It must be confessed, this is my first post to any blog, ever. I know, I'm a cave man. So for my 'test post', I thought I'd try a little mathematical riddle that may or may not be familiar...
Fast Fred and Slow Susan propose to have a race; a marathon of 100km. Given their (literal) track records, the outcome seems utterly predictable – Fred will easily outpace the slower Susan. As a result, he gives her a head start. Since she is 32 years old, he will allow her a 32 km head start before he even begins to run. Fred knows he runs at twice Susan’s pace, so that any head start of less than 50km should pose no problem for him. So that he may know when to begin, they will both carry phones with GPS devices, and be in constant contact.
The race begins, and Fred faithfully waits until Susan’s little blip shows her 32km along the route. Then he begins. After very little time (since Fred is so fast), he notices that he himself has come to the 32km mark. “Susan”, he says into his phone “I’ve reached your 32km already”.
“One point for you,” she replies “but I am still ahead; I have reached the 48km mark… I’m nearly half done!” Fred fixes the 48km mark in his mind's eye and presses on.
Before long, he reaches the 48km marker. “Susan”, he says “I’m at 48km now, and I’ll pass you before long”.
“two points for you, but I’m still ahead,” she replies “56km and counting… do you suppose you’ll have passed me by the time you reach 56 km?” Something bothers Fred about her comment, but he puts it our of his head and runs, the destination of 56km firm in his mind.
He soon reaches the 56km mark, and again calls his opponent “Susan, I’ve reached 56km”
“Very quick, three points” she says “and have you passed me yet?” Fred snorts in response. “I’m at 60km now. Do you suppose you’ll have passed me by the time you reach 60?”
“Surely, you’ll have gained some ground by then” Fred replies.
“Of course. In the time it takes you to reach my current spot, I’ll have moved on ahead. In fact, EVERY time you reach a spot where I’ve already been, I’ll have moved on further! Even when I'm saying 'a million points for you', I'll still be in the lead! We can keep this up forever, and I guess you can never catch me. It looks as though the race is mine Fred, you may as well give up!”
The easy question is, who will win the race? The harder question is: Why do we have a never-ending sequence of events in which Susan is winning, when it seems that Fred should win? The original version of this little riddle is called Zeno’s paradox, but I challenge you to use your head before your web browser to find the answer. So much more satisfying!
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6 comments:
This problem used to bother me as a teenager before I'd ever heard of Xeno's Paradox. I would wave my hand in front of my face and wonder how it got from point A to point B without "jumping" space. When I did hear of Xeno's Paradox, I still didn't know what to do about it. And after spending my morning trying to figure it out -- I have to conclude that I still don't know what to do with this problem!
I tried looking at the problem with convering series, and calculus -- but couldn't quite figure it right. I can see that the change in distance is always 1/2 the previous delta d, and assuming constant velocity that means that the time between increments is approaching zero, as is the change in distance traveled.
It seems that Fred is hindered by having infinitely less time to move in, but that's just restating the problem -- isn't it?
In relation to Xeno's paradox, I would like to believe that motion is discrete at the Planck length, thereby making this a finite problem and eliminating the difficulty. Of course, I am not a physicist so maybe this is cheating.
Is that like saying that there is a smallest unit of time and space that can be divided? I've heard this suggestion before, but I'm in no position to evaluate it.
The answer I'd intended is that the infinitely many moments in which Susan is winning all occur within finite time; all before they reach 64km (Susan's distances are indeed halving; her progress would read 32, 48, 56, 60, 62, 61, 61.5...). I like the punch line that you can add up infinitely many numbers and only reach a finite sum (64). So if Susan were to actually keep 'awarding points' to Fred, she'd end up speaking infinitely fast!
As for Planck length, that opens up a whole new can of worms! As you say, Susan's successive moments would ground out, and end at some crazily large number instead of going on infinitely.
I've always disliked the idea that far enough down, our reality is just graph paper, and the particles are little marbles that teleport between consecutive cross-hairs without traversing the intervening distance. From what I understand, many learned people believe that is just how it is, but still a few do not. Perhaps that's that's the germ of another post...
Found this "authoritative-sounding" text on a forum...
original post on physforum.com
"This seems to be a common misconception among people, that the Planck length is somehow an automatic discretisation length. It's not. The length arises due to consideration of the relative strengths between the quantum forces and gravity. In any quantum system, the shorter the distances you're considering the higher the energy. At 'large' distances (ie 100+ times the Planck length) gravity is so weak compared to the other 3 forces you can ignore it when doing quantum calculations (ie no inclusion of the graviton in Feynman diagrams). However, at the Planck length the gravitational strength is enough that it's no longer considered weak and you must factor it into your quantum models. We presently have trouble doing this.
So the issue is not that distances below the Planck length have no meaning, but that all theories we have at present like the Standard Model we know will be completely meaningless for such distances. For theories which have made an effort to describe gravity quantum mechanically (ie string theory) distances in the range 0.1~100 Planck lengths are the usual lengths worked on. Compact spaces are of the order 1 Planck length but have complicated structure of size smaller than a Planck length, because strings are able to move around in that range."
Hmm, that "effective range of gravity" is news to me, but does sound like it makes sense. I had some more historical reasons to feel justified in my scepticism. I think Planck's constant was originally applied to energy, and was derived purely empirically. I think it was later attributed to the amount of energy required to make an electron change its orbit?
Another metaphor I like would be something to do with resonance... a guitar string will only vibrate at certain (discrete) wavelengths despite being a continuous piece of string. I don't know TOO much about it, but I thought that sounded plausibly applicable to energy (maybe the essential nature of energy has to do with vibration), but to extend it to other units like distance and time seems at best convenient, and maybe not that illuminating.
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