The paradox that Zeno postulated is not really a paradox at all when you carefully consider the individual tenets of each side of the equation.
The original paradox states (approximately) that if a tortoise wishes to travel from point A to point B it must first traverse a given distance of (let us say) halfway. In front of the tortoise will now be a shorter distance that remains to be traversed. If the tortoise then traverses half of that distance, there remains yet another shorter half-length of the distance to travel. In theory, the distance between points A and B contains an infinite number of these half distances. Ergo, to traverse them would require an infinite amount of time regardless of your velocity. This is the pivotal error in this paradox.
Yes, there are an infinite number of Euclidean points in between points A and B. For that matter there are an infinite number of mathematical points between any two points you can locate. This is an artifact of Euclidean geometry. The nonmaterial nature of Euclidean points allows them to mathematically manifest in such a fashion. However, in the material world there is no such theoretical luxury as the ability to identify an infinite number of points. To identify all of them would take an infinite amount of time (is there some resonance here?), and is therefore, unachievable.
We live in a delimited material system that has a finite structure. This finite aspect is what overturns Zeno’s paradox. There is a fundamental graininess to our material world. It is one of the great quests of particle physics to identify the fundamental building blocks of our material universe. Some say that quarks are those components. Others are now leaning towards string and superstring theory, to define the basic grain structure of reality. Whatever may be the case, a given distance is physically delimited and cannot contain an infinite amount of any dimensional quantity. It is for this reason that we are able to walk across the room or travel anywhere else for that matter. We live in a world of finite distances and finite quantities of matter.
As a scientist, the realization of the flaw in Zeno’s paradox was a personal epiphany. It demonstrated to me how scientific analysis must be set aside in the accomplishment of everyday tasks. You may spend an infinite amount of time theorizing, but the job just will not get done. This is where I first located the correct and central function of faith and its role in my own character. I must have faith that I can traverse what is otherwise an infinite number of scientific points in space, even if I only wish to make it out the door in the morning.
For further raging, ranting and raving on this topic please post a reply.
PS: Thank you Democritus for creating a situation where this was remotely relevant to anything. You too CalMeacham.
Q: How do you separate physics majors from math majors?
A: Put them at one end of a room, and a bunch of females at the other end. The math majors won’t go to them, since they know they can only get twice as close, twice as close, ad infinitum. The physics majors will go, because they know they’ll get close enough.
Of course, I majored in both physics and math, so I’d be paralyzed with indecision
I think that merely stating that we cannot comprehend an infinite number of points doesn’t not automatically mean it doesn’t exist.
But…now that we get into it. I’ve never accepted the geometrical idea that a point can be zero dimensions and yet, put enough of them together and you get a line. A line is a one dimensional figure that, with enough of them, creates a plane. Etc, etc. The whole idea behind a figure being in a certain dimension is that it is completely impossible to put even an infinite amount of them together and get anything more than the same number of dimensions you started with.
With that in mind, every point contains an infinite amount of points, and every one of those points comtains an infinite amound of points. In zero dimensions, area has no meaning. So while we can certainly claim that between point A and point B there are an infinite number of points, that, in itself, is a ridiculous and impossible claim.
OTOH, mathematically, there are different levels to infinity. Infinity squared is larger than infinity even though they’re both really the same. So, while there might be an infinite number of points, perhaps the answer is that we can traverse infinity faster than space can create it.
And to think I’m not even smoking anything right now.
Not to mention any physics major would be familiar with Feynman, and would therefore distract the math majors with beer and graphing calculators while they wooed the ladies.
This all depends on which school of philosophy you’re from. IIRC Descartes “proved” something almost exactly like that. If I weren’y busy cooking dinner I’d look it up.
Zenster, thanks for posting that. In regards to your last statement, I totally agree and it was, again, Descartes, using intuition and deduction and especially methodic doubt who was a master of this.
Screw dinner. I just did a little reading. It was Kant who I was thinking of. Regarding the Copernican Revolution: Kant believed that the mind brought the objects and truths into reality. That our perception of them was actually the key. Magda, funny as it sounds, was actually close this when she mailed Zenster Zen and the Art of Motorcycle Maintenance. Pirsig, or Phædrus or whomever you call him, said this. His concept of Quality was that it comes before rationalization or comprehension, but it argues this same point. If we can’t achieve it through rationalization or Quality, then it must not exist.
Anyway, I’d better get back to the stove…hope nothing’s burning.
The resolution of Zeno’s paradox does NOT depend upon space (or time) being discrete. The paradox rests upon the (erroneous) assumption that the sum of an infinite (or unending) sequence of positive terms is infinite (or grows without bound).
In order for an arrow to reach its target, it must travel half the distance; this takes some time, say t[sub]0[/sub]. In order to travel half the remaining distance it takes some more time, say t[sub]1[/sub], etc. All of the t[sub]n[/sub] are positive terms. By assumption the sum t[sub]0[/sub] + t[sub]1[/sub] + … grows without bound. Therefore the arrow never reaches its target. The resolution does not depend upon time or space being discrete.
Once we realize that the infinite series has a finite sum, namely 2 t[sub]0[/sub], the paradox goes away.
The first person to ask “So, does 0.999… = 1?” is going to get hurt.
Ordinarily I try to avoid posting when other people have already written what I would (which is why I haven’t posted to the “Klaatu Kikto Barata” thread), but I just HAD to get my sig line in here somehow.
My refutation when introduced to it in high school was like this:
A. Between any two points, there are an infinite number of points.
B. By observation, any motion passes between two points, and motion happens.
C. Therefore, any motion, by definition, passes through an infinite number of points.
D. If it were impossible to move through an infinite number of points, Zeno’s object (arrow, tortoise, whatever) would never be able to move through the first half of the distance to begin with.
Basically, if Zeno is correct, no movement is ever possible, but by observation, it is possible. Therefore he is not correct–motion always occurs through an infinite number of points in space.
Of course, the infinite sum argument is better, but I didn’t have calculus yet.
enderw23 and dr. matrix are right; just because the universe has inherent graininess doesn’t mean that smaller distances are undefined. the smallest structures in the universe may be 10E-30 m or some such, but smaller distances are still definable. the finite sum of infinite series is a much better refutation.
however, i believe enderw23 is mistaken about the nature of infinity. it is not merely ‘a very large number’. if it were, then you are correct, you couldn’t make a line out of points. infinity is more properly defined as ‘the vastest, hugest, most incredibly unimaginalby large number ever multiplied by a still larger number a ridiculously huge number of times’. and of course, that still doesn’t do it justice. it is not like a billion in that it has no concrete value. there is always something hugely bigger possible, and then you can take that number and raise it to its own power. it is impossible for mere humans to visualize in a way similar to trying to imagine fourth dimentional objects. so don’t even try it or you’ll hurt yourself. as such, it is not limited to behaving like other numbers, and so a line may indeed be made of an infinite number of points.
