Zeno's paradoxes in the modern era
Zeno made several arguments against the possibility of change and this sparked various responses from his contemporaries (Aristotle, for example) and from philosophers in the twentieth century (Russell, for example).
My question is related to what their standing is today?
Does anyone philosopher still think that they prove that change is impossible?
Or have the philosophers and mathematicians solved the paradoxes.
For a summary of the paradoxes read these links:
https://plato.stanford.edu/entries/paradox-zeno/#ParMothttps://en.wikipedia.org/wiki/Zeno%27s_paradoxes
https://en.wikipedia.org/wiki/Zeno%27s_paradoxes
My question is related to what their standing is today?
Does anyone philosopher still think that they prove that change is impossible?
Or have the philosophers and mathematicians solved the paradoxes.
For a summary of the paradoxes read these links:
https://plato.stanford.edu/entries/paradox-zeno/#ParMothttps://en.wikipedia.org/wiki/Zeno%27s_paradoxes
https://en.wikipedia.org/wiki/Zeno%27s_paradoxes
Comments (130)
1. Prove that in order for one's position to change, one must first do an infinite number of 'things'
2. Assert that one cannot do an infinite number of 'things' in a finite time.
From 1 and 2, conclude that one's position cannot change.
The resolution is to observe that, if 'things' is defined in the way that it needs to be in order for the proof of 1 to succeed, there is no reason to accept assertion 2.
I don't think modern mathematics or philosophy, or even calculus, is needed in order to perform that analysis and identify the reliance on the nebulous notion of 'things'. Aristotle's propositional logic suffices.
The current status of the (veridical) paradoxes is what it always has been: they eloquently demonstrate that, when one reasons to a conclusion that contradicts what one confidently observes to be the case, there must be a flaw in the reasoning, and one has to carefully examine it in order to locate it.
"However, Aristotle did not make such a move. Instead he drew a sharp distinction between what he termed a ‘continuous’ line and a line divided into parts. Consider a simple division of a line into two: on the one hand there is the undivided line, and on the other the line with a mid-point selected as the boundary of the two halves. Aristotle claims that these are two distinct things: and that the latter is only ‘potentially’ derivable from the former. Next, Aristotle takes the common-sense view that time is like a geometric line, and considers the time it takes to complete the run. We can again distinguish the two cases: there is the continuous interval from start to finish, and there is the interval divided into Zeno’s infinity of half-runs. The former is ‘potentially infinite’ in the sense that it could be divided into the latter ‘actual infinity’. Here’s the crucial step: Aristotle thinks that since these intervals are geometrically distinct they must be physically distinct. But how could that be? He claims that the runner must do something at the end of each half-run to make it distinct from the next: she must stop, making the run itself discontinuous. (It’s not clear why some other action wouldn’t suffice to divide the interval.) Then Aristotle’s full answer to the paradox is that the question of whether the infinite series of runs is possible or not is ambiguous: the potentially infinite series of halves in a continuous run is possible, while an actual infinity of discontinuous half runs is not—Zeno does identify an impossibility, but it does not describe the usual way of running down tracks!"
Why do you think Aristotle invented potential infinite to get out of the paradox?
Because, while the logical discipline that he had developed was sufficient to identify the flaw in Zeno's reasoning, Aristotle did not spot how that could be done. So he instead opted for a much more elaborate and philosophically controversial approach.
Nevertheless, his counter-argument looks reasonable to me. However, it doesn't seem to me that Aristotle's notions of 'potential infinite' and 'actual infinite' are essential to his chosen rebuttal. It suffices for him to observe that running down the track and marking the ends of every sub-interval is different from running down the track without doing that, and the runner that gets from A to B does the latter.
No.
It doesn't solve the problem. It makes the problem so small so that it is no longer visible.
Wolfram.
I don’t think that’s true of all his paradoxes. It’s not about time but about completing (or even starting) a supertask.
From the Wikipedia article:
Perhaps the flaw is the premise that motion is continuous? Perhaps motion is possible precisely because distance isn’t infinitely divisible.
For the same reason we can’t assert that we can move through the divisions between 0m and 0.5m in n seconds, between 0.5m and 0.75m in n/2 seconds, and so on.
To escape the paradox one has to change something fundamental about the way we think of motion.
Yep. That's not a problem.
½+¼+?+...=1.
How does that not solve the paradox?
Odd.
Where's that fit? Are you wanting to treat distance over time as discontinuous? That'd be novel.
I'm claiming that it's impossible to count in order the rational numbers between 0 and 1 and that for the exact same reason it's impossible to pass through in order the rational-numbered distances between 0m and 1m.
Up to A-Level, yes. And calculus can't help in this case. That the sum of a geometric series is finite can't show that it's possible to count in order the rational numbers between 0 and 1 and for the exact same reason can't show that it's possible to pass through in order the rational-numbered distances between 0m and 1m.
Returning to my previous point, Wolfram's argument is that if it takes n seconds to move from 0m to 0.5m and n/2 seconds to move from 0.5m to 0.75m, and so on, then it will take 2n seconds to move from 0m to 1m. But this is akin to arguing that if it takes n seconds to count in order the rational numbers between 0 and 0.5 and n/2 seconds to count in order the rational numbers between 0.5 and 0.75, and so on, then it will take 2n seconds to count in order the rational numbers between 0 and 1. Such an argument begs the question because Zeno's argument is that it's impossible to count in order the rational numbers between 0 and 1 (or move from 0m to 1m) tout court – i.e. its impossibility has nothing to do with it allegedly taking an infinite amount of time.
That there's a finite sum to a geometric series of time intervals is a red herring.
Tough. It is possible to travel a metre. I've even done it a few times. Bet you have, too. SO there is something wrong with your account.
Now, why do you feel the need to replace the geometric sequence in Zeno with the rational numbers?
If it's impossible to pass through in order the rational-numbered distances between 0m and 1m and if it's possible to travel from 0m to 1m then when we travel from 0m to 1m we don't pass through in order the rational-numbered distances between 0m and 1m.
It doesn't need to be every rational number. It can be every 1/(2n). Or Zeno's paradox can consider every rational numbered-distance rather than just every 1/(2n) metre. It makes no difference, so yours is a strange response. Zeno isn't saying that you have to pass through the 1/2- and 1/4-way points but not the 1/5-way point. :brow:
However, if I find any fault in Zeno it's that he made a choice, a choice in favor of math and logic and declared motion as an illusion. Since Zeno actually doesn't say why he made that choice, I think he was, and is still, fooling around with us.
We can, by that reason or to be truthful, lack of reason, declare observation trumps logic and math and we'd still be on an equal footing as the great Zeno himself.
The better way out of the paradox is to realize that both [logic/math] and observation are true but that there's an explanation as to why they don't agree.
I think calculus does the trick.
I don't think it has do with time but the distance too is a geometric series, no? What do you get if you add all the terms in the series (1/2 + 1/4 + 1/8 + 1/16 +...}? 1 no? If the math shows anything it's that adding smaller and smaller quantities to a number doesn't actually result in an unmanageable infinity (Zeno would've loved that). Rather the sum tends to a finite limit - exactly what we need to resolve the paradox.
I believe I've read Aristotle's work quite well, and thanks for the reference.
Quoting Walter Pound
Aristotle demonstrates that change (becoming) is fundamentally incompatible with being (represented as a describable state. If change is described in states of being, there would be one state followed by a different state. To account for the change between them we'd have to posit another state as intermediary. But this would just introduce another different state, so we'd need to posit more states to account for the change, resulting in an infinite regress of described states, without any real change. So he concludes that a state of being is fundamentally distinct from becoming, change.
Accordingly he divides reality into two distinct aspects, form and matter. Form is described as actual, active, while matter is described as potency, or potential. All reality is composed of these two aspects, and the separation is theoretical only. The difficult part to understand is the distinction between "forms" in the physical world, and "forms" in the human mind. In the physical world, forms are what have actual existence, and are actively changing. In the human mind forms have actual existence, as what is real to the mind, but they exist as formulae which are described, or defined. states of being. So there is an incompatibility between what is "actual" within the human mind, and what is "actual" in the physical world. This is what creates paradoxes like Zeno's. In the physical world, the forms of existence are actively changing and this is fundamentally incompatible with the forms by which the human mind describes physical existence, as states of being.
Quoting Michael
That's right, there can be no definite order to the rational numbers, because any "first" number is arbitrary and randomly chosen. The better question is whether the principles which assert zero as a rational number are truly consistent, and this questions the validity of negative integers. But that's off topic of the thread.
It doesn't resolve the dichotomy paradox.
Or the paradox shows that reality isn't as we think it is, e.g. space isn't infinitely divisible and/or motion isn't continuous.
"Space is infinitely divisible" is theory. So, right, when that theory leads you to conclude something obviously absurd, you don't go with the absurdity. You realize you screwed up somewhere.
If motion is continuous (and space infinitely divisible) then I must pass through each 1/(2n) unit of distance (even if I don't stop at them), and surely that counts as a "task".
Quoting Terrapin Station
Ah, I see. Misunderstood you.
I suppose. Is there a problem?
The problem is that it is no more possible to pass through each 1/(2n) unit of distance than it is to count each 1/(2n). It's a task that can't even start.
I don't know what you mean by a mathematical task. Walking 1m requires physically passing through the 0.5m mark, and before that physically passing through the 0.25m mark, and so on. If space is infinitely divisible and motion continuous then each 1/(2n)m mark physically exists and must be physically passed through.
You don't need to figure it out. You need to actually do it, whatever's going on in your head. Just as you there's no first number to count to, making counting them impossible, there's no first distance to move to, making moving through them impossible.
Because the principle behind counting each 1/(2n) number is the same as passing each 1/(2n)m mark.
I'm not saying you have to figure it out. I'm saying that, assuming the infinite divisibility of space and continuous motion, each 1/(2n)m mark must be physically passed in ascending order, but that because there's no first 1/(2n)m mark, movement cannot start, just as because there's no first 1/(2n) number one cannot start to count each 1/(2n) number in ascending order.
I don't see why I should accept that the mathematical problem of infinite divisibility should prevent movement from starting.
But you accept that it prevents counting from starting? What is the difference between counting each 1/(2n) number and moving through each 1/(2n)m mark? Each is a physical event. We can even tie them together and say that a machine "counts" each time it passes through a 1/(2n)m mark. If such a movement is possible then such a counting is possible, but we know that such a counting is impossible and so it must be that such a movement is impossible.
The thing that makes counting impossible is the thing that makes movement impossible, so it doesn't matter that motion doesn't depend on counting. What it depends on is having completed a sequential series of events with no first event – which doesn't make any sense. A sequential series of events with no first event cannot be started.
At any moment in time, try to simultaneously observe both the position and motion of a moving object. In order to maximise one's observational precision of the object's position, one has to pay more attention to features in the visual field that establish the object's position. But this comes at the cost of being less precise when judging the state of the object's motion.
Of course, this experiment when interpreted along classical lines does not demonstrate the uncertainty principle, for it merely demonstrates one's ignorance of the total state of the arrow. But this is an irrelevant argument in the case of Zeno's paradox, for the arguments Zeno presents are phenomenological arguments that appeal only to thought experiments or practical demonstrations in which the motion of the arrow is temporarily ignored while it's position is determined and vice versa. I'm simply saying we have every reason to be phenomenologically suspicious of being able to imagine, or be literally aware of, an arrow's position simultaneous with it's state of motion.
Imagine if we had taken a video-recording of the object's motion in order to establish a per-frame analysis of the object's positions over time. No per-frame analysis will tell us about the object's motion, since for that we need to look at inter-frame differences which is a feature not present in individual frames. This is again, analagous to the uncertainty principle in that motion and position are estimated, or rather constructed, with respect to incompatible features.
The distances aren't just theoretical. When walking a metre the 0.5m mark is an actual point in space that has to be physically passed, as is the 0.25m mark before that, the 0.125m mark before that, and so on.
I don't see any reason to think that the physical act of passing the 0.5m mark is any different in kind to the physical act of saying "0.5", so I don't see any reason to think that the physical act of passing each 1/(2n)m mark can be done without there being a first mark to physically pass but that the physical act of saying each 1/(2n) number can't be done because there is no first number to say.
Such a motion is impossible even if you don't need to count each distance. The logic behind passing each 1/(2n)m mark is the same logic behind counting each 1/(2n) number; it's a sequential series of events with no start and so cannot be started.
You don't need to accept that at all. It just follows from claiming that motion is continuous and so that to get from A to B you have to pass each 1/(2n) mark. This is a sequential series of events with no start, and so logically can't be started.
Quoting Luke
I don't know what you mean by this.
The uncertainty principle is derived from the Fourier transform which involves the problem of "the start" (or however you want to call it), in the sense of a time period, which is similar to what Michael is arguing. A time period is defined by frequency, but the shorter the time period, the less accurate is the determination of frequency. The problem is reciprocal, if the time period is too short we can't determine the frequency, if we can't determine the frequency the time period is indefinite. "The start" is the first time period, and the shorter that time period is, the more indefinite any determination made from it is. This is very similar to the problem of acceleration. If a thing is at rest at one moment, then accelerating at the next moment, there must be a time of infinite acceleration.
It depends on what you mean by "determine". The mathematics of quantum uncertainty refers, at least according to its most literal interpretation, to the [I]logical inconsistency[/i] of two or more propositions, in this case that a particle simultaneously possesses a precise position and a precise momentum.
According to this interpretation, Zeno's paradox is a valid argument, and might even be useful in intuitively explicating some of the principle of quantum mechanics, but nevertheless does not prohibit motion, because Zeno's paradox is understood as referring to the modification of a particle so as for it to have a precise position, at the expense of the precision of it's state of motion.
Personally i don't think appealing to physics or mathematics is ultimately relevant in solving the paradox but that Quantum mechanics complements the vagueness of our phenomenological intuitions in many respects.
When I imagine zeno's paradox, I tend to imagine an arrow travelling for a bit and then I stop it momentarily in my imagination and say to myself "This is now the arrow's position. Now how did it get here?". But of course I am not allowed to mentally stop the arrow from moving, for I would no longer thinking of a moving arrow.
Is it even possible to imagine a moving object that has a precise velocity and/or position? Personally I don't think so. I always find myself either fantasising that I have mentally stopped the arrow in order to measure it's position, or that I am entirely ignoring it's position when thinking about it's motion.
Okay, consider this scenario:
1. So the distance between Mars and Earth is a finite distance of X feet.
2. The ship that we use to go there is a finite size of Y feet.
3. The time it takes to travel the finite distance of X feet is a finite amount of time.
4. The finite distance of X feet is infinitely divisible.
5. The time it takes to travel a finite distance of X feet is also infinitely divisible.
Therefore, the ship changing its location is impossible?
Isn't the mistake in interpreting 4. to mean that a finite size of X feet, since it is infinitely divisible, is also infinitely long if each part of the distance is decreasing and continuing to decrease infinitely small rate? Consider, that the time it takes to travel those smaller distances is also changing at the same rate.
Isn't this just confusing different kinds of infinity?
I don't think so.
The size of X distance and the time it takes to travel X distance is still finite.
Would this mean that the infinitely small thing takes an infinitely long time to travel a finite distance?
(since we have already assumed that the time taken to travel a finite distance is also finite, this seems false.)
Or would such an infinitely small thing have to travel at infinite speed to travel through a finite distance?
There's no need to change the speed of anything. Zeno's paradox just starts with common intuitions about movement through space.
This was a bit of a throwaway line for a separate argument (to my 'motion doesn't require counting' argument) that occurred to me late last night of trying to turn Zeno's logic against itself. I'm not sure whether it works, but I was thinking something along these lines:
If you identify counting with motion, or if you define counting as a necessarily physical act (e.g. machine countability), then how can the physical act of halving a distance be achieved if you accept Zeno's conclusion that motion is impossible? Halving distances is required for Zeno's argument, yet it is apparently an impossible act. Therefore, either mathematical tasks are not necessarily physical acts or else halving distances is impossible.
Then it doesn't depend on it.
Maths is just language. We can choose any description we want for movement. You choose descriptions that do not work, and conclude that what we see is impossible. It's wrong-headed.
Or it could all be holographic. Or a product of software.
Which do you favor?
There's a difference between the act of halving a number and there being half a number. We can say that there is an infinite number of rationals between 0 and 1 but that it is impossible to count them all. We can say that the space between two points is infinitely divisible but that it is impossible to move through them all.
That there's a half-way point between any A and any B is, allegedly, a physical fact about space rather than just a constructed mathematical premise. So I have no idea what you're trying to say here. That there isn't an infinite series of half-way points between A and B except as a matter of language?
Why doesn't it work? If space really is infinitely divisible and if motion really is continuous then movement requires passing though each 1/(2n)m position, which is a sequential series with no start - and a series with no start by definition can't be started.
Sounds odd. It's a bit like saying that since there is no highest number to count to, I can't ride a bike. How does one have any effect on the other?
I'm not saying that. I'm saying that because there's no first 1/(2n)m position to pass through - a physical fact about space (assuming infinite divisibility) - then movement cannot start.
There isn't. Just as there's no first rational number after 0. This isn't just some epistemic problem where we're unable to calculate the first rational number - there just isn't a first rational number. And there just isn't a first 1/n metre position to pass through.
Unless movement is discrete.
It doesn't require counting or determining a first position. It requires there being a first position. But if motion is continuous then there isn't a first position.
I'm not saying that motion is impossible because space is infinitely divisible. I'm saying that continuous motion is impossible because continuous motion entails having started a sequential series with no start, which is contradictory.
So because motion is possible it isn't continuous (even if space is infinitely divisible).
I thought the problem was in determining the first step after the start of the sequence (or after 0)?
The arguments that they give are supposed to be deductive arguments that demonstrate the impossibility of change, but pure reason alone cannot tell us what the actual state of affairs is. Pure reason alone can only tell us whether something is logically possible or impossible.
The question then is this: is there any logical contradiction in proposing that change occurs?
If there is no logical impossibility in proposing that change occurs, then no argument from pure reason alone could ever demonstrate the impossibility of change and Zeno and Parmenides have pursued a fool's errand.
Thus, the question of whether change occurs or not can only be proven empirically.
It’s not about determining the first step but about there (not) being one.
Thinking about this leads me to what I think is a fairly precise mathematical statement of the controversial assumption that Zeno's argument makes. It is this:
(A) An object O cannot move from location A to a different location B on a path Y unless, for any countable subset S of points in Y, there exists an order-preserving map from the natural numbers onto S.
In Zeno's example, S is the set of points at proportions 2^-k along the track Y, for natural number k. But there are infinitely many different types of S that present the same problem. For any real number x>1, the set of numbers x^-k for natural number k does the same thing. The sequences 1/k, 1/k^2, 1/k^3 and so on do it too. In fact, take any monotonic-decreasing function f:R+ -> [0,1]. Then the set S = f(N) provides the required "blockage".
There are maps from N onto any countable set S in Y, but they do not preserve order. For example, the map f: k |-> 2^-k has 1/2 = f(1) < f(0) = 1, reversing the order that 0<1.
The question is, why should we accept assumption (A)?
As (I think) I said earlier, we don't need calculus to dissolve Zeno's problem. All we need do is identify the questionable assumption on which it relies.
Because if space is infinitely divisible then there exists such a subset and if motion is continuous then it must pass through each member sequentially? I don’t see how that can be avoided.
"Another response—given by Aristotle himself—is to point out that as we divide the distances run, we should also divide the total time taken: there is 1/2 the time for the final 1/2, a 1/4 of the time for the previous 1/4, an 1/8 of the time for the 1/8 of the run and so on. Thus each fractional distance has just the right fraction of the finite total time for Atalanta to complete it, and thus the distance can be completed in a finite time. Aristotle felt that this reply should satisfy Zeno, however he also realized (Physics, 263a15) that it could not be the end of the matter. For now we are saying that the time Atalanta takes to reach the bus stop is composed of an infinite number of finite pieces—…, 1/8, 1/4, and 1/2 of the total time—and isn’t that an infinite time?
Of course, one could again claim that some infinite sums have finite totals, and in particular that the sum of these pieces is 1× the total time, which is of course finite (and again a complete solution would demand a rigorous account of infinite summation, like Cauchy’s)."
Does this part of the article answer your objection?
No, because the point I’m making has nothing to do with time.
I agree with the first point. For the second point, we need to be careful about what we mean by 'sequentially'. If we mean that we pass through x before y iff x
How exactly could one show that one can’t start counting each rational number between 0 and 1 from smallest to largest? The reasoning will be the same as that. Or are you saying that even this is a questionable assumption?
Proving that there is no smallest will do it. Like this: Assume there is a smallest, call it x. Then x must equal 2^-M for some M. But 2^-(M+1) is less than that and is also in S, which contradicts our assumption that x was the smallest. Hence there can be no smallest.
But that proof doesn't do anything to support a belief that it is impossible to move from A to B.
Quoting Michael
The single act of passing all members of S is the single act of traversing track Y from A to B. In doing so, object O will pass each member, in order. The act starts at time 0 with object O at location A.
So we have an act that does everything we need, and which has a beginning at time 0. Observing that there is no smallest element of S and hence no 'first passing of a member of S' does nothing to obstruct that. It just demonstrates that the notion 'the first passing of a member of S' is empty, just as the notion 'the present king of France' or 'the beginning of this circle' is.
Yes, I agree this is where the problem is right here. If the arrow is moving, there is no such thing as its position. That's why Aristotle sought to create a separation between these two, "becoming" and "being". The arrow paradox is clearly based in this problem. "Moving" implies that time is passing, "having a position", if we're not referring to something motionless, requires a moment at which time is not passing. So the arrow cannot be moving if it has a position. Simply put, there is an incompatibility between "having a position", and "moving", these two would be contradictory. So we cannot describe the same situation in these two contradictory ways.
Quoting Luke
The problem is that there is an incompatibility between moving, and having a position. The two are contradictory. So it doesn't make sense to describe the movement of a thing in terms of position. And if, or when we do, such as to say that a thing moves from position A to position B, then we are not saying anything about the movement itself, only that it was at position A and is now at position B. To describe the movement is to describe how it got from A to B. But to say it moved from A to B is not to describe the movement, which is how it got from A to B.
But this is just like saying that the single act of counting all members of S is the single act of counting the rationals from 0 to 1. It’s a nonsense assertion. You can’t just say that this task happens and so it doesn’t matter that there is no first rational number, and so you can’t just say that continuous movement happens and so it doesn’t matter that there is no first position to move through.
I chose to use the word 'passing' rather than 'counting', with intent. There is a critical difference between 'counting' and 'passing'.
To me, 'counting, in order' [you didn't say 'in order' but it was implied] means identifying an order-preserving map from the natural numbers to the set S. That is impossible.
'Passing, in order', means identifying an order-preserving map f: [0,1] -> [0,1] such that the map obtained by restricting the domain of f to f^-1(S) is order-preserving. That is easily done.
It is the insistence that the points must not only be Passed In Order, but also Counted In Order that is equivalent to assumption (A), and which is unacceptable (and I would say also unintuitive, but intuition is in the eye of the beholder).
I’m not saying that. I’m saying that passing in order is a sequential series of events with no start and so cannot be started. My mention of counting is an analogy to show why one cannot start a sequential series of events with no start. We all agree that we can’t count the rational numbers in order but we don’t all agree that we can’t pass the rational divisions between two points in order, even though it’s no different in kind.
'Sequential' is the problem word here. I would say that passing in order is not sequential, because the events are not sequential if we use the usual meaning of being in order-preserving bijection with the natural numbers.
We cannot 'start' a sequence that has no start because by definition a sequence has a start - the item that is the image of natural number 0 under the bijection. But the set S, with the natural order, is not a sequence, so there is no sequence to be started. To turn S into a sequence we need to change the order. But since order-preservation is required, that cannot be done, hence there is no sequence, and no problem.
What is this dichotomy paradox?
"Ordered." Rational and real numbers can be ordered, just in the way that you describe - indeed, that is how they are usually ordered. But they cannot be put into a sequence in that order.
'Series', like 'sequence' is a technical mathematical term. Usually it is used to describe the sequence of partial sums of a sequence, although sometimes it is used just as a synonym for 'sequence'. Either way, as we observed above that since the set of events S is not a sequence, under the natural order, neither is it a series.
All we can say is that it is a totally ordered set. To be a series or sequence, and to have a 'start' it would have to be a 'well-ordered set'.
But totally-ordered sets are not all well-ordered, and sets of the form S are examples of that.
I would say that the object O passes the points in S in order. Intuitively, we feel that that implies that there must be a first passing, a 'start'. But that is only an intuition, not a logical consequence of any of the properties of the objects being considered. Sometimes our intuitions lead us astray, and this is such a case.
That's why the Zeno paradox is veridical rather than falsidical. It conflicts with our intuitions, but not with logic. Hence we have to conclude that, in this case, our intuitions are wrong.
Let me get this straight, make sure I understand.
Zeno's paradox can be stated as follows 'In order to travel a distance of 1 meter, you must first travel 0.5 meters, in order to travel a distance of 0.5 meters, you must travel 0.25 meters. For any distance you can travel, you must first travel half that distance. Thus you cannot travel the distance.'
Fleshing out the paradox entails fleshing out the relationship between the thought exercise of division and the impossibility of travelling the distance. Michael's version seems to be:
(1) If journey did not have a beginning, it could not have occurred.
Your response to this is:
(1A) For the purposes of the paradox, a journey is a sequence of distances which must be travelled. This is [math]\big[\frac{1}{2n}\big]_{n=\infty\rightarrow 1}[/math] (abusing notation but I think it makes sense). The beginning of the journey would be the least element of the set - the smallest distance travelled - since this set has no least element in the ordering described, the usual notion that a journey must have a beginning (the journey presumably being the sequence of distances travelled in their usual ordering) is not in play. In effect, this is a confusion of two distinct concepts - the well ordering of travelled locations in typical journeys, and the mere total ordering of journeys constructed through the thought experiment. The same distances are considered, but under different orderings. IE
[math]\big[\frac{1}{2n}\big]_{n=\infty\rightarrow 1}\neq\big[\frac{1}{2n}\big]_{n=1\rightarrow \infty}[/math]
even though they are equal as raw sets. Equality of sets does not imply equality of ordered sets.
Do you think your response also addresses the case where we replace (1) with (2):
(2) The number of distances travelled is infinite, and we cannot do an infinite task.
?
Also - and I'm not sure if this pedantic or crucial - I would say the journey does have a beginning, and it is the spacetime event that is the location A at the time that is the infimum of all times at which the object O is anywhere in Y - {A,B} (recall that Y is the locus of the object O). Similarly, the end of the journey is location B at the supremum of that set of times.
In that sense, the journey has a well-defined beginning.
It is the set of passings of the waypoints in S that has no beginning. But that set of passings is not the whole journey.
Quoting fdrake
Yes, if 'journey' is used to refer only to the passings of waypoints in S, rather than the usual meaning of the whole path Y, that is central to where Zeno goes wrong.
Quoting fdrake
I think this has even more problems than (1). The term 'task' is dragged up out of nowhere, with no clear meaning or relation to the problem. Nor is any support provided for the claim that we cannot do an infinite task - a claim that seems very unintuitive to me.
If space (or the number line) is continuous, and motion is analogously continuous, then there shouldn't be a first position. Our inability to define the value of a first position is what we should expect. This should prove that continuous motion is possible rather than impossible.
That's a lot clearer to me, thanks.
Even if you want to talk about the movement from A to B being continuous the half-way point between them is a discrete point that actually exists and passing through it is a discrete event that actually happens.
If our concept of numbers is derived from geometrical intuition, then any attempt to address Zeno's paradox via appeal to the resulting mathematics is equivalent to trying to make sense of the A series of time via the B series.
Not only does this way of thinking make Zeno's paradox even more bewildering and intractable, it isn't acceptable from the point of view of the intuitionist, who identifies the construction of numbers not with sketches on paper, but with the very phenomenal passage of time.
From this perspective, first-position events are constructed by the very act of starting to count. Zeno's paradox can instead be interpreted as a paradox concerning the conceptual issue of what it means to distinguish spatial positions that represent different times, such as the hands of a clock.
Quantified measurements of position information appeal to machinery, and hence to Computable Analysis that recognizes only a countable set of numbers, namely the computable numbers which represent equivalent-classes of Turing computable total functions.
Computable analysis reveals that the ordering of any two computable numbers is decidable provided the numbers are in fact different. But due to the negative result of the Halting Problem, there is no universal algorithmic test for deciding whether two computable numbers are in fact different or equal. Consequently the very notion of either difference or equality with regards to nearby positions or times is not a mathematically meaningful a priori notion for the constructivist.