Didn't see anything there that says I don't understand mathematics. However,I am good at recognizing falsity when I see it though. Whether that falsity is mathematically proven or not doesn't really matter too much to me.

Not at all. We can reason about infinity without actually doing anything an infinite number of times. If someone (God, perhaps) were to add up Banno's infinitely many fractions or carry out my multiplication to infinitely many decimal places, then the result would be 1 in either case. — aletheist

The problem is, that this feat of adding infinitely many fractions would never be finished, so 1 would never be reached. Therefore it is false to say that it equals one. It does not. You can claim Banno's cheat all you want, nevertheless you're still claiming that a falsity is true.

From your Wikipedia page Banno: "As n approaches infinity, sn tends to aproach 1." Does that means =1 to you?

Not playing, Meta. Go read a maths book.

That would be really really boring. I already know you can do whatever you want with maths, just make it up as you go, and prove whatever you want to prove.

The problem is, that this feat of adding infinitely many fractions would never be finished, so 1 would never be reached. Therefore it is false to say that it equals one. — Metaphysician Undercover

Mathematics is entirely a matter of necessary reasoning about hypothetical states of affairs. There is no falsity whatsoever in saying that if someone were to add infinitely many fractions in a particular series, then the result would be 1. The fact that no one can actually add infinitely many fractions is completely irrelevant.

Mathematics is entirely a matter of necessary reasoning about hypothetical states of affairs. There is no falsity whatsoever in saying that if someone were to add infinitely many fractions in a particular series, then the result would be 1. The fact that no one can actually add infinitely many fractions is completely irrelevant. — aletheist

No, the fact that it is impossible to add infinitely many fractions is relevant. Because your premise is "if someone were to add infinitely many fractions...". But this is an impossibility. Therefore your premise is false. It is stating "if someone were to do something impossible..." Your premise is false therefore your conclusion is equally false.

I already know you can do whatever you want with maths, just make it up as you go, and prove whatever you want to prove. — Metaphysician Undercover

It is really no different from philosophy in this regard; it all boils down to one's assumptions. To get us back on topic, Zeno's alleged paradox exploits this by smuggling in the idea that any finite interval of space consists of infinitely many individual points, such that one must somehow pass through them all in order to get from one place to another. Once we dispense with that misconception and recognize that space is continuous, and the only actual points are the ones that we arbitrarily define, the paradox dissolves.

You are confusing actual possibility with logical possibility. Mathematics deals with the latter, not the former. It is indeed actually impossible to add infinitely many fractions, but it is not logically impossible.

There's a difference between moving from the starting point to the end point, and reciting all the distances moved along the way.
Especially if every such recitation takes the same amount of time (or longer than some specific non-zero amount of time).
Of course, if every such recitation took a duration proportional to the corresponding distance, then the moving and the reciting would be more alike.

Anyway, either both distance and duration are discrete, or both are not.
Modeling both with the continuum works, and has the bonus of numbers like π and e. (Y)

Anyway, either both distance and duration are discrete, or both are not. Modeling both with the continuum works, and has the bonus of numbers like π and e. — jorndoe

Even the real numbers do not constitute a true continuum, because they still amount to an aggregate of discrete individuals. However, I agree that this psuedo-continuum is an adequate mathematical model of continuous phenomena (like space and time) for most practical purposes.

↪Metaphysician Undercover
No; because history shows you cannot understand mathematics.

https://en.wikipedia.org/wiki/1/2_%2B_1/4_%2B_1/8_%2B_1/16_%2B_⋯ — Banno

(Y) Though the proof is a bit short. :D

Before you folk wander too far away, have you noticed the similarity between the OP here and over at http://thephilosophyforum.com/discussion/1037/fallacies-malady-or-remedy ?

It is really no different from philosophy in this regard; it all boils down to one's assumptions. To get us back on topic, Zeno's alleged paradox exploits this by smuggling in the idea that any finite interval of space consists of infinitely many individual points, such that one must somehow pass through them all in order to get from one place to another. Once we dispense with that misconception and recognize that space is continuous, and the only actual points are the ones that we arbitrarily define, the paradox dissolves. — aletheist

I don't think there is any issue with points in this paradox. I believe the problem is quite similar to how TheMadFool states it. The issue is the assumption that space is infinitely divisible. Zeno assumes that in theory, space is infinitely divisible. However, in practise space is not infinitely divisible. The conclusion makes a statement about what can and can't be done in practise. The theory is wrong, and therefore cannot be successfully applied in practise.

You are confusing actual possibility with logical possibility. Mathematics deals with the latter, not the former. It is indeed actually impossible to add infinitely many fractions, but it is not logically impossible. — aletheist

Sure, mathematics is logic, but valid logic does not necessitate a true conclusion. The premises must be judged. So I am judging the premise with respect to what is actually possible. Your premise states "if someone does a thing which is actually impossible to do...". So I judge it as an impossibility and therefore a falsity. And so I judge your conclusion as a falsity as well.

Logic cannot be wrong; but it can be inappropriate. Given a paradox, one ought look for a better grammar, a new logic; rejecting logic as "wrong" shows a profound misunderstanding. Logic is mere grammatical structure. — Banno

What Meta does is simply refuse to accept the grammatical structure that allows the dissolution of the problem.

Good for him. But then the problem becomes his, not ours.

Flies and fly bottles and such. Or leaving the troll to look elsewhere for sustenance. Some conversations can only serve as examples of how not to have a conversation.

The issue is the assumption that space is infinitely divisible. Zeno assumes that in theory, space is infinitely divisible. However, in practise space is not infinitely divisible. — Metaphysician Undercover

You keep confusing potentiality with actuality. Either space is infinitely divisible or it is not. Whether anyone can actually divide space into infinitely many parts is completely irrelevant - only whether it could potentially be divided into infinitely many parts. Zeno's clever strategy was to exploit this confusion. Although space is indeed infinitely divisible, it is not infinitely divided. Although there are infinitely many potential locations between two defined locations A and B, there is no need to account for all of those intermediate points in actually moving from point A to point B, because they are not actual points.

Your premise states "if someone does a thing which is actually impossible to do...". So I judge it as an impossibility and therefore a falsity. And so I judge your conclusion as a falsity as well. — Metaphysician Undercover

You are misstating my premiss. It is not, "if someone does a thing which is actually impossible to do ..." It is, rather, "if someone were to do a thing which is actually (but not logically) impossible to do ..." The conditional nature of the whole proposition is key here.

One problem with summation of the geometric sequence solution:

The half (1/2) is an arbitrary choice. I don't know if Zeno actually said ''half'' or not. Anyway the sum of the sequence 1/2, 1/4, 1/8,... IS 1.

However let us choose another fraction instead of half. Say one-third. That is to say before we reach from point A to pointB we have to reach one-third(1/3) of the way and before that 1/3 of 1/3 and so one.

The sum of 1/3, 1/9, 1/27, 1/81...sequence is NOT 1. It is half(1/2).

You can try that with other fractions too. The sum doesn't equal 1.

Therefore the paradox remains unresolved as far as math is concerned.

So after one gets half way, one continues on to the end.

8-)

Can you see that the series ½ + ¼ + ⅛... is infinity long, and yet adds to 1? — Banno

For practical purposes we say it "equals" 1, but it actually only approaches 1, as it's clear 1 is the boundary, but to say it "equals' 1 the same as 2+2 "equals" 4 is equivocation. So, I don't buy into your mathematical solution even if math could dictate metaphysics.

This is just to point out that all movement is impossible, which was the asserted conclusion to begin with. That is, it's not just impossible under this paradox to get to the arbitrary finish line, but it's impossible to get to any arbitrary point, meaning any movement whatsover is impossible.

meaning any movement whatsover is impossible — Hanover

But we can move. We do it all the time.

Here's the problem: Infinity is ultimately an incoherent concept and should it be inserted into any discussion, confusion will follow if you think it through.

Of course we move. Such is the paradox. My point was that your pointing out that we can't get to the halfway point under the paradox, much less the finish line, really can be reduced to the claim that we can't get to any point at all. That means we can't move at all.

So after one gets half way, one continues on to the end. — Banno

But I'm summing the entire distance from A to B. The only difference is I'm moving a third of the distance now.

How do you explain that walking in thirds takes me only to the half-way point?

Ok

You dropped the italicised bit in.

One can count the rational numbers without putting them in sequence.

Just list the fractions between one and two; 3/2, 4/3, 5/3, 5/4, 7/4... — Banno

It was explicitly mentioned several times, and implied any time it wasn't, that the counting is sequential, given that it's an analogy to the movement between two points, which would involve an object passing sequentially through each rationally-numbered coordinate between them.

And so by the same logic, if we can't even start sequentially counting every rational number, because there isn't a first number, an object can't even start sequentially moving through each rationally-numbered coordinate, because there isn't a first coordinate.

Although there are infinitely many potential locations between two defined locations A and B, there is no need to account for all of those intermediate points in actually moving from point A to point B, because they are not actual points. — aletheist

And so what is the first potential rationally-numbered coordinate that an object must pass through in its movement from A to B?

Your talk about how actually plotting the coordinates is an arbitrary decision which will use some decided-upon minimum unit completely misses the point of the paradox. Although we might actually only plot every 1/18th of a unit in our coordinate system (and so count 1/18 as the first rational number), the object must still pass through what would have been the 1/36th unit had we plotted that, and so on.

This is why continuous movement in continuous space should be impossible. An object would have to sequentially pass through every potential rationally- (even real-) numbered coordinate to get from A to B, just as continuous counting in a continuous number line is impossible, as we'd have to sequentially count every potential rationally- (even real-) numbered coordinate between A and B.

If the counting is to be possible it must be that the number line (or just the counting) is discrete; we have some minimum fraction to work with (say, 1/18). And so if movement is possible it must be that space (or just the movement) is discrete; the object has some minimum fraction to work with. But it's because we want to maintain both continuous space and continuous movement that the paradox arises.

The only reason we cannot do it is because we could never count infinitely fast which is what we would need to be able to do to complete the series. — John

Then leaving aside the fundamental logical problem as explained above, this would entail that the only way an object can sequentially move through every rationally-numbered coordinate between two points is if it could move infinitely fast. But, of course, objects don't move infinitely fast.

If that article is to be believed, then everyone is missing the point by offering physical and mathematical solutions. I brought it up earlier so that I could make explicit that I was aware that I was not actually addressing the paradox by indulging in physics talk.

The article suggested that it was a matter of metaphysics, and identity. One is one, two is two, just tautologically. They're crisp and well defined. One cannot derive two from one, and if you could it would always lead to an indeterminate, like zero or infinity.

Wosret is right, in that Zeno was uninterested in the mathematics of infinity. There is a correct solution to the problem mathematically, but this only means that the paradox wasn't designed for a post-calculus world. Zeno makes obvious mistakes in his paradoxes due to his ignorance of the existence of different types of infinity, but the original paradox is Paramendian - one cannot become many. Parmenides viewed the world as a sphere containing no differentiation - all was of the same substance. From Wikipedia: "[What exists] is now, all at once, one and continuous... Nor is it divisible, since it is all alike; nor is there any more or less of it in one place which might prevent it from holding together, but all is full of what is." (B 8.5–6, 8.22–24). From this he asserts that time doesn't exist, since change from one state to another is impossible given this framework for understanding the world.

This view seems very strange if taken at face value - after all, we can obviously divide space and time into measurable quantities. The only way I have been able to make sense of it is by viewing it as an imprecise conception of the Primordial Existential Question, or why anything exists at all. For Parmenides, the appearance of the world is an illusion hiding the underlying singularity of all things. So viewing it in this way, Zeno's paradox can be reformulated as something like: "How can we proceed from timelessness to time, or from dimensionlessness to dimension, if we view reality as truly timeless and dimensionless?" The point is not to demand an explanation of how one moment becomes another or how space can be divided, but rather to fundamentally question time and space. In other words, we cannot use a yardstick when the concept of space is not yet defined.

It was explicitly mentioned several times, and implied any time it wasn't, that the counting is sequential, given that it's an analogy to the movement between two points, which would involve an object passing sequentially through each rationally-numbered coordinate between them. — Michael

An analogy can only be useful for illustrating an argument, and you have yet to offer an argument. You assert that moving from place to place is possible if and only if one can utter all rational numbers between 1 and 2 in sequence and in finite time, but you haven't offered an argument for this assertion.