Alex asked:

Has Zeno's paradox been solved? Or has it been shown that it's not a problem any more? I've only
found information concerning the problem with nothing providing a (clear) solution.

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There are two aspects to this problem, Alex, and it may be said that under one of these Zeno's
problem has been solved while under the other it is essentially insoluble. Under one aspect, the
solution was devised in the 17th century, when the calculus of motion was invented. The way it works
is not easy to describe in words (a diagram would be easier), but if you can visualise a slope
ascending steadily in accordance with the coordinates of Achilles and the turtle, you can see that we
are dealing with converging fractions; what is missing from this is the timecoordinate. But this is
simple for us; so if you now add an arbitrary slope to represent time, all three lines will eventually
intersect at one point which fixes the 'instant' where Achilles overtakes the turtle.

From this you can see that the 'defect' of the riddles, as they stood back in the Greek days, was a
defect in their mathematics, namely the inability of the Greeks to account for irrational numbers (the
point of intersection is such an irrational). We are more generous nowadays, because it does not
matter greatly to us whether or not we can assign a precise numerical value to this point; it is
sufficient to know that such a point exists and we can easily devise an algorithm to locate it, to any
number of decimal places we wish to allow.

As a logical problem, however, the problem is not up for solution at all. It was originally devised as a
'visualisation' of the single-block universe of Parmenides. Zeno determined to show that all movement
is an illusion under the logic which appertains to his master's philosophy, of which the main tenet is:
'What is, is.' Something that is, cannot not be. If you pursue this thought to its (logical) consequence,
you can show that the idea of something coming into existence or even changing from one state of
existence into another defeats our logical faculty. The same applies, naturally, to converging
fractions. In logic, any two lines representing, as here, Achilles and the tortoise, go on forever without
touching, although this assumes of course that there is no smallest possible quantity. But since logic
is not concerned with quantity, the division involved in these fractions has no good reason ever to
stop. Today we would of course dispute this, for in a quantitative argument the fractions would
coincide at some point or other, because there must be an end eventually to divisibility. But with this
argument we are back at cartesian coordinates.

As framed by Zeno, the riddle remains a riddle. The only option, if you are dissatisfied with its logical
aspect, is to assert that logic is inapplicable to the riddle, that Zeno in fact proposed a physical
argument and was using logic only because the state of Greek mathematics was such as to permit
the riddle being posed in such a way. Personally I'm inclined to accept this point of view, but you
might try the argument on others to see if they can think of something to dislodge this proposition.

Jürgen Lawrenz

Sydney

Yes. Here's one solution: If you have a group of teenagers, say, at a school dance, and the girls are
on one side of the room and the boys on the other, and they go halfway toward each other... then
halfway again... and so forth... perhaps they will never meet. But at some point they'll be close
enough for all practical purposes (haha, the joke solution). But seriously, what you have is infinities
cancelling out, when you look at the paradox. That is, the distance, say, gets smaller by half each
time, and there is an infinite series of those fractions. But that distance is being traversed in shorter
and shorter times also. So as the distance decreases so does the time to traverse it, and as it
approaches zero, so does the time taken. So the total time is finite. Another way to look at it is to take
the sum of the set: 1/2+1/4+1/8... what is that sum? It is one unit. This is such an old problem. For
this and other paradoxes, take a look here: http://home.earthlink.net/~djbach/paradox.html.

Steven Ravett Brown