[Danny_Ocean]

The situation, then, is this. Any proper variant of Zeno's paradox, such as the above thought experiment, provides a mathematical and logical account of the physical process of movement through space (or time), and argues that it is impossible for Achilles to win (or move at all). So, in order for this paradox to be resolved, one needs to either show something wrong with the math or logic (which calculus and series-based solutions do for the improper variant, but not for the proper variant), or show why this mathematical analysis cannot be used in our physical world. As suggested below, maybe space and time are not so that between any two points one can always find another point, which would indeed prevent this analysis to go through, and possibly our naive conceptions of space and time are mistaken in other ways as well. But calculus and series-based proposals do not challenge any of our conceptions of space and time in any way, as they are purely mathematical analyses that say nothing about the nature of space and time at all. Hence, these kinds of solutions do not resolve the paradox in this second way either. In short, there is nothing in calculus or series-based solutions that prevent the infinite sequences to crop up that lead to the whole paradox. So, as such, they do not resolve the paradoxes.


Issues with the issues with the proposed calculus-based solution:

If we more closely examine the thought experiment, it is clear that Achilles naming the positions "first", "second", and so forth, is a nonphysical/mathematical act rather than a physical act; as an illustration, try getting your friend to say the word "Bob" on the 1/2 second mark, then the 1/4 second mark, and so on... you just can't do it. Consequently, the "counting process" is a mathematical process, while the "catching up with the turtle" is a physical process. As with most attempts to peddle Zeno's paradoxes, the central element is the conflation of these two processes. But they are simply not to be identified. The mathematical "counting process" goes on to infinity, and this is never something one could complete. However the physical "catching up with the turtle" process is something that can be completed. This is shown by an elementary application of limiting process theory, with time as a parameter.

These considerations (one must divorce the mathematical and physical processes at hand) also apply to the paradox as given in the "much more stinging" form: "for Achilles to capture the tortoise will require him to go beyond, and hence to finish, going through a series that has no finish, which is logically impossible". Here the word finish has been confusingly used for both the physical process and the mathematical process in an effort to conflate the two.

The issue with the statement "Indeed, the problem with the calculus and other series-based solutions is that these kinds of solutions beg the question. They assume that one can finish a limiting process, but this is exactly what Zeno questioned." is similar. They (the vast majority) do not assume that one can finish the limiting (mathematical) process, and they do not need to. To finish the physical process it is not required to finish the associated mathematical (limiting) process. The two processes are completely different in nature, and divorcing the two is essential if one is to resolve the paradox.

The mention of time does not make the paradox into a strawman, and telling someone they can't mention time in their solution is extremely unfair, because the problem is posed in the form of the physical, and consideration of time is implicit in any consideration of the physical. Just because someone worded the problem without using the phrase "time" does not make it illegal to use the word "time" in the solution.


Are space and time infinitely divisible?
Another proposed solution to some of the paradoxes is to consider that space and time are not infinitely divisible. Just because our number system enables us to give a number between any two numbers, it does not necessarily follow that there is a point in space between any two different points in space, and the same goes for time.

If space-time is not infinitely divisible (and thus not perfectly continuous), it is "discrete" (composed of “lumps” and “jumps”, as is experimentally observed in the field of quantum physics e.g. electron orbitals jumping from one level to another). This means that motion is, at the smallest physical level, a series of jumps from one quantum space-time coordinate to the next, each occurring over distance and time intervals that are not divisible into smaller measures.

Thus the total number of quantum jumps made while traversing from point A to point B is finite, and therefore there is no paradox.


Does motion involve a sequence of points?
Augustine of Hippo was the first to posit that time has no precise "moments," in his 4th century C.E. text, Confessions. In Book XI, section XI, paragraph 13, Augustine says, "truly, no time is completely present," and in Book XI, section XV, paragraph 20, Augustine says "the present, however, takes up no space."

Some people, including Peter Lynds, have proposed a solution based on this ancient premise. Lynds posits that the paradoxes arise because people have wrongly assumed that an object in motion has a determined relative position at any instant in time, thus rendering the body's motion static at that instant and enabling the impossible situation of the paradoxes to be derived. Lynds asserts that the correct resolution of the paradox lies in the realisation of the absence of an instant in time underlying a body's motion, and that regardless of however small the time interval, it is still always moving and its position constantly changing, so can never be determined at a time. Consequently, a body cannot be thought of as having a determined position at a particular instant in time while in motion, nor be fractionally dissected as such, as is assumed in the paradoxes (and their historically accepted solutions).


Conceptual and semantical approaches
Another approach is to deny that our conceptual account of motion as point-by-point movement through continuous space-time needs to match exactly with anything in the real world altogether. Thus, one could deny that time and space are ontological entities. That is, maybe we should acknowledge our Platonic view of reality, and say that time and space are simply conceptual constructs humans use to measure change, that the terms (space and time), though nouns, do not refer to any entities nor containers for entities, and that no thing is being divided up when one talks about "segments" of space or "points" in time.

Similarly, one can say that the number of "acts" involved in anything is merely a matter of human convention and labeling. In the constant-pace scenario, one could consider the whole sequence to be one "act," ten "acts," or an infinite number of "acts." No matter how the events are labeled, the tortoise will follow the same trajectory over time, and all of the acts will be "finished" by the time the tortoise reaches the finish line. Thus, the labeling of acts is arbitrary and has nothing to do with the underlying physical process being described and that it is possible to "finish" an infinite sequence of acts.

From the philosophical standpoint of Bergsonian space-time, the paradox is resolved as follows. The steps of the paradox as presented above can be summarised as:

There are an infinite number of positions defined by any finite movement.
Let movement from one position to the next be called an 'act'.
An infinite number of acts cannot be completed in a finite amount of time.
An infinite number of acts cannot even be started.
Thus movement cannot be started or completed.
Movement is an illusion.
Moving backwards, any claims about the nature of illusions or acts are intrinsically claims about the nature of experience. According to Bergson's conception of time, all moments of time are comprised of a mixture of both a 'snap-shot' extrinsic property and a durational intensive property, which are irreducible to one another.[2]

The arrow paradox makes an argument that considers only time as a measurable, extensive, homogeneous construct that can be modeled spatially (the above diagram of it with lines being a good example). Thus a conclusion concerning the nature of experience is not warranted by an incomplete proof of only partial properties of time. The point is that 'acts' are experiential in nature.


The notion of different orders of infinity:
Some people state that the dichotomy paradox merely makes the point that the points on a continuum cannot be counted — that from any point, there is no next point to proceed to. However, it is not clear how this comment resolves the paradox. Indeed, as one variant of Zeno's paradox would state: if there is no next point, how can one even move at all? Also, it is not clear what this comment has to do with different orders of infinity: the rational numbers are countable, i.e. of the same order of infinity as the natural numbers, but on the rational number line, there is for any rational number still no next rational number either.


Status of the paradoxes today:
Mathematicians thought they had done away with Zeno's paradoxes with the invention of the calculus and methods of handling infinite sequences by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century, and then again when certain problems with their methods were resolved by the reformulation of the calculus and infinite series methods in the 19th century. Most philosophers, and certainly scientists, generally agree with the mathematical results.


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