I get this line from your reference to Hawking's talk. It says a lot in few words.. So much and so quickly that most won't recognize what just went by. But imo worth repeating here.

"A qualitative understanding of the laws, has been the aim of
philosophers and scientists, from Aristotle onwards. But it was
Newton's Principia Mathematica in 1687, containing his theory of
universal gravitation, that made the laws quantitative and precise."

From qualitative to quantitative. From hard or squishy, moist or dry, warm/cool, larger/smaller, bright/dull, & etc. - from that to numbers. Perhaps it was the constraints of the circumstances of the talk itself that let Hawking slight Galileo here....

But you apparently missed this line:
"and the analogy of Gödel’s theorem."

I.e., Godel is for mathematicians.

That the car was "going at a constant speed" is just an assumption, so it may not be the truth of the matter. — Metaphysician Undercover

Are you gong to argue that the car is not moving at any speed during its traverse of the distance A to B? Because it would seem to be the case that notwithstanding variations in speed (itself something to be defined) it must be moving at some speed.

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.

But I am going to ask you to write something - anything - that is true. — tim wood

Sorry tim, but if I already knew the truth, then I wouldn't be looking for it, would I?

Are you gong to argue that the car is not moving at any speed during its traverse of the distance A to B? — tim wood

Yes, that's about it. Speed is what we assign to the car, it is what we say about it, it has speed. In philosophy we must maintain the distinction between what we say about the thing, and what is really the case, to allow for the real possibility that what we say about the thing might actually be a falsity. If the property which we assign to the thing, "speed", in this example, has faults within its conception (contradictions for example), then despite the fact that it has become acceptable to say this, the concept is defective, and it is really not true to be attributing that property.

This is the point. When we use math to figure out things like instantaneous velocity, the volume of a supposed infinitely small tube, etc., — Metaphysician Undercover

This is exactly where you go off the rails. There is no "infinitely small tube." This is your ignorance speaking again.

it is implied that we know things about reality which we do not. This is a falsely supported certitude. — Metaphysician Undercover

Nobody is making any claims about reality. Gabriel's horn is a strictly mathematical example.

If the property which we assign to the thing, "speed", in this example, has faults within its conception (contradictions for example), — Metaphysician Undercover

What contradiction? 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. And yes, care to distinguish the description from the thing described, but that too within a small margin of error. The only contradictions that can arise are in the description - the thing both is and is not as we describe it. Or observation; the thing is observed to behave in contradictory ways. So which, and how?

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

Are tangent and instantaneous rate of change not the same thing? — Ryan O'Connor

No they're not. In functions from the reals to the reals (ignoring the multivariable case where this terminology doesn't make sense) the instantaneous rate of change is the slope of the tangent line.

If you reject the notion of instantaneous rate of change, how can you not reject the notion of tangents? — Ryan O'Connor

I don't, you and @Meta do. I'm agreeing that IN REALITY there may not be such a thing. But in math, there most definitely is. Again. you are equivocating math and physics. I'll stipulate to @Meta and your point that instantaneous rates are murky in physics. In math they're perfectly well defined.

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. — Ryan O'Connor

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

https://en.wikipedia.org/wiki/Geometric_series

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. — Ryan O'Connor

On the contrary, 1/3 is the critical case here. If you can't agree that 1/3 = .333... then you fail freshman calculus and high school algebra too. There's nothing more to talk about then.

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. — Ryan O'Connor

You know you keep making claims totally contrary to photographic technology. A single photon is not sufficient to make an impression on any film stock or digital sensor in existence. So you're just flat out wrong here. And if you don't open shutter at all, no photons will come in; and to get sufficient photons in to make an impression on a light-sensitive medium, you need to keep the shutter open for a period of time. Of course I include an electronic shutter that merely activates the sensor for a period of time.

And if no further photons hit the film we have an image with absolutely no blurriness. — Ryan O'Connor

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. You're just wrong. Read a calculus book or work through the proof I linked (twice now) on Wiki.

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. — Ryan O'Connor

I see no benefit to this point at all. What of it?

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. — Ryan O'Connor

Then why did you give the impression you didn't?

(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. — Ryan O'Connor

I don't think we're having the same conversation anymore. But regarding your claim about video, I invite you to wave your arms in front of your webcam while recording, then play it back frame-by-frame. You'll see motion blur. I know this because I've seen it many times. In fact if you are making a video with the intention of capturing still frames you have to make sure to stay motionless for a few moments at a time to avoid motion blur.

Zeno's entire concern briefly demonstrated:

To go from A to B you have to go half that distance, since the distance is distance as space. Half of AB is a distance, otherwise you are at the half point instantaneously. So he must go the quarter. But the quarter is spatial it too has a half point mark, ... And a computer can run this activity to an uncountable infinity. There is no paradox as to how motion starts from energy. The question is how it is that the supertask is done in finite bounds (time and space)

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). 

A particle is not the system. It passes through the system

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

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?

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.

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. — Ryan O'Connor

Not all contemporary math. In complex analysis one may move to the Riemann sphere and take the north pole as infinity, but I stick to the complex plane and use expressions like "unbounded" instead. Now, set theory is another animal altogether. Not my cup o' tea, but fishfry is an excellent in-house expert.

Not all contemporary math. — jgill

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

We measure the car at 60mph and maybe that's accurate to within a small margin of error. — tim wood

I said "faults", and I used "contradiction" as an example of a fault. That there is a "margin of error" is another indication of fault. When a small margin of error is ignored or neglected, as if it doesn't exist, one can fall for a paradox like Zeno's, where that small margin of error is infinitely magnified to produce the appearance of contradiction.

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? — Ryan O'Connor

I really don't get your question. I was talking about points, not particles, so your question has some underlying presumptions which I don't follow.

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

I addressed this already. The so-called "underlying quantum fields" are models produced from observations of particles, and are meant to model the interactions of particles. It is implied that there is an underlying substratum which validates this modeling, but the modeling itself, the quantum field theory, does not represent the underlying substratum, it represents the interaction of particles. Until we get accurate and precise modeling of the particles any speculation concerning the substratum is not well informed.

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). — Ryan O'Connor

OK, so we say that the rule calls for the computer to carry out an endless, or infinite process. We know that the computer cannot succeed in carrying out this request, because it will wear out first, so all the time spent will be wasted, for the computer to be trying to carry out a process it can't. 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.

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. 

Infinity creates a situation in this question about space that make the question of continua difficult (The reason being that discrete apace is an oxymoron). However in numeral mathematics infinities work ok. Paul Cohen found contradictory proofs in infinite mathematics in the 1960's but the subject simply is not understood properly enough. Maybe a theory of everything which provides the connection between discrete apace (a point) and finite geometry (solids) be found. But nonetheless banishing infinity from mathematics is a move of an ostrich

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

I said "faults", and I used "contradiction" as an example of a fault. That there is a "margin of error" is another indication of fault. When a small margin of error is ignored or neglected, as if it doesn't exist, one can fall for a paradox like Zeno's, where that small margin of error is infinitely magnified to produce the appearance of contradiction. — Metaphysician Undercover

You want to maintain clarity as to the distinction between description/observation and the thing described/observed. Good! How fast is our car traveling - the one we describe/observe as maintaining to within the limits of accuracy a steady speed of 60 mph?

And a point about Zeno I do not see you mention. If Achilleus, or the Arrow, were to stop at each of the waypoints, then that would be a problem. But he/it doesn't, so the issue of passing particular points is no different from passing any point, and yet all those other points are never mentioned. Why is that, do you suppose? Achilleus - or the Arrow - seems to have no problem whatever passing those. Zeno's then, just an entanglement with words.

I had my weekly chat with an old colleague (math prof) living in a retirement home today and brought the subject of instantaneous velocity up. He has his doubts about the existence, as do some of you, and for the same reasons. It doesn't bother me either way. I'll use the term since others will know what I am talking about: a certain mathematical limit.

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. — Ryan O'Connor

And what the heck does that mean? 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.

How many grains of rice in a barrel? How many of wheat in a wheat field? The point is that we set eyes on very large numbers of things all the time. Granted not the aleph-C of the number line. But I far more easily take in a one inch line than the limits of a several thousand acre wheat field. You say I don't actually see all the numbers? I certainly do not see all the grains. Is that your test? Which point don't I see - assuming points can be seen? And even if they cannot be seen, that does not make them not actual. And that whether you take them as on the line or only in the mind that puts them there.

You seem to be confusing senses of existence and actuality.

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. — Ryan O'Connor

I think we're done then.

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.

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. — Ryan O'Connor

[/quote]

We're just not having the same conversation.

Forget about cameras, sensors, and speedometers - it all boils down to the question of whether a line can be constructed by assembling points. — Ryan O'Connor

The question doesn't come up in math. We use the phrase "the real line" as an alternate way of saying, "the set of real numbers," but everything can be done without reference to geometry.

Your earlier post indicated that you agree that this is a mystery (given the orthodox views)? Why not consider alternate views? — Ryan O'Connor

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 nonstanard analysis. So in fact I'm very open to alternative versions of math, but I don't see that you've presented one. The problem with finitism is that you can't get a decent theory of the real numbers off the ground.

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

"The philosopher and theologian are conscious of infinity, but from the mathematician's view they do no use it so much as admire it. The mathematician also admits infinity; the great David Hilbert said of it that in all ages this thought has stirred man's imagination most profoundly, and he described the work of G. Cantor as introducing man to the Paradise of the Infinite. But the mathematician also uses infinities..." Leo Zippin

"Every since we first sought number in the object, the series of numbers has begun with 1. Making zero the first of numbers means no longer abstracting them from the object" Jean Piaget

"The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality." Hermann Minkowski, 1909

"At a time when Minkowski was giving the geometrical interpretation of special relativity by extending the Euclidean three-space to a quasi-Euclidean four-space that included time, Einstein was already aware that this is not valid, because it excludes the phenomenon of gravitation. He was still far from the study of curvilinear coordinates and Riemannian geometry, and the heavy mathematical apparatus entailed"
https://en.wikipedia.org/wiki/Pseudo-Euclidean_space
https://en.wikipedia.org/wiki/Minkowski_space#cite_ref-14

https://www.quora.com/How-can-I-a-non-mathematician-wrap-my-mind-around-the-Axiom-of-Choice: The reason the Axiom of Choice is (somewhat) controversial is that while it allows us to prove some very useful mathematical statements, it also allows us to prove some less intuitive statements (e.g., the Banach-Tarski paradox).

And finally:

"As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality." Einstein

(Don't forget Kant's second antimony: (On Atomism)
Thesis:
Every composite substance in the world is made up of simple parts, and nothing anywhere exists save the simple or what is composed of the simple.
Anti-thesis:
No composite thing in the world is made up of simple parts, and there nowhere exists in the world anything simple)

If we can rearrange volume and thusly double the volume
And we can paint an infinite space with finite paint,

Then it's clear humans struggle with the union of mass and volume

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. — Ryan O'Connor

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. Here's an example. The ancient Greeks used circles to model the movement of the planets, and Aristotle proposed that the orbits were eternal circular motions. It turned out that these models were wrong, therefore it was not true for them to have been saying that the orbits were circles even though this concept was employed and enabled prediction.

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). — Ryan O'Connor

Again, there is an issue of truth here. If the process is terminated then it is untrue to say that it is potentially infinite. 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. Therefore t is false to say that the potentially infinite process is useful.

But nonetheless banishing infinity from mathematics is a move of an ostrich — Gregory

No, the opposite is the case. Ignoring the fact that infinities in mathematics is a very real problem, is the type of ignorance which is analogous with the ostrich move.

But he/it doesn't, so the issue of passing particular points is no different from passing any point, and yet all those other points are never mentioned. Why is that, do you suppose? Achilleus - or the Arrow - seems to have no problem whatever passing those. Zeno's then, just an entanglement with words. — tim wood

What do you think "passing a point" means? Do you mean to say that there are physical points out there, which the arrow can be seen flying by? If so, then you ought to be able to show empirically, the physical existence of such points, and I don't think there will be an infinity of them. If these points are just imaginary, then the arrow doesn't really fly by them and you have created a false scenario, by describing the arrow as flying by points.

I propose that the truth is that the points are imaginary. If this is the case, then any method of measuring motion, velocity and such; which employs points, is really giving us a false measurement. We might be able to find real physical points, which if they exist, would validate such a method, but then these points would not be infinite, so that scenario with infinite would become irrelevant, because we'd have to make a new method of measuring velocity based on empirically verified points. As I explained to you earlier, this is pretty much what relativity theory does, but each empirically verified point turns out to be a different frame of reference, and that the points are at rest relative to each other is very doubtful due to the observed phenomenon of spatial expansion.

Aristotle said infinity is "one". Modern mathematics say it's two (countable, uncountable) with perhaps subdivisions. A Philosophy Overdose audio on youtube said that Paul Cowen proved and then disproved that there is an infinity set between the two of Cantor. This is a challenge for mathematicians and they are not going to listen to a philosopher like you

Daoism at its founding taught that being and non-being create each other. Such dialectical logic is all over philosophy. Yet mathematics is a field that does not use such ideas

https://www.harinam.com/tao-te-ching-verse-2-being-and-non-being-create-each-other/

"Paul Cowen proved and then disproved that there is an infinite set in between the two of Cantor"

The continuum hypothesis has arguements for and against it too, as can be seen in the WIkipedia article on it. Organizing these ideas into something consistent is going to take work by hundreds of mathematicians and many many years of toil

Philosophy says that a point is a negation of a line, but if the points are ordered they create a double negative, and thus the positive of the line. The line does the same with the solid, and the solid in turn abrogates what comes before and creates dimensions greater than three. Philosophy and mathematics come onto these questions from very different perspectives and we philosophers can't expect our explanations and demands to be accepted by those with high math skills