My problem with infinity
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Woken up plagued by the concept of infinity again. Precursor to a difficult day ahead methinks. Infinity in my mind is that place you never get to. Try counting to infinity and you won't get there. Purists might argue that you would if you spent an infinite amount of time counting. But, seemingly we do get there. The abstraction I have in my head is this: A clock face with a straight line through it, going straight through the 6 and 12 and extending where else, but to infinity. Draw another parallel line to the left of the clock. Then, when the hour hand is on the 9 it points to the nearest point on the second line. As time progresses, the point on the second line moves up, heading straight for infinity. Now by rights, it should never get there, but it does, 3 hours later. It got there I believe at 3 hours - (1/infinity). It's a theoretical place, but it's also real because we just went through it to get to midday. So, where is it? I've expressed my concerns to my wife but she just gives me that slightly desperate lonely look. Oh dear, deep questions about the Universe always leave me feeling edgy.
Regards, Rob Philpott.
I guess this would be a bad time to mention that there are an infinite number of fractions of a second in a second. ;P
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How do you mean the point on the second line 'got there'? I think I'm starting to see your idea, but I need some elucidation. Just started reading Gödel, Escher, Bach for the second time in a row, and it deals extensively with, many of the more paradoxical aspects of infinity, starting with Zeno's Paradox, which is probably some kind of cousin to Rob's Paradox. I avoided getting too involved in the math during my first pass, already committing myself to an immediate second pass to better appreciate the finer points.
It does indeed sound like a variant of Zeno's paradox: one expression of which is that a runner has to pass through an infinite amount of points to arrive at a destination (since any given distance, however small, contains an infinite number of points), so how does he manage to do that? One view to dissolve the paradox is that nothing in the universe is truly continuous: all existence, all energy transfer, and hence all movement as well, actually occurs in quanta and therefore nothing actually "passes through all the points". So the imaginary clock line was pointing to some very very high point on the line to the left of the clock at some quantified moment, and in the next, it was pointing straight upward....
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Woken up plagued by the concept of infinity again. Precursor to a difficult day ahead methinks. Infinity in my mind is that place you never get to. Try counting to infinity and you won't get there. Purists might argue that you would if you spent an infinite amount of time counting. But, seemingly we do get there. The abstraction I have in my head is this: A clock face with a straight line through it, going straight through the 6 and 12 and extending where else, but to infinity. Draw another parallel line to the left of the clock. Then, when the hour hand is on the 9 it points to the nearest point on the second line. As time progresses, the point on the second line moves up, heading straight for infinity. Now by rights, it should never get there, but it does, 3 hours later. It got there I believe at 3 hours - (1/infinity). It's a theoretical place, but it's also real because we just went through it to get to midday. So, where is it? I've expressed my concerns to my wife but she just gives me that slightly desperate lonely look. Oh dear, deep questions about the Universe always leave me feeling edgy.
Regards, Rob Philpott.
You may be interested in this Zeno's Paradox[^] If you have studied calculus in high school or college, you may remember that Calculus and other forms of math based on calculus rely on the fact that the sum of an infinite series does have a finite value. When I first was given this as an explanation in high school, it took me a while to accept it. Part of the confusion is that the assumptions require a change of reference to a frame of reference that is not really achievable. Measurement from what frame of reference permits infinitely small measurements? It seems to me that the answer is none. So the idea of always being 1/2 the distance left is also purely imaginary. Mental exercises that stimulate ideas a fun nevertheless! :)
"Coding for fun and profit ... mostly fun"
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Agreed, parallel lines do not intersect. Although I believe you could argue they still intesect at infinity. My problem is the instant when two intersecting lines suddenly become parallel due to rotation.
Regards, Rob Philpott.
Parallel lines can intersect, depending on the geometry. Take, for example, lines of longitude. The paradox you present here is very similar, though less intuitive as we are not used to thinking of time as being 'curved' (like the earth is in the previous example).
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The second point is just the point on the left hand line which the hour hand points to. It's hyperthetical. And my problem is that it reaches infinity in a finite amount of time. Zeno's Paradox. Not sure I like the sound of that. Anything with Paradox in the title suggests headaches.
Regards, Rob Philpott.
"Anything with Paradox in the title suggests headaches." -- Aye! Just look at the Birthday Paradox... :doh: X|
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Woken up plagued by the concept of infinity again. Precursor to a difficult day ahead methinks. Infinity in my mind is that place you never get to. Try counting to infinity and you won't get there. Purists might argue that you would if you spent an infinite amount of time counting. But, seemingly we do get there. The abstraction I have in my head is this: A clock face with a straight line through it, going straight through the 6 and 12 and extending where else, but to infinity. Draw another parallel line to the left of the clock. Then, when the hour hand is on the 9 it points to the nearest point on the second line. As time progresses, the point on the second line moves up, heading straight for infinity. Now by rights, it should never get there, but it does, 3 hours later. It got there I believe at 3 hours - (1/infinity). It's a theoretical place, but it's also real because we just went through it to get to midday. So, where is it? I've expressed my concerns to my wife but she just gives me that slightly desperate lonely look. Oh dear, deep questions about the Universe always leave me feeling edgy.
Regards, Rob Philpott.
Do parallel lines ever meet? No, but as you look at them from a distance as they appear to converge then the greater the distance the smaller the "care factor". Thoughts of infinity reminded me of the following mathematical problem: Let x = 0.999999... (recurring) Then 10x = 9.999999... (recurring) Then 10x - x = 9.999999... - 0.999999... So 9x = 9 So x = 1 which contradicts the first statement. :wtf:
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Woken up plagued by the concept of infinity again. Precursor to a difficult day ahead methinks. Infinity in my mind is that place you never get to. Try counting to infinity and you won't get there. Purists might argue that you would if you spent an infinite amount of time counting. But, seemingly we do get there. The abstraction I have in my head is this: A clock face with a straight line through it, going straight through the 6 and 12 and extending where else, but to infinity. Draw another parallel line to the left of the clock. Then, when the hour hand is on the 9 it points to the nearest point on the second line. As time progresses, the point on the second line moves up, heading straight for infinity. Now by rights, it should never get there, but it does, 3 hours later. It got there I believe at 3 hours - (1/infinity). It's a theoretical place, but it's also real because we just went through it to get to midday. So, where is it? I've expressed my concerns to my wife but she just gives me that slightly desperate lonely look. Oh dear, deep questions about the Universe always leave me feeling edgy.
Regards, Rob Philpott.
It didn't get there - it just pointed there, where ever the hour hand points it is always pointing at infinity.... Regards, DoubleWord
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That has a simple solution :) Draw a circle, follow it on the edge. You will never get to the end, but does that mean the circle has no edge. Circles has no end, so they are infinite, but only in one direction. You can go outside the circle beyond its end. Same is the earth, go only to the east (flying suppose) and you will never finish your journey to the end of the earth. So earth is infinite, and yet you have an upper end and can go to space. So the globes are infinite in two directions but yet, you can go beyond its end, going up :) The universe is infinite in three direction, so it does not have an edge of any of the three directions, yet it has an edge, but you cannot reach it, since you are rotating in three dimensinal figure :) That's how I explain it :) Infinity anyway doesn't exist in practise, it is mathematical method and math is only instrument used to understand our world, it doesn't need to be real :)
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hehe, sooner or later we will find that everything real is finite so no problem for you mate.
Alex C. D.
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Woken up plagued by the concept of infinity again. Precursor to a difficult day ahead methinks. Infinity in my mind is that place you never get to. Try counting to infinity and you won't get there. Purists might argue that you would if you spent an infinite amount of time counting. But, seemingly we do get there. The abstraction I have in my head is this: A clock face with a straight line through it, going straight through the 6 and 12 and extending where else, but to infinity. Draw another parallel line to the left of the clock. Then, when the hour hand is on the 9 it points to the nearest point on the second line. As time progresses, the point on the second line moves up, heading straight for infinity. Now by rights, it should never get there, but it does, 3 hours later. It got there I believe at 3 hours - (1/infinity). It's a theoretical place, but it's also real because we just went through it to get to midday. So, where is it? I've expressed my concerns to my wife but she just gives me that slightly desperate lonely look. Oh dear, deep questions about the Universe always leave me feeling edgy.
Regards, Rob Philpott.
I don't know what you're on, but I want some.
But fortunately we have the nanny-state politicians who can step in to protect us poor stupid consumers, most of whom would not know a JVM from a frozen chicken. Bruce Pierson
Because programming is an art, not a science. Marc Clifton
I gave up when I couldn't spell "egg". Justine Allen -
Do parallel lines ever meet? No, but as you look at them from a distance as they appear to converge then the greater the distance the smaller the "care factor". Thoughts of infinity reminded me of the following mathematical problem: Let x = 0.999999... (recurring) Then 10x = 9.999999... (recurring) Then 10x - x = 9.999999... - 0.999999... So 9x = 9 So x = 1 which contradicts the first statement. :wtf:
But that is no contradiction becuase 0.999… = 1 (http://en.wikipedia.org/wiki/0.999...).
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daveauld wrote:
how can there be an edge? there is more space beyond......
<geek mode> The latest thinking is that space is finite in three dimensions. Four dimensional space-time is collapsed into a finite three-dimensional volume by the effect of gravity. As an example, concrete over the whole world (ignore the environmental impact for now), and then smooth out all the mountains with a really big bit of sandpaper. The surface of the earth (a two dimensional object) is finite (in three dimensions) but has no edges. Now exapnd that concept from a two-dimensional object in the three dimensions into a three-dimensional object in a four dimensions. </geek mode>
Electron Shepherd wrote:
Now exapnd that concept from a two-dimensional object in the three dimensions into a three-dimensional object in a four dimensions.
I've alwayes tried to picture that in my mind without success. I've always been amazed by this idea and hopefully when I become dust, I will fully understand it.
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Woken up plagued by the concept of infinity again. Precursor to a difficult day ahead methinks. Infinity in my mind is that place you never get to. Try counting to infinity and you won't get there. Purists might argue that you would if you spent an infinite amount of time counting. But, seemingly we do get there. The abstraction I have in my head is this: A clock face with a straight line through it, going straight through the 6 and 12 and extending where else, but to infinity. Draw another parallel line to the left of the clock. Then, when the hour hand is on the 9 it points to the nearest point on the second line. As time progresses, the point on the second line moves up, heading straight for infinity. Now by rights, it should never get there, but it does, 3 hours later. It got there I believe at 3 hours - (1/infinity). It's a theoretical place, but it's also real because we just went through it to get to midday. So, where is it? I've expressed my concerns to my wife but she just gives me that slightly desperate lonely look. Oh dear, deep questions about the Universe always leave me feeling edgy.
Regards, Rob Philpott.
I beleive infinity does not exist, whatever can't be reached isn't there. If it isn't ther it does not exist. I beleive what happens is that we call infinite whatever we are not capable of understanding, which does not mean it is what we beleive it is. Infinity = Unknown
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I don't know what you're on, but I want some.
But fortunately we have the nanny-state politicians who can step in to protect us poor stupid consumers, most of whom would not know a JVM from a frozen chicken. Bruce Pierson
Because programming is an art, not a science. Marc Clifton
I gave up when I couldn't spell "egg". Justine AllenTry reading that when you are on something, it should really boost things up.
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The second point is just the point on the left hand line which the hour hand points to. It's hyperthetical. And my problem is that it reaches infinity in a finite amount of time. Zeno's Paradox. Not sure I like the sound of that. Anything with Paradox in the title suggests headaches.
Regards, Rob Philpott.
A paradox is merely an indication that one or more of your premises, which are asserted to be true, are actually false, or that the apparent contradiction exposed is just that: apparent, but explainable. Think of this as a problem of velocity. Your second point moves up that line accelerating exponentially (logarithmically?). Maybe when it reaches the speed of light, it creates a worm hole and transfers to the first line. Heh.
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That is not infinity, that is undefined. If you replace the 0 with an x, you can argue about what happens once x creeps closer to 0, but that's about it.
-- Kein Mitleid Für Die Mehrheit
It's Cantor work. If you have a casino roulette and you divide it to all the real numnbers in [0, 1), that is from 0 to 1 not including 1. Then the probability of getting any given number in this roulette is 0. that is: 1/infinity = 0 (By extension, any number divided by infinity is 0, Cantor, was the first to use it outside a Limit). So, one can say, "hey if go ahead and clear it for infinity then I get: infinity = 1/0". Sadly, it's not defined (undefined) in real numbers, but it's currentrly accepted that 1/0 = infinity in riemman sphere (that is complex numbers plus infinity) So today saying that 1/0 is undefined (in the reals) is like saying that 5 - 9 is undefined (in the naturals). http://www.wolframalpha.com/input/?i=1/0[^] http://mathworld.wolfram.com/DivisionbyZero.html[^] And yes infinity is a number, it's a number defined as the quantity of natural numbers, and it's the first transfinite number. By the way, I didn't understand that of the clock ;P . But I can tell that zeno's paradox doesn't give me bad dreams. Of course it resembles quantum theory, if something gives me headache is quantum gravitation, but that's another tale. ---------------- Beyond the scope of the conversation (ie. ignore if you don't undestand a bit) To save some of the load of the lounge... I wrote more in my member's blog... here the permalink: http://www.codeproject.com/Messages/3322477/Infinity-and-paradox.aspx There I wrote there a paradox, I don't know the name of that paradox, or where in Internet (other than there) I can get a cite to it? I got it in a book, I don't recall the name, and I was unable to find it... so I wrote it down there. Theraot
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daveauld wrote:
how can there be an edge? there is more space beyond......
<geek mode> The latest thinking is that space is finite in three dimensions. Four dimensional space-time is collapsed into a finite three-dimensional volume by the effect of gravity. As an example, concrete over the whole world (ignore the environmental impact for now), and then smooth out all the mountains with a really big bit of sandpaper. The surface of the earth (a two dimensional object) is finite (in three dimensions) but has no edges. Now exapnd that concept from a two-dimensional object in the three dimensions into a three-dimensional object in a four dimensions. </geek mode>
Just a few thoughts I had, but this isn't something I've studied formally: - The universe had a beginning, and as such it must've had a finite size to start with, no? Maybe it had infinite density, but I don't understand how if something had a finite size, it could have an infinite density. - In the storage of a computer, a three dimensional array, lets say 5X5X5, still looks like a single dimensional array of 125 elements. If space in the universe models that of the insides of a computer (scary oversimplification), then all the dimensions (up to and even including string theory), could really be boiled down to one. Somebody educate me... - You wouldn't have to sandpaper the earth - it's already extremely smooth. If you were to shrink planet earth down to the size of a billiard ball, it would be thousands of times smoother than a regularly manufactured billiard ball. No offense meant to any billiard ball manufacturers here... - Is the surface of the earth two dimensions? I wouldn't think so, but I could be wrong. - The process of making a basketball in a factory involves creating a strip of rubber, and then cutting out contours so that the rubber is stitched together. The strip of rubber must still be three dimensions before it's sewn together to become a basketball, no? I don't believe in infinity. Not quite sure what the difference between theory and fact is, perhaps it has to be observed, no? How, by definition, could infinity be observed? Richard http://www.RichardRogers.ca/ (IM) rogersrr_ca@hotmail.com
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Woken up plagued by the concept of infinity again. Precursor to a difficult day ahead methinks. Infinity in my mind is that place you never get to. Try counting to infinity and you won't get there. Purists might argue that you would if you spent an infinite amount of time counting. But, seemingly we do get there. The abstraction I have in my head is this: A clock face with a straight line through it, going straight through the 6 and 12 and extending where else, but to infinity. Draw another parallel line to the left of the clock. Then, when the hour hand is on the 9 it points to the nearest point on the second line. As time progresses, the point on the second line moves up, heading straight for infinity. Now by rights, it should never get there, but it does, 3 hours later. It got there I believe at 3 hours - (1/infinity). It's a theoretical place, but it's also real because we just went through it to get to midday. So, where is it? I've expressed my concerns to my wife but she just gives me that slightly desperate lonely look. Oh dear, deep questions about the Universe always leave me feeling edgy.
Regards, Rob Philpott.
"I've expressed my concerns to my wife but she just gives me that slightly desperate lonely look." Take her out for a nice dinner and an evening on the town. Leave the kids at home with a sitter. Tell her "Infinity is the realm of the divine" and that she is beautiful and that you will always love her. Jeepers! If she knew then what she knows now ('bout you, and it sounds like she suspects something)... that just might be reason enough for her to find a way to communicate with her former self, and go out with your college roommate instead. ;P
-Bob
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But that is no contradiction becuase 0.999… = 1 (http://en.wikipedia.org/wiki/0.999...).
It it an interesting article. There is a "fault" of sorts with the proof (if it can be called that) that I gave. Initially, I quoted: Let x = 0.999... (recurring) Then 10x = 9.999... (recurring) The second line is actually false. 10x = 9.999.........0 as any number multiplied by 10 has a zero as its final digit. This means 10x - x = 9.999.........0 - 0.999... which is NOT 9. If 9x <> 9, then x <> 1. Of course, there is an infinite number of 9's after the decimal point and before that problematic zero so I guess I have a problem with infinity (and beyond!). :wtf:
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It's Cantor work. If you have a casino roulette and you divide it to all the real numnbers in [0, 1), that is from 0 to 1 not including 1. Then the probability of getting any given number in this roulette is 0. that is: 1/infinity = 0 (By extension, any number divided by infinity is 0, Cantor, was the first to use it outside a Limit). So, one can say, "hey if go ahead and clear it for infinity then I get: infinity = 1/0". Sadly, it's not defined (undefined) in real numbers, but it's currentrly accepted that 1/0 = infinity in riemman sphere (that is complex numbers plus infinity) So today saying that 1/0 is undefined (in the reals) is like saying that 5 - 9 is undefined (in the naturals). http://www.wolframalpha.com/input/?i=1/0[^] http://mathworld.wolfram.com/DivisionbyZero.html[^] And yes infinity is a number, it's a number defined as the quantity of natural numbers, and it's the first transfinite number. By the way, I didn't understand that of the clock ;P . But I can tell that zeno's paradox doesn't give me bad dreams. Of course it resembles quantum theory, if something gives me headache is quantum gravitation, but that's another tale. ---------------- Beyond the scope of the conversation (ie. ignore if you don't undestand a bit) To save some of the load of the lounge... I wrote more in my member's blog... here the permalink: http://www.codeproject.com/Messages/3322477/Infinity-and-paradox.aspx There I wrote there a paradox, I don't know the name of that paradox, or where in Internet (other than there) I can get a cite to it? I got it in a book, I don't recall the name, and I was unable to find it... so I wrote it down there. Theraot
Yeah sure, but then you'd have to extend the algebra to support the concept of division by zero, and all the other weird things that creeps into as the result of allowing division by zero. I was referring to the math that most people know and care about. This seems to be highlevel math for math grad students. :)
-- Kein Mitleid Für Die Mehrheit