Mathematical Paradox

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A frog jumping is just a metaphor. And a good one to quickly visualize the set up the problem. But once your imaginary frog starts getting down to smaller than millimeter distances you dispense with the frog in your mind's eye and think of it as a black dot jumping forward on graph paper eternally halving its distance to the edge of the page, and for the squares on the graph paper representing ever smaller units of measurement.

I don't think it would. If it is going 1/2 of the speed it previously went, it would gradually narrow down. 1/2, 1/4, ect, until is wouldn't be able to make a significant difference, and would just look as if it was standing still.

How much pi do you need? Is related.

http://blogs.scientificamerican.com/observations/how-much-pi-do-you-need/

I think the answer is yes, if you'll grant me a little snark.

Whole numbers are only a very gross grasp on the real world, transcendental measurements are all that exist in the real world, there is never a 50% reduction in distance in the physical world that is known. Therefore the modeling system is "too clean" to provide an accurate description. This allows our imaginary frog to behave exactly as we see fit.

You might put the frog inside a black hole, where he is to escape from, with the desperate and sad effect that he can only approach the event horizon yet never reach it. This of course assumes that a frog inside of black hole is still a frog and is capable of moving toward a goal.

I think the OP may have been thinking along the lines that the entity doing the "jumping" (frog, nanorobot, or bacterium, or whatever) would make random errors in calculating its jumps. Further-that that randomness itself can be quantified.

Kinda like the concept of "standard deviation" (the average that student scores on a test deviate from the mean of all the scores).

A perfect Euclidean frog would NEVER reach the opposite wall.

But our imperfect frog (with its quantified amount of imperfection)halves its distance with each jump, but it makes some average error in making these "halved" distances. And that these errors can be graphed in some way to allow you to predict when the "frog" actually does hit the wall due to these errors.

Lets say that the frog is off by an average of one percent each jump, and that that amount of error can be either direction.

Not sure how it would work, but maybe you could graph (or calculate with an equation) when a frog of say five percent average deviation would hit the wall....as a opposed to a later model frog with a more accurate one percent average deviation. That is- how many jumps it would take for a frog of given jump accuracy ability to reach the opposite wall.

Our frog is what happens when an unstoppable frog and an immovable frog have babies.

i think if there is a paradox associated with the OP, it would be that something can be moving with a positive velocity forever without ever covering a finite distance.

it stems from the illusory notion that reality can be quantified.
for example, there is the notion of the "point". if a point has no dimensions, then the "next" or "adjacent" point to it in a theoretical line is zero distance away from it, therefore it is the same point. there fore all points in a line are the same point, and therefore a theoretical line is an impossibility.
yet the notion of a line is fundamental to the notion of a plane which is fundamental to the notion of volume or space, so everything is an illusion since all the points that comprise it do not exist in reality.

S = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + 1/256 + . . .
2S = 2 x (1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + 1/256 + . . . )
2S = 2/2 + 2/4 + 2/8 + 2/16 + 2/32 + 2/64 + 2/128 + 2/256 + . . .
2S = 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + 1/256 + . . .
2S = 1 + S
S = 1

1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + 1/256 + . . . = 1

If each jump takes constant time, then not taking the Planck Length into account, it would take infinite time.

However, the OP's statement of the problem is highly flawed in that it introduces a step that takes real time and thus misses the point of the so-called paradox.

For a clearer presentation, consider the classic example of an arrow flying at a target. It covers half the distance in half the time, then a quarter of the distance in a quarter of the time, then an eighth of the distance in an eighth of the time, and so on.

Some who have little understanding of math see this as meaning it never gets to the target. In reality, the arrow maintains its velocity (in real life it will lose a little due to drag) and will hit the target right on time.