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Zeno's paradox when seen from the perspective of pure mathematics is easily dealt with using the limit concept, but giving it an anthropomorphic twist makes it absurd. And the idea of a first non-zero coordinate shows a very limited knowledge of mathematics.

Would you agree a wise intellectual warrior should first know his enemy before striking? — jgill

Since both you and @fishfry reacted the same way, I've learned that I shouldn't try to make a point in the form of a question. I know that there is no first non-zero coordinate on the real number line, that's exactly what I was trying to highlight. Let me try again. Before she arrives at x=1, do you believe that she must first cross all points between 0 and 1? And before she arrives at x=0.5, do you believe that she must first cross all points between 0 and 0.5? If so, then we can take this reasoning to its limit and say that to move she must first reach the first non-zero coordinate. And if there is no first non-zero coordinate, then she cannot move. This is Zeno's Argument which is what trying to highlight, but apparently didn't convey very well.

In the Numberphile on Zeno's Paradox, James Grime says the following:

"I want to give you the mathematician's point of view for this, because, well, some say that the mathematicians have sorted this out........So something like this-- an infinite sum-- behaves well when, if you take the sum and then you keep adding one term at a time, so you've got lots of different sums getting closer and closer to your answer. If that's the case, if your partial sums--that's what they're called-- are getting closer and closer to a value, then we say that's a well-behaved sum, and at infinity, it is equal to it exactly. And it's not just getting closer and closer but not quite reaching. It is actually the whole thing properly."

Is your view that the problem (of Atalanta travelling from x=0 to x=1) is resolved by completing an infinite process?

I think the most compelling solution to Zeno's Paradox (of Achilles and the tortoise) that is often presented is by looking at the situation holistically. If you ask what are the velocities of Achilles and the tortoise you can work backwards to calculate the instant when Achilles passes the tortoise. It seems so simple when you think of it this way. This is the type of thinking that I'm promoting: starting with the whole and working backwards to determine instants.

its demonstratable that space is infinitely packed. — Gregory

It is? Can you point me to the demonstration? This may be going off on a tangent, but the Bekenstein bound poses an upper limit on the information that can be contained within a given volume. Or are you talking about mathematical space being infinitely packed?

Quantum physicists descriptions of reality are not necessarily accurate. They have to fit true philosophy. — Gregory

I believe that physics, math, and philosophy all inform each other. Physicists don't have it all figured out, but philosophers need to update their views based on the latest advancements in physics (and math).

@fishfry I'm trying to let you off the hook on talking to me about the nature of continua but you keep coming back for more. I'm not complaining!

If we allow that the real line is made of points (which are just real numbers) then the answer is that there is no first nonzero positive real number. That is the answer, so your claim that "we cannot answer this question" is false. — fishfry

Exactly. And since there is no first nonzero positive real then she cannot take her first step. Her journey doesn't begin. Motion is impossible. When I said that 'we cannot answer this question' I should have said 'we cannot answer this question satisfactorily.'

Now who's claiming a line is made of points? You are the one doing that! — fishfry

My view doesn't exclude points. Let me try this again but with a much more digestible image.



I suspect you would say that you see infinite points there. I don't. In this picture I see 5 points, each connected by lines. That's it. WYSIWYG. And if we relate this back to Atalanta's story, I would say that we have still photos of her at 0, 1/3, 1/2, 3/4, and 1. That's all we've got. We are not justified to say that at some time she was at 1/10 or at 7/8 because we don't have the photographs. At one moment we see her at 0, we blink, and when we open our eyes she's at 1/3. Motion happens when we blink, when we are not looking. And I've said it before but perhaps it might sink in this time that this is consistent with QM. If we are continuously observing a quantum system it will not evolve. It will only evolve when it is not being observed. Change happens when we blink. Not surprisingly, it's called the Quantum Zeno Effect.

that doesn't entitle you to mischaracterize the math of the real numbers. There is no smallest positive real number. — fishfry

Perhaps my point didn't come across. All I was trying to say was that 'there is no smallest positive real number' means that Atalanta cannot decide what point to go to first.

In your model of a continuum, whatever it is, is there a smallest real number? — fishfry

In my view, we cannot speak of the 'completely measured continuum' which is what I would call 'the real number line'. We cannot do so because it would require infinite measurements and that's impossible. Instead, we must only speak of systems which we can actually look at (at least in principle), like the one above. And in that case, the smallest positive number is precisely 1/3.

There is no problem. Mathematical analysis took care of that years ago. — jgill

Atalanta is walking from x=0 to x=1. What is the first non-zero coordinate that she walks to? I'd like to know how mathematical analysis solved this.

As long as we think that a line is composed of points we cannot answer this question. Motion is only possible when she jumps from one point to another. And what I'm saying is that she doesn't jump over infinite points, she jumps over a continuum.

You're throwing ideas out there that are philosophical and calling them science. — Gregory

No, I'm saying that my philosophy is in agreement with modern physics.

What we know is what our senses say, yet you say our senses and the paradigms they are in are close to having ZERO accuracy. Why trust your use of measuring equipment then? We have no real understanding of what stuff looks like at the quantum level. — Gregory

We know much more than what our measuring equipment says. Our measuring equipment gives us data, physics gives us models (that match the data and make predictions). And with these models we can talk sensibly (to some extent) about what reality is like at the quantum level.

But your solution to Zeno's paradox is that the quantum world doesn't exist unless we measure it. Which, well, is more paradoxical than what Zeno proved......And you are saying that there is nothing there at all unless we measure it. What kind of nonsense. — Gregory

No, unmeasured particles exist in a superposition state (a state of potential). This is the interpretation of QM that physicists are taught in school. It's not paradoxical (there are no contradictions), it's just weird because it violates our classical intuitions. In fact, I believe it's nature's way of cleverly avoiding paradox/contradiction/singularities.

Think about a video game, when Mario is dilly-dallying collecting coins is it reasonable to ask what Bowser is doing at that moment? No, that part of the world just has not been resolved. All we could say is that Bowser is somewhere in his castle waiting for Mario. He has the potential to be in one corner of the room, and the potential to be in the other corner of the room. We can describe his position only as a potential state. Your console is unable to resolve Mario's universe everywhere and at all times so it only resolves what you're looking at. Why would our universe be any different?

I live and breathe physics simulations so I can't help from seeing our universe as a big simulation. And since I don't think any computer can have infinite computing capacity, the computer of our universe has got to cut corners somewhere. And it does it remarkably well. Essentially what you are craving is an infinite computer. You are craving a supernatural world.

And by the way, measurement is not something exclusive to humans.

You should try posting an unpopular opinion in the politics-related threads around here. Namecalling is all they've got. — fishfry

I wouldn't dare, I've already had my fair share of backlash in the math group.

I pointed out that video is actually a sequence of stills... — fishfry

I don't agree with this. You're missing the key ingredient: time. A video is a sequence of stills, each held for some interval of time.

even a still captures motion because it records photons over a nonzero interval of time. — fishfry

I don't completely agree with this. It's true that macroscopic stills capture some motion but when we look at the microscopic level, it is possible for a still to capture zero motion (e.g. a single photon hitting a quantum sensor). And in a simulation (which is what I'm most familiar with) the stills capture zero motion (note: the velocity field seems to reflect instantaneous velocity but let's not go there).

So we're talking past each other. Perhaps we can find agreement at least in that. — fishfry

I don't really agree with this. I held the 'point-based' view for most of my life so I think I understand your position, I just don't think it's correct. I think the problem is that you don't understand my position. This is not necessarily your fault, in fact since nobody else here seems to understand my position it's probably my fault.

it's not the job of math to solve Zeno's paradoxes; and even if it is, it's not an interest of mine. I wish you the best with your efforts in that direction but I can't help. — fishfry

Understood. And to be realistic, since you haven't understood my other arguments I've presented I think it would be a waste of your time.

dt is a differential form. — fishfry

You're right. This was purely a laziness thing because I didn't want to find the symbol for delta. Thanks for the correction.

I'm not trying to solve the nature of the continuum here. — fishfry

Okay, given that our discussions have been focused on the nature of a continuum, I think this is a pretty good reason to wrap them up.

This is why I called you an idealist. — Gregory

Name calling doesn't win debates.

There is no true ontology in that field of research and if you think your body is primarily empty space I'd have to say you have a cognitive distortion — Gregory

Space is far from empty. Fields permeate all of space.

Aristotle's "solution" was that the whole exists prior to parts. To which I ask:

1) do the parts of the whole exist

2) how many are there of these parts. — Gregory

1) The parts only exist when measured. If nobody is looking, it is meaningless to speak of parts (just as it is meaningless to say that Schrödinger's cat has a definite state).

2) There are only ever finitely many parts. The number depends on how many measurements you've made.

I communicated this to @fryfish a while back on this thread but I don't think you were active at that time so I'll repeat the idea. Your view is that a string is created by assembling a whole bunch of nothing (i.e. 0D points). As long as you have enough nothing then you can eventually produce something. It's crazy.

With Aristotle's view, we start with something, we start with the string. How many endpoints does the string have, let's say 2 for simplicity. Can we create more endpoints? Sure, just cut it more and more times. The more cuts you make, the more endpoints you have. How many endpoints do you have? Well it depends on how many cuts you make, but certainly you will have some finite amount. Now, how many times do you need to cut a string until it disappears - until it's just a whole bunch of 0D points (i.e. nothing)? Does infinity do the trick for you? If you are open minded you have to acknowledge that there's a problem here. And if we can't go from a continuum to infinite points, why would we think that we can go from infinite points to a continuum? 

I've noticed that when I hit the  button and enter Ryan, your handle doesn't come up. Do you know why that is? Moderators, any clues? — fishfry

I suspect it's the apostrophe...it's always given me computer problems. I've posted a comment in the 'Feedback' section, hopefully a moderator will allow me to change my name.

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That is a cool video, thanks! I've worked for many years in simulating physical systems (in fact I was simulating rotating fan blades today) and when it comes to transient simulations, it's important to select a timestep sufficiently small to adequately resolve the transient fluctuations. Too large and you can incorrectly predict the rotational velocity of a helicopter's blades, you might even predict that they're not moving. As such, it is natural to then want a smaller timestep. But one thing that we never want is to set the timestep = 0 seconds. Not only would the equations blow up, but no matter how many timesteps we run we would never get past the initial state. Motion is impossible when dt=0. So what we do is pick a dt small enough to sufficiently resolve the frequencies of interest. And when we create an animation based on those frames we hold each frame for dt seconds before advancing to the next frame.

And what I haven't been able to get across to you is that a frame held for 0 seconds is totally different than a frame held for dt seconds. The 0 second frame is like a point in time while the dt second frame is like a timeline. We make movies, not by assembling 'points' but by assembling 'lines'. If all we have are points, then we'll never construct anything with more extension than a point. It is because of this sort of reasoning why I earlier called the 'real number line' the 'real number point'.

Now if we go the other way around, we don't face the same problem. If we start with a line we can easily produce points, just as we can produce new endpoints when we cut a string. When we start with continua we get both continua and points. It's a much richer foundation that avoids the paradox.

Why is this my problem, or math's problem? — fishfry

I'm going to come back to this. The point-based view has reached its limit, it will never be able to solve Zeno's Paradox. But Zeno's Paradox is that loose thread on our sweater that we should pull because shedding ourselves of our inconsistently knitted sweater will reveal that underneath the sweater we're wearing a killer tuxedo.

This is math's problem because mathematicians are the only ones who can 'climb mountains'. I'm just a Joe-schmo who's pointing at some unexplored mound that's hidden behind clouds saying 'that could be a bigger mountain'. 

Hi Moderators, when others try to mention me with @ my name doesn't come up. I suspect that it's due to the apostrophe in my name. Can I change my name?

It is a lot more satisfying to do this than to argue indirectly, IMO. — jgill

Agreed. It is easy to talk about doing something (e.g. completing an infinite process), it's another thing to actually do it. I think your story is quite fitting because I believe that when we construct objects by assembling points, we never get anywhere. We're always left with a 0D object...an empty set of sorts.

I can offer a couple of reading suggestions based on these two comments. — aletheist

Thanks for the recommendations, both seem like interesting reads and your paper seems especially relevant.

The greatest error of modernity is saying that the world is information and is not as it appears to us. The world we see transcends any interpretation of QM and psychological studies on mind-matter interaction. What you see is what there is. There is more there, but not less. Any other position is insanity. Zeno's paradox will never have a complete solution, but it is a sign of a healthy position to be comfortable with a paradox — Gregory

I don't think anyone is saying that there only exists information, certainly I'm not saying that. I'm also not denying 'what you see is what there is'. The problem is that you and others are incorrectly filling in the gaps between quantum measurements to assume that a particle is travelling a definite path, and by making this assumption you fall victim to Zeno's Paradoxes. Zeno's Paradox does have a simple solution, Aristotle's solution which I have been promoting here. I don't think anybody has provided a single valid criticism of this view, other than 'it doesn't jive with my classical intuitions'. You need to update your intuitions based on modern physics. Lastly, paradox is a sign that progress can be made, we should not just accept it as being beautiful mystery that will permanently be out of reach. We should be motivated by it.

But what good is that? (in reference to talking about a potentially infinite process) — Metaphysician Undercover

I've said this many times, but I'll say it one more time. We can talk (using finite statements and operations) about potentially infinite processes without ever initiating (let alone completing) the potentially infinite process. That's what calculus is all about (in my view). If you avoid infinity altogether (i.e. actual and potential) then there is no way to get calculus off the ground. Let go of executing the potentially infinite process and embrace the finite statements that talk about the potentially infinite process.

This is probably the crux. "Math does not have to be practical". — Metaphysician Undercover

Maybe we are using 'practical' differently. I mean practical as having an applied benefit, like engineering better gadgets. If you consider the proof of Fermat's Last Theorem 'practical', then by your definition I think all of math is practical.

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I find your philosophy of mathematics to be quite disjoint from the act of doing mathematics. Your view does not explain equations or geometry or axioms, it doesn't provide any explanatory power. It only attempts to delegitimize powerful and proven tools (e.g. calculus).

Your ontology is weak. You say a table is one, yet it has 4 legs and a top piece. What number of parts do these have? This process is infinite and it takes a delicate balance to understand all it's intricacies. Motion passes through infinity and the finite, but you want to reduce the question to Aristotle's lame argument,: namely that parts are only potentially there. Bringing in QM isn't going to help your case mr. idealist. The world is real. "Ignore the world and the world will come to you" — Gregory

Newton's world was one where the present took the center stage, time marched forward instant by instant, and everything always held a definite state.

Einstein's world is one where simultaneity and the present are relative. This view challenged presentism, making eternalism and the block universe (i.e. the whole universe spanning all time and space) an attractive view.

Bohr's world is one where objects only have definite states when they're measured.

My views does not ignore reality, rather is agrees with the latest developments in physics. Perhaps you're stuck in a classical world, a world plagued by singularities. It at least seems that way given your affinity for infinity. As it is when we grapple with singularities, I doubt that you have a compelling solution to Zeno's Paradox.

Your question about a table seems silly. If you want to talk about 'the whole' you have to talk about the wave function of a whole system, not a table. If you don't like the word 'potential existence', would 'superposition' be more appealing to you?

Consider this: in between any two distinct states of a system lies an unobserved wave function. If we continue to observe the system it will not evolve to a different state (Quantum Zeno Effect). Motion happens when we are not looking. Why are you assuming (as Newton did) that everything always has a definite state?

even pi can only be understood as part of a finite number — Gregory

I'm not sure how to respond to most of your post, but as for pi we typically understand it precisely using some algorithm described with finite characters.

We already discussed the difference between the rule ("program" in this case) which sets out, or dictates the process, and the process itself. If the process is interrupted, it ends, and is therefore not infinite. The rule ("program") is never infinite, nor is it potentially infinite, it's a finite, written statement of instruction, like "pi", and "sqrt (2)" are finite statements, even though they may be apprehended as implying a potentially infinite process. — Metaphysician Undercover

Hmm, maybe I wasn't consistent, let me try again.
-The aforementioned program is written with finite characters.
-The execution of the program is potentially infinite.
-The complete output of the program would be actually infinite (if it existed).
-The program spits out numbers as it is being executed, so it doesn't need to be terminated to get something useful from it.
-We can discuss the execution of the program without ever running it (e.g. we can say 'if I executed the program, it would be potentially infinite)

In the end, I think you're splitting hairs here. What's your point?

I don't see how you can say this. Pi says that there is a relationship between a circle's circumference and diameter. This information is totally useless if you do not proceed with a truncated version of the seemingly infinite process, such as 3.14. "The solution to the problem is pi" doesn't do anything practical, for anyone, if you cannot put a number to pi. — Metaphysician Undercover

π is often written as the solution to a problem - for one it's what they say is the volume of Gabriel's Horn. And there are many cases where π gets cancelled out, for example cos(2π)=1, so there's no need to evaluate it. Also, who said math had to be practical?

That leaves the question, when there is no interval of time, is it meaningful to speak of time or anything that requires time. — tim wood

We can certainly talk about a system at a particular instant of time. For example, we can talk about the time on our clock, and the position and shape of objects at that instant. What we can't talk about is rate of change.

If it makes sense to speak of a time as a "time" when there is no motion, then that same notion applied to a car implies that it's spending most of its time, even while in motion, not in motion. — tim wood

Let's say that every instant has a corresponding number on my stopwatch. I believe that the numbers on my stopwatch can only ever be computable numbers. Since traditionally we say that the computable numbers are countably infinite, the duration at which the car is not in motion has measure 0. Therefore the car is spending all of it's time in motion. What you must appreciate is that there is 'something' in between the instants and it is during this 'something' when the car moves.

I've looked into constructivism a bit, which is enjoying a modern resurgence due to the influence of computer science and computerized mathematical proof assistants. But you would reject even that. — fishfry

I certainly like the idea of constructivism in that it is necessary to construct a mathematical object to prove that it exists, and my understanding is that constructivists reject actual infinity. However, I haven't investigated the details of constructivism to know where my views differ. It's on my to do list....and as of now, so too is the Peirceian viewpoint.

But why am I supposed to be burdened for finding a mathematical viewpoint other than the standard one accepted by almost all the world's mathematicians except for those pesky constructivists and rare finitists and ultrafinitists? Why is this my problem, or math's problem? — fishfry

This is a great question. Nothing is your problem unless you want it to be. But here's my response to your 'all the world's mathematician's' comment: the world's mathematicians are all climbing up the 'point-based' mountain (and certainly doing good work) but it's a competitive space and (I think) most contributions these days are in highly specialized niche areas. I'm here pointing out that there's another place to climb. I can't promise that it's a mountain (but it might be) but I can say that very few people are climbing it. What's the harm in taking a quick look?

I suppose I owe you that. I'm going to find it depressing and frustrating but I'll take a run at it sometime. — fishfry

...ah okay, so the harm is potential depression and frustration. I'm sure it is frustrating talking math with a non-mathematician, but don't be so certain about it being depressing. There's a chance that you'll like the ideas.

Why isn't Aristotle's solution just circular because it makes the results of a mathematical construction prior to the construction itself? — magritte

I'm not a physicist but this is my understanding of how our world works. A wave function of the universe (describing potential/probabilities) exists spanning all of time. When a measurement is made, that wave function locally collapses to a distinct state. So we start with a probabilistic description of the universe which exists prior to the individual moments.

The construction itself certainly must come first. What we disagree on is what that construction is. You see the individual measured points in time being fundamental while I see the continuum wave function being fundamental. Zeno's paradox highlights the weaknesses in your point-based view. In short, we can't build a timeline from points in time.

If I give up points as bounds, then how would I have anything but an endless line? — magritte

We don't need points to bound the open interval (-∞,+∞).

There is an issue of truth here. There is something there causing the form, and the concept of "field" attempts to account for whatever it is. If the concepts employed are inadequate, then it's not true to say that this is what is there. — Metaphysician Undercover

Sure, physicists could be wrong but that doesn't mean you should stick with outdated information (which could also be wrong).

If the process is terminated then it is untrue to say that it is potentially infinite. — Metaphysician Undercover

A program written to spit out the natural numbers one at a time is potentially infinite, regardless of whether it's been executed or interrupted.

And if we know that in every instance when such a process is useful, it is actually terminated, then we also know that it is false to say that a potentially infinite process is useful, because it is only by terminating that process, thereby making it other than potentially infinite, that it is made useful. — Metaphysician Undercover

If you have ever seen π as the solution to a problem (instead of, say, 3.1415) then the process hasn't been terminated, it hasn't even been initiated. It's incorrect to say that potentially infinite processes are only useful when prematurely terminated.

We measure the car at 60mph and maybe that's accurate to within a small margin of error. But at no point during its 60 mph run is its speed zero. — tim wood

There are two ways to describe the car's motion.
1) Over any non-zero interval of time the car's average velocity is 60mph.
2) At any instant the car's instantaneous velocity is 60mph.

I believe that the latter description does not make sense since velocity is a measure of change over time. But this doesn't imply that I believe that there is some interval for which the car's average velocity is 0mph.

It seems to me that with any line I look at I'm looking at an infinite number of points. Not potential points, but actual points - that is, to the degree that one, or any, point is actual. — tim wood

This is how we were taught to think, but is it possible that you're just looking at a line? Do you think it's even possible to see a single point? If not, what about 2 points? 3? If something magical happens at infinity when they form a continuum, why not skip the magic and just start with a continuum?

Just out of curiosity though, how do you develop a theory of the real numbers without infinite sets? Even the constructivists allow infinite sets, just not noncomputable ones......The problem with finitism is that you can't get a decent theory of the real numbers off the ground. — fishfry

I think this is the wrong question to ask. We know what we want (a consistent foundation for calculus) and we think we know how to get it (with real numbers and infinite sets). But real number might not be the answer.

I think a much better question to ask is 'can we build a consistent foundation for calculus without real numbers and infinite sets?' I believe the answer is yes. Now, this doesn't imply that the alternative requires scrapping everything about real numbers. For example, I believe the alternative would have still have the area of a unit circle being pi, it would just mean something different.

If I see a coherent one presented I'll engage with it. In the past I've engaged extensively with constructivists on this site and learned a lot about the contemporary incarnations of that viewpoint. I've also studied the hyperreals of nonstandard analysis. So in fact I'm very open to alternative versions of math, but I don't see that you've presented one. — fishfry

Like the real numbers, the fundamental objects of the hyperreals are points/numbers. A continuum is constructed by assembling infinite points, each with a corresponding number. And so the hyperreals gets no closer to answering the question 'how can a collection of points be assembled to form a line?' I haven't investigated the constructivists' methods to produce an informed comment but from what I've seen it looks like more of the same. Points/numbers are treated as fundamental.

I don't have a formal theory but I do have an intuition on how the alternative would look. In line with Aristotle's solution to Zeno's Paradox, my view has continua (not points) being fundamental. I think the mathematics of calculus would be almost entirely unaffected in moving to a continuum-based view, but before we could even talk about it, you'd have to first be open to (at least temporarily) shedding your point-based biases and look at graphs in a new light. I would love to hear your criticisms on my earlier post to you where I drew the polynomial in an unorthodox way.

It's a useful concept, especially in an applied sense like engineering.

I'm not proposing we ban Infinity altogether. I'm proposing that we restrict ourselves to only use Infinity in a potential sense.

What is real and fundamental in quantum physics is the points where particles appear. — Metaphysician Undercover

I believe physicists think particles are only states of quantum fields. If so, you should not be thinking of the point where a particle appears, but instead the continuum where the quantum field exists.

The so-called "underlying quantum fields" are models produced from observations of particles, and are meant to model the interactions of particles. — Metaphysician Undercover

If you place iron filings over a magnetic field the filings will take a form in line with the field. While it's true that we only see the filings, it is untrue to say that the field is just a model. It's real. The same goes for quantum fields.

So if we turn to study that rule, should we not put our efforts into avoiding this rule, making it so that the rule never comes up, because it's like a trap which the computer will fall into? Therefore instead of pretending to be having success at carrying out infinite processes, which is self-deception, we should be looking at ways to make sure that such rules are banished. — Metaphysician Undercover

No. If we terminate the potentially infinite process we still get something useful (e.g. the rational approximation of pi on your calculator is a useful button). Also, I would argue that calculus is the study of these 'rules' and calculus is arguably the most useful branch of mathematics. What I agree with you on is that we should not try to carry out the infinite process to completion...that is a fruitless endeavor. 

Not all contemporary math. — jgill

You're right, I shouldn't have generalized.

You're approach is typology but you haven't said anything about the system works. (Topology says how you get results)

An object is bounded by points and a finite surface area. This is how continua is defined. The infinity is in the paths within these bounds, because parts, motions, and paths are uncountably infinite with it — Gregory

Perhaps I am misusing terms. Certainly, when I say graphs are topological, I don't use it in the standard sense of the word topology. Instead, I'm using it in a looser sense, simply that the properties (of interest) of the graph are maintained through continuous deformations. The results are largely unchanged in reinterpreting graphs as topological objects (in the sense of topology mentioned above).

As for continua, I simply mean objects with extension.

The arrow moves through any point with forward velocity so it's never ever at rest — Gregory

This is challenged by Zeno's Arrow Paradox.

You are making a claim about reality (i.e. it's made of events of information). Aristotle slumped into this when he said parts are potential. What exists is the whole composed of all it parts, which are bounded by points (finite) and limit in space (finite). A material body doesn't have math in it. We use imperfect mathematical formulations to understand to described in the field of physics. You can't draw philosophical conclusions from physics is the conclusion. You fell for the Parmedian world view by trying to figure out the logic of his disciple — Gregory

The whole need not be bounded by points. Think of the open interval (-1,1). I think physics can and should inform our philosophy and you're right that I'm influenced by Aristotle and Zeno.

The Zeno effect and the anti-Zeno effect refer to how observation changes eternal states. The ancient Arrow paradox is just used to illustrated the effect and the effect does not resolve the Arrow paradox because its not specifically related to it — Gregory

I haven't looked for a reference but I assume they named the Quantum-Zeno effect because of the Zeno's Paradox (especially Zeno's Arrow Paradox). If so, I don't see how it's unrelated.

I'm agreeing that IN REALITY there may not be such a thing. But in math, there most definitely is. — fishfry

My view is that actual infinity should not be permitted in math any more so that than it is permitted in physics, but that's just my view and it's contrary to contemporary math so I'm willing to leave it at that.

You can't maintain your credibility while arguing against freshman calculus. here's the proof "somewhere," a somewhere I already linked to earlier. — fishfry

The sum is defined as $s=a\frac{1-r^{n+1}}{1-r}$

There is a difference between substituting a finite number in for n and substituting infinity in for n. Obviously the latter is not allowed, so we talk about convergence instead. If we ignore the fact that we're talking about convergence when talking about the sum of an infinite series then we are essentially substituting infinity in for n. It is a matter of convention that we say that geometric series sums to a number. More formally, we should say that it converges to a number. I think 0.333... converges to 1/3 is a valid statement.

You know you keep making claims totally contrary to photographic technology.....There are no single-photon detectors outside of physics labs. But again, I don't know why you're belaboring this point. If you don't think that 1/3 = .333... AND you agree that you are making a mathematical point, then there is no conversation to be had. — fishfry

Ha! I knew you were going to say this. That's why I had switched to a quantum sensor. With an SLR camera I agree that every photo has some degree of blurriness, but with quantum sensors that's not necessarily the case. There is no law which states that we can't know the position of a particle with perfect precision. Also, I don't think particles are points, but instead excited states of quantum fields.  

I don't think we're having the same conversation anymore. — fishfry

Forget about cameras, sensors, and speedometers - it all boils down to the question of whether a line can be constructed by assembling points. Your earlier post indicated that you agree that this is a mystery (given the orthodox views)? Why not consider alternate views?

What is real and fundamental in quantum physics is the points where particles appear. — Metaphysician Undercover

My impression is that you're a finitist, so I presume that you believe our universe had a beginning of time. If particles are fundamental, they must have existed at that initial moment, right? Were they concentrated at a point? I take it that you think a measurement involves the interaction of particles, so at the initial instant wouldn't they all be measuring each other? If so, how would they ever move, given the quantum Zeno effect?

Consider this: "QFT treats particles as excited states (also called quanta) of their underlying quantum fields, which are more fundamental than the particles." source
 

You can study that rule, but you cannot study the process dictated by that rule, because it does not exist. — Metaphysician Undercover

I think you're splitting hairs here. By rule I assume you mean the 'computer program' and by process I assume you mean 'the execution of the computer program'. If so, then we are in agreement, we can study the rule (i.e. the computer program). 

The tangent line IS defined as the limit of the secant (for a function from the reals to the reals). — fishfry

Are tangent and instantaneous rate of change not the same thing? If you reject the notion of instantaneous rate of change, how can you not reject the notion of tangents?

Are you talking about Mathematics or some kind of notion of reality that goes beyond math? — fishfry

I'm talking about mathematics. I understand that 0.333... converges to 1/3, but it is only (a useful) convention which states that convergence and equality are the same. If you're saying that it's proved somewhere that the two terms are equivalent then let's leave it at that.

I don't understand why you like one infinite representation rather than another, but you are riding a hobby horse and making no rhetorical points with me at all. You're wrong on the math and confused on the metaphysics. — fishfry

The Stern-Brocot string for any rational number has finite characters. I don't accept the claim that that LL = LLLRrepeated since LL corresponds to a position in the tree and LLLRrepeated corresponds to a path along the tree. However, let's not waste any more time on 1/3.

Why are you saying this? I assume you must know it's wrong, no mechanical device is capable of exposing a light-sensitive medium for a true instant. Are you speaking metaphorically? If you set your hypothetical camera to an instantaneous shutter speed no photons could get in and the image would be blank. — fishfry

As with your speedometer argument, your addition of a shutter only complicates the issue without providing any further explanatory power. We add a shutter to cameras only to limit the amount of light that the film is exposed to. Without a shutter we can (at least in principle) still take photographs. Remove the shutter and the instant the first photon hits the film we have a photograph. And if no further photons hit the film we have an image with absolutely no blurriness. This image captures no motion. It is not a video by any definition. I was anticipating you challenging the practicality of such a photograph, which is why I went quantum, but perhaps that's not necessary.

video is nothing more than a series of stills, whether analog or digital frames. — fishfry

I understand how videos and flipbooks work, and yes we use stills to create them, but the magic ingredient which you are ignoring is time (specifically non-zero intervals of time). We hold each frame for 1/24 seconds before advancing to the next frame.

We're not going to solve Zeno's paradoxes here. — fishfry

(I believe) Zeno's paradox was already informally solved by Aristotle. I'm just defending his view. And loosely speaking the view is simply that we start with videos, not stills. With videos we not only can capture motion but we can also pause the video to extract a still. With only stills, motion is not possible, as per Zeno. Zeno's paradox is important and seen as unresolved because the notion of stills being fundamental is deeply rooted in our beliefs.

Are you just sharing a nice quote from Hawking, or is this supposed to support your argument? If the former, thanks! If the latter, please provide a little explanation.

"Quantum indeterminacy is the apparent necessary incompleteness in the description of a physical system" [ source ]

Doesn't that sound a little like Gödel's incompleteness theorems?

the square root operation is closed over real numbers — TheMadFool

I can accept that the square root operation is closed over the 'reals', but that doesn't mean it's closed over the real numbers. I don't deny the value of irrationals or question the centuries of work on calculus, I'm just questioning whether we actually know that irrationals are numbers.

The op is asking a philosophical question about the existence of certain mathematical objects, not whether those mathematical objects have mathematical existence. — Metaphysician Undercover

A few have pointed to my original question so let me try to clarify. My view of existence encompasses mathematical existence. While I have doubts about the existence of irrational numbers, I don't doubt the value of our descriptions of irrational 'numbers' (e.g. Dedekind Cuts, Cauchy Sequences, infinite series, and so on). I know this seems like a contradictory statement since conventionally they're all equivalent. But I see an infinite series as an algorithm (described completely with finite characters) which if executed to completion would output an irrational number (described completely with infinitely many characters). Since that output could never be generated, the output (the number) cannot exist but the algorithm certainly can, for example on my laptop. This is why I think the number 3 can exist but not the 'number' sqrt(2). We never actually work with irrational 'numbers', we only work with their algorithms or rational number approximations. So why do we even need to assume that irrational 'numbers' exist? Why not assume that irrationals are the algorithms that we actually work with?

I think one issue is that some non-computable irrational 'numbers' don't have algorithms...

Discrete curves? — Gregory

No, continuous curves.

Insofar as the car is moving and never while it is moving not moving, then any method of description that stops it is simply not reflecting reality, but maybe if anything, something other than reality. — tim wood

What you don't realize is that it's your description which stops the car from moving. That's Zeno's paradox. Motion is impossible if time is just a collection of instants. My description allows the car to move because I'm allowing for time to be more than just instants. In between the measured instants lies an unmeasured wave function within which motion is possible. Motion happens when we're not looking. It's demonstrated by the Quantum Zeno Effect.

Photographs are not instantaneous. The shutter stays open for a period of time, usually a fraction of a second.......any measurement is taken over time. There's no difference between photo and video. Video after all is just a collection of still images, either analog or digital frames. And a single photo is taken over a period of time, namely the shutter speed. — fishfry

Imagine a dark room and a quantum sensor (which I'll call a film to stick with the photography analogy). The moment a single photon hits the 'film' we have a 'photograph'. This photograph has no blurriness and captures no motion. It is a photograph, not a video. There is no physical law which states that we cannot know the position of a particle with perfect precision. It's just that if we do know the particle's position precisely, we can't know anything about its momentum (velocity). Static measurement (e.g. position) and dynamic measurements (e.g. velocity) are both valid, important, and distinct from each other.

Videos are not just a collection of still images. They are a collection of still images where each image is displayed for some non-zero duration. If each image was displayed for 0 seconds then you wouldn't get a video, no matter how many images you pile on. It all comes back to 'how can you form a line from a collection of points?' The answer is that you can't. But you can easily go the other way. You can easily cut a line to form a point. You can easily pause a video to produce a still image.
 

The person who puts one's efforts into pointing at the problems in existing systems need not be the one who produces the repair........You demonstrated that you do not grasp the need for the point to be prior to the line, therefore your claim that it would result in a weaker mathematics is based in misunderstanding. — Metaphysician Undercover

I think you are indeed pointing at the problem but when you start talking about your solution involving multiple time dimensions and spatial expansion, I find it hard to follow. It all seems like mumbo jumbo. Give me something concrete to chew on. Does your philosophy produce any graphs or equations? How does your philosophy make sense of the infinities in calculus?

What quantum physics demonstrates to us is that points have real existence, and continuities are constructed. — Metaphysician Undercover

I don't agree with this claim so I'd like to see your evidence that supports it. What is fundamental in quantum physics is the wave function, a continuum. Definite states (like points) only emerge when a measurement is made.

I don't see how you get from points to continua. — Metaphysician Undercover

I'm not going from points to continua. I'm going from continua to points. My graphs and videos aren't seeming to help here so let me expand on this quantum analogy. I'm not a quantum physicist so take this with a grain of salt.

Assume that there exists a wave function of the universe that spans all of time. This is the fundamental object of our universe and it is a continuum. And until the wave function is measured it is meaningless to talk about who lived when and where because a wave function does not describe what is, it describes what could be. It is only when you make a measurement that all of the potential states collapse into a single actual state. When I say that points are emergent, I mean that they only emerge when we make a measurement. We cannot say things like 'there are infinite points on this line' because we have not actually placed infinite points on the paper...what we placed on the paper was a line.

Now let's say that in '2+2', the '+' represents a process. — Metaphysician Undercover

Put it this way: a computer program that calculates 2+2 is what I mean by 'process' and such a program can be studied (even if the program is never executed).

Continua is infinitely pointed. So it has instants all over it. — Gregory

This is the view that we've been indoctrinated with. We can't help from thinking that points (i.e. instants) are fundamental and so we believe that continua are just a collection of points. Zeno's paradoxes reveal that this view has serious problems. Instead, consider the alternative: that continua are fundamental and points are emergent. Points only emerge when a measurement is made. When we draw a graph, we only label a few points where our curves intersect. Those are the only points on the graph. Don't look in between the curves and conclude that there exist uncountably-infinite points there...what lies in between the curves are continua.

How I see it, we need to say "the infinite" is on one side and "the finite" is on the other and motion is movement between them — Gregory

I think your intuition is close. But I would say that motion happens in the chasm of space (filled with infinite potential) in between "the finite".

Idn. I've been recently working on this question from the angle of non-Euclidean geometry. I'm trying to understand what space even is — Gregory

When you draw a graph, do you think you actually draw infinite points placed perfectly on the page? Or do you place a few points on the page accurately and then imprecisely connect the dots. What I am proposing is in line with how we've always drawn graphs. And if we're honest with ourselves, it's the only way to draw graphs. It would take an infinite being to draw a graph with perfect precision. Let's not make math dependent on the existence of an infinite being.

A car traveling 60 mph down the road. Is anyone here going to suggest that at any time, however defined, that car is not moving, or, that there some time, some moment, when it is by no test whatsoever distinguishable from a parked car? The moving car is a reality. — tim wood

At every time interval we will find the car moving. At every instant in time the car's motion is indistinguishable from that of a parked car. It's Zeno's Arrow Paradox. Did you know that if a quantum system is continuously being observed that it will not evolve? It's called the Quantum Zeno Effect. As confirmed by experiment, motion (i.e. change) only happens when we're not looking. It happens in between the measurements. It happens in between the measured instants in time. And what exists in between the instants? - continua.