recently I saw the following: n/inf. = 0, which leads to n = 0 * inf. (Maybe one should treat 'inf.' like 0: don't divide by it!)
Heh, heh, you must still know what you are allowed to do: e.g. claiming x = y and "proving" it by multiplying both sides with 0 isn't a legal method either d:) Be careful with 0 as well as with +/ inf. IIRC and if no mathematician is listening, we have divided by infinity quite frequently in limit problems (?= Grenzübergänge). As said above ...
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Right, Inf as 0 is not allowed in most algebra manipulation.
Furthermore, le product '0 x inf' is 'undefined'.
The only arithmetic allowed with any non zero number are :
n + inf = +inf , n  inf = inf and n * inf = inf
And any expression like inf/inf, 0*inf, infinf, etc have no meaning in algebra and therefore are 'undefinited' such as any 0/0 ou n/0 notation.
In the other hand, "dividing by infinite" is quite a current operation in analysis of convergence, divergence and limits of function and series.
In this case :
lim(1/x) = 0 when x > inf has a meaning.
Such as
lim(1/x) = +inf when x > O+ and lim(1/x)=inf when x > 0
lim(x^x) = 1 when x > 0
lim(x*sin(1/x))=1 when x > +inf
etc.
Please note that
n / inf = 0 is true for any (real) number different of ±inf.
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My concept of infinity is based upon a single book I read a number of years ago, and those concepts may have been completely bogus or outdated by now. But...
You need to be specific about which infinity you are talking about, since there an infinite number of them. The smallest infinity is the set of rational numbers. Due to a limited character set I'll call that I0. Then there is the infinity of irrational numbers, I1. Greater than that is the number of curves in a plane, I2, &c. I0+I0 = I0, I0*I0 = I0, I0/I0 = I0, I0I0 = I0, I0^I0 = I1.
A quick check on Wikipedia shows I'm not too far off base:
Cardinality_of_the_continuum
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"The smallest infinity is the set of rational numbers."
Nonsense, the number of primes is also infinite and certainly smaller than the number of rational numbers. Probably some ratio of Big O over Ln O or some such algebraic manipulation, but it is smaller.
Someone wiser may argue otherwise.
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It's possible to find a onetoone mapping between the set of prime numbers and the set of positive integers (first prime, second prime, etc.). There's a onetoone mapping between the positive integers and the integers. There's a onetoone mapping between the integers and the rational numbers. Hence there's a onetoone mapping between the prime numbers and the rational numbers, and the two sets are of the same size.
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I have heard this arguement before and readily accept it for any multiple and most countable infinities. However, Prime numbers are so sparse in comparison to natural numbers that this arguement is very hard to accept. As you progress into the larger primes, they tend to become very scarce and hence my own feeling that their infinite number must be less than the infinity of the natural numbers.
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Fortunately, infinity is so mind numbingly huge that scarcity hardly seems to count for much. As long as there's a proof (and there is) that there is always a larger prime number than the last one you found, there's a onetoone mapping with rational numbers.
What I have a hard time getting my head around is the sudden change when you get to irrational numbers. The idea that you are at infinity tothepowerof infinity since the numbers extend both in growing magnitude, as well as a complete infinity between any two numbers.
Curves on a plane I have only the most feeble grasp of.
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While primes are countable, they are barely so as there are spaces where there are billions (and even larger digits of numbers) of natural numbers in between MOST primes. That is why I am skeptical of the infinity of natural numbers being = the infinity of primes.
If you were to graph or show a density graph, the primes would be like grass in the desert.
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Think about it going the other way.
Among the rational numbers, integers are so sparse that they almost vanish. Yet there are as many integers as there are rational numbers.
Once you've removed all the nonintegers from the rational numbers, removing the composite numbers from the remainder is barely a trim.
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The 'paradox' of infinity is that ratios become meaningless
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Very good analogy, yet I would argue, integers are evenly spaced ie I know when the next one will appear. That isn't true for primes. The next prime may be a million, a billion, or a trillion digits away. There is no definite regular pattern on when I can expect the next one.
I guess that I may have issues with the proof of countability. I don't fully grasp (as I don't readily agree with) the idea that if I can count both of them, that makes them equal.
It takes me forever to count to one, if I count ratios, yet, that infinity is = to the infinity of integers, a much sparser set. I still say the sets are not equal, yet the countability therom says they are the same.
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Hi Ron,
You are running up against the 'paradox' and extrapolating the observable 'morphology' of numbers in finite numberspace. But the extrapolation is not valid. Infinite numberspace is sort of like accepting Quantum Electro Dynamics: it is not in your physical experience...
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For infinity you have to think "out of the box". The simplest analogy I am sure everyone knows is that a 1 to 1 correspondence holds for natural numbers and even numbers, and therefore their sizes are equal, yet every logical person would think that the size of natural numbers is twice that of even numbers. The point is: when we think of this, we are trapped into the idea that infinity has a boundary, and no matter where you place the boundary, this "twice as big" holds true. Only have you begun to realize that infinity has no boundary can you get out of the box.
KC
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hello,
a good read on this:
http://www.ccs3.lanl.gov/megamath/workbk/infinity/inhotel.html
there is also a redone version of the hotel california song which is hotel infinity...
cyrille
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Plus, you may already know this, but primes are spaced fairly densely. They are dense enough that the sum of their reciprocals is infinite. That's not true for, say, the squares.
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Can you explain why squares would not have an infinite sum?
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This is all I get:
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This made it work.
Of course, it's better to do it with a calculator.
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There are a lot of reasons.
By the integral test, the integral of 1/x^2, from 1 to infinity, is [1/x], which equals 1. That is, it converges, so the integer sum must, too.
Also by comparison with the sum of the reciprocal triangle numbers. 1, 1+2, 1+2+3, etc. The triangle numbers are n(n+1)/2, which increases like n^2, so if one sum converges, so does the other.
2/(n(n+1)) = 2/n 2/(n+1)
This means
1/1 + 1/3 + 1/6 + 1/10 +...
=
2/1  2/2 + 2/2  2/3 + 2/3  2/4 + 2/4  2/5 + ...
This sum telescopes to
2/1  2/(n+1)
As n tends to infinity, the sum tends to 2.
In fact, the reciprocal squares are LESS than the reciprocal triangle numbers, so this proves that their sum must be less than 2.
Finally, Euler found the sum of reciprocal squares exactly. The sum is pi^2/6. The Hp50g has this result build in, btw. It's called the Basel problem if you'd like to know more. And, in fact, it was along this similar line of investigation that led Euler to prove that the sum of the reciprocal primes is infinite.
Edited: 1 Oct 2010, 8:23 p.m.
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I misunderstood you. I thought you were saying that the sum of squares, not the sum of (inverse squares) was finite.
But nice explanation anyway!
Best regards,
Bill
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Actually, Euler had argued about that, too! Example, you could argue that the sum of the powers of 2 is 1. He thought that if you took positive numbers until you "went past" infinity, you could end up back at negative numbers.
That sort of playing fast and loose with infinity is what led later mathematicians to try to treat the concept more rigorously.
In the case of the sum of the squares (not their reciprocals), you could use the analytic continuation of the Zeta function, at 2. In that case, the sum would merely be zero.
Edited: 3 Oct 2010, 5:12 p.m.
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My view on this in explorationmode at this time. Good timing for the question, then :)
Infinity may just be a theoretical/mathematical construct with no basis in reality  if the universe is discrete in energy, space and time and it does not have infinite length in any dimension...
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One of the neatest descriptions of infinity I have heard for the lay person came in a radio program covering teaching kids and how there are many different ways to learn and understand a topic. (The following is my interpretation of the program I heard many years ago and hopefully conveys the essence of what I heard but, for sure, does not have the same details as the original story)
Part of the story described a teacher asking her class to tell her what infinity was and to describe it. She was going for the standard answers that include some form or idea of forever or unbounded when one kid tells her that infinity is like a box of rice. She steered the child back toward her answer told him that while there are a lot of grains of rice in the box the number certainly isn't infinite and if we wanted to, we could certainly count them all.
After class the radio interviewer asks the child about his answer and finds out that the specific box the child was thinking of has a picture on it of a person holding that same box of rice; and, of course, the picture of the box had a picture of a person holding a box of rice, etc. etc. This kid probably understood infinity better than any of his peers but because it wasn't what was expected in the context, it was not acknowledged as correct.
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Quote:
One of the neatest descriptions of infinity...
My favorite allows the user/student to actually see infinity.
Take any object (say a box of rice or a grain) and drop it. For the object reach the ground it must first travel half the distance. After traveling half the distance, it must half the remaining distance before reaching the ground, and so on an infinite number of times. The object only reaches the ground after an infinite number of half steps.
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That's a great visual of an infinite series, like the repeating decimal 0.999999...  we must accept that this is equal to 1,
just as is the sum (1/2 + 1/4 + 1/8 + ...) though we "know" it never gets there!!
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My favorite version of that one involves a party where an engineer, a mathematician and a third person who poses a challenge about approaching Raquel Welch, standing across the room...
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I was looking for a link to that joke. The one I found had the Devil messing with the engineer and mathematician.
The other image I recall from my youth is the barber shop where my dad took me had mirrors on opposing walls and looking into one of those mirrors definately gave you a sense of infinity. I'm sure the barbers who stood there all day were used to it bit it always mesmerized me m
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I have read two books on infinity recently. One is Naming Infinity: A True Story of Religious Mysticism & Mathematical Creativity By Loren Graham, science historian & JeanMichel Kantor, French mathematician. Very interesting book. A brief excerpt from a review:
"There was a “crisis” in mathematics near the end of the 19th century. The problem was an old one, how to deal with infinity, since once it is considered many paradoxes develop. The crisis was when it became clear than if infinity could not be defined rationally then many other mathematical objects, i.e., sets, could not be defined. This brought the entire structure of mathematics itself into question."
The other is The Infinities by John Banville. It is not at all mathematical, but is one of the best books I've ever read, with every sentence carefully crafted to be beautiful. I couldn't do it justice here in a few words, but there is a review of it in the 2/24/2010 LA Times.
Ed
Edited: 29 Sept 2010, 2:03 p.m. after one or more responses were posted
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Ed, your formatting is making your lines approach infinity :)
Oh, you fixed it. Not so funny now. :(
Edited: 29 Sept 2010, 2:04 p.m.
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I'm gonna go way off base here:
I'm not convinced that irrational numbers exist.
Someone who wanted to dispute that might say, "Oh, well, you must not understand the proof that the squareroot of 2 is irrational. Because it is very well established that that number is irrational, and therefore that irrational numbers exist."
But that's not true. There is a proof that there is no rational number that, when squared, gives 2. That does NOT prove that the squareroot of 2 is irrational. A third possibility is that there is NO number that equals the squareroot of 2 (if only rational numbers exist).
Of course, there are numbers that, when squared, are approximately 2. And they can get better and better as they get more decimal digits. As far as that is concerned, we might as well say the squareroot of 2 exists. Also, we can treat it formally  eg., (xsqrt(2))(x+sqrt(2))= x^22, whether or not sqrt(2) is a "number" or a symbol or whatever.
So, okay, we can ACT like irrational numbers exist. But do they really?
I saw a mathematician who worked on some sort of computer math (using a computer to find proofs, I think) who argued that irrational numbers don't exist from a completely different point of view: It could be possible to make some "ultimate computer" for this universe, by using every single particle in the universe to make it. This computer can only work with so many finite digits. It might be enormous, but there is a limit. Maybe you can calculate 5 trillion digits of pi, maybe you can calculate a trillion trillion digits of pi, but you might not be able to calculate a trillion trillion trillion trillion digits.
(Yes, there are formulas to calculate digits of pi without calculating preceding digits. But this ultimate computer presumably has a memory limit, so to calculate more digits would require dropping others. So even if you could calculate the trillion trillion trillion trillion trillionth + 1 digit, you might not be able to hold all those digits *simultaneously*)
If you can't know all those digits at once, is it meaningful that they "exist" in some philosophical sense?
You can actually use those infinite cardinal numbers in proofs. It's possible to show the cardinality of rational numbers is less than the cardinality of real numbers, hence irrational numbers exist. Or that the cardinality of algebraic numbers is less than the cardinality of real numbers, hence transcendental numbers exist.
Well, this isn't a way out of the above paradoxes. It just links the concept of infinite cardinal numbers to the concept of irrational numbers. We can ACT like infinite cardinal numbers exist, just like we can ACT like irrational numbers can exist. And if you think that one exists, you probably should think that the other exists, too. But infinite cardinal numbers are probably even less intuitive than irrational numbers, so I don't think looking at this issue from the former perspective have any persuasive power for the latter perspective.
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Smaller and larger infinities? Onetoone mapping? You guys are all nuts. Excepting Don Shepherd's and Bill Platt's posts*.
Glad I'm a cimple engineer.
*OK, I'll throw in Egan Ford's box of rice.
[Edited to add footnote]
Edited: 29 Sept 2010, 10:03 p.m.
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I don't think human beings can comprehend infinite, and possible even finite existance. Let's say that the universe is finite. It ends. To our minds, something must exist beyhond that sudden end. If it's a wall, something is on the other side of the wall. There can't be just "nothing", because that nothing has to exist. The nothing is something. To me this is perhaps harder to understand than an infinite universe that never ends. But, still, how can it go on forever? (Simple concepts, avoiding curves in spacetime) Numbers don't end: they're not finite. There can be a finite number on an infinite background, but they don't really "end".
As a psychiatrist, we have similar problems with our own existance. When people commit suicide, they generally are trying to "escape" what is perceived as an inescapable situation of unhappiness or suffering. But because they don't escapte "to" something, or in that sense, "from" something, they end up simply not existing, which goes against the initial intent. They don't preserve any sense of "they". Humans have a difficult time with the concept of mortality... I would argue that it is the hardest thing we have to deal with in our lives, our own mortality, but I believe worse than that, the mortality of the people around us. I believe, and this is supported by verses in the JudeoChristian Bible, that if people didn't die, we'd have no religions anyway. The central and primary purpose of religion is to try to cope with mortality.
Longwinded way to say that if we understood these concepts mathematically, we might even provide religion some answers they've been looking for (those whose religions allow them to look.)
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fascinating wanderings this thread is taking  so glad about that, makes me feel there's still a certain hope  even if small  to get to understand things, even those so elusive as these.
Sometimes we're trapped in our own definitions, too rigid to allow a simple understanding. It's ok to be rigurous as long as you know when that's not required. Infinity (and nonexistence) are perhaps two of those, notwithstanding their mathematical treatment.
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Michael,
This is getting "infinitely" off topic but....
I almost completely agree with what you said about the the basis for religion except that I think that the need for religion is more fundamentally weaved into human brain structure. I'd say (and I'm not the only one) that it's a lot like music, which serves no obvious purpose (some may argue with this) except that virtually all cultures embrace some form of it and most individuals enjoy at least some of it. Without death I think that humans would still have religion, but it wouldn't be taken so seriously  it might be even more like music.
Georg Cantor's work on infinite sets was indeed threatening to religious leaders at the time. They no doubt saw that understanding infinity took away some of the need for what they preached. Maybe if everyone in the world studied his work and later theories (like Gödel's) we'd have less intense religions, fewer wars, a more sustainable human population on the planet and probably a lot of smart people too!
Katie
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Quote:
The central and primary purpose of religion is to try to cope with mortality.
Quote:
I almost completely agree with what you said about the the basis for religion... Without death I think that humans would still have religion, but it wouldn't be taken so seriously  it might be even more like music.
This is only true if you ignore the concept of revelation, both experiential and historical, which actually form the primary bases for both Judaism and Christianity.
Quote:
...except that I think that the need for religion is more fundamentally weaved into human brain structure. I'd say (and I'm not the only one) that it's a lot like music, which serves no obvious purpose (some may argue with this) except that virtually all cultures embrace some form of it and most individuals enjoy at least some of it.
As a Christian, I believe this imprint in the brain structure is part of what is meant by "in the image of God He created him." [Genesis 1:27]
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But what if natural numbers don't exist either?
Show me two identical objects and I'll argue that they are distinct. In that case, what is the meaning of any natural number? It's just an abstraction we make up in order to validate the abstract process of counting.
So the argument of the "existence" or not of numbers, rational or otherwise, seems to me to be semantic.
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Maybe I do believe in irrational numbers. I was thinking, the proof that the square root of 2 cannot be rational isn't enough, because...
Quote: Of the three ways in which men think that they acquire knowledge of things—authority, reasoning, and experience—only the last is effective and able to bring peace to the intellect.
Roger Bacon
Now I guess I'm not believing in "rigor".
But can you test that the square root of 2 is irrational, experimentally? Yes  by checking that the decimal digits don't repeat.
Someone is bound to chime in here, You can NOT test that they NEVER repeat. Yeah, so what? You can test for thirty, a hundred, a million places.
If someone proved pi was irrational, but we calculated it out as
3.14159266666666666666666666....
with 6's out to a billion places and more, I think we should be quite skeptical of the proof, no matter how well reasoned and rigorous it seems. (unless someone was able to then prove why it should be so nearly rational  example, sum 7 here agrees with pi to over 400 places, but there is a good reason for that (the sum is related to an approximation to the error function integral)
Edited: 1 Oct 2010, 10:35 a.m.
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Quote:
But what if natural numbers don't exist either?
Show me two identical objects and I'll argue that they are distinct. In that case, what is the meaning of any natural number? It's just an abstraction we make up in order to validate the abstract process of counting.
So the argument of the "existence" or not of numbers, rational or otherwise, seems to me to be semantic.
I'm not so sure what to say to this! It's a mind bender. But natural numbers sure seem to exist, don't they? They do to me, anyway. They don't even seem to be abstract. Like, if I think of 2, I'm not necessarily thinking of 2 apples or two rocks or any other set of things.
But maybe it's just because I've used them for so long that I'm used to that conceptual model.
It does occur to me that, supposedly, all electrons (for example) are literally indistinguishable. If they are not, that could have actual physical consequences, in terms of particle statistics
On the other hand, 4yearold children don't learn to count by counting electrons.
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Subatomic particles may be indistinguishable, which destroys my argument, but I don't know if that can be proven beyond doubt. So much of this "knowledge" depends on mathematics, which is an abstraction. The real world, as others have mentioned, doesn't conform well to math (ie, the apparent quantum nature of space, and even of time, another concept which may not really exist), although these theories are also derived mathematically.
Ok, my head is hurting now.
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Subatomic particles, e.g. electrons, *are* indistinguishable. This fact has a lot of nice observable consequences going pretty far in particle physics, and can be explained best by mathematical (here: quantum mechanical) models. Nature as a whole conforms rather well to mathematics, being no wonder, since mathematics was developed to describe nature ("máthema" = ancient Greek for "science, learning", so "mathemátikos" = "concerning science, scientific"; "phýsis" = ancient Greek for "nature", "physikós" = natural).
The honourable members of this forum, being technical people in their majority, focus on the shortcomings this model has in particular applications. This is quite reasonable (= "logikós") and will drive further progress in modeling.
But it's nothing new: As for each and every model you have to know and consider its limitations. So far, however, this model describes the laws of nature pretty well over 40 orders of magnitude. So I would keep it until somebody shows me something better d8)
Edited for error correction.
Edited: 5 Oct 2010, 2:24 a.m.
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Crawl, I disagree. You're comparing a mathematical concept (that of continuity) with the real world. In the real world, continuity doesn't exist. Each and everything can be broken down to smaller entities only a finite number of times until you get to a smallest entity which cannot be divided any further.* So the real world is finite in every respect.
Writing down an irrational number with decimal digits is nothing more then a rational approximation. We can write down the number that, when squared, gives 2, only in a formal way, e.g. as sqrt(2). Or the ratio between the diameter and circumsphere of a perfect circle as PI. In the real world, there is no such thing as a perfect circle.
* This might not be true for Time, but that's a totally different story.
Edited: 1 Oct 2010, 3:29 a.m.
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Quote:
Each and everything can be broken down to smaller entities only a finite number of times until you get to a smallest entity which cannot be divided any further.
Historically we have discovered:
Particles visible to the naked eye
Particles visible under a microscope
Molecules
Atoms
Protons, neutrons, and electrons
Various subatomic particles
Other hypothetical particles have been postulated
So how can we be so sure the discoveries won't go on and on?
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We know, that there is a smallest distance, the Plank Distance. And even if we find our way down the particle path, we will always end up in rational space because a given particle is divided into a finite number of subparticles. Energy can't be divided, either, to arbitrary levels, it's all discrete and not continuous.
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I'm obviously playing Devil's Advocate here, so there's no need to try to prove me wrong. But I think you're misunderstanding my argument. (there are two arguments anyway)
The argument at no point compares the square root of 2 or pi to anything in the real world  the length of a line drawn with a pencil, or an interval of time measured with a stop watch. It only has to do with calculating the digits. Calculating the digits is independent of any real world application.
And there are many ways of calculating those digits. Bracketing the solution of x^2=2, starting with 0 and 2 as the initial end points. Using the binomial expansion, 1+1/21/8+1/165/128+.... Taking the log with the series 11/2+1/3..., dividing by 2, and exponentiating with e^x=1+x+2^2/2+.... There is a digit by digit method that is similar to long division.
There are probably many, many more. It turns out that ALL of those calculations will give the exact digits, now matter how far you go. How do we know that they do?
There are two ways to know: You can actually do the calculation yourself, and see that the digits are all the same.
Or you can have faith that since they represent the same real thing, they must be equal in all ways. Can you imagine if they did NOT give the same result? I'm not saying that I seriously think something like that could ever happen, I'm just asking that you try to imagine it.
However, at some point, you cannot calculate more digits. At that point, all that's left is faith.
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Quote:
However, at some point, you cannot calculate more digits. At that point, all that's left is faith.
I do have faith, at least in logic. ;)
The point is that a human brain cannot "begreifen"* an abstract concept without resorting to some real world analogy. It's astonishing how far mankind has got with such a limited resource. ;)
* German for "understand". Means "put your hands on" something as Walter has already pointed out.
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Quote:
The point is that a human brain cannot "begreifen"* an abstract concept without resorting to some real world analogy
There must be something wrong with my brain then, as I can "comprehend" the concept of infinity with no fundamental jitters to my existencial experience.
Seriously, why does it have to be "experienced" to be understood??
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Seriously, why does it have to be "experienced" to be understood??
It's a question of what you mean with "understand". As a mathematician, I've learned to just follow the rules of logic and apply them to some proven theorems and/or axioms to get to a new level of information. An entity like infinity or zero is just an object with given properties to which a set of rules applies. No need to have a "higher" understanding of its concept. If you ask a math guy "Where is infinity" he'll ask you back "How is 'where?' defined?".
On the other hand, math has its origins in the real world (astronomy, geography, physics). So its not a surprise that we try to apply mathematical concepts to the real world. This must fail for some of them, as infinity or continuity.
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I'm obviously playing Devil's Advocate here, so there's no need to try to prove me wrong. But I think you're misunderstanding my argument. (there are two arguments anyway)
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However, at some point, you cannot calculate more digits. At that point, all that's left is faith.
You are confusing two concepts (1) does the sqrt(2) EXIST and (2) can it be CALCULATED. If you accept that the real numbers are continuous then the regular proof that sqrt(2) is irrational will show that the number exists. You're argument is essentially "irrational numbers can't be computed, therefore they do not exist."
So it comes down to whether we are dealing with reality or the abstract model of mathematics  "faith" as you put it.
Dave
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It depends on what you mean by continuous. For any two rational numbers, an infinite number of rational numbers exist between them. That's true whether or not irrational numbers exist*.
Now, maybe what you're talking about are Dedekind cuts. If irrational numbers don't exist, it is possible to divide the real number line into two pieces such that the both the upper and lower piece are both open on both ends. I'm not sure why that should be something we can't accept (I believe the common assumption, that it is always possible to have one set be closed, is accepted merely as an axiom, not something that can be proved). Dedekind cuts were intended, I think, to give some grounding to irrational numbers (and without necessarily relying on a sequence of infinite (rational) operations), which were presumed to exist. If we don't presume they exist, we don't need to accept it.
*And if we're thinking about the ultimate computer issue, maybe real numbers are not continuous. A real ultimate computer could handle who knows how many digits, but to make things easy let's say it can only handle 4. So there's 1.414 and 1.415, but nothing in between. Then 1.414^2=1.999, and 1.415^2=2.002, and there literally is no number between them that could possibly be the square root of 2.000 (or, for that matter, of 2.001).
Edited: 3 Oct 2010, 4:51 p.m.
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You are confusing two concepts (1) does the sqrt(2) EXIST and (2) can it be CALCULATED.
Also, I'm not necessarily confusing those concepts. There have been historically people who have argued that something does not exist if it cannot be constructed. It is a minority view, granted, but I don't think it's something that can be said to have actually been disproved.
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But the square root of two can be constructed very easily. Draw two line segments of unit length that are perpendicular and share an endpoint. Now draw a third line segment connecting the other endpoints of the first two. The length of this third line segment is exactly the square root of two.
Measuring it exactly is another matter.
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I said before I was kind of playing Devil's Advocate and just wanted to see how far this argument can be taken. But it can be taken some interesting places.
Even if I concede that irrational numbers exist, do I have to concede that nonComputable numbers exist? I don't think I have to. And that relates back to the original topic of the thread. The only thing that makes real numbers have the cardinality of the continuum is noncomputable numbers. Even computable transcendental numbers have cardinality alephnull. So it seems that it is possible to reject noncomputable numbers and the theory of transfinite numbers, since they are interrelated like that.
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If you can't know all those digits at once, is it meaningful that they "exist" in some philosophical sense?
Of course they "exist" in a mathematical sense. Even if you can't compute all of them and store them in memory all at once, you can, on demand, compute any digit of it that you like. If I ask you today to compute the 10^100th digit of pi, and then ask you to do the same thing again at some arbitrary point in the future, you will get the same result, assuming that you don't make a mistake in the calculation. There is a single, unique value for pi. If that value didn't "exist", you wouldn't be able to reproduce it on demand.
If you're going to argue that pi or sqrt(2) don't exist, why shouldn't the same argument apply to 1/7? The fact that a placevalue expansion of a rational number falls into a repeating sequence doesn't make it any more or less reproducible than pi or sqrt(2).
Edited: 5 Oct 2010, 1:54 a.m.
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I don't agree that the rational decimal expansion case is equivalent to the irrational case. It takes very few bytes of data to store all the decimal digits of 1/9. A rule like
Nth decimal digit of 1/9 = 1
suffices.
If you were skeptical that that worked, you could use long division (which does not require storing previous digits) and calculate each digit, even past the limit of digits your computer can store, and verify them. Even though you dropped early digits to calculate more, you can still use the general rule to get them.
The only way to store all the decimal digits of pi is to *actually* store them, so when you run out of space, you can't check more. If your computer's limit is 100^100 digits, and you want to calculate the 100^100+1, maybe you can by dropping the first digit. But since the only way you know the digits to pi is by storing them, now that you dropped the first digit, until you recalculate it, no one in the entire universe knows that pi~3.
(Remember, this is the ULTIMATE computer. You can't write pi~3 on a scrap of paper, because the material of that paper would have been better utilized in the computer. Likewise, a human can't just memorize it  at best, the digits a human can remember would count towards the limit of the computer)
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The only way to store all the decimal digits of pi is to *actually* store them,
I strongly disagree. A program that can produce the n'th digit of pi on demand is externally indistinguishable from a program that produces the n'th digit from storage. It's just a form of data compression; from an informationtheoretic point of view there is no difference between compressing pi into a short algorithm or compressing your shopping list into a selfextracting ZIP file, except that your shopping list is presumably of finite length. This also means that the actual information content of pi is very small.
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Okay, if it is trivial to produce the 100^100+1 digit of pi on demand, then what is it?
I happen to know that the 100^100+1 digit of 1/9 is 1.
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But what is the 100^100+1th digit of 9/9?
 Pauli
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That's a good one d8)
@Crawl:
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Okay, if it is trivial to produce the 100^100+1 digit of pi on demand, then what is it?
If I got it right then Eric didn't claim this being trivial, but more compact than storing all the digits of pi. OTOH you need *very* few bytes for storing 1/9 as decimal number. Way back they taught us to use a "period bar" for such numbers, e.g. "zero point period one" for 0.111...
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I guess he's trying to get around the limited storage of the ultimate computer, but that doesn't work, because it also has some finite calculation speed. How many digits of pi can be calculated before the heat death of the universe?
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Please check this thread for the relevance of this.
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I guess this quote is the relevant one (at least it seems to be most relevant to what I was saying)
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BBP algorithms have been proven mathematical correct. They can be used to find specific Pi digits without the overhead of computing all digits.
The record holders used BBP to validate some digits, but not all (it'd take far too long). It's reasonable to assume that if you got the last 100 or so correct then the path to get there must also be correct. Got faith? :)
I'm sure there is a computable probability that there is a wrong digit somewhere. However, the percentage of people that care round to zero.
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:) The most relevant part IMHO was that some 42 digits of pi are sufficient for all real world problems in this very universe. Everything else is just l'art pour l'art.
FWIW
Edited: 8 Oct 2010, 3:53 p.m.
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I didn't say it could be done quickly. In terms of information theory, though, it is equivalent, and thus pi has only very slightly more information content than 1/9.
We're straying from my point, which was that one can't credibly claim that irrational numbers don't exist just because one can't write down an exact decimal expansion of one on a finite piece of paper, or store that decimal expansion in a finite amount of computer memory.
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Quote:
The only way to store all the decimal digits of pi is to *actually* store them, so when you run out of space, you can't check more. If your computer's limit is 100^100 digits, and you want to calculate the 100^100+1, maybe you can by dropping the first digit. But since the only way you know the digits to pi is by storing them, now that you dropped the first digit, until you recalculate it, no one in the entire universe knows that pi~3.
Doesn't this argument also knock out a bunch of rational numbers? What about 1/N, where N is an integer with 100^100+1 decimal digits?
Heck, what about an integer with 100^100+1 digits? Does that number exist?
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Heck, what about an integer with 100^100+1 digits? Does that number exist?
For sure. It's equal to 1E200+1 ... d:)
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Do numbers exist? Are they tangible, are they reality? Are they real? They represent a count of tangible (real) objects, but are they real in themselves?
Similarly words only represent real objects. The word "tree" is not real, but the tree itself is.
The word "infinity" is not real, but infinity itself is.
Have we been so conditioned by a world focussed on finite things, that it's become difficult for us to fathom infinity? Because we are so used to things having a beginning and end, we find it difficult to imagine an infinite space. We want to start measuring here and end measuring it there. That's why we have difficulty understanding an infinite being God who declared his infiniteness by the words that say an infinite amount: "I AM".
If only we grew up with infinity as a natural part of life, we might understand it more, & be more comfortable with it.
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Your word reaches the duality language/consciousness.
Is human being's spirit able to make concepts without any language?
In my humble opinion, there is no way to do so.
Then, infinity exists in language, and thereafter/therefore in our minds. But the fact 'reality' is over our ability for us to reach it, totally.
Infinity is perhaps, so, the word used to mask our inability.
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My concept of 'infinity' is the same as the one I have of 'time' before the big bang or 'space' outside the universe: none! I think it is impossible for human brains (or any brains for that matter) to understand either.
Religion is a means to explain things people don't understand (yet), death being only one of them. I favor finding a scientific explanation, go on searching for a scientific explanation, while simply accepting it's not yet found or coming to terms with the fact that an explanation can not be given, as it is the case with 'infinity', 'time' before the big bang, 'space' outside the universe or the 'meaning of life'
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Mathematics and physics are full of mindboggling models being hard to understand for our simple brains. In German, one word translating "understanding" is "begreifen"  meaning you can *touch it with your hands*. So, ideas or concepts like infinity (or QED or curved space or dimensions > 3 etc.) are nonunderstandable per definition. This may save us some fruitless discussions  so far I don't know another language providing a similar term, but that's most probably due to my 1/inf knowledge, sorry folks ;)
Death is another topic: you can "understand" it easily in the sense explained above, but a large fraction of mankind refuses to accept it as the final limit of a person. Must have a psychological (i.e. not logical, irrational) reason.
Back to topic: Of all simple models for the idea of infinity, I like the one based on two parallel mirrors best  it's pricipally the same as the rice box, BTW. The one with the falling object is just cheating people "knowing" the concept of infinity already  the same is well known as the race of Achilles vs. a turtle ;)
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Language (specially literal ones like the German example you provide) is the biggest hindrance to understading infinity. Abstraction is required, therefore math!
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Quote: Of all simple models for the idea of infinity, I like the one based on two parallel mirrors best
I always liked playing with parallel mirrors and looking into them, but it's only a good model in theory with ideal mirrors. In reality you always look in the mirrors under a certain angle so that the tunnel of reflections curves away from you  you're not able to actually 'see' infinity which is inevitably behind the curve. The image you see also turns gradually darker, thereby obstructing your view on infinity in a second way.
Quote: Abstraction is required, therefore math!
The mathematical definition and abstraction of infinity is something entirely different from 'having a concept of' or 'understanding' infinity  it's the exact opposite! Being able to define something and work with it doesn't mean anyone understands it.
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In reality you always look in the mirrors under a certain angle so that the tunnel of reflections curves away from you  you're not able to actually 'see' infinity which is inevitably behind the curve.
Things they are a'changing ;) even reality does ;))
You may either drill a keyhole in the center of one mirror or place a little webcam there. Enjoy!
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Things they are a'changing ;) even reality does ;))
You may either drill a keyhole in the center of one mirror or place a little webcam there. Enjoy!
Won't work! The webcam can see its own reflection but there it stops because the webcam's sensor or lens does not reflect the reflected image back.
If you argue one could put a semipermeable mirror in front of the cam the reflected image gets darker and darker and fades away very soon. Again, in theory this would go on and on, in reality you can not see infinity.
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George,
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You may either drill a keyhole in the center of one mirror or place a little webcam there.
Won't work! The webcam can see its own reflection but there it stops because the webcam's sensor or lens does not reflect the reflected image back.
OK, I solemnly withdraw the webcam, though not completely convinced because I thought of some wide angle lens there. Anyway, how about the first suggestion?
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George,
OK, I solemnly withdraw the webcam, though not completely convinced because I thought of some wide angle lens there. Anyway, how about the first suggestion?
Are you trying to have fun with me? For being reflected on its own path, the light beam has to be perpendicular to the mirror. It's not a question of a wide lens, the reflection will always have to happen exactly in the middle at the position where the eye, the sensor or whatever is, in order to stay on this path to infinity. In reality, the eye/sensor will therefor inevitably break the infinite reflection path. The barber shop mirror game only works because you look from slightly off center, you always see a tunnel that curves away from the ideal straight line.
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See Douglas Hofstadter's book "I am a Strange Loop" chapter 5 "On video feedback". He describes the phenomenon he observed when he used a video camera taking a picture of the television it was connected to. In other words the video is a closed feedback loop. Focusing on the center of the television screen is not very interesting. When the camera is focused more towards the frame (his television had a metal strip along the edge) of the television he was able to create both finite and infinite "corridors". The angle of the video camera was critical when the image shifted from finite to infinite.
It sounds quite interesting but I have not tried it myself yet.
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It sounds quite interesting but I have not tried it myself yet.
I used to teach video back in the days of B&W cameras and monitors and Panasonic Portapacks. And I used to show my students video feedback as one of the first things, it was a lot of fun.
Also, back in the day, I came up with a way to do this all inside of a computer, no camera needed: Write a bit of very fast code that takes an area of memory and displays that on a monitor bitbybit or characterby character treating memory in the same arrangement as the screen, say 25x80 characters. Now write some more code that allows you to change the starting location of this mapping so that you can sort of "walk around" in memory and see where you're walking on the screen. Walk into the area being mapped and you've essentially got video feedback. It's looks and acts just like the real thing.
I wrote this for several early PC's. You had to write this in machine language because the goal was to get this all done within the time between screen refreshes. That usually limited the amount of memory you could map so you couldn't use the whole monitor but that was just as well, as this was just a window into memory anyway.
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Very clever implementation of the same concept. I haven't done machine code in the last 20 years,so it would be quite a challenge for me personally, but it will go on my never ending todo list.
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If you try this (it is trivial  just connect your video cam to your TV set and aim the camera at the TV screen  relatively straight on), after you watch the first "zoom" to infinity (that's the impression I get), ROTATE the camera (one partturn twist)  you will then see a spiral to infinity. Keep rotating, playing with the amount of rotation  it's almost addictive.
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The video loop overcomes the problem of the nonreflecting eye in the center, a straight tunnel to infinity is possible with this. Also could one adjust the amplification of the looped signal for it not to get darker and darker with every step.
But a third problem keeps us from actually seeing infinity: after every loop, the image has to be at least one pixel smaller than the one before or the effect would actually not be visible. And those onepixelsteps are finite on a real world monitor. Again, this tunnel of reflections is a nice way in theory, but fails in reality to make 'infinity' visible.
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IMHO, for pure entertainment, zooming into the Mandelbrot set, is the best experience that can elicit the concept of infinity.
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Quote:
Are you trying to have fun with me?
Not at all, George d: I wanted to offer you a personal view as perpendicular as possible. So other effects will darken your picture earlier than the edge of the mirror. But Don's method may be another way to get it, though more high tech d:)
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Being able to define something and work with it doesn't mean anyone understands it.
True, but that is how it starts. IMHO, the study of anything complex starts with learning what it does and not what it is. Eventually after doing you may (if you are lucky) achieve an understanding (or what it is). Next, you write it down, and win some type of prize.
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Next, you write it down, and win some type of prize.
Laugh Out Loud!
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True, but that is how it starts. IMHO, the study of anything complex starts with learning what it does and not what it is. Eventually after doing you may (if you are lucky) achieve an understanding (or what it is). Next, you write it down, and win some type of prize.
Working with it or winning a prize for it is not what I mean by 'having a concept of' or 'understanding'.
As you say 'if you are lucky' you achieve an understanding. But we're not always lucky. There are many things in life that we use but don't (yet) understand or are unable to explain.
The human brain, besides infinity, is one of these... ;)
Edited: 3 Oct 2010, 10:44 a.m.
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This topic just seems to keep going on for ever, and ever, and ever, and ...
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This topic just seems to keep going on for ever, and ever, and ever, and ...
Tending to infinity, actually ... :)
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I can't wait to see how it ends. :)
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Just continue watching ... 8)
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All this infinity discussion reminds me of a hops induced discussion we had one day after classes at the university.
We were trying to figure out if Einstein's hair was always so messed up because he was driving in his convertible at the speed of light so he could turn on his headlights and see if he saw them when we pondered this: increasing velocity means we are moving through more space in less time. We postulated that if we were moving at a velocity of positive infinity, we would be moving through all space in no time and there would be nowhere left to go so, ergo, we must be standing still. < i did warn you %^) > Thinking back, I don't ever recall wondering about the velocity of minus infinity???
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This thread is like the Energizer Bunny. It just keeps going, and going, and going... ad infinitum.
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All right, I do have something to add.
Today my 11 year old is talking about zero and says that it is in the middle of the numbers because there are an infinite numbers above and below it.
Then he thinks a little and postulates that every number is in the middle of the set of numbers because there are always infinite numbers above and below every number. (At this point I'm beaming proud rays that he can even think about stuff like this <VBG> )
I don't have an answer (don't think there really is one) but I thought it was yet another interesting ponderance on the topic.
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He sounds like a bright kid! Could you please have him start working on these questions, plaguing me since my teens (in my 50's now): Where is the universe (all galaxies, space to the "edge") situated? Why is this what there is (apparently)? Any insight greatly appreciated from a fresh young mind.
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I asked my smart dad what was beyond the furtherest star, he said more space. It took me two weeks to wrap my head around that idea. Sam @ 82
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Very, "Men in Black" kind of question. It's really intriguing to think that there might be multiple universes... maybe this one is in some alien kid's locker. Maybe ours is is just a science project...
