12
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Is there a distance so small it can't be further divided?
Is there a distance so small it can’t be further divided?
The modern solution to this problem is the use of infinitesimals, as used by Leibniz and Newton in their development of the calculus. ...
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12
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Is there a distance so small it can't be further divided?
The time it takes for the arrow to reach one half of the distance, is one half of the time.
So the total length traveled by the arrow is one half the distance, plus one half of one half, plus one half ...
- 573
7
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Accepted
Is Aristotle's resolution of Zeno's paradoxes vindicated by motion in the intuitionistic continuum?
There are several notions of intuitionistic continuum, the closest ones to Aristotle's are Brouwer's "fluid continuum", and especially late Weyl’s version of it since On the New Foundational Crisis of ...
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4
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Is there a distance so small it can't be further divided?
There does not have to be some distance so small that it can't be divided in half, to solve the paradox.
The infinitesimals referred to by Mark Andrews become the differentials in calculus as first ...
- 7,772
4
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What is the difference between Zeno's "Dichotomy" and Richardson's "Coast of England" paradox?
The issue is not time but whether the mathematical series which models the sequence is convergent or divergent.
In the case of Zeno's arrow paradox, in which he argues that the arrow never gets there, ...
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Which ancient Greeks are known to have commentated on Zeno's Paradoxes?
The prime critic was Aristotle : 'Physics', VI.
Plato does not set out Zeno's arguments but in his dialogue, 'Parmenides', there is in connexion with the paradoxes some argument or interplay between ...
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Is there a distance so small it can't be further divided?
I do not wish to leave this post in original format as comments have made clear that my assumption was not totally accurate, see below.
In physics the shortest possibly lenght is called the Planck ...
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3
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Accepted
How do empiricists explain zeno's paradox(es)?
Not everyone agrees that the paradoxes have been solved, but they aren't strictly empirical problems either.
There's a sizable school of thought, myself mostly included, holding that calculus ...
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2
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What resolves Zeno's argument for the non-existence of place?
Aristotle resolves the argument in Physics bk. Δ On Place, ch. 3 (210b):
Zeno's problem—that if Place is something it must be in something—is not difficult to solve. There is nothing to prevent the ...
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Why is Aristotle's objection not considered a resolution to Zeno's paradox?
The most clear way for me to resolve this famous-seemingly-deep-hard paradox lies in the modern infinite calculus which can be clearly expressed using Leibnitz dx symbol. So in summary, Zeno "...
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Why is Aristotle's objection not considered a resolution to Zeno's paradox?
There is a difference between a point and an indivisible. I guess the important thing is to consider the distinction. The easy way out here is to consider a point as an object without magnitude and ...
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What are possible resolutions of the length unit paradox stemming from Zeno's Paradoxes?
Most of the Zeno-like paradoxes involve an infinite sequence of events buried in the language used in the problem. Your example appears to go down the path of doubling the distance between the lines ...
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2
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How can one explain this conflict between philosophy and classical physics?
The task the OP presents is similar to the Achilles and the Tortoise presented by Zeno in Plato's Parmenides. A task which can be completed is divided into an infinite series of smaller tasks which ...
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Does the uncertainty principle resolve Zeno’s arrow paradox?
Zeno's arrow paradox is a redefinition of "motion": Quantum physics is not required to deal with Zeno's arrow paradox. The statement of the "paradox" works by invoking the idea of "motion" while only ...
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How can one explain this conflict between philosophy and classical physics?
In my opinion, the underlying fallacy of your argument is that you assume that because you have described the motions as divided into an infinite sequence of sub-motions, then somehow they actually ...
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Zeno's stadium paradox: If space is not continuous or discrete, what is it?
The problem will all paradoxes of this particular sort is actually a problem of language: we don't have a good way (in spoken language or specialized languages like mathematics) to conceptualize ...
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2
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Why is Diogenes the Cynic's solution to Zeno's Dichotomy Paradox insufficient?
My two cents.
Diogenes solution was a practical démonstration that refuted Zeno's claim. Diogenes demonstrated in practice that Zeno's theoretical construction is not a correct description of this ...
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2
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A new challenge to physical reality
According to calculus: The sum of (½)^n from n=0 until infinity is 2. It is the sum of the geometric series with q =1/2, a standard example for a convergent series.
You can easily check by running in ...
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2
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Is there a distance so small it can't be further divided?
ε
Epsilon is the smallest value that can be added to a number such that n+ε>n. In practice I have mostly seen it used in computer modeling to solve the case that when you have material that is 1 ...
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2
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Is there a distance so small it can't be further divided?
As far as natural numbers go, when they are represented as distances, they all admit of being broken in half. Zero is the sort-of-exception, except that 0/2 is determinately evaluable (unlike division ...
- 12.4k
2
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Is there a distance so small it can't be further divided?
I think there are two questions here: Zeno's Paradox, and the smallest indivisible unit of length.
Smallest Unit of Length
If you want an answer from physics, rather than philosophy, you're talking ...
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2
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Is there a distance so small it can't be further divided?
"Is there a distance so small it can't be further divided?" I searched through the existing answers and discovered that neither "atomism" nor "Democritus" are mentioned. ...
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Is special relativity immune to the paradox of Achilles?
Yes, there are n things wrong with your argument, where I estimate n to have an upper bound of around 10. Here are some of them...
None of the versions of Zeno's paradox about Achilles that I have ...
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A new challenge to physical reality
While the other answers here have discussed how calculus addresses the issues arising from infinities and infinitesimals here, there is another issue here regarding the physical soundness of Zeno's ...
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Accepted
A new challenge to physical reality
Short Answer
Mathematics describes the universe, and Xeno's ignorance of calculus does not determine how the universe works. It would be disconcerting if an infinite series of diminishing lengths ...
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A new challenge to physical reality
The paradox assumes that we can do as small movements as we want. Our muscles are not able to produce movements smaller than some multiple of the Planck constant. Assuming this the paradox vanishes.
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Experiencing and sensing time dilation when a person dies and the logic of
It seems that our experience of the passage of time in the absence of any visual cues is linked to the rate at which certain periodic processes take place in the brain. Crudely, if you wanted to make ...
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1
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Zeno's stadium paradox: If space is not continuous or discrete, what is it?
It is not a paradox at all
At the time Zeno lived, coordinate systems and frames of reference were not yet being invented and studied. You really do not need relativity to solve this. All you need is ...
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vote
Accepted
Who said 'You can't step into the same river once'?
It is attributed by Aristotle in the Metaphysics to Cratylus, a follower of Heraclitus. Reference: Aristotle, Metaphysics, 4.5 1010a10-15.
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