What do you make of 1/3 = .333...? Can't you distinguish between a number and one of its representations that you don't happen to like? After all 1/3 is just a shorthand for the grade school division algorithm for 3 divided into 1. — fishfry

I believe that's a false equality. The correct statement is "the potentially infinite process defined by 0.333... converges to the number 1/3" not "the number 0.333... equals the number 1/3". Decimal notation is flawed in that it cannot be used to precisely represent some rational numbers, like 1/3. If we want a number system which can give a precise notation for any rational number, we should use Stern-Brocot strings, where 1/3 = LL.

Yeah yeah. One of Zeno's complaints. If you look at the arrow at a particular instant it's not moving. How does it know what to do next in terms of direction and speed? Not a bad question actually, one that I won't be able to answer here. — fishfry

If photographs can't capture motion but videos can, why not conclude that motion happens in the videos? The reason why we are reluctant to come to this conclusion is because we reject the notion of videos being fundamental.

We want points (photographs) to be fundamental and continua (videos) to be composite and as long as we hold this view we will not find a satisfactory resolution to Zeno's paradoxes. If you flip things upside down and see continua as fundamental and points as emergent, then everything makes perfect sense. There's no problem with pausing a video to produce a static image.

Suppose there were such a thing as an instant of time, modeled by a real number on the number line. Dimensionless and with zero length. So the arrow is there at a particular instant, frozen in time, motionless. Where does its momentum live? How does it know where to go next? Does it have, say, "metadata," a data structure attached to it that says, "Go due east at 5mph?" You can see that this is problematic. — fishfry

It is clear that you appreciate the profoundness of Zeno's Paradox. Zeno presented these paradoxes in response to the criticisms from the 'one from many' camp calling his views ridiculous. Why not consider the 'many from one' view that he supported? He was wayyy ahead of his time so his view did seem to have problems of their own...but in light of modern advancements in physics his view no longer seems crazy.

It still has nothing to do with what I originally said, which is that you don't need calculus to determine the instantaneous velocity of a moving object. And I'll concede that by instantaneous I only mean "occurring over a really short time interval." I have to say I'm not nearly as invested in this point as the number of words written so far, I should probably stop. — fishfry

Don't stop here, you may just be on your way to becoming a crank! With this admission you have placed yourself on a slippery slope. Instantaneous velocity is no different from the tangent of a function at a point. Do you accept that the derivative corresponds to a limiting process of secants rather than the output of a completed infinite process (i.e. tangent at a point)?

An object moves with constant velocity. Does it have a velocity at a given instant? — fishfry

Only if you consider 0/0 a valid velocity.

I'm not looking for people to buy in, I'm looking for truth. If others are looking for the same thing, they might like to join me. — Metaphysician Undercover

Never in the history of mathematics or physics has discovering truth set us backwards. The fact that your philosophy would result in a weaker mathematics is a red flag that you're on the wrong track. Don't get me wrong, I agree that there's a problem, I just don't agree with your resolution.

I made this video on my proposed resolution to Zeno's Paradox. What do you think?

So we ought to conclude that "objects" and "processes" are distinct categories. — Metaphysician Undercover

I think you're taking my words too literally. Clearly, the process of taking my dog for a walk is not an object with mass and momentum. When I say that processes are valid objects of mathematics, I simply mean that they can be studied in themselves, just as one might write a book entitled 'The Art of Dog Walking'.

Continua based constructions are based on an uncountable infinite amount being manipulate to a finite result. That is new. That's exactly what Newton did.

I don't think you solved Zeno's paradox because you're putting the infinite quantity into philosophically blurry box and focusing just on finite results. Zeno said "the finite" and "the infinite" we're inherently exclusive of each other and so he concluded that idealism was true (a form of idealism wherein there are no 1s and 1s to be added into two)

I don't think you solved Zeno's paradox because you're putting the infinite quantity into philosophically blurry box and focusing just on finite results. — Gregory

Isn't that what we do with quantum mechanics? We have finite results corresponding to our actual measurements and everything between the measurements is a 'blurry' superposition? Why must everything be 'sharp'?

That's just how math is

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[in reference me me asking why everything must be sharp] That's just how math is — Gregory

It's true that in my video I showed a blurry curve in between the measurements, but prior to that I showed the curve being 'topological' [ I use 'topological' only in the sense of the graph having properties which are preserved through continuous deformations]. If instead of saying that I'm 'blurring' the graph, what if we say that I'm reinterpreting it from being a geometric object to a topological object? With this view, your argument can't simply be 'that's just how math is' because topology is certainly acceptable math.

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

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.

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.

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Continua is infinitely pointed. So it has instants all over it. If things move at the top level but fundamentally unmoved in subdivisions an object can't move at all. 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

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".

Time for you to develop a new axiomatic system, then, that leads to "Truth". — jgill

It's not time for me to do that, I'm not a mathematician. There's something called the division of labour. The person who puts one's efforts into pointing at the problems in existing systems need not be the one who produces the repair. Of course the people using the system would probably not like the person pointing and would have the attitude of 'if you think you can produce a better system than ours, then do it'.

Pi is a finite number because it's inbetween 3 and 4. But if the length of a circumference is multiplied by pi than you have a length with space corresponding to each number, so the circle has infinite space within a definite finite limit (like being inbetween 3 and 4). Aristotle never understood this stuff — Gregory

At least I'm not alone then, because I haven't got a clue what you're saying.

The fact that your philosophy would result in a weaker mathematics is a red flag that you're on the wrong track. — Ryan O'Connor

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

I made this video on my proposed resolution to Zeno's Paradox. What do you think? — Ryan O'Connor

I don't see how you get from points to continua. You show measurement points, then you assume that there is some sort of continuum connecting those points. The problem I see, is if certain measurement points are actually possible, then these must be represented as real points which the moving object actually traverses and can be measured at. That's why I give priority to these points, as the real features. The supposed continuum might not have any sort of linear existence at all, in fact we might not have the vaguest idea of how the points are related to each other in the underlying substratum of reality, which produces the appearance of a continuum. For all we know, the object might appear at point A, then completely disappear, and then reappear at point B a moment later, and this is what appears to us as motion.

The reason why I say that priority must be given to the points, is that whatever it is about the underlying substratum which produces the appearance of continuity, this 'power' must be constrained by possible points of appearance. If there wasn't such constraints then we'd have the problem of infinite points where the object could be measured. Furthermore, the nature of spatial expansion demonstrates that there must be points where expansion is centered.

So I find the video mostly acceptable, but what you are really showing is a points based motion, points where the object might be measured to be at, and you are assuming that there is some sort of continuum which underlies the points and connects them. Therefore all you need to do to be consistent with my perspective, is put the points as primary, being the real constraints of real space, and allow that whatever continuum emerges from existence at the points it is a creation produced from the relationships between the points, and this set of relationships comprises the substratum.

When I say that processes are valid objects of mathematics, I simply mean that they can be studied in themselves, just as one might write a book entitled 'The Art of Dog Walking'. — Ryan O'Connor

I have doubt in the truth of this. Are processes valid objects of mathematics, or ought they be relegated to physics? Let's start with something simple, assume that a number is an object of quantitative value. So '4' represents such an object, it must be a static and unchanging value to maintain its validity, therefore it cannot be a process. Now let's say that in '2+2', the '+' represents a process. So the inquiry is whether the process represented by '+' is a valid mathematical object to be studied by mathematics. We need to determine what the '+' means. What does it mean to add one unchanging quantitative value signified by '2', to another? Mathematics does not answer this inquiry, it just makes an assumption about how processes like these affect quantitative values. And we can see the same with the other processes, multiplication, division, etc., these process affect quantitative values, but if quantitative values are what are properly referred to as objects, then these processes are something different.

the curves are continua. — Ryan O'Connor

Discrete curves?

What does it mean to add one unchanging quantitative value signified by '2', to another? Mathematics does not answer this inquiry, — Metaphysician Undercover

Uhhh

Furthermore, the nature of spatial expansion demonstrates that there must points where expansion is centered. — Metaphysician Undercover

I thought you were Aristotilean. You must be aware that Aristotle rejected points (infinitesimals) and instants

You must be aware that Aristotle rejected points (infinitesimals) and instants — Gregory

Aristotle also posited eternal circular motion, which is nonsense.

Eternal circular motion is fine. What is stupid is what you said about math not defining what addition means

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You take a snapshot of a moving car. You look at the photo and ask, "How fast was it going?"

You take a snapshot of a moving car. You look at the photo and ask, "How fast was it going?" — jgill

You know, this is more tricky than it looks. Suppose you have a long exposure time. Then you'll see a blurred image, and you can work backwards to determine the velocity. Photographs are not instantaneous. The shutter stays open for a period of time, usually a fraction of a second. During that time the film or digital sensor collects photons. So there's an element of time involved even in a photo. If the object is moving slowly relative to the shutter speed you won't see blur, but in theory the blur is always there. If I can choose the shutter speed I can always tell you how fast the object was moving by analyzing the blur.

@Ryan same point to you. In fact your earlier point is correct, 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.

Eternal circular motion is fine. — Gregory

OK then, show me this perpetual motion which you know about.

In fact your earlier point is correct, any measurement is taken over time. — fishfry

That's why velocity is always an average, requiring at least two temporal points. Duration is derived, just like distance is. To infer an instantaneous velocity requires a second derivation.

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I apologize for calling your statement stupid. I had just had a fight with someone and your comment annoyed me. It seems you are always debating fishfry or someone about numbers and there relation to Kantian synthesis vs an analytic view. To me that's just a discussion about psychology and mathematics does truly take care to define what addition, subtraction, multiplication, and division are. I have not seen where you have a unique insight into the issue. But on eternal motion, Einstein and countless physicists believed in it. Eternal inflation, the "big bounce" , and all these ideas are just noting more than versions of it. I'm very aware of Aristotle's arguments about an accidental series (one that stands on its own) and an essential series (one with supernatural support). I've discussed this with Thomists who have PhD's. There is no consensus on philosophy on this. I think it's a physics mathematica question and that calling on supernatural support is unnecessary since I can describe it in terms of physics. But this isn't the thread to go into that, since I see no connection between it and continua

Why does a segment with a length of finite digits change into a length multiplied by pi (pi×2×r) when the segment is made into a circle? The circumference will have digits going to infinity while as a segment it did not? This must be readily explained in mathematics but I don't remember ever seeing an explanation on it

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↪jgill
And you answer? — tim wood

I just brought it up as a topic. fishfry is of course correct. :smile:

The circumference will have digits going to infinity — Gregory

This is indeed a puzzle.

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).

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.
 

I don't see how QM indeterminacy can be fitted into mathematics at its foundation