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ThinkingSapien
Guest
i think you are applying the wrong mathimatical tool to the problem. Stick with linear methods for linear problems and it will all work out.
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Nihilist
Guest
Berkely wasn’t a physicist at all- but basically he argued that observed reality is the ‘real thing’, and that whatever we might think ‘should’ happen, on the basis of theory, is less real than what is observed to happen.
Zeno would say- for such-and-such a theory, motion is impossible. Therefore my perceptions are wrong.
Berkely would say- I observe motion, therefore it is real. But such-and-such a theory would render motion impossible. Therefore, the theory is wrong (or irrelevant).
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Oreoracle
Guest
As an aside, I once read from a math historian who said that Zeno’s point was satire meant to illustrate the need to deal with infinite series. The Greeks had a well-known aversion to infinity, but Zeno’s Paradox was meant to show that they needed to grapple with infinite series and sequences eventually if they wanted to explain even mundane phenomena such as motion.
So Zeno actually didn’t buy into the argument he espoused. He was saying, “Look guys, we need better mathematics or else we could make silly arguments like this.”
Dismissing motion just because it’s imperceptible could be problematic though. Take Schrodinger’s Cat for example. Small, imperceptible motion could result in large-scale changes. If we dismiss the imperceptible motions, we are at a loss for explaining the large-scale events.
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aragonjohn1
Guest
no, that is, not yet, possible according to known laws in the physical Universe. but though, nature and energy can change form in order for there to be a change in "un-ordered"and elemental matter there first has to be a purposeful act or reflection to co-order or make an object entirely. though even fortuitously depends on the locale where You are originating individually in fact. adjustably yes and no
God bless
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thinkandmull
Guest
I’m still not satisfied. The fact that the infinite equals 1 is the very mystery I am addressing. And length HAS to be infinitely divisible, otherwise the length would be composed of things that have no length. I am pretty sure there s no answer to this
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Nihilist
Guest
Hmm… I must give my opinion that this is a mistaken characterization of Zeno. He was an Eleatic, and disciple of Parmenides- and is portrayed as a spokeman of Eleatic monism in Plato’s ‘Parmenides’ dialogue- which our best source about him. We know about his paradoxes through Aristotle and others. (The other important Eleatic was Melissus of Samos)
Essentially, Parmenides (as we know from his own writings) was a radical monist, who believed that the truly existent was basically static and unchanging, and that what we see (the changing, movement, the temporal), was an illusion. Zeno attempted to support Parmenides’ thesis by demonstrating that what we see (e.g. motion) is logically impossible.
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Jaaanosik
Guest
It appears that the length is not infinitely divisible. The Planck length is the limit in our space-time.
One might argue that there is no length in between two points separated by the Planck length.
A particle at point A; then there is a discrete jump to point B. The point B is the Planck length away from point A and it takes the Planck time for the jump to happen. It appears there is nothing in between.
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Oreoracle
Guest
Well I suppose Zeno was a nutter after all. Thanks for the clarification.
The infinite doesn’t equal 1, rather an infinite number of terms can add up to 1. Here’s the basic idea:
We have the series 1=1/2+1/4+1/8… To justify the fact that this is indeed equal to 1, we have to clarify what an infinite series is actually meant to represent. In other words, we have to define what “1/2+1/4+1/8…” means. One way to do this is to think of how you would approximate such a series if it did exist. Since the terms get smaller as we progress though the series, a reasonable method of approximation would be to add up the first several terms. The sum of the first n terms is called the “n-th partial sum”. If we choose a large enough n, we’ll be close to the true value of the series since the contributions of the remaining terms will be small.
But maybe we’re more ambitious than that, and we want to see where better and better approximations will take us. To this end we calculate the first partial sum, the second, then the third, and so on. The first few partial sums of this series are 1/2, 3/4, 7/8, and so on. In fact, the n-th partial sum will be (2^n-1)/(2^n). Our numerator and denominator will get bigger over time, with the numerator being only 1 less than the denominator. Thus it’s easy to see that these partial sums will approach 1 over time.
In calculus, one formalizes what “approach” means through the concept of limits. For example, we would say that the limit of the sequence of partial sums as n approaches infinity is 1. There is a formal definition of what a limit is that is available on Wikipedia if you’re interested. But basically that’s it: We define an infinite series to be the limit of its partial sums. This resolves Zeno’s Paradox and many other paradoxes.
Planck length and Planck time refer to the smallest measurable units of length and time. Physicists aren’t suggesting that smaller intervals of distance and time don’t exist.
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Jaaanosik
Guest
Some do suggest that. The issue is that nobody knows what
s there... Thats the edge of the reality.
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yppop
Guest
This paradox presented in the OP was created by Zeno of Elea. The implied conclusion is: “If space is continuous (infinitely divisible), there can be no motion; and if there is motion space cannot be continuous”.
Of course there is motion, and space certainly seems to be continuous, so the consensus of the other Greek philosophers (and all mathematicians that followed) was that Zeno couldn’t be right. So we were stuck with the hard to imagine reality of motion in continuous space. No fear, mathematicians to the rescue with the concept of a limit as Oreoacle showed in post #2 with the sum of the infinite series 1/2+1/4+1/8+… is a finite number 1. This makes mathematicians happy even though philosophically we can’t ever reach infinity so we will always be a thin hair short of 1.
However the mathematicians forged on, happy with their handling of very small quantities (the infinitesimal, delta-x, epsilons) to suit each situation. The physicists were also happy with continuous space until the science reached into the very small scales of the quantum and infinities began to show up in their calculations. To deal with the limit to which space has to be shrunk, physicists invented their own schemes such as renormalization, string theory, or quantum loops, all of which impose a limit on how far space can be shrunken. What this implies is that space is not continuous. The space that defines the dimension of the universe must be discrete as implied by the geometric series, 1/2+1/4+1/8+…, except in discrete space the series ends when it reaches the Planck Length, ~10^(-35) meters.
Bahman, my sometime adversary, by virtue of presenting a proliferation of crazy ideas sometimes comes up with one that is surprisingly deep. His thread "Consciousness is Primary" contained an interesting insight in the form of an operator P’P operating on an initial state S to change it into a subsequent state S’. I wanted to discuss this before the thread disappeared.
In post #17 of this thread, he wrote, “…. There is no guarantee that there exist a unique map between the form that we perceive and what exist out there.” I infer from this that he imagines two levels of reality in which the mechanisms of reality are different and that what we experience is a manifestation of “what exist out there”. If that is true then I contend that at the ground of reality there is no motion of matter through space, as Zeno demonstrated and Berkeley contended. And what we observe as motion of matter through continuous space is, at the ground of reality, incremental reconfiguration of discrete space, just as we observe motion of simulated matter on the screen you are looking at, as the result of incremental reconfiguration of pixels.
Yppop
Of course there is motion, and space certainly seems to be continuous, so the consensus of the other Greek philosophers (and all mathematicians that followed) was that Zeno couldn’t be right. So we were stuck with the hard to imagine reality of motion in continuous space. No fear, mathematicians to the rescue with the concept of a limit as Oreoacle showed in post #2 with the sum of the infinite series 1/2+1/4+1/8+… is a finite number 1. This makes mathematicians happy even though philosophically we can’t ever reach infinity so we will always be a thin hair short of 1.
However the mathematicians forged on, happy with their handling of very small quantities (the infinitesimal, delta-x, epsilons) to suit each situation. The physicists were also happy with continuous space until the science reached into the very small scales of the quantum and infinities began to show up in their calculations. To deal with the limit to which space has to be shrunk, physicists invented their own schemes such as renormalization, string theory, or quantum loops, all of which impose a limit on how far space can be shrunken. What this implies is that space is not continuous. The space that defines the dimension of the universe must be discrete as implied by the geometric series, 1/2+1/4+1/8+…, except in discrete space the series ends when it reaches the Planck Length, ~10^(-35) meters.
Bahman, my sometime adversary, by virtue of presenting a proliferation of crazy ideas sometimes comes up with one that is surprisingly deep. His thread "Consciousness is Primary" contained an interesting insight in the form of an operator P’P operating on an initial state S to change it into a subsequent state S’. I wanted to discuss this before the thread disappeared.
In post #17 of this thread, he wrote, “…. There is no guarantee that there exist a unique map between the form that we perceive and what exist out there.” I infer from this that he imagines two levels of reality in which the mechanisms of reality are different and that what we experience is a manifestation of “what exist out there”. If that is true then I contend that at the ground of reality there is no motion of matter through space, as Zeno demonstrated and Berkeley contended. And what we observe as motion of matter through continuous space is, at the ground of reality, incremental reconfiguration of discrete space, just as we observe motion of simulated matter on the screen you are looking at, as the result of incremental reconfiguration of pixels.
Yppop
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thinkandmull
Guest
I would have told Zeno that even in his “illusion” of an external world his paradox applies so whats the point of saying the world doesn’t exist
Also, even if there is no space between two points, those points have length, otherwise the lengthless would compose that with length. Has this been addressed yet.
The Bing Bang theory is based on the idea that the universe started in a singularity that was not continuous. Interesting
Also, even if there is no space between two points, those points have length, otherwise the lengthless would compose that with length. Has this been addressed yet.
The Bing Bang theory is based on the idea that the universe started in a singularity that was not continuous. Interesting
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Linusthe2nd
Guest
The secret is not to pay any attention to Sophists who always try to tie you up in " logical " knots. You get from A to B every second of your life, so the Sophist must be wrong - even if you cannot explain why.
Linus2nd
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Oreoracle
Guest
There is always a distance between two distinct points, and the “length” of a point is zero. More precisely, they don’t have length at all, just as lines don’t have area and surfaces don’t have volume.
What you are suggesting is that we can glue points together to make a curve, and since curves have length, individual points must have length. This is on the right track, but it’s misleading. For one thing, the “length” of a point cannot be a real number. To see this, take any line segment and find its midpoint. This breaks the segment into two segments. Now find the midpoint of this segment, and do the same for the smaller segment this creates, and so on. There are clearly infinitely many points in a line then, so the length would be infinite if the length of a point were a real number, because distances are additive. There are versions of calculus which let you treat lengths as infinitesimals (infinitely small numbers), and this is a better approach, though it’s not the traditional one.
But I think there is really a more fundamental problem with your approach, which is the assumption that we can take a point and connect the “next” point to it when forming a curve. This assumes that the number of points is countably infinite, when in reality they are uncountably infinite. This means there is no such thing as a pair of adjacent points in a curve (more formally, it means that you cannot make a one-to-one correspondence between the set of points and the set of positive integers).
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yppop
Guest
With discrete space the length of a line is created simply by adding up the gaps between the rational numbers. A rational number is defined as the ratio of whole numbers (fractions). They have the property that their decimal equivalent ends in an infinite string of zeroes such as ¼ = 0.25000000; repeats a single integer forever such as 1/3 = 0.333333333333333…; or repeats the same group of integers forever such as 1/7 = 0.142857214285721428572… . There is an infinity of rational numbers, called aleph 0.
With continuous space, the length of the line is created by filling all the gaps between the rational numbers with an infinitely infinite number of non-rational numbers. The infinity associated with the real numbers (rational + non-rational) is called aleph 1. Aleph 1 = 2^ aleph 0.
Consider the hydrogen atom, a proton consisting of 3-point particle (having no physical dimension) quarks surrounded by a single point particle electron. The volume of the hydrogen atom is 30,000 times the size of the proton. In other word, the hydrogen is mostly empty space. To explain a structure in which the material parts are immersed in an ocean of empty space, it seems to me that it is plausible to infer that if the electron and the quarks are constructed from discrete space, the “empty space” must be continuous. A typical atom is 50,000 times the size of its nucleus, hence we and the universe are composed mostly of empty (continuous) space.
If discrete space informs matter, then what is the continuous space in which it is immersed? There can be only one answer to such a construction: discrete space represents the substance of matter and continuous space represents the spiritual component. We and the entire universe are organized hylomorphically.
For those of you, whose impression upon reading this post is, “Yppop is crazy!” here are some quotations from people (with far greater credentials that me) who have similar thoughts:
Yppop
With continuous space, the length of the line is created by filling all the gaps between the rational numbers with an infinitely infinite number of non-rational numbers. The infinity associated with the real numbers (rational + non-rational) is called aleph 1. Aleph 1 = 2^ aleph 0.
Consider the hydrogen atom, a proton consisting of 3-point particle (having no physical dimension) quarks surrounded by a single point particle electron. The volume of the hydrogen atom is 30,000 times the size of the proton. In other word, the hydrogen is mostly empty space. To explain a structure in which the material parts are immersed in an ocean of empty space, it seems to me that it is plausible to infer that if the electron and the quarks are constructed from discrete space, the “empty space” must be continuous. A typical atom is 50,000 times the size of its nucleus, hence we and the universe are composed mostly of empty (continuous) space.
If discrete space informs matter, then what is the continuous space in which it is immersed? There can be only one answer to such a construction: discrete space represents the substance of matter and continuous space represents the spiritual component. We and the entire universe are organized hylomorphically.
For those of you, whose impression upon reading this post is, “Yppop is crazy!” here are some quotations from people (with far greater credentials that me) who have similar thoughts:
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*1. If physical space has at all a real existence it is not necessary for it to be continuous; many of its properties would remain the same even if it were discontinuous. And if we knew for certain that physical space was discontinuous there would be nothing to prevent us, in case we were so desired, from filling up its gaps, in thought, and thus making it continuous; this filling up would consist in a creation of new point-individuals and would have to be effected in accordance with the above principle. (of continuity) (Richard Dedekind - World of Mathematics, pg 530)
2. Nevertheless, there are some intriguing hints that this particular universe may in fact be a discrete digital universe, not a continuous analog universe the way most people would expect. In fact these ideas actually go back to Democritus, who argues that matter must be discrete, and to Zeno, who even had the audacity to suggest that continuous space and time were self-contradictory impossibilities. Through the years I’ve noticed many times, as an armchair physicist, places where physical calculations diverge to infinity at extremely small distances. Physicists are adept at not asking the wrong question, one that gives an infinite answer. But, I’m a mathematician, and each time I ' wonder if Nature wasn’t really trying to tell us something, that the real numbers and continuity are a sham, and that infinitesimal small distances do not exist! – (Gregor Chaitin - Meta Math, The Quest For Omega – pg. 91)
3. “What we observe as material bodies and forces are nothing but shapes and variations in the structure of space. Particles are just appearances.” (Erwin Schroedinger - Life and Thought,1989)
4. “ But if the ultimate model for the universe is to be as simple as possible, then it seems much more plausible that both space and its contents should somehow be made of the same stuff—so that in a sense space becomes the only thing in the universe.” ( Stephen Wolfram - A new Kind of Science, pg. 474)*- Status