Solution to Zeno's Paradox
*Zeno's Paradox (actually the first of four) purports to show that motion is impossible.
If you want to walk from one point to another, you must first cross half the distance, then to cross the remaining distance, you must first cross half of the remainder, and so on.
If the distance you want to walk is 1 kilometre long, you will first reach 1/2 km from your start, then 3/4 km from your start (1/2 + 1/4), then 3/4 km (1/2 + 1/4 + 1/8), then 7/8 and so on. After you have taken N steps of the journey, you will have gone a distance equal to 1-(1/2)N kilometres. No matter how big N becomes (that is, no matter how many successive half-distances you cover), this distance will always be less than 1 and you will never arrive at your destination.
(Another permutation of Zeno's First Paradox seemingly prevents you from leaving your starting point. Before you can cross a given distance you must first cross half the distance, but before you can do that you must cross half that half-distance, ad infinitum.)
The solution I propose involves observing that Zeno's First Paradox assumes the person or object doing the moving is infinitely small, eg. a point (or a line perpendicular to the direction of travel).
Say your walking stride length is 1 metre, and for convenience the distance to your destination is 10 metres.
Using these figures, it's easy to see that it will take you 10 strides to reach your destination. Working through the paradox, to reach 10 m you must first reach 5 m, which you easily can cover in 5 strides. From this point to reach half the remainder (7.5 m from the start), you only need 3 more strides. In fact, with 3 more strides you exceed that distance. To fit with the paradox, you must reduce your stride length and keep on doing it so that you don't exceed each subsequent half-distance. And you must do so an infinite number of times in order to never reach your destination.
However, this is only possible with an object of zero size. If we started with an object with finite (positive) size, like a tortoise or a human, and measured the stride lengths from one heel to the next, at some point in our stride-length-reducing exercise the heel of a foot would exceed the length that is left to travel, so that the tortoise or human can be said to have arrived at the destination.
Interestingly, by merely changing how we measure stride-lengths in this scenario, we can illustrate the non-arrival of a zero-sized object. By measuring toe to toe rather than heel to heel, we in effect have a zero-sized object which, with ever reducing stride lengths, never arrives at its intended destination.
A funny way of describing this is that in order to 'prove' that motion is impossible using Zeno's Paradox, you must purposefully try not to reach your destination (that is, by successively reducing your stride length).
(You can perform this thought experiment using a line segment of finite size. Imagine you have a 1 metre long ruler. For the first 5 m, you 'walk' or flip the ruler end to end to cover the distance.
Then for the next 2.5 m you can flip the ruler for two-lengths, then angle the ruler so that it covers the remaining 1/2 metre without exceeding it. And so on. Over time, the final angle of the rulers gets more and more oblique as the final 'stride' distance gets progressively smaller.)
Therefore an object with a finite size can traverse a finite distance, but an object that is infinitely small cannot do the same.
This solution has a couple of interesting ramifications. The first is that since a point is infinitely small, its 'stride length' is infinitely reducible and it can satisfy Zeno's First Paradox, namely a point cannot move. Points can mark positions but they cannot traverse a finite distance.
Conversely, a moving object cannot be a point. If you observe an object travelling from one point to another, the object must have a finite size, no matter how small. Motion is a property of finiteness.
Stimulus: "The Infinite Book", by John D. Barrow. The description of Zeno's paradox above is largely drawn from pages 20-21.
kyy 2007-04-02 1542 z+10
*Note: This is version 1.1 of the solution. The first two comments below refer to version 1.0 of the solution, which published by kyy on 2006-03-21 1332 z+9.
posted by kyy at 5:26 PM
23 Comments:
Hi again, once again, the assumption that objects are finite needs to be rigorously established in order that your argument is sound. We cannot use assumptions to rigorously justify our assumptions.
I quote:
"Therefore an object with a finite size can traverse a finite distance, but an object that is infinitely small cannot do the same."
Criticism:
I cannot say that this is true as well. Define what is infinitely small? Are we defining it to be the value that calculus uses? If so, then that is contradictory to the fact that many curves etc, are composed of such infinisimals, but can be finite, i.e a circle. If you mean infinitely small, but not zero, it must therefore be some finite albiet value smaller than any other. Since it is not 0, then,
consider infinite small = e.
e > 0, thus 2e > 0, 3e > 0, etc.
We restrict ourselves to using integers alone.
If 2e < 0,
then, e < 0. Which contradicts our initial premise. Thus, we see that infinitely small values can nontheless be greater than a given positive value. Neither can e = 0,
since we assumed that it is an infinitely small value, but nontheless not= 0. For if it were 0, then 0 + 0 + 0 ... = 0, then it is static. But since e > 0, then
e + e > 0, e + e + e ... > 0, so it does traverse some finite distance > 0. Since any distance = 0, is static, and any distance > 0, is motion, or travel, then it does indeed traverse such a finite length.
I quote:
"This solution has a couple of interesting ramifications. The first is that since a point is infinitely small, its 'stride length' is infinitely reducible and it can satisfy Zeno's First Paradox, namely a point cannot move. Points can mark positions but they cannot traverse a finite distance.
Criticism:
By my analysis that infinitely small strides or points, can be greater than some positive integral length, or some length, it shows that points can traverse a finite distance. If we considered points to be the make-up of a number line, the number line represents a spatial distance, and points do exist inside them, as certain fixed numbers. And by Cantor, et al, there exists an infinite number of such points that corresponds to numbers.
philosphers_are_boring_fucks' (henceforth 'pabf') comments got me thinking about what it means to be infinitely small (or infinitesimal).
Basically, something which is 'infinitely small' is a quantity that is smaller (by a finite margin) than any quantity you can think of, even if you are given an infinite number of opportunities of coming up with a quantity which you consider to be the smallest.
Like a couple of school kids trying to outdo each other through coming up with the smallest number (like that's ever happened), the infinitely small outdoes them both by describing a number (really, a process of coming up with a number) which is ALWAYS smaller than whatever figure their overactive minds can conjure.
Returning to the task at hand, is it true what pabf says: an infinitesimal may be infinitely small, but it ain't nothing.
Well, it would seem from the following that the answer is: it's possible.
Say, you're the modest kind of kid that likes to think in whole numbers (as opposed to proportions). So, given a chance, you (let's call you 'Kid A') always come up with a number that's one smaller than whatever your opposition comes up with.
It's clear that your 'number' (or number-generating process) will always produce a number that's smaller than the other kid (let's call him Fred). Fred's a normal kid, so he comes up with numbers like 'point zero zero zero - to a hundred zeros - one', then 'point zero to a million squillion zeros, then one', and so on.
Very quickly, you're gonna end up in negativeland. But no-one said anything about negative numbers not being allowed. By schoolyard rules,no rule may be introduced after the start of mental jousting, so an infinitely small number can be a negative number.
Interesting as this may be, we'll have to leave negativeland forthwith and return to the land of the living, cos eventually we'd like to discuss real-world objects with non-negative quantities like size.
There's a new kid in school, and to make things interesting Kid A and Fred invite him to their weekly geekfest. The new kid, Ah Choo, is a little older and wiser. Ah Choo's just moved here and due to differences in school systems, he's been held back half a year. (Plus where he comes from, they do calculus in primary.) He's hungry, literally, and eager to bring it.
By this time Kid A is getting bored with whole numbers, and so gives proportions a go. And they introduce a new rule: no negative numbers.
Fred loses the toss and is sent in to bat. Since Fred is nobody's biatch, he starts off with 'point - reverse googol - one.'
Kid A: What's that?
Fred: It's one with a googol of zeroes after it, but reverse it and put a point in front of it.
AC: Nice.
Kid A: My number is always half as small as whatever Fred says.
AC: Mine's also always smaller than yours, but by the same margin as the distance of your numbers to the origin.
Kid A: So you mean zero.
AC: Not just zero, but ALWAYS zero.
So zero can be an infinitely small/er number, in the sense that we can come up with (1) a method of producing a number that is smaller than any number, ad infinitum, (2) such that the numbers produced by that method is always zero.
Another way of looking at this is that 'infinitely small' describes a way of producing numbers that are smaller than any other given (positive finite) number, even if you are have an infinite number of opportunities to come up with a smaller given number (which is, of course, unsurprising since zero is smaller than ALL positive numbers regardless of size).
'Zero is an infinitesimal, so what?' i hear you say. Well for one thing, this explains why my maths (yes, over here we really do say and spell it like that) teacher described a point as infinitely small. While this is correct, it would be more precise (though no less accurate) to say that a point has zero size.
Returning to our friend Zeno, where my analysis says:
"The solution I propose involves observing that Zeno's First Paradox assumes the person or object doing the moving is infinitely small, eg. a point."
it would be more precise to say:
"The solution I propose involves observing that Zeno's First Paradox assumes the person or object doing the moving is so infinitely small that is has zero size, ie. a point (or a line perpendicular to the direction of travel)."
And to further clarify my analysis, after the sentence:
"And you must do so an infinite number of times in order to never reach your destination"
I would add the following passage:
"However, this is only possible with an object of zero size. If we started with an object with finite (positive) size, like a tortoise or a human, and measured the stride lengths from one heel to the next, at some point in our reducing-stride-length exercise a foot (being of non-reducing, finite size) would exceed the length that is left to travel, so that the tortoise or human can be said to have arrived at the destination.
Interestingly, by merely changing how we measure stride-lengths in this scenario, we can illustrate the non-arrival of a zero-sized object. By measuring toe to toe rather than heel to heel, we in effect have a zero-sized object which, with ever reducing stride lengths, never arrives at its intended destination."
To sum up then, Zeno's First Paradox holds true only for zero-sized objects, which is why it is a 'paradox' for real world participants.
Of course, we could always introduce a new rule prior to our next sandpit bout.
kyy 2007-04-02 1542 z+10
Sup muh nigga, long time no see. This is philosophers_are_boring_fucks.
kyy may i ask that you supply rigorous proof for any assumptions used in your next replies?
"So zero can be an infinitely small/er number, in the sense that we can come up with (1) a method of
producing a number that is smaller than any number, ad infinitum, (2) such that the numbers produced by that method is always zero."
"Another way of looking at this is that 'infinitely small' describes a way of producing numbers that are smaller than any other given (positive finite) number, even if you are have an infinite number of opportunities to come up with a smaller given number (which is, of course, unsurprising since zero is smaller than ALL positive numbers regardless of size)."
I argee with 1, but i do not see how your method 2, which is a recursive method of letting the denominator of a fraction tend towards infinity, which is still greater than 0.
The definition of zero is the absence of quantity. If there is no quantity, then there is no quantity to be measured. On the other hand, the infinisimal is a quantity, as your own definition shows, by adding the 1 at the end of the ever increasing 0 decimal number. If it were 0, it wouldn't make sense to add the 1, since that 1 denotes the presence of a quantity that is to measured, while 0 is not.
(http://dictionary.reference.com/search?q=zero)
Therefore, there is nothing to be measured, while on the other hand, the infinitesimal can be measured in a few mathematical ways.
Ie by non-standard analysis, Weierstrass, etc.
While on the other hand, the infinitesimal is an object that is smaller than every other standard number, but greater than 0.
(There is a difference between standard numbers and non-standard numbers. This is equivalent to the concept of different orders of infinities, ie countable or uncountable infinities in set theory. Therefore as defined in wikipedia, and other places:
A definition
An infinitesimal number is a nonstandard number whose modulus is less than any nonzero positive standard number.
What standard and nonstandard refer to depends on the chosen context.)
Correct me if i am wrong, but i take you to mean that just because an infinitesimal is smaller than every other "standard" number, as is 0, they are one and the same. But this is false.
This is because, it is mathematically defined to be smaller than every other standard number, but not smaller than every other non-standard number (that is the different orders of smaller and smaller infinisimals). And the crucial point here is that there is no smallest non-standard number in this order of every decreasing smaller infinitesimals. So there is no number to which we can say, "this number is smaller than every non-standard or standard number like 0 is."
Also, if you are right, then it must mean that all infinitesimals are 0, which is false, since the definition in infinitesimal analysis is that,
0 < x < 1/n, for all n.
There maybe other minor details to add into that for full rigor, but for the moment that is what i know.
If we say that 0 =< x < 1/n, for all n, the calculus which bases itself on the derivative, will be made false. For reasons i will now show below.
We cannot also conclude that they are equal just because we have a similarity in part of their definition. This is the same as concluding that 1 is a prime, because we define primes as a number which is only divisible by itself and 1. 1 satisfies this definition but is not included for various number theoretical reasons.
(As a site note, if you are not fully convinced that the premise:
"If a mathematical object which is equal in part to another mathematical object, with respect to its definition, then the 2 mathematical objects are equal."
This is a generalisation of your assumption. I say in part because that isnt the only definition of 0, there are others like division by 0, mulitiplication, addition, and not to mention various uses of 0 inside mathematical formulaes that)
Counter-example:
1) The triangle is defined as that which has 3 sides, so is the isoceles triangle, and the equilateral triangle. Therefore, they are one and the same thing. But this is false, since they are not the same.
There are other more counter-examples if you tried to find, since the premise itself was false.)
However if infinisimal = 0, then infinisimal calculus is flawed.
The definition of a derivative:
derivative of y at x: dy/dx
So if dx = 0, then dy/0 is undefined, which means that the definition of a derivative is flawed. But the definition of a derivative is derived from the properties of a mathematical slope.
http://en.wikipedia.org/wiki/Derivative#Definition_via_difference_quotients
In the derivative definition, you find the h in the denominator, h is the infinisimal, if h = 0, then we are dividing by zero, which means the gradient of a slope is in essense always undefined, but this is false.
If we simply apply the same reasoning to a straight line y = x, the gradient is clearly 1. So its computable, or defined, but on the other hand, using your equality of the infinisimal = 0 into calculus, we get that the graident = undefined.
I have not the expertise to continue finding more counter-examples in math showing why that equality infinisimal = 0 is wrong, but if we argee that mathematics is correct, then your definition is wrong.
So considering the lengthy reasons above, your premise that:
infinitesimal = 0
is false.
Therefore, referring to your original post, and my own reply:
"Therefore an object with a finite size can traverse a finite distance, but an object that is infinitely small cannot do the same."
This cannot be established, because of the reasons above. Therefore an object which is infinitesimal can traverse a finite distance.
But besides all those assumptions, i believe your point in talking about stride lengths:
"Using these figures, it's easy to see that it will take you 10 strides to reach your destination. Working through the paradox, to reach 10 m you must first reach 5 m, which you easily can cover in 5 strides. From this point to reach half the remainder (7.5 m from the start), you only need 3 more strides. In fact, with 3 more strides you exceed that distance. To fit with the paradox, you must reduce your stride length and keep on doing it so that you don't exceed each subsequent half-distance. And you must do so an infinite number of times in order to never reach your destination."
is based on the assumption that the strides move discretely. As in, when one stride is completed, the next stride necessarily starts off where the previous stride ended.
If we assumed it was continuous, then stride lengths do not matter.
However, motion of the stride lengths, on the surface, is the point of zeno's paradox. We either assume that motion is continuous or it is discrete. But in trying to solve Zeno, you have already assumed that motion of the strides, is discrete. Therefore you used an assumption, to contradict an assumption. So your solution is not correct.
"However, this is only possible with an object of zero size. If we started with an object with finite (positive) size, like a tortoise or a human, and measured the stride lengths from one heel to the next, at some point in our stride-length-reducing exercise the heel of a foot would exceed the length that is left to travel, so that the tortoise or human can be said to have arrived at the destination."
As you said, the stride will exceed the length that is left to travel. But you assume that the next stride will necessarily start where the previous stride left of. Or a step by step motion.
If i am not wrong, but when the distance needed to traverse is smaller than the stride length, then we can easily traverse that length, because the stride length is longer. So for example, when we are given a distance, 10 cm, and the stride length is 3 cm, then we when we begin to move, we place 1 stride or foot, from the 0 to 3 interval on the 10 cm distance. So when zeno says, you need to travel 1/128 the distance of 10 cm first, before 1/64, then by your argument, we have already done so, because our stride length exceeds those tiny distances.
The fallacy i want to point out to make it clear, is that, as you said, zeno requires the objects to be point-sized, or 0 in order for the paradox to work. But in order to derive this, requires the assumption that infinitesimal objects = 0. The reasons you gave for them contain assumptions and errors which i have shown. Therefore, it is left that we consider objects that are not point-sized. As you said, finite objects necessarily have stride lengths, which will necessarily exceed the distance that zeno requires to be traversed before the next one, by virtue of the stride lengths being greater. However, in doing so you assume that motion is discrete, or step by step, that is why each stride ends up at the end point of the previous one, in order that your stride length argument have meaning. But the point of Zeno was that in assuming that motion is continuous or infinite divisible, we are lead to consider distances that require to be reached initially before the next one, and this process of reaching the next one is an infinite process. Therefore, in assuming that motion is discrete, or done in a step by step fashion but not continuously, you have contradicted the initial assumption of zeno that motion is continuous, and derived a contradiction of zeno using that flawed assumption.
I am very sorry for being so long and boring, but i hope i am clear. Ultimately, zeno is asking us to imagine that the strides move forward from B to A, in a motion such that the group of points that define the boundary of this stride, say
(....)
B .................... A
(....)
B .................... A
(....)
B .................... A
etc
But of course, this is just a finite representation of the infinite number of points from B to A, which the stride must traverse each point in order to move continuous as zeno assumes.
While you assume it is like this:
(....)
B .................... A
(....)
B .................... A
(....)
B .................... A
Where B to A is filled with an infinite number of points, and the stride is of the same length as the diagram above.
Now since Zeno asks us to let all motion be continuous, or all object's motion be continuous, then
the motion of the strides from B to A needs to continually progress from B to A, in a point by point manner. Then zeno's paradox is still alive, for there are an infinite number of points, but the task was done in a finite time.
Fundamentally speaking, if all objects are assumed by Zeno to move continuously, then since your stride length is composed of objects, then they must move continuously from B to A. Then for them to suddenly move 1 stride, and another stride, is to neglect his assumption altogether. Since we have traversed many points at once, instead of traversing each point per time.
Or to put into a more clearer form:
If first judge whether zeno is right, then he is telling us that all motion is continuous. So therefore, if he is right, then there is no discrete, step by step motion.
Therefore if we assumed discreteness, then we contradict zeno. So to assume continuity of the distance we are dividing, is to contradict ourselves.
Thus the argument is logically unsound.
Thanks for bearing with me on this tedious and boring piece.
There is something wrong with the diagrams, ignore the .. to the left of the strides aaa, its just meant to represent the blanks. So here are the new diagrams:
Let aaa be the stride length. Where B to A is always a continuous length.
..aaa
B ............... A
...aaa
B ............... A
....aaa
B ............... A
(Motion is continuous)
..aaa
B ............... A
.....aaa
B ............... A
........aaa
B ............... A
(Motion is discrete)
a few corrections for clarity:
"I argee with 1, but i do not see how your method 2, which is a recursive method ..."
should read:
"I argee with 1, but i do not see how your method, which is a recursive method ..."
to add some more comments on method 2, the contradictions and counter examples from assuming infinitesimal = 0 applies to your method 2 as well, ie it is false.
more comments on this:
"The definition of zero is the absence of quantity. If there is no quantity, then there is no quantity to be measured. On the other hand, the infinisimal is a quantity, as your own definition shows, by adding the 1 at the end ..."
if you see the definition, 1,2, and some others, are conventional definitions, ie argeed upon by convention, like units etc, while 5,6 are more existential definitions, ie the property of quantity implies existence of that property, which in turn allows for a division of units on that quantity, or existing object. Which is contrary to the definition of 0, and as i have said, the existential aspect of it. Conventions can be made up, ie whatever unit we please that is reasonable for the given measure is just a symbol for the chosen unit existing quantity, but existential definitions point out that it is possible to do such a process on a given object, which is not made up, but a given property of the object itself.
http://dictionary.reference.com/search?q=measure
correction:
"If i am not wrong, but when the distance needed to traverse is smaller than the stride length, then we can easily traverse that length, ..."
read as:
"If i am not wrong, when the ..."
This comment has been removed by the author.
dear peter aka philosophers are boring fucks,
firstly, i must thank you for continued patronage of my little essay. i never thought, when i first wrote this, that anyone would read it, let alone mount a sustained, spirited defence of their own criticism. (This is the problem with blogs, you try to keep it simple, but always there's someone, somewhere, who cannot resist the urge to make it unnecessarily complex. but i digress...)
lest this discussion get mired in comments about comments about comments and so on, i will respond to those of your criticisms that directly address my resolution.
re discrete vs continuous motion
the scenario i described above using a ruler making strides can be easily transmogrified into one involving a ball moving continuously (slipping, sliding or rolling, it doesn't matter), witness:
"Say the ball's diameter is 1 metre, and for convenience the distance to its destination is 10 metres.
Using these figures, it's easy to see that the ball will need to cover a distance 10 times its diameter to reach its destination.
Working through the paradox, to reach 10 m the ball must first reach 5 m, which the ball can easily can cover in half the time it would take to cover 10 m.
From this halfway point to reach half the remainder (7.5 m from the start), the ball needs another 3 seconds, assuming it takes 1 second to cover a metre. In fact, with these 3 more seconds the ball will exceed that distance. To fit with the paradox, the ball must halve its distance of travel, and keep on doing so so that it doesn't exceed each subsequent half-distance. And the ball must do so an infinite number of times in order to never reach its destination."
Voila! Continuous motion covered.
In case it's not obvious from reading my resolution, i'm trying to keep things simple for my intended audience of laypersons, rather than specialists or people with some degree of mathematical training like you. (standard vs nonstandard numbers - sheesh!)
Let's see: what other criticisms did you bring up.
Oh yes, the quibble about infinitesimals not being zero.
Of course infinitesimals and zero are not the same (duh). Zero, as you correctly point out, is 'nothing', and infinitesimals, though small, is clearly 'something', ie. not 'nothing'.
However, for the purpose of resolving Zeno's first paradox, this distinction is unnecessary given that my resolution turns on the distinction between the finite and the non-finite, including both zero, infinitesimals and other as yet unstudied mathematical entities.
But to make specialists like you happy, i could change the line:
"However, this is only possible with an object of zero size."
to
"However, this is only possible with an object of zero size, or an object of infinitely small size."
Whatever.
My conjecture here is that objects of infinitely small size do not exist (except in mathematics and logic). And neither do objects of zero size.
The 'paradox' of this particular Zeno mind-twister relies on specialists (like him and you) using simple language to trick ordinary people (like me and my audience) into confusing the elegant world of mathematics with the gritty world of finite objects which we live and breathe, just because those two somewhat similar, but actually very different worlds, can be described using the same language.
Infinitely reducing stride lengths (or 'distances travelled' if you wanna be picky) may be possible in the mathematical paradise described by Zeno, but in our world even concepts like continuous motion hit a physical barrier (planck length, speed of light, etc).
So while a mythical Achilles may elegantly halve his stride length ad infinitum so that he never reaches his destination, an Achilles of flesh and bone must decide, when his stride-length reaches the planck length, say
-- or whatever future post-quantum
minimal length our descendents will
uncover -- between stopping just shy of the destination, or leaping across that minimal length to reach it.
If at the physical minimum our hero of finite proportions chooses to stay put, he violates the axiom of infinitely halving distances required to prove the correctness of zeno's assertion that motion is impossible.
That is to say, motion IS impossible if you, while trying to get to your destination, continually halve your stride-length/speed/whateva until you get to the physical minimum length, at which point you decide to stop rather than traverse the minimum finite distance.
If, on the other hand, our finitely conceptualised hero chooses to reach the destination, he also violates zeno, because this time he travels a distance more than half the previous distance travelled.
Either way, Zeno Paradox No 1 is debunked.
Peter or PABF: In case you still think it's possible to have infinitely or zero-sized objects, please describe to me how you would go about 'observing' (rather than 'imagining') an object that is smaller than the planck length. (This should be quite interesting.)
Hello again. No problem because i want to see whether your proof is valid or not. You have to know what constitutes a valid inference, rather than an invalid inference. You see, you are still using the same assumptions to prove your conclusion. This is called circular reasoning.
http://en.wikipedia.org/wiki/Begging_the_question
In logical terms, that is an invalid inference, because the conclusion is not necessarily true at all, since you presuppose that which you are trying to prove. Anybody can prove whatever they want with circular reasoning, i can prove zeno is right by circular reasoning as well.
First assume motion is continuous, therefore zeno is correct. Q.E.D
So now let me show you your assumptions again:
First assumption:
1a) When you try to disprove zeno using your stride length argument, you assumed that the steps can exceed a given distance, by zeno's requirement.
1b) This is because you assumed that these steps can be completed in the given time t.
1c) But in fact, zeno says the opposite, because given a time t, you can never even begin to cross any distance, because in order to even begin crossing any distance along a continuous number line, you have to cross an infinitely dense region of points.
1c1) In other words, you have to complete an infinite number of steps, in a finite time.
1d) Now, in order to make any such a step, requires this step to move along this continuous number line, where each instant in time, as this steps moves towards a distance, has a unique coordinate respective to that number line.
For example, let the line distance =d, then let the step = f, where f < d.
Then this 3-d space we are in, is assumed by zeno, to be continuous. (It is a contradiction to assume it is first discrete, to derive your disproof, since that is an invalid logical step.)
So the starting point of this object will be:
(0,0,0)
Then it will move to:
(0,0,f)
After time t. Where the speed is f/t.
Your assumption: The step will move to f, after t.
Zeno's assumption: The step cannot even move at all, because in order to do so, requires completing an infinite number of tasks, in a finite time.
So you see that you assumed that the step can move at all, while zeno says the step can't move at all.
So you have a contradiction, because you have used an assumption, while disproving zeno's premise. Ie circular reasoning.
Second assumption:
2) Assume that the mathematical continuity does not have any correlation with physical reality.
2a)Therefore zeno is false, because continuity cannot exist physically.
Problems with inductive hypothesizes used in disproving a logical deductive claim.
==================================
Again, same invalid inference. Do you have proof that mathematical continuity does not exist in the physical world? If you don't have logical or philosophical proof or mathematical, then your proof isn't really a valid proof. If you use scientific models, or hypothesizes to disprove the principle of continuity, you are then using more assumptions to disprove a claim.
1)First of all, it is not proven that the given scientific theories we have currently are rigorous proven.
2) This is because, inductive hypothesizes are only probably true , rather than certainly true.
3) Reason being, induction requires observation of cases, and unless observation is complete, can we say that the hypothesis is a certain truth.
4) This is because, there might exist a case, whereby it contradicts the original inductive hypothesis.
5) Since our knowledge of the universe is incomplete, we cannot know for sure if there might exist a counter-example, or no counter-examples to our original hypothesis.
6) Therefore, inductive hypothesizes without complete knowledge of the universe, is not certainly true, in logical terms.
This isn't just my idea, but its a very known philosophical fact. We have Descrates giving his demon skeptical argument to throw doubt into observational claims. Then we have Hume's analysis of inductive inferences, showing why it can not succeed to give us certain truths, due to the argument i gave. Then we have Karl Popper's falsification principle for inductive statements, which states that inductive hypothesizes must be those which are falsifiable, then there's Berkeley's critique of observer dependent objects, etc. In other words, they will in the end be false, due to a counter-example.
Third assumption:
3) Assume that physical reality is finitely divisible. Ie some planck sized object is not divisible.
3a) Then clearly zeno is false, since zeno assumed continuous, while you assume discrete.
Again for the same reasons as to why you cannot inductive hypothesizes to disprove a metaphysical claim, otherwise those who use calculus based reasoning, have immediately solved zeno. Since infinite tasks are completable in a finite time.
All i see is re-descriptions of your claims, but there is no solid proof of your claims, for the mentioned reasons.
To clarify things, the reason motion is impossible at all, is because an infinite number of tasks is required to be completed, in a finite time. Because to move to any distance at all, requires crossing an infinite number of points.
So since if we assume zeno is true, then you cannot assume that an object moved, for you would be making an invalid logical inference.
Next, another point zeno makes is that there is no point that is next to starting point. This is because there is always an infinite amount of points in between any points.
This is due to Cantor, who showed that the algebraic numbers are cannot be well-ordered. Well-ordered means, we can number of the first algebraic number as 1, then the second alg num as 2, and so on.
So since they cannot be well-ordered, there is no first number that is next to the starting point.
If there was a first number that was next to the starting point, then that we can immediately number that number as 1. Which would contradict the mathematical axioms.
So you cannot say any object or stride-length etc, moves at all, because it can't even start at all, due to the lack of having a first point.
Remarks
=======
This is to make things clear:
1) You said that i said that 0 sized objects exist, but that was nothing anywhere that led me to such a conclusion, nor did i say it.
2) Im not nitpicking when i see that your usage of terms, in order to justify some conclusion is incorrect. For that is simply the nature of correct reasoning, if you frame an argument using incorrect usages of words, then we reach a contradictory conclusion.
3) Quote: "My conjecture here is that objects of infinitely small size do not exist (except in mathematics and logic). And neither do objects of zero size."
Again, circular reasoning. You can't use an assumption to disprove or prove something, an expect that proof or disproof to be rigorous.
4) "Peter or PABF: In case you still think it's possible to have infinitely or zero-sized objects, please describe to me how you would go about 'observing' (rather than 'imagining') an object that is smaller than the planck length. (This should be quite interesting.)"
Again, you assume that inductive hypothesizes are deductive or logical truths, which is totally contrary to what people have been saying in philosophy for 2000 years.
Therefore it is possible that infinisimal objects can exist, even though science gives us limits to the measure we can probe into. But just because i can't show you how that is to be done, doesn't mean those objects don't exist. Anymore than me not being able to show you how to solve fusion power doesn't mean it doesn't exist, since by assuming scientific facts about the sun and stars, it does. And also that science as we know it now, is not indubitable or free of doubt, therefore we cannot use its "facts" to justify our conclusions. This is due to skeptical arguments for sensory data, arguments by Parimendes, Hume's fork, Popper falsification principle, etc.
So to conclude, you haven't actually contradicted me, you only used your assumptions about motion and science to disprove what i have said. Which as i have said, is an invalid form of logical inference. Therefore your argument is not a logical disproof of zeno.
I highly suggest you read:
http://plato.stanford.edu/entries/paradox-zeno/#Dic
http://en.wikipedia.org/wiki/Fallacy
http://en.wikipedia.org/wiki/Inference
No offense, if you keep trying to repeat your arguments using the same assumptions, there is no point in critiquing you any longer, for i will be wasting my time repeating myself over the same points, when it is clear to any logical person that what you are saying is circular reasoning, and not a valid justification. After all, we are out for the truth isn't it? So therefore use valid inferences, and do not use arguments which have assumptions in them.
This is much clearer argument than above, i hope:
To clarify the point using cartesian coordinates in that space juxtaposed with that object in motion along the distance d:
i) First of all, in order to move along d, means we traverse either an infinite number of points, or a finite number.
ii) Now if we assume zeno's condition, ie continuity, then we cannot have a space that is finitely composed.
iii) So when object O goes along d, the object must go along an infinite number of points.
vi) So if we say O can go along d, then we assume that:
First assumption:
viA) Crossing an infinite number of things is possible, in the finite time t.
Proof:
viA1) If crossing an infinite number of points isnt possible in a finite time, then we either do not even begin to move, or the time taken is infinitely long.
ivA2) Since it is clear that we assumed it had to move, and that
time is finitely long, therefore we
must assume that viA is true.
Q.E.D
Second assumption:
ivB) If first point next to the starting point A, exists in d, then motion is possible, since there is a continuity from A to A1.
Proof:
ivB1) If there existed no first point next to A, then exists a distance from A, to A1, which is not composed of points.
ivB2) Therefore, line A to A1, is not further divisible into divisions, and/or non-existent.
First case:
ivB3) It cannot be non-existent, otherwise the distance between A to A1 would be 0, which implies that A = A1. Or equivalently: (0,0,A) = (0,0,A1) in 3-d cartesian space.
If A1 = A, then A1 is the starting points, rather than the first point from A.
Q.E.D
Second case:
ivB3.1) If line A to A1 is not further divisible, then we contradict (ii), or zeno's continuity condition.
ivB3.2) Therefore we must assume that A1 is the first point to A.
Q.E.D
Synthesis of the preceding arguments:
So all i have proven is that we assume these 2 assumptions, when we assume that O goes long d in continuous space. (iii)
v) There are 2 definitions of infinity:
vA) Infinity can never be completed in a finite time.
Potential infinity.
vB) Infinity can be completed in a finite time.
Absolute infinity.
vi) Now zeno's condition (ii) means that the definition required is vA.
Proof:
viA) Since O goes long d in a finite time t, and since space is continuous, therefore an infinite number of points have been crossed in a finite time.
viB) Therefore it is possible to complete an infinite number of things in a finite time.
Q.E.D
vii) So the contradiction in your argument is this:
viiA) First of all, you assume that there exists a first point A1, next to A, such that O goes from A to A1.
viiB) But as we have seen, continuous space does not possess a first point A1.
viiB1) The argument due to Cantor, is that the number line cannot be corresponded to the natural integers.
For example, Let the first algebraic number = a1, and the next, a2, etc.
To be well-ordered means:
(a1 <-> 1), (a <-> 2), (aN <-> N)
So if the number line is well-ordered there exists a first point. If it isn't then there does not exist a first point.
viiB2) Therefore, you contradict yourself, since you assumed that there exists a first point, and you assumed that there does not exist a first point.
Q.E.D
viii) The second mistake, is that you use an invalid inference rule, or fallacious rule, to justify your conclusion. The fallacy being, you used an assumption to justify your conclusion.
viiiA) First of all, you assumed that infinity is completed in a finite time. But i do not see a proof of this in your thought experiment. You merely assumed it and continued imagining with that assumed condition. But you didn't justify it.
viiiB) Therefore, your conclusion is invalid, because of this invalid step as well.
Q.E.D
So this cannot be a valid or rigorous disproof of zeno's paradox.
Im going to remark on your use of QM, and those comments on assuming some mathematical space over another, to show you your mistakes in saying them, and why its invalid as a rigorous disproof of zeno's paradox.
It is contradictory to assume that QM's mathematical formulation exists in physical reality, while assuming that mathematical continuity does not exist. This is because GM assumes continuity exists in the physical world, and it is not even known whether QM is the ultimate theory of physics, since QM fails to include gravitation into its framework. Therefore, to simply assume one over the other, because what is the most convenient inductive framework, is just as invalid as a logical deductive step, as assuming the opposite, and to claim the conclusion is consistent and logically proven. They even use as you call them, mathematical constructs of the mind, in deducing the Planck length. So i think its highly strange to consider one mathematical system that exists in physical reality, and another not to exist, and furthermore, to fit these assumptions in order to "prove" your conclusion that mathematical continuity does not exist, and to do so without proof as to why one exists over the other.
If you claim observation or experimental observation as proof, then we have a highly paradoxical situation that has been noted since the conception of QM and GM together, that the 2 theories give us good observational results in the same universe, but their assumptions of physical space, ie discrete or continuous is in contradiction to each other! Therefore, that isn't a good reason to choose one mathematical space over another, unless there is a valid proof that is non-contradictory, which your QM argument isn't.
Also you can't truthfully claim that you know for certain QM gives us the ultimate insight into reality, because we don't know if this is just another QM era in physics waiting to be taken over by some new theory that properly includes gravity and the other 3 forces. Even the experts in the field acknowledge that, that is why string theory is there, because they need to synthesis gravity and the other 3 forces. Even in the beginning of QM, nobody just accepted it blindly. Einstein, a pioneer in QM, understood that it didn't represent the ultimate theory, that is why we have the EPR paradox, and hidden variable theories of QM. Then we have Feynman who maintained an open mind, who invented Quantum chronodynamics, saying that he doesn't know if QM is the ultimate theory (you can hear it for yourself by searching youtube.com for "richard feymann" and hear his talk.) Then we have Henri Poincare who said that scientific frameworks were chosen according to how well and effective they described the world, ie none is more true than another, except which works best. He was the inventor of chaos theory , and some extremely general mathematical theories.
So i hope you learn that what you are doing is accepting things as true, without a valid and rigorous proof. Any tom dick and harry can use assumptions to "prove" something, like how pigs could fly assuming that they used to be able to fly with wings attatched, but since the first humans evolved, those wings disappeared. Then he would assume this and that or ad hoc arguments to cover your requirements, but still he would be right, because you can't say for sure he's wrong. So in order to say for sure, requires that the conclusion be free of assumptions, and it is non-contradictory. Since your disproof has both, your conclusion is not rigorous, or water-tight, or certain, or a necessary truth at all.
Important corrections:
Change:
vi) Now zeno's condition (ii) means that the definition required is vA.
to:
vi) Now zeno's condition (ii) means that the definition required is vB.
Change:
If A1 = A, then A1 is the starting points, rather than the first point from A.
to:
If A1 = A, then A1 is the starting point, rather than the first point from A.
Change:
ivB3.2) Therefore we must assume that A1 is the first point to A.
Q.E.D
to:
ivB4) Therefore we must assume that A1 is the first point to A.
Q.E.D
Change:
viiB2) Therefore, you contradict yourself, since you assumed that there exists a first point, and you assumed that there does not exist a first point.
to:
viiB2) Therefore, you contradict yourself, since you assumed that there exists a first point, and you assumed that there does not exist a first point, at the same time.
Just to make sure you dont confuse my meaning and structure.
ok i get your point about my reasoning being circular, and also about the impossibility of motion because (according to zeno) you can't even make the very first step.
i'll have to think about this a little more and get back to you.
thanks for sticking with it.
ok now i've thought of a way of explaining my position.
when i was writing my original exposition i went through a long period of uhming and aahing about whether to call it a 'solution' or a 'resolution', for the (probably obvious) reason that (i) i am not trained in logico-mathematical proof, and (ii) the resolution that i propose does not prove or disprove the paradox, it merely describes why there is a paradox. (in fact, as you'll see below, my account assumes zeno's reasoning as logically sound, within the framework of a formal system, though it does not accord reality).
eventually i settled on 'solution' because i thought that if i called it a 'resolution' i would have less visitors to the blog (ie. more people would search for "zeno solution" than "zeno resolution").
to make it easier for you, i'm following on from your comments in para (3) from your last post.
ok so here's what i think is happening.
on one hand, you have zeno using words like achilles, a tortoise, distance, travel and starting point, finish line, etc.
now these words have ordinary meanings in the real world. achilles is a human, if i walk from my front door to my car, i could describe myself as having travelled that distance. i could call the front door point A and the car point B.
conveniently for zeno, these words also have meanings in the world of mathematics. here i'm drawing a line between our physical world, and mathematics, which is a formal system we devised originally to solve problems that we encounter in our physical world.
in the formal world, words like point and distance have meanings which are analogous to concepts in the real world.
but like all analogies (even good ones) there is a point at which the analogy breaks down.
so for example, for a real world concept like the distance from my front door to my car, there's probably no real need to define it as any more than the 'shortest length of a string going from one to the other'.
however, if you wanna solve other more interesting (and probably more useful) problems, you'll want to have a definition, a formal definition, which is amenable to all the useful things we've found out from counting numbers, ie. mathematics, which happen to be very useful for describing the motion of apples dropping on people's heads and for sending spacecraft to the moon.
so you end up with a formal definition of 'distance' as a 'line of shortest length between any two points' and a definition of 'line' a 'finite or infinite two dimensional segment containing an infinite number of points', and a definition of 'point' as a 'marker of position having no size', or similar.
these formal definitions have evolved by analogising real world concepts into formal systems so we can manipulate those formal objects using rules and other formal relationships that we've discovered.
the thing with formal systems is, they work fine within the confines of their system (although even this is suspect, eg. godel).
however, there are limits to their usefulness as analogies of the real world.
what zeno's paradox shows is the boundary between these two similar, but quite distinct worlds.
in the real world, motion is possible. it happens every day, every where.
in the formal world of zeno's paradox motion is impossible.
when confronted with this paradox, zeno's response is all motion, no matter how apparent, is an illusion.
however, you'll forgive me if i believe my senses more than the conclusions of a dead, though smart, greek philosopher.*
if you, like me, believe what you experience everyday, then there's something clearly wrong with zeno's reasoning or assumptions.
personally i think his reasoning is sound.
this then leaves his assumptions. there are not many.
the only one that's salient is that zeno assumes or implies that distances are infinitely divisible.
i suppose you could add the operation of halving distances, but this is covered by infinite divisibility.
(i used to think that objects of zero size are implied from infinite divisibility, but i think it's safe to say that it isn't necessarily so.)
now given our experience with subatomic physics we know that there are many fundamental particles which are so small, they are treated as points of zero size in calculations.
in the formal world of mathematics, there is no reason why there should be a limit to divisibility. in fact putting an artificial limit to divisibility limits the utility of mathematics.
in zeno's paradox the ball is 'obstructed' by having to cover an infinite number of half-distances, having to cover an infinite number of points, or completing an infinite number of tasks in a finite amount of time.
however, in the informal world in which we live, such a thing poses no problems. if you want to move a ball a distance of 10 metres, you just give it sufficient force to roll or slide it that distance.
this means that zeno's first paradox reveals, in regular american parlance, a 'disconnect' between our world and the formal world of maths.
and in my resolution (which i will shortly revise in order to remove the circular reasoning), is designed to show that the disconnect is rooted in the formal definitions of point and distance.
to clarify, what i am suggesting is that given that motion is possible in the real world, and that zeno's reasoning is sound (given the formal definitions of points and infinite divisibility), the irresistible conclusion is that in the real world infinite divisibility of distance is not possible.
so at some very fundamental scale in the informal real world, we get to minimal lengths (which at our present level of scientific knowledge is the planck length).
at this level we can say stuff that doesn't make sense in the rarefied formal world of numbers, stuff like if achilles is at the starting point, the very next point is one planck length away.
also we can say that achilles moves by 'jumping' from one planck length to another, without being able to exist in a position between them.
this kind of discreteness or granularity to otherwise seemingly continuous phenomena may seem counter-intuitive to everyday experience (hence the paradox).
but is it really? no need for sophisticated quantum mechanical examples, just look at the humble ladder.
the height of a person standing on a ladder is discrete, in that he/she can only be at certain heights and not others, as determined by the location of the rungs.
however, just because the person's height on a ladder is discrete, does not stop us from talking about the distance between two rungs, or the half distance, or the half-distance of the half-distance, and so on.
indeed, we can measure the distance between two rungs by referrence to smaller distances such as the centimetre.
that is to say, the discreteness of certain real world phenomena does not invalidate the 'continuousness' of its analogous concept in the formal world.
in fact, it's not surprising that informal discreteness can happily coexist of formal continuousness.
the formal world which we have developed has come about through utility (mathematics included - the utility is just about discovering and proving new relationships between formal concepts).
a formal system which disallowed half distances between the rungs of some fundamental ladder would not be very useful for other things which we might like to investigate.
(if you think having such limits in formal systems is weird, have a look at the history of zero and negative numbers.)
who knows? maybe distances shorter than a planck length is useful for describing some other physical phenomena.
but i digress.
to sum up then, zeno's reasoning is sound, ie. motion is impossible if you're talking about infinite divisibility of distances.
however, the fact that motion is possible in our world means real objects are not encumbered by such abstractions, ie. there is a failure in the analogy between the formal and the informal.
and in resolving this 'disconnect', my conjecture is that, given the reasoning in zeno's paradox is sound, that in our informal world there is at some fundamental level a minimal distance for moving objects.
i cannot prove that there exists a minimal distance for moving objects, nor have i intended to. i am neither a theoretical scientist, nor an experimental one. nor am i a mathematician.
i am merely pointing out what seems to me to be the logical conclusion given the starting points.
the trouble you're having with my account, if i may be so bold, is that my exposition, though logical, does not sit neatly within either the formal system (ie. it does not prove or disprove or come to some other formal terminus), nor does it read like a scientific paper. (*hence its lumping into philosophy which, i must say, also bores me.)
it is, indeed, all conjecture, but then again, where would we be without conjectures to spur more persistent minds toward finding elegant proofs? (for a start, we probably wouldn't be having this conversation.)
the interesting thing about this exposition is that if it's true, then all motion (in the real world), however apparently continuous, is discrete at the most fundamental level.
(it's also highly possible that time is also discrete, but that's another question).
that motion occurs in discrete jumps might not sound very interesting, since we live in a post quantum mechanical world.
however, because there can be no 'position' for a moving object to exist at between the two ends of the minimal length, this means that an object moving at that level effectively disappears, only to reappear at the next physical point.
an object moving in such a fashion 'winks' in and out of existence like the light from an artificial satellite pulsing as it flies across the night sky (except we're talking about matter disappearing and reappearing, not light).
in fact, for a moving object there can be no 'positions' between the start of its motion and its next point of call.
so while we may be able to measure or at least deduce the length of the distance between two physical minimal points (and so in a way expose ourselves once more to zeno since that length contains an infinite number of formal points, ie zero size).
to me, that's a pretty interesting to result.
granted, i have drifted somewhat from my original resolution involving objects of zero size.
however, i'm writing this off the top of my head so maybe, after some further thought, i can come to other conclusions about what this means.
(if you wanna stay in the formal world, i supposed you can use calculus or summing series to 'prove' that it's possible to complete an infinite number of tasks in a finite time period. or you can use cantor to show that an infinitely long line has the same number of points as a line of finite length.
probably these things are different sides of the same multidimensional coin rather than 'proofs' of zeno per se. i dunno, like i said, i'm not a mathematician.
i have to say tho, it's kind of cool that in the end, it's a toss up which world is weirder, the formal one which we've devised, or the informal one in which we live.)
Hitting a Zeno in One: Solution to the Dichotomy
my previous comment used the real world to resolve zeno's paradox.
i will now use only formal arguments to destroy his Dichotomy Paradox, which is a more economical variant of the achilles and tortoise paradox, and hence a tougher nut to crack.
this description of the paradox is from wikipedia.
suppose homer wants to catch a stationary bus.
before he can get there, he must get halfway there. before he can get halfway there, he must get a quarter of the way there. before traveling a quarter, he must travel one-eighth; before an eighth, one-sixteenth; and so on.
because you can keep dividing the half-distances to infinity, homer is faced with an infinite number of distances to traverse.
according to zeno this is not possible, so homer is unable even to start moving.
aristotle put it in one line:
'That which is in locomotion must arrive at the half-way stage before it arrives at the goal.' [Aristotle, Physics, v1:9 239b10]
Solution
according to zeno and as per aristotle's statement above, any object in motion must arrive at the half-way mark before it reaches the finish.
since NOTHING CAN BE SAID TO 'ARRIVE AT' OR 'GET TO' A PLACE WITHOUT HAVING FIRST TRAVERSED SOME DISTANCE TO THAT PLACE, to arrive at the half-way mark the object must have been able to traverse that half-distance.
(if one does not admit this concept of 'arrival', then 'motion' in general has no meaning.)
this being so, an object which is able to traverse a distance from one point to another is necessarily able to first travel half that distance.
since inherent in the formulation of each half-distance is the ability for the object to have traversed that distance PRIOR TO the formulation of the next one, we can logically conclude that an object can indeed traverse an infinite number of smaller and smaller half-distances.
The Zeno-In-One Solution
That which is in motion must necessarily be able to arrive at the half-way stage before arriving at the goal, no matter how many successive half-stages there are.
Hello again, there are still problems with your solution.
1) Zeno doesn't claim that motion is impossible, ie totally physical unreal, but he claims that continuous or discrete motion is an illusion of our minds, rather than a real physical fact of the world. So using a definition of motion, to prove or disprove zeno, would then have to clarify whether that motion is an illusionary motion caused by the mind, or a real physical fact. Now, what you have just done is merely shown that zeno assumes motion, in order to deduce his contradiction. But that doesn't disprove zeno at all, because that is simply a logical method called reductio ad adsurdum, which means, we assume that a statement is true or false, then derive a contradiction from that assumption, which leads to the proof of the negation of our initial assumption. This is what mathematicians have been doing for thousands of years. Thus, that is not a logical way to disprove zeno.
2) Zeno in fact talks about continuous motion being contradictory, rather than discrete motion. So even if you disproved the dichotomy, you haven't really proven that motion of some sort exists. There are many articles which deal with the conditions or assumptions that Zeno is dealing with, so your solutions have only so far addressed 1 point, which is trying to prove discrete or continuous motion exists, no need to mention your assumption that reality is discrete, which has actually been addressed by another of Zeno's paradox called the Stadium. The other assumption which anyone has to address in order to fully resolve Zeno is:
a) Proving or disproving that an infinite number of tasks is completable in a finite time.
So together with the proof or disproof of the type of motion (continuous or discrete), with a, can we maybe then say the solution is rigorous. Unfortunately, yours has not addressed this.
3) Your reply has many speculations, which do not contribute to the rigorous proof or disproof of zeno. What i understood you as meaning, is that you are trying to tell me that you actually have a rigorous disproof of the Achilles paradox, but it isn't, for reasons i have made clear. You can have your beliefs no doubt, but it seems to me you are trying to convince people that you have disproved zeno, which is certainly not the case.
4) There is a misunderstanding in saying that im deriding your solution because it is not formalised, or because it is not properly written out like a scientific paper, because your as i said, i understood you as trying to convince people that your solution is rigorous, and i explained using the mathematical formalism where your answers or details in those answers were wrong. For example, you said an infinitesimal was 0, but if we are to believe mathematical formalism of infinitesimals, then your assumption is false. Then you used that assumption to derive a contradiction in zeno. Thus that was simply to point your mistake in your reasoning. Again, i repeat, my understanding of your intention of making this solution is that you really believe that it is a fully rigorous solution, so my intention was to show you your mistakes, if you do not claim it anymore to be rigorous, then "solution to zeno's paradox" seems exaggerated. Btw, i found your site on wikipedia's zeno page,which has been removed now.
5) Yes, anyone can discuss the implications of discrete motion IF it were real, this has been done many times already in quantum mechanics, or philosophy (ie mereology), since you are interested in this, i recommend:
http://www.abstractatom.com
That site is quite rigorous, and talks about the existence of whole and parts, and whether the exist in physical reality, via some philosophical convention known as metaphysical realism, and the like.
Also: http://people.umass.edu/schaffer/
http://maverickphilosopher.powerblogs.com/posts/1192157247.shtml
This guy has summarised the problems and assumptions of Zeno's paradoxes quite nicely, which concludes with a true understanding of what Zeno's paradoxes really mean, which differs from your understanding.
Discussions about the relation of mathematical formalism with the physical world are:
a) The philosophy of Mathematics by Oxford press
Which is a really good collection of papers that provide the different viewpoints of philosophers on the nature of mathematics
b) The unreasonableness of mathematics by Nobert Weiner
This discussed from an emotional or subjective viewpoint about the question of why mathematics is so successful, if we are to believe that mathematics is merely a human creation.
c) Science without numbers by Hartley Field
This book is fucking good, basically he replaces the abstract objects of mathematics, in newtonian physics, with real physical objects, that the symbols refer to, and builds newtonian physics from this nominalistic viewpoint, which is what its called. Yes newtonian physics is "dead", but the application of his method is profound because it may possibly be generalised to any physical theory, thus removing platonism from physics. Then you can read about a program called finitism by Kronecker, which is related to Hartley's program, that wants to remove any mention of objects which are not integers in mathematics. (you can search that in wikipedia for more info) Which in effect tries to make math more physically real, by relation to integers which are more intuitive.
To sum this interesting part, go to wikipedia's philosophy of mathematics, and you will find a good summary of the different viewpoints.
Im sorry, but if we are to discuss about the IF's about this or that, it will be very long, so to cut short, i can only criticise or provide info.
peter,
zeno's paradox DOES actually contend that motion is impossible. just google 'motion is impossible' and see what pops up.
the paradox, dear readers, is that what appears to be established by logical methods (motion is impossible) appears to contradict observable reality (motion is possible)
even if what you say is true, that the paradox addresses the impossibility of completing an infinite number of tasks within a finite amount of time, this has already been established by others.
for example, if we let each 'task' be represented by a point, then i refer you to proofs that establish that any line of finite length (representing a finite time period) consists of an infinite number of points or 'tasks'.
you said: "Now, what you have just done is merely shown that zeno assumes motion, in order to deduce his contradiction. But that doesn't disprove zeno at all, because that is simply a logical method called reductio ad adsurdum..."
ah, now we get to the meat of it.
what i am saying is that zeno, by phrasing his paradox in the way that he has, has given our dear readers only part of the picture.
i am not merely pointing out that zeno assumes motion (duh).
i am pointing out that his assumption of motion has important consequences which he has failed to elaborate - probably because if he were to do so his paradox would fall in a heap.
maybe a second go will clear things up...
zeno says:
1.to arrive at a place, one must have travelled half the distance (the 'halfway mark') to that place.
2. but to arrive at the halfway mark, one must have travelled half that half-distance to the halfway mark. and so on to an infinite number of half-distances.
3. since it is impossible to complete an infinite number of tasks in a finite amount of time*, motion is not possible at all.
[*this is an assumption by zeno which has been easily established as false - see what i said above about proofs of a finite line containing an infinite number of points.]
what zeno has (cleverly) left out is a little nub of information between (1) and (2), namely:
"to arrive at the halfway mark, one must necessarily have travelled a certain distance to that mark, BECAUSE 'arrival' at a place is meaningless unless it assumes that one has gone some distance in order to get there."
that being so, zeno's paradox is killed at its root, because it will then read:
"1.to arrive at a place, one must have travelled half the distance (the 'halfway mark') to that place.
but to say that one has 'arrived at' (or 'reached' etc) the halfway mark necessarily implies one must have travelled the half-distance to the mark.
since the first half-distance covers all subsequent half-distances, TRAVERSING THE FIRST HALF-DISTANCE NECESSARILY IMPLIES TRAVERSING ALL SUBSEQUENT HALF-DISTANCES, regardless of the number of such half-distances.
therefore the paradox does not arise."
in fact, you could come up with a simpler and more powerful rebuttal which kills the paradox prior to (1), namely
GENERAL SOLUTION TO ZENO'S FIRST PARADOX
"Any point that one may be said to arrive at necessarily implies that one has travelled some distance prior to arriving at that point, which in turn logically means that one has travelled any number of shorter segments of that distance to the point."
at this point, my friend, i would like to extend my thanks and gratitude for your comments to my solution - it has certainly been the better for it.
however, given that you appear to be reiterating your arguments from your earlier comments instead of addressing my latest comments, it would seem pointless for this discussion to continue.
so with that i bid you adieu.
(of course, please feel free, to comment on my rebuttals of zeno's other paradoxes.)
peter (or would you prefer Mr Lynds?)
it would appear you have been a very busy boy, what with having your girlfriend post 'independent' news releases, creating fake aliases to promote your work, and (horror of horrors!) deleting my links on wikipedia while adding and leaving your links (two can play this game, ad infinitum LOL).
oh dear, commenting on my blog must be such a come down from crashing on the couches of rock stars.
i at least finished my bachelor(s).
do be a dear and drop me a line when you publish your theory of everything, won't you? i expect no less than an autographed copy of your tome delivered to my door, post haste.
cheerio!
im not even peter lynds, i just highlighted him because he has a good point. i would dare you to post your "solution" of zeno, to Mind, Sorites, or any highly reputible philosophy journal of high standing, and then tell me their replies. A solution of zeno's paradox isnt some child's toy, or infant abc's, but is a 2000 year old problem which requires rigorous explanation which you lack as i already mentioned. If you dont post it to the journal, ill ask my professor, or Hilary Putnam who is a reputable philosopher to judge your solution, which would most probably be lacking. I hope you dont delude yourself into thinking that you "solved" zeno when you clearly didn't
If I move a finger 1cm or indeed ANY distance, science says by way of using calculus (infinite-series) to calculate its movement (and to resolve Zeno's Paradoxes), that it goes through infinite contiguous little steps -- which leads to an explanation of how we traverse infinite steps in finite time. Hence Achilles gets to catch the tortoise (so goes the theory).
What bodily processes (electrical, chemical, neurological) begins, maintains and stops that infinite PHYSICAL stepped movement?
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I have a very simple solution to Zeno's paradox.
In fact its not even a paradox, its an absurd notion.
We have to accept / understand these 2 rules.
1. An object at 0 speed already covers a certain distance (length)
(Even a micro organism or a dust)
2. No objects can travel below the speed at which its already occupying.
A train which is 1km long cannot travel below this 1km because at 0 speed it has covered 1km distance.
To subdivide this further below 0 is like saying one can travel back to the past.
Therefore this paradox is result of an absurd notion.
I think the solution is none of the above, but rather the following:-
No object or particle in uniform motion can have a precisely defined position. There is no such thing as an instant of time, only an interval. A million instants combined is still an instant with no duration, so an instant is a meaningless idea. There are only time intervals.
No matter how infinitessimal the time interval, the object in motion traverses some distance, so it's actual position is always uncertain.
Zeno's paradoxes assume precise and certain positions. But this assumption is false. For that reason the paradoxes fail.
The proof that Zeno ' s paradox is untrue is proven by Relativity. If you measure an arrow at rest compared to an arrow in relative motion, the arrow will be length contracted in every axis that is in relative motion. Just because the arrow appears to have no relative movement in a snapshot just means the observer didn't take careful enough measurements. Case closed.
There is one problem with your must travel half way ad infinitum is that space is formed from bits, Planck lengths. Movement at the most fundamental level involves quantum jumps of one Planck length at a time, discrete jumps similar to how electrons change orbits in discrete jumps. There is no half way between quanta. Read my other refutation of Zeno ' s paradox below using relativity.
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