Warning: this is a long post dealing with an abstract concept, and will probably bore the pants off most anyone. You have been warned.
I have a question that’s been flying around in my head ever since I first thought of it about 8 years ago. It has to do with the concept of infinity. It just didn’t make any sense to me. Sure, it’s a useful mathematical concept for various reasons, but in the physical world, is there infinity or just indefinity? I thought about this for about a year until I came up with a very interesting thought experiment:
Think of a movie director’s clapper thing. I don’t know what it’s called, but you know what I’m talking about. It’s the thing that gets clapped shut to make that nifty noise when they call “action!” I’ve heard this was originally so that the sound track could be acurately aligned with the video track later. The point is that it makes that nifty sound because even though the bar on top of it is hinged to the bottom on one side (the fulcrum), when the two are brought together the entire contact surfaces of the top and bottom meet simultaneously.
Now apply this to an imaginary device that consists of two rays originating at the same point but pointing in different directions. Now imagine that this point is a hinge, so the angle between the two rays can be changed at will. Now imagine decreasing this angle to the point that the two rays physically meet. Now remember that all* points along both rays will contact the other ray simultaneously. The nature of a ray (mathematically speaking), is that it continues for infinity in one direction, making it infinitely long. Now, for the ends of these rays to meet, they have to travel an infinite distance, at an infinite speed, implying (at least to me), that they would never meet at the ends. An infinite distance can never be covered, and an infinite speed could never be attained. However, near the fulcrum, there would indeed be a measureable speed and distance. You could watch the two rays approach each other, and they would eventually hit and be in contact.
To me, this seems totally at odds with the entire idea of the rays being infinite in the first place. The idea that they are indefinitely long is more in line with the experiment, though I’m really not sure if there’s a distinction between the two.
Any thoughts on this one? Like I said, I’ve been thinking about it for a long time and am still no closer to deciding whether this thought experiment is at all relevant to the concept of infinity. Thanks in advance for any ideas.
(keep in mind that in this discussion we are moving mathematical objects, not physical objects, so speed of light limitations do not apply)
At any point along the ray you choose, no matter how far away from the end point, the point will have a measurable speed and distance. The farther out you go, the greater the speed and the greater the distance. At every point, the ratio of the two will be the same.
Let’s say the rays collapse in 1 second. Where the rays are one meter apart, the point moves a 1 meter/second. Where the rays are 1,000,000 meters apart, the point moves at 1,000,000 meters / second.
Call the distance of the point from the endpoint x1.
The limit of the distance between the rays at distance x1 as x1 -> infinity is infinity.
The limit of the speed between the rays at distance x1 as x1 -> infinity is infinity.
The limit of the time until points of the rays at distance x1 touch as x1 -> infinity is 1 second.
Limits allow us to make infinite calculations that have finite results, the basis for calculus (but maybe we are getting off the thread onto that “Why teach calculus” thread).
I have to confess you kinda lost me there. I understand physics pretty well, I like to think of myself as an advanced layman. Mathematics, however, is something else entirely. I don’t see how your answer addresses the question, which I guess I could have been more clear about.
My problem is that points near the fulcrum have a measureable set of attributes, yadda yadda yadda. However, the idea that all points along the rays connect the other ray simultaneously doesn’t jive with the idea that they are infinitely long, in my own mind. If they’re infinitely long, then when do the points at the infinite end of the rays meet, and how the heck did they get there?
(I realize that there’s no “infinite end”, it’s a contradiction of terms. I just can’t think of any other way to put it.)
I have also pondered this question.
If my understanding is correct, There is a problem with your analogy in that your rays have a beginning. An infinite ray, or line, would have no beginning or end.
Also, no matter how far you move along these parallel lines, the distance to where they meet remains the same. Infinite in both directions.
Thanks, Eclipsee, for recalling a headache I thought I was rid of.
Peace,
mangeorge
Work like you don’t need the money…
Love like you’ve never been hurt…
Dance like nobody’s watching! …Unknown
Think of a movie director’s clapper thing. I don’t know what it’s called, but you know what I’m talking about. It’s the thing that gets clapped shut to make that nifty noise when they call “action!” I’ve heard this was originally so that the sound track could be acurately aligned with the video track later. The point is that it makes that nifty sound because even though the bar on top of it is hinged to the bottom on one side (the fulcrum), when the two are brought together the entire contact surfaces of the top and bottom meet simultaneously.
I understand the analogy, but pulling this out of context,
The cracking noise generated by slapping the two sides together is pretty much immaterial.
Movie scenes aren’t always shot with sound, so the clacking is never recorded in the first place. (and if shot with sound, closed quietly so as not to sound)
What counts is the timing shown on the board, usually expressed in hours:minutes:seconds:frames format.
And formats vary for synching, so all of the times represented are merely a way of expressing the concept of an absolute time for later synching, that may have to be extensively juggled to achieve an actual synch, that is good enough to fool as being correct.
It depends on perception, and perception can be decieved easily, as you were in the above example.
When you deal with time or infinity, you’re dealing with an abstract concept.
You can never nail it down, because the conception itself uses imagination to construct it.
So any explanation of infinity is as good as any other.
Sorry, E, to have answered what I thought was your question and not what you thought. I still don’t understand what you mean; you painted an image, but really never stated the question.
Since there appears to be only a problem with the “infinite end”, and no such end exists, I just don’t see any problem at all.
Mangeorge, a “line” has no ends. A “ray” is half of a line – it has one endpoint, and is infinite in the other direction.
That’s right, infinity is an abstract concept (as is a ray for that matter). You shouldn’t think of infinity as an actual number, but as a direction that numbers can tend to. In your example, the points on the rays that approach infinite distance from the origin, approach infinite speed when the rays are brought together. The key word is “approach”.
Actually, I don’t find this scenario all that mind melting, because it obviously is irrelevant to the physical world. One interesting paradox, that is a bit mind melting, is “Achilles and the Tortoise”. (I think this may have been mentioned in another thread, but I didn’t read it, so I’ll arrogantly assume no one else did either.)
Achilles is in a race with a tortoise. Magnanimously, he allows the tortoise a head start. Clearly, before he can pass the tortoise, he must cut that head start in half. When he accomplishes that, he must close to within half the remaining distance, etc. He must always catch halfway up with the tortoise before he can pass it. There are an infinite number of these half intervals, and it takes a finite time to cross each one. Achilles will therefore perpetually be catching up, never passing.
Sometimes, this paradox is used as proof that there must be a “quantum” distance, a distance so short that it is no longer possible to halve it. A moving body will jump from point to point across these distances without having to traverse the points in between. If that doesn’t melt your mind, I don’t know what will. Well, without resorting to chemicals that is.
It can be easy to add up an infinite number of vanishingly small quantities and come up with a finite sum.
Once you have gotten the hang of performing an infinite number of operations by appropriately transforming the problem, you can put the chemicals away.
The speed of thought is faster than the speed of light.
That aside, nothing in the real world is truly infinite. If I am wrong about this, someone please correct me.
–mangeorge
I know you were kidding, but just in case you wanted to know, the answer is no.
It can be shown that the natural numbers are on the same degree of infinity as the integers, plus the zero. Both sets are considered countably infinite.
It terrifies me to even be IN a mathematical discussion, but here goes: I’m still stuck on the example of the “clapper.” Everyone but me seems comfortable with the premise that the top and bottom surfaces meet at the same instant at every point along the way. How can that be? Why don’t scissors do the same thing? Forgive my ignorance; Please help me to understand this.
With scissors, the points that meet are not along the line from the fulcrum to the tip. What you see with scissors are effectively two line segments rotated around a common point (which is NOT on the line segment). The line segments point of intersection changes as you rotate.
If you haven’t seen a “clapper”, think of an infinitely stiff, thin, and straight book with just two pages (it does not have to be infinitely high or wide). When the pages are closed, each point on one page touches a point on the other page (in fact, it is the same point, since we are infinitely thin and have effectively collapsed into a single page).
The moment we open the book even a little, the only parts of the pages that touch are along the line that joins the two pages. The pages are farther apart at the outer edges, but you can get as close as you care to by appoaching the line segment where they join.
Uh, hmmm… Am I really up to this right now? What the heck…
Beeruser:
[QUOTE}The speed of thought is faster than the speed of light.[/QUOTE]
Whatever mechanisms are involved in processing thought I had understood (I’m willing to learn otherwise) to be dependent on the biochemical transactions of the nervous system. While neither chemist nor biologist, I did struggle through some undergraduate courses in both (long enough ago to have possibly forgotten crucial concepts), and I had soldiered on through my subsequent life believing that the chains of exchanges of various atoms between the participating molecules would not likely be consumated at a speed faster than that at which an electron can move.
That sounds like the “My Infinity is Bigger than Your Infinity” (Way Off Topic!)
Which is bigger, the set of even integers (…, -4, -2, 0, 2, 4, …) bigger than the set of all integers (…,-2,-1, 0, 1, 2,…).
At first glance it looks obvious, the bottom set is bigger. For every number in the top set, you have two in the bottom set. But we had a few folks in my class argued that infinity IS infinity, it doesn’t matter. You can just as easily pair each number together (using zero as the center reference (ie. …,-4 to -2, -2 to -1, 0 to 0, 2 to 1 etc.) and get the same damn results. I believed the obvious at the beginning of the year, but after going through the Calculus insanity (I took that class the same year as HL Math) I don’t know. Hopital’s Rule for limits and series hammered the final nail in the coffin of my mathematical sanity. It didn’t really help that the teacher never EVER answered the infinity question.
Anyway, this question is just theory and in theory, the rays “connect” simultaneously. Forget maximum speed, distance and all the like.
In fact the fact it doesn’t have endpoints wouldn’t really matter in this realm of geometry.
Now if you must add those pesky laws of physics into the equation, In addition to requireing a good deal of energy to close the clacker, I imagine the thing would not be completly stiff. It would be bent slightly as the edges get pushed out. The clacker would be forever closing in my opinion if you cosider the top speed being the speed of light.
Thanks, Jens. I get it now.
Now for my 2 cents worth on infinity.
The concept of infinity has always intrigued me: I’m uncomfortable with it, but more uneasy with the alternative…is there one?
Also, can anyone explain to me how time and space are really the same thing? I see the relationship, but don’t quite grasp the idea
completely.
I guess the alternative to infinity would be finity. I am not being sarcastic here, I really don’t get what you mean by alternative.
While time and space are related, and they both involve dimensions that can be measured, and if nothing in space ever changed time would be meaningless, and if you had no time you certainly could not explore space, I was not even aware that they were “the same thing”.
I readily confess my ignorance. I guess what I’m saying is that if time is really just a way of measuring spatial relationships, then in a sense time and space are different ways of expressing the same thing…true or not?
Some people talk about a finite universe. How can that be?
“Time is money, and money takes up space.”
–from Relatively Speaking, by A. Epstein
