Does calculus solve Zeno's Paradoxes fully, or is there still a conflict proposed by them that is not answered. I am mostly just curious about the first two paradoxes (The Dichotomy Paradox, and The Achilles and the Tortoise Paradox). I have read about how it fixes it, but it still seems to be a problem, maybe I just don't fully understand the solution. Thank you so much!
In my own opinion, modern real analysis (the rigorous theory of the real numbers and calculus) does definitely solve the matter once and for all -- mathematically. We can sum an infinite series and get a finite result; and we can do so with absolute logical rigor. At least if we believe in set theory. However I do not believe that the matter is nearly as clear regarding the physical universe. We have no evidence that spacetime is a continuum; it might be discrete for all we know. And even if spacetime is a continuum, accurately modeled by the real numbers, it's not clear to me how we can hope to carry out an infinite summation in finite time. In other words, to get from point A to point B I have to first traverse half the distance, then half the remaining distance, etc. Mathematically we know that 1/2 + 1/4 + ... = 1. But what can possibly be the physical interpretation of adding up infinitely many finite intervals of time? In my own personal opinion I really do not believe Zeno's paradoxes have been satisfactorily resolved in the physical realm. I'm pretty sure this is a minority opinion.
Well...that's easy; if you have a finite temporal interval 1/2, and you add 1/4 and 1/8 and 1/16 and so on, then you get a temporal interval of 1. I really don't understand what "physical interpretation" is supposed to mean in this context. If 2+2=4, then adding 2 apples to 2 apples gives me 4 apples; if a geometric series converges to a sum, then adding up temporal intervals equal in magnitude to the terms of that series gives me a temporal interval equal to that sum.
But only at the infinith (I believe that that is not a word) step would 1/2+1/4+... = 1 , and by placing a limit on 1/2 + 1/4 +... at 1, that might clear it up. Yet it seems strange and almost ridiculous to assert that infinite events can happen in any amount of time. Wouldn't that mean that the same number of events (infinity) happen in 1 second as in 1,000 years. would this force things to travel at infinite speed? and remain still at the same time?
1/2 isn't an "event", it's an interval of time, just like every other term in the series. The series sums to 1, and thus the total interval of time is 1. There is no profound philosophical truth lurking here; it's just like adding 2 apples to 2 apples (actually, it's like adding a half of an apple to a quarter of an apple...) The premise of Zeno's paradox is that it would supposedly take infinite time in order to move a given distance. Analysis demonstrates conclusively that Zeno's conclusion does not follow from his premise; therefore, the paradox is resolved.
the individual movements of the whatever that happen withing 1/2 a second are events, and if the same amount happen between 1/2 a second and a decade, then how does motion happen?
Yes. Well, people wouldn't use the word infinity. While they would say the number of events is infinite, that adjective applies to a wide class of cardinal numbers. The specific number is the "cardinality of the continuum", and is the cardinality of the set of real numbers. It is also equal to [itex]2^{\aleph_0}[/itex]. If you assume the Continuum Hypothesis (as I will, for simplicity of notation), it is equal to [itex]\aleph_1[/itex]. As Achilles runs, he is indeed travelling at [itex]\aleph_1[/itex] spatial events per second. However, this fact doesn't tell us anything about how many meters per second he may be traveling, other than it's not zero. I don't see where you got this idea.
Of course we have evidence: continuum physics works. SR, GR, QFT all have continuum models of space-time. Discrete models of space-time have predicted things we might see, but haven't. While they still remain consistent with data (by making our model finer and finer as we fail to observe the effects of a coarse spacetime), such predictions are by nature less specific, and so every piece of evidence that agrees with the more restrictive model of continuum physics is a piece of evidence that favors continuum space-time over discrete space-time.
I've seen two versions of the paradox presented. The first version is "we've identified an infinite sequence of events -- how can infinitely many events happen in a finite amount of time?" Of course, argument from incredulity is not an argument. One needs to argue why one thinks infinitely many events implies an infinite amount of time. As I've not seen people argue this point coherently, I can't really say what misconception people have. However, Calculus addresses this point by adding up the time between events and verifying that it really is finite. The second version is "we've identified an infinite sequence of events, how can there be more events beyond infinity?" While this could also be an argument from incredulity, I believe it is better attributable to the hogwash about the infinite that tends circulate around. Again, I've never really seen the point argued, just asserted, and the problem is the assertion is patently wrong. In the analysis, we can plainly see that there are events beyond the infinite sequence identified as Achilles catches up to the Tortoise. I suspect the psychological problem is that when they identify the infinite sequence of events, they promptly forget the picture where it's clear what's going on, and switch to a new picture of evenly spaced points marching on "forever", and implicitly assume any other events must occur within the span covered by that "forever" and thus cannot be after them. But in any case, ordinal numbers give a natural extension of the notion of sequence that retains the notion "after each point, there's a next point" that goes beyond counting with the natural numbers. And more generally, there is the notion of an order type. But it's far less complicated if you simply don't forget the original physical picture.
Are you arguing that a) The Continuum hypothesis has a definite truth value in the physical world; which implies its truth value can be discovered by physical experiment; and b) There are transfinite cardinalities of "spatial events" in the physical universe? These are quite radical claims. That's why I expressed my doubt. If one believes that spacetime is modeled by the real numbers, one then has to look for analogs of the objects of ZFC in the physical world; and one has to then stand by the assertion that the questions of set theory are actually questions about the real world, and subject to physical experiment. In other words if you believe that Zeno's paradoxes are solved in the physical worldd by virtue of being solved in real analysis; then you will have to defend a whole host of very difficult propositions. Do you believe that there are analogs of all the transfinite cardinals in the physical world? After all, we can just keep taking power sets of these aleph-1 objects that you've already posited. Do you think there are inaccessible cardinals? And that this question is the proper study of physicists? Do you think the Axiom of Choice is true in the physical universe? If it is, then you must accept Banach-Tarski. If not, physics has to throw out a lot of theorems of modern math. I hope I'm expressing why I think claiming that Zeno is solved in the physical world is a very difficult claim to support. That claim implies that the real numbers are physical things. And the reals bring all the baggage of set theory.
No; I find the the CH wholly irrelevant to physical concerns, and thus am free to make a choice which simplifies notation. This is certainly the statement of our best physical theories, and I have no reason to go against them on this point. And while it is surely true that only a finite amount of information extracted from their full complexity actually "matter", that just means it would be an interesting and possibly useful exercise to work out the properties of this information and come up with a 'background-free' description that does not make reference to the manifold of events. e.g. I believe Einstein's hole argument shows the only relevant physical information about a collection of point particles that pass through a 'hole' in space-time consists of a finite graph with various labels on its edges and vertices. But asserting that a continuum of events is actually wrong is in direct conflict with the scientific evidence we have to date. I don't see the problem with there being [itex]\aleph_2[/itex]-many regions of spacetime or [itex]\aleph_3[/itex]-many classes of regions. Sure: 0 and [itex]\aleph_0[/itex], for example. But more seriously, I think models of set theory with inaccessible cardinals have inaccessible cardinals, and those without, don't. If the presence of an inaccessible cardinal in the mathematical theory had observable consequences, then of course it's a question in the proper study of physicists. I'm not aware of any consequences, though. I think the axiom of choice is true in our successful physical theories of the universe. I'm not sure what else your question would mean. And of course I accept Banach-Tarski -- it would be rather silly to think that measuring non-measurable sets would make sense. Honestly, I think it's more surprising that you can't do the same thing in one dimension. (Aside: I've recently encountered the theorem that if you believe all sets are measurable in ZF, then it follows that you can partition the real line into more parts than it has points) No they don't. The reals are a tiny fragment of 'all the baggage of set theory' -- their first-order theory isn't even complicated enough for GĂ¶del's incompleteness theorem to apply. Similarly, the math actually used in our physical theories -- or even by most mathematicians -- is only a small fragment of the whole of ZFC.
I'm not seeing the paradox, something moves across an infinite number of intervals which get very small. If something is moveing a constant speed s = m/h, then the time it spends traveling across each interval gets smaller and smaller but the overall speed of the object is the same...
Yeah, if you travel 1 m/s, it takes 1/2 s to travel 1/2 m, 1/4 s to travel 1/4 m, 1/8 s to travel 1/8 m and so on. So no matter how many intervals you divide a second or the distance into, in the end you add them all back together and still get the same answer. It's like asking if I divide a second up into infinitely small time intervals and then add up an infinite amount of those time intervals, how can an infinite amount of time intervals fit in one second? Well, no matter how many you divide them up into, you put them back together and you are back with what you started with.
I don't understand how you can say that you have a choice in the matter. If you believe (as you have said) that a moving object passes through [itex]\mathfrak{c}[/itex] spatial events; then the actual cardinality of that number is the proper subject of physical investigation. There is one and only one correct answer. Do you believe physical cardinalities are like the popular explanation of light: a wave on Monday, a particle on Tuesday, depending on how you look at it? My understanding is that physicists don't worry about that because 1) They understand that ZFC is math, not physics; and 2) Deep down they know they are on shaky ground claiming there are [itex]\mathfrak{c}[/itex] of anything in the physical world. I just don't see how you can claim there are [itex]\mathfrak{c}[/itex] of something, yet believe that the actual cardinality is a matter of personal preference or choice. Likewise with the Axiom of Choice and the B-T paradox. It's true of the mathematical continuum. I doubt most physicists would say its true of the physical world. But you seem to have accepted (without evidence) that they are physically true. Am I understanding you correctly?
Is it meaningful to say these are physical "paradoxes"? We simply observe that it is in fact possible to go from point A to point B. So to me the only question is whether our mathematical model has the logical consistency to give the same answer no matter which way we describe the motion. The fact that it can is more a validation of our model than a resolution of a nonexistent physical paradox. It might be relevant to contrast this to the Banach-Tarski paradox which says that a ball can be decomposed into a finite number of subsets which can be rearranged by rigid motions and reassembled into a much larger ball. This is also not a physical paradox because it has never been observed in nature. The subsets are too complicated, and we know that real stuff is made out of particles, not continuum. In this case, we just have to accept that this is a quirk of the continuum, and we adjust the theory of measure so that we never speak of measuring such complicated sets.
Exactly. The solution to someone arguing in favor of Zeno's paradoxes is stand up and silently walk out of the room, just as Diogenes reportedly did. Problem solved. From the wikipedia article on the subject of Zeno's paradoxes, According to Simplicius, Diogenes the Cynic said nothing upon hearing Zeno's arguments, but stood up and walked, in order to demonstrate the falsity of Zeno's conclusions. To fully solve any of the paradoxes, however, one needs to show what is wrong with the argument, not just the conclusions.This is just wrong. In the sciences, the burden of proof is always on the presenter of a new idea. One way of disproving a conjecture about the physical world is by a physical experiment that shows the premise to be false. There is no reason to look at the logic. The burden is on the proponents of the idea to determine where they went wrong and to rectify their concepts with physical reality. Diogenes falsified Zeno's paradoxes within minutes of its initial presentation. Why philosophers worry about this problem thousands of years later is beyond me.
Are you asking if an asymptote ever touches the X axis? No, it doesn't. In the reality of observations this is different due to there being a smallest possible size/distance. You can't have a distance less than a planck length.
Zeno's paradoxes demonstrate -- on the assumption that he hasn't made a logical error -- that the impossibility of motion is a prediction of our methods of analysis. Falsifying the conclusion would mean one or more of its hypotheses is wrong. Which claim do you wish to reject? Are you to assert that when walking from here to there that you don't pass through the point half-way in-between? That a physical theory is disallowed model space as a continuum? Something else? But, fortunately, his arguments are somewhere between incoherent and logically flawed, so we don't have to sacrifice any of the hypotheses.
What physical experiment could distinguish between the alternatives? Without an observable consequence, I'm free to choose however I like between indistinguishable alternatives. It's not really any different than my freedom to choose whatever coordinate chart I like to do my calculations.
I must say I'm surprised that anyone educated in either math or physics could believe that the objects characterized by the ZFC axioms have any basis in physical reality. Infinite sets? Nonmeasurable sets? Undefinable sets? The uncountable reals, their powerset, the powerset of the powerset, ad infinitum? And I also believe that in order to think that 1 + 1/2 + 1/4 + ... = 1 in the physical world requires that one also accepts the physical reality of the objects of ZFC (whatever they are. They're very far from the sets envisioned naively by Cantor.) But clearly you are one such. And I have no arguments beyond those I've already made. So I'm afraid I have to leave it at that.