Bell's Theorem

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So the accuracy of the measuring apparatus is always suited to the purpose it is designed for, and it is judged by its usefulness not for truth or falsity.
Precisely. And the purpose of the 10 million different measuring apparatuses (apparati?) is to measure velocity. So QED we are measuring velocity. And so the statement is true per CToT. We are not dealing with your metaphysical notions of truth or falsity here. And of course it is not 10 million. Duh.

Now let's move over to our acceleration issue.

I agree that for many practical purposes the use of averages is completely acceptable.
First of all it is clear that you are OK that we can measure average velocity.

But the acceleration itself, which is the cause of the body's motion has already occurred by the time the body is moving.
Acceleration does not cause anything. No wonder you are confused. Acceleration is a change in the velocity of an object. An object can undergo acceleration by being acted on by a force (F = ma) or by being affected by the curvature of spacetime.

We drop our bowling ball. After one second we determine that the velocity is ~9.8 meters per second (m/s). (I'm using the "~" here to mean average). After two seconds our velocity is ~19.6 m/s. After 3 seconds the velocity of our bowling ball is ~29.4 m/s.

Hmm something is going on here. Let's look more closely - let's chop up time a bit more finely - 10 times per second. After 0.1 seconds our ball is going ~0.98 m/s. After 0.2 seconds it's going ~1.96 m/s, etc. And lo and behold, after one second the velocity is ~9.8 m/s.

No matter how finely we chop up time - or how many different ways we chop up time - we get the same results. So this is a true statement:
The velocity of our object is increasing by 9.8 m/s every second within the limits of accuracy of our measuring devices.
Again, we are using CToT, not your metaphysical notions of truth.
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So, the high speed cameral has limitations, and when we get to situations with things accelerating at an extremely rapid rate, in an extremely short period of time, as in the case of high energy physics, the high speed camera is inadequate. And, the fact that the assumption of "constant acceleration" is adequate and useful at low rates of acceleration where a small error is insignificant, is not proof that it would be adequate for high rates of acceleration where the small error would be greatly amplified.

I didn't give this bit the attention it deserves. You said "the fact that the assumption of "constant acceleration" is adequate and useful at low rates of acceleration" - that's wonderful! If you agree that it's useful and adequate enough at low rates of acceleration, then you've accepted the only thing I really wanted you to. Gravity accelerates things at 9m/s/s, on planet earth, at least for the low rates of acceleration that we measured.

You go on to talk about other instances of acceleration that aren't directly caused by gravity, which I think it's fair to say is beside the point. The conversation is about how gravity accelerates things, not about how your leg muscles accelerate your own body.

You and I both agree, 9.8m/s/s is an adequate and useful idea of how gravity accelerates objects, on earth and for low speeds. And in fact Newtonian physics, which has pretty much the same simplistic vision of gravity as that, was enough to get human beings on the moon! How wonderful.

9.8 m/s/s isn't some perfect magical truth. It's an approximation that works, that we derived by simply looking at the world and taking notes. If you agree that it's useful and accurate in the contexts we generally use it, then you agree with me.
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Woops, mispost
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I was not any good at calculus, but I think calculus is what you are talking about. So question to you, MU: do you buy calculus? Or is that flawed and misleading?

As I said, I think calculus is very useful in very many situations. However, its usefulness has limitations. and when it is employed beyond these limitations it is misleading. This I believe is the case in modern high energy physics, it is employed beyond its limits. And, I believe It is misleading because people like you will argue that the problem which has not been resolved, the problem I referred to in the exchanges between Newton, Leibniz, and Berkeley, has actually been resolved.

This is why calculus is misleading, it has produced a very acceptable work-around for the problems first exposed as Zeno's paradoxes, which is very useful in a wide range of practises. However, since it does not actually resolve the problems of Zeno's paradoxes, these problems reappear, as the uncertainty principle for example, when we reach the limits of its applicability. If one insists that the problems have been resolved, then the true nature of the uncertainty principle will not be understood.

you're asking the right questions, except instead of saying "let's look at the data and check if the acceleration is going up and down wildly" you're just saying "oh well we can't know for sure so I give up, there's nothing left to discover."

Don't give up so quick, we have a lot of data from the camera. I mean, if you WANT to remain ignorant of the pattern of how things fall by gravity, then by all means give up here. But the rest of the world is operating on many centuries worth of physics past the point that you give up.

I'm not ready to give up. However, I'm already fully aware of the process you are laying out, and completely understand and respect its usefulness. Therefore I am bored and ready to move on. I can tell you however, that there is a point in time when we can know for sure that the acceleration is going up wildly, and that is at the 'zero point' in time, when motion starts.

So I will ask you now, are you fully aware and respectful of the problem that I am talking about? If so, then lets move directly to that specific problem and address it directly. In your example, there is a 'zero point' in time, the time when motion is supposed to have begun. So let's bring this zero point into your numerical expressions, and produce a "slice in time" which is the period between -.1s and +.1s. Do you agree that the averaging technique will not give a good representation of this time period? If you agree, then how do you propose that we deal with this period of time?

I didn't give this bit the attention it deserves. You said "the fact that the assumption of "constant acceleration" is adequate and useful at low rates of acceleration" - that's wonderful! If you agree that it's useful and adequate enough at low rates of acceleration, then you've accepted the only thing I really wanted you to. Gravity accelerates things at 9m/s/s, on planet earth, at least for the low rates of acceleration that we measured.

The problem though is that we have no way to measure the rate of gravitational acceleration at the precise moment that a thing starts to fall, and it actually may be completely different from your calculated rate.

You go on to talk about other instances of acceleration that aren't directly caused by gravity, which I think it's fair to say is beside the point. The conversation is about how gravity accelerates things, not about how your leg muscles accelerate your own body.

No, I started the conversation, as a discussion about the problem of measuring acceleration in general, that's why I referred a number of times to the effects of this problem on quantum mechanics, as the uncertainty principle. It was Eric I believe, who started talking about gravity as a specific example of acceleration, and then you. But that was brought up as an example of acceleration. It appears like you just do not want to look at the problem I mentioned.

Precisely. And the purpose of the 10 million different measuring apparatuses (apparati?) is to measure velocity. So QED we are measuring velocity. And so the statement is true per CToT. We are not dealing with your metaphysical notions of truth or falsity here. And of course it is not 10 million. Duh.

My spell check did not like "apparati". Anyway, I apprehend a slight mistake here. "The purpose of the measuring apparatus is to measure velocity" is true by coherency theory of truth, not by CToT. This is the categorical separation I referred to, and to mix them up is known as a category mistake. To state the "purpose of x is..." is to make a statement which is true or false by a stated definition, not by correspondence.

Acceleration does not cause anything. No wonder you are confused. Acceleration is a change in the velocity of an object. An object can undergo acceleration by being acted on by a force (F = ma) or by being affected by the curvature of spacetime.

I might agree to this, but you are just drawing us further away from the possibility of any truth by CToT. If acceleration is not considered to be the cause of change in velocity, being the intermediary between the prior motion and the posterior motion, and instead is just a calculated change in motion, then there is nothing real in the world which "acceleration" refers to. We can see the same issue with "energy", we can say that the word refers to something real in the world, or we can say that it's just something calculated according to a formula. You seem to be choosing the latter, which denies the possibility of correspondence truth in this subject.

No matter how finely we chop up time - or how many different ways we chop up time - we get the same results. So this is a true statement:
The velocity of our object is increasing by 9.8 m/s every second within the limits of accuracy of our measuring devices.
Again, we are using CToT, not your metaphysical notions of truth.

This is not truth by correspondence theory, it is true by coherence theory. The velocity determined is correct by the method of calculation, but this does not necessarily correspond to anything real.
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So let's bring this zero point into your numerical expressions, and produce a "slice in time" which is the period between -.1s and +.1s. Do you agree that the averaging technique will not give a good representation of this time period? If you agree, then how do you propose that we deal with this period of time?

Why would it fail to give a good representation? The only problem with our high speed camera data for this moment in time is that it has limited resolution, so we wouldn't necessarily be able to see how it starts moving at that moment in time (I've been rounding previous measurements of distance to 2 decimal places to sort of mimick the problem of camera resolution).

We'd have to film it up close and make sure everything is much more precise at the moment the thing is dropped. But there's no problem with it conceptually. First we get the average velocity between -.1s and .1s, and then we can look at how that velocity changed over that time frame by dividing that into even smaller segments, and then even smaller ones if we still have more questions.
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This is why calculus is misleading, it has produced a very acceptable work-around for the problems first exposed as Zeno's paradoxes, which is very useful in a wide range of practises. However, since it does not actually resolve the problems of Zeno's paradoxes, these problems reappear, as the uncertainty principle for example, when we reach the limits of its applicability. If one insists that the problems have been resolved, then the true nature of the uncertainty principle will not be understood.

Is calculus used to solve problems? (Answer yes or no.) And just what are the problems of Zeno's paradoxes? Achilleus gets where he's going, and faster than the tortoise. The arrow flies through the air and so forth. As to the arguments themselves. they all involve some faulty assumption. So where are the paradoxes?
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Why would it fail to give a good representation? The only problem with our high speed camera data for this moment in time is that it has limited resolution, so we wouldn't necessarily be able to see how it starts moving at that moment in time (I've been rounding previous measurements of distance to 2 decimal places to sort of mimick the problem of camera resolution).

The camera takes two shots, one at -.1s, and one at +.1 seconds. You produce the average, the speed for that time period, but this is obviously not a good representation. In reality the thing is moving in half that time period, and not moving in the other half. Furthermore, in the half that it is moving, it's average speed must be twice as fast as what your average says for that time period. I think that's a very significant, and in some applications, potentially a very important difference. If we add to this, the fact that by the special theory of relativity simultaneity is relative, there is the potential for even more significant inaccuracy, and uncertainty.

Is calculus used to solve problems?

Yes

And just what are the problems of Zeno's paradoxes?

Try Wikipedia.

Achilleus gets where he's going, and faster than the tortoise. The arrow flies through the air and so forth.

Not according to the logic applied to the premises. That's why the term "paradox" is used.

As to the arguments themselves. they all involve some faulty assumption.

Yes, they are faulty assumptions about the continuity of space and time, which are still held. You are on the right track here. Do you see that these same faulty assumptions are still held today? Next, can you apprehend that improved mathematical axioms will not resolve the the problems created by these faulty assumptions. No matter how good the logic, false premises will always leave the conclusions unsound.

So the issue is that space and time are understood as infinitely divisible continuums, or one continuum, and so division of them, or it, may be completely arbitrary. This does not correspond with reality, hence Zeno's paradoxes. The proposed solution was "infinitesimals", but these were arbitrary, and therefore still not consistent with reality. Calculus bring "infinite" right into the mathematics, and this is a form of indefiniteness, hence uncertainty.
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You produce the average, the speed for that time period, but this is obviously not a good representation.

A good representation of what? You keep saying things like "inadequate" or "not a good representation". Some measurements are adequate for some purposes and inadequate for other purposes. You can't just raw say it's inadequate, it can only be inadequate in relation to some goal.

Now it's not like you gave me a specific goal and I said "all we need to do is measure the location at these points in time". In fact measuring them at those points in time was YOUR suggestion, not mine. Don't tell me it's inadequate - tell yourself. If you want me to help you get adequate measurements to accomplish some goal, then all you have to do is tell me the goal and ask.
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So the issue is that space and time are understood as infinitely divisible continuums, or one continuum, and so division of them, or it, may be completely arbitrary. This does not correspond with reality, hence Zeno's paradoxes. The proposed solution was "infinitesimals", but these were arbitrary, and therefore still not consistent with reality. Calculus bring "infinite" right into the mathematics, and this is a form of indefiniteness, hence uncertainty.

I am not aware of anyone of note who "understands" space and time as infinitely divisible. Infinite divisibility a convenient fiction in calculus to solve problems, and the solutions obtained seeming true and valid. As such a tool and a good and effective one.

But what does Zeno do, as e.g., with the arrow? He supposes (reasonably for him we may suppose) there is an interval of time so short that within it the arrow is not moving. And just here some care with language is needed. Tell a boy to make his bed and the interval of time in which nothing moves could be days. At the other end are Planck distances and times. But to my way of thinking, there is a concept of minimum (and maximum) time and size - perhaps scale a better word than size - beyond the limits of which the matter in question does not exist. For example, the smallest unit of water is a molecule of H2O, smaller than that and water does not exist. As to time, the smallest unit is still an interval, and shorter than that and time itself ceases to be. And of course within an interval, the arrow moves. As to the logic, grant me the premises I want and I will undertake to prove, logically, anything you like.

And so far I do not think I have written anything you do not know perfectly well, or disagree with. What, then, do you dislike? That some people using calculus may think the world is itself infinitely divisible? There is no accounting for what some people think - a point Aristotle himself made, as I'm sure you also know perfectly well. That leaves right thinking, itself not always easy to do or recognize. But that aided by keeping in mind that all the rules, laws, theories, and mathematics just attempts at representations of the world itself (-as-it-is-in-itself) expressed in terms of what people can understand. And it appears that good progress is made along those lines.

I submit to you, then, in all respect, that your difficulties stem from supposing that some people take some ideas as literal and absolute truth, when in fact they're just aids in thinking.

As to infinities in mathematics, that a whole other topic, and the which well known and understood by mathematicians, and useful. Same page here? Or different pages - or books!?
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A good representation of what? You keep saying things like "inadequate" or "not a good representation". Some measurements are adequate for some purposes and inadequate for other purposes. You can't just raw say it's inadequate, it can only be inadequate in relation to some goal.

I mean "a good representation" of what actually happened, as i said, the goal is truth, in the sense of correspondence.

Now it's not like you gave me a specific goal and I said "all we need to do is measure the location at these points in time".

I gave the goal, truth. That's why Eric got sidetracked and started talking about correspondence theory of truth.

In fact measuring them at those points in time was YOUR suggestion, not mine. Don't tell me it's inadequate - tell yourself.

No, I'm telling you it's inadequate. I specifically requested those points in time to demonstrate to you, the inadequacy of your technique. Of course you would not suggest those points, because that time period, the time when acceleration starts, cannot be adequately represented by your technique. And this is a very real problem for high energy physics. I'm starting to think, that just like I was fully acquainted with your measuring technique, of using averages, you were actually fully aware of the problem I am talking about. And, like tim, you simply want to ignore it, and deny that it is a problem.

So now you intentionally avoid that specified time period saying, 'that's not my problem, it's your problem, because I have no interest in that time period. My averaging method serves my purpose, and I do not care if it doesn't serve yours. So keep your problem to yourself, and don't try to make your problem my problem.' But I'm not saying it's your problem or mine, I'm saying it's a problem with the technique. It's the technique's problem.

Infinite divisibility a convenient fiction in calculus...

I see, I need to say no more on this issue, you've stated my case for me.

He supposes (reasonably for him we may suppose) there is an interval of time so short that within it the arrow is not moving.

This is a misrepresentation. He is not talking about a short interval of time, he is talking about a point in time. I mean, you can say that you do not believe that there is such a thing as points in time, therefore this assumption is wrong, but the problem is that we always use, and refer to, points in time when making any temporal measurements, as the start and end points of the measured period. The start point divides time into prior and posterior, such that there is no duration within that point.

And so far I do not think I have written anything you do not know perfectly well, or disagree with.

As stated I disagree with your representation of Zeno's arrow paradox. He is very clearly talking about points in time, not infinitesimal intervals of time. And, your statement, that there is no such thing as a point in time, does not negate the fact that we use points in time for all of our measurements of time. So, you might insist that points in time are not real, they are "a convenient fiction", but as the premise for temporal measurements, you are then insisting that all temporal measurements are unsound conclusions.

Suppose objects are moving relative to each other. And, we can describe the spatial relations between objects. Would you not agree that any specific spatial relations would only exist at "a point" in time? The objects are moving, so any interval of time would not provide determinate relations. So the reason for the assumption of points in time is to provide for "truth" in spatial relations of moving objects. Without these points there is no such truth. Now, not only are temporal measurements unsound, but spatial measurements as well.

But that aided by keeping in mind that all the rules, laws, theories, and mathematics just attempts at representations of the world itself (-as-it-is-in-itself) expressed in terms of what people can understand.

OK, so if you believe that mathematics attempts at representations of the world, and you also apprehend that calculus is based in "a convenient fiction, then it ought to be a no-brainer for you to see the deficiencies of calculus which I am pointing to. Simply put, it fails to do what mathematics "attempts" to do, in your words, give us a representation of the world. It just gives us a convenient fiction.
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It just gives us a convenient fiction.
And you and I, and I suspect you and most people, attach an altogether different significance to what you call the "deficiencies." And yes, people often ignorantly refer to "points" in time. But calculus usually refers to the value of a variable as some input approaches a limit - no infinities, although they're approached, and no "points in time." And if Zeno wants to think in terms of points in time, what is that to us beyond an historical oddity - however reasonable it may have seemed to him at the time? And to be sure, "point in time" is easy to say, but were there actually such a thing, a durationless interval, then atomic motion would stop and everything on the instant collapse.

I think we're at an impasse. i think you hold that nothing can be measured exactly, of things subject to measurement, and thus all knowledge of such things is deficient and flawed. I myself would say that of such things, exact measurement is impossible in principle and thus we do the best we can, which is usually pretty well, and this not a failure or a deficiency, but instead a success.
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No, I'm telling you it's inadequate. I specifically requested those points in time to demonstrate to you, the inadequacy of your technique.

So now you intentionally avoid that specified time period saying, 'that's not my problem, it's your problem, because I have no interest in that time period.

Ok so you realise it was your idea to do that, so let me just reiterate how inappropriate is for you to complain to me about how bad your idea was.

If you ask me to figure out a way to get an answer, I can tell you, and THEN we can go into if the technique is adequate or not. Until then, your own problem with your own technique is something for you to work on with yourself, and it's not a criticism of me or any idea I've had.

I'm completely happy to look at that time period too, you just never asked me a question about it. Instead of asking, you started telling me what I would do. You're doing things in the wrong order and being too hasty, making careless assumptions again. Slow down.
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I myself would say that of such things, exact measurement is impossible in principle and thus we do the best we can, which is usually pretty well, and this not a failure or a deficiency, but instead a success.

:100: +/-0.000000001
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And you and I, and I suspect you and most people, attach an altogether different significance to what you call the "deficiencies." And yes, people often ignorantly refer to "points" in time. But calculus usually refers to the value of a variable as some input approaches a limit - no infinities, although they're approached, and no "points in time." And if Zeno wants to think in terms of points in time, what is that to us beyond an historical oddity - however reasonable it may have seemed to him at the time? And to be sure, "point in time" is easy to say, but were there actually such a thing, a durationless interval, then atomic motion would stop and everything on the instant collapse.

Sure, call it a "limit" instead of a point if you want, that doesn't change what it refers to, and that is a point of division, which separates one period of time from another. And as I said, the problem is not specifically that such limits or points are unreal, the problem is that the concept is applied as if they are real. So you can argue all you want, that there is no real problem because we all know that such limits are not real, but then the problem is the hypocrisy with which the concept is applied, as if the limit is real.

I think we're at an impasse. i think you hold that nothing can be measured exactly, of things subject to measurement, and thus all knowledge of such things is deficient and flawed. I

No, this is not what I'm arguing. I am pointing out the flaws and deficiencies and indicating that I believe a better system is possible. Whether or not exact measurement is possible is completely irrelevant, What is relevant is whether it is possible to improve the current technique. So, unless you can demonstrate that it is impossible to find a better system than the use of limits, then my activity of pointing to the flaws in this system and suggesting that we find a way to change this system, is very reasonable activity. Don't you agree, that pointing to the flaws and deficiencies of a technique, and indicating that these ought to be rectified, is a very good thing to do, even if those who currently use the technique tend to feel insult, offence, and so they strongly defend the technique which they use?

If you ask me to figure out a way to get an answer, I can tell you, and THEN we can go into if the technique is adequate or not. Until then, your own problem with your own technique is something for you to work on with yourself, and it's not a criticism of me or any idea I've had.

I am not criticizing you, or any idea you've had, I am criticizing the technique you are demonstrating. I am explaining that your technique for modeling acceleration through the application of averages, is inadequate for representing the most significant and important aspect of the concept of "acceleration", and this is the point in time at which acceleration begins.

Now it seems to me that we disagree as to whether this truly is the most significant and important aspect of "acceleration". Your model places this point as outside the representation, a limit which is approached, as tim states above. So you are inclined to say it's not my problem, that point in time is outside of "acceleration" as I model acceleration. It's not a part of "acceleration" so your criticism is irrelevant. I insist that it is an integral part of your model of "acceleration", significant, important, and necessary to your representation, so this is a requirement. The application of the limit is the primary premise.

I'm completely happy to look at that time period too, you just never asked me a question about it. Instead of asking, you started telling me what I would do. You're doing things in the wrong order and being too hasty, making careless assumptions again. Slow down.

Well, if you had been more attentive to what I wrote, you would have seen that this is the question I was asking about, when the mention of "acceleration" first came up, and we could have gotten right to the problem, without us wasting each other's time for the last nine days or so.

I see now, that my posts were addressed to EricH, as Eric is the one who brought up acceleration in the midst of our discussion concerning the possibility of medium-free waves. Perhaps you missed those posts, so I'll reproduce them now.

That something is "accelerating" requires a multitude of measurements of velocity, and each measurement of velocity requires multiple determinations of spatial-temporal location.

The concept of "acceleration" involves a fundamental philosophical problem. Acceleration is the rate of increase of velocity. So if an object goes from being at rest, to moving, there is a brief period of time where its "acceleration" is necessarily infinite. This is a fundamental measurement problem, and another form of the same problem is at the heart of the uncertainty principle of quantum physics, as the uncertainty relation between time and energy in the Fourier transform.

This problem was exposed by Aristotle as the incompatibility between the concept of "being" (static) and the concept of "becoming" (active). The way that modern physics deals with this problem, through the application of calculus does not resolve the problem. It simply veils the problem by allowing the unintelligible issue, infinity, to be present within the mathematical representation.

Now, the very same philosophical problem which Newton and his contemporaries had to deal with in the relationship between bodies, becomes paramount in modern physics in its relationships of energy. The issue though, is that Newton and his contemporaries were dealing with relatively long durations of time, so the methods of calculus were adequate for covering up this problem which only increases as the period of time is shortened. Now physicists are dealing with extremely short durations of time, so the uncertainty becomes very relevant and significant. That's what the time/energy uncertainty indicates, the shorter the time period, the more uncertain any determination of energy will be.

Accordingly, using the current mathematical conventions, such calculations of acceleration will never be done "beyond all reasonable doubt", because the current convention is to allow the unintelligible (infinite) to be a part of the mathematical representation..

You see, I have always been asking about that time period, and the whole interim has simply been a diversion. Do you see why it appears to me like you are simply avoiding the issue? You say "slow down", but we are discussing the opposite, acceleration. So unless you can show how your actions of attempting to decelerate the discussion are relevant, then I can only see your digressions as intentional diversions.
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I am pointing out the flaws and deficiencies and indicating that I believe a better system is possible.
And at the risk of trying your patience, what exactly are those flaws and deficiencies which justify your calling the "system" hypocritical? The reason I ask is that in sum you appear to be criticizing a tool, a tool which given appropriate inputs delivers results to an arbitrary degree of precision.

And the words "flawed" and "deficient" and "hypocritical" and their like, are themselves as you use them flawed, deficient, and hypocritical. Because to be sure, the tool can be none of these things, but only just itself. So why the freighted language? And my guess is that for you the invective has become the substance,

Seen the correct way, calculus, e.g., is neither flawed nor deficient, and certainly in no way hypocritical. Instead it is exact. In a sense then it is either all right or all wrong, and because all that it does is just what it does, then it must be all right. Further, since it gives answers to an arbitrary degree of precision, it is therefore in itself altogether correct.

It is as if you held a hammer in your hand and said of it that it was flawed, deficient, and hypocritical. All it is, is a hammer. If it is the only hammer you have, then either find or build one more suited to your tastes or learn to use it. If I may finish with a metaphor, it is as if you wanted a hammer and chisel that themselves would reveal the angel in the block of marble, and the tools you had in hand flawed and deficient and even hypocritical because they cannot. Fact is, there is no such hammer or chisel, but the user of them produces the angel, and even that no angel but a representation.
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And at the risk of trying your patience, what exactly are those flaws and deficiencies which justify your calling the "system" hypocritical? The reason I ask is that in sum you appear to be criticizing a tool, a tool which given appropriate inputs delivers results to an arbitrary degree of precision.

The flaws I spent the last week and a half explaining. And it isn't the system, which I say is hypocritical, but it's people like you who recognize the faulty assumptions (arbitrary points in time for example) inherent within the system, then insist that there are no deficiencies to the system, who I say are hypocritical.

Seen the correct way, calculus, e.g., is neither flawed nor deficient, and certainly in no way hypocritical. Instead it is exact. In a sense then it is either all right or all wrong, and because all that it does is just what it does, then it must be all right. Further, since it gives answers to an arbitrary degree of precision, it is therefore in itself altogether correct.

This is very unsound logic. A system which uses numerous different axioms must be either all right or all wrong? That doesn't even make sense. And you say "it is exact", but also to "an arbitrary degree of precision". Making exactness arbitrary leaves "exact" as completely meaningless. So you are saying absolutely nothing here. And your "metaphor" is even worse, being in no apparent way, analogous.
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Sorry, it's not me with points in time, as you read and replied to just a couple of posts ago, nor anyone I know of, unless they're speaking very informally. We agreed ( I thought) that exact measurement is not possible. So there is no "system" such as you hope for. That leaves the exactness of calculus. By that I meant that calculus is exact in what it does. Same formulas same inputs yield the same answers. And the answers work! What is the deficiency in a tool that always works, works consistently and accurately, and is correct - unless it is your idea of how it should work, or that it does not accomplish the impossible?

Clearly you want ideals and they don't exist except in fairy tales.
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By that I meant that calculus is exact in what it does. Same formulas same inputs yield the same answers.

Right, 2+2 always equals 4. There's is no doubt about that. But how this relates to the physical world is another issue altogether. I'm interested in the latter, not the former, because the former is very boring to me. And your suggestion, "it works", is equally boring.
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And yes, people often ignorantly refer to "points" in time. But calculus usually refers to the value of a variable as some input approaches a limit - no infinities, although they're approached, and no "points in time".

Don't be so critical. I've used "points in time" frequently in complex dynamical systems. And in complex analysis, a contour in the complex plane, z(t)=x(t)+iy(t), has a value for t=.5, e.g. And you think physicists don't use points? Do you think that limits define points, or points define limits? As for the infinite, there is indeed a "point at infinity" in complex variable theory.

So, unless you can demonstrate that it is impossible to find a better system than the use of limits, then my activity of pointing to the flaws in this system and suggesting that we find a way to change this system, is very reasonable activity

This might have made a semblance of sense had you been present when my math genealogical ancestor, Karl Weierstrass, and Cauchy were pulling together the common definition of limit two hundred years ago. You could have presented them with your clearly defined objections to their work and been present for their reactions. Oh, to have been a fly on the wall. :cool:
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The notion of "points in time" is rarely debated in science and math to the best of my knowledge. Do dimensionless points exist? This is more a philosophical issue than a mathematical one. Does a point on a ruler really exist? It certainly corresponds to a real number, but there are those who question the existence of irrationals. So, what happens to that point? It exists for some but does not for others?

Calculus is fundamental to the major branch of mathematics called analysis, founded on the idea of limits. Ordinarily, it assumes the existence of these points regardless of whether one speaks of rulers or time scales. Real analysis, the underlying structure of calculus, contains the axiom of completeness, which means points exist as viable entities. But some would say temporal points are different.

I thought of the famous debate between Einstein and the popular philosopher, Bergson, a hundred years ago. However, their issues revolved about whether time itself was independent of human experience. I'm not sure instants in time came up. But more recently a non-academic wrote a paper on time and physics in which he argues against any sort of "instant" in time. Some thought him brilliant, but others thought him a conveyor of nonsense. The latter is the more popular among physicists.

The point at infinity in complex analysis is the north pole of the Riemann sphere. No matter where you go in the plane as you move out away from the zero point the projection onto the sphere moves toward its north pole. So, in this sense, there really is a "point at infinity". :cool:
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Would you agree with me that "point in time" is at best a locution to convey informally in language an aspect of a technique useful in math, and not otherwise real? If "point" implies no dimension and the dimension in question is time, then "point in time" implying no-time time? Certainly nothing can happen at a point in time, happening requiring duration, yes? Even being suspect at a point in time: being requiring at some level activity, and thus duration.

And thus any argument wherever found that adduces points in time, if not in the useful mathematical sense, is either nonsensical or relying on a sense of "point" that is in fact not a point at all.
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↪jgill
Would you agree with me that "point in time" is at best a locution to convey informally in language an aspect of a technique useful in math, and not otherwise real?

Would you say an interval of time is real? Consider the intervals [0,1] or (0,1). Each requires end points, one includes its end points and the other does not. What are these points? Fictions designed for the sake of argument? This makes intervals of time as suspect as their end points.

Bergson compared the unfolding of time as a tape steadily rolling off one drum and onto the other. So the word "duration" implies an infinite interval. Thus any notion of time's flow excludes finite intervals. The exact duration of an event is as much a non-real artifice as a point in time.

A photo of Zeno's arrow, frozen in flight, implies that relying on a dimensionless point excludes the recognition of the arrow's momentum. So, yes, points in space and in time's continuum are mostly mathematical objects. I say mostly since I consider them to be metaphysical "objects", outside the realm of physics - which is what many physicists concluded after reading Lynds' paper on the non-existence of points in time. Neither provable or excludable. Just fodder for endless, non-productive philosophical discussions.

Incidentally, the appearance of Lynds' article (2003) sealed the fate of Foundations of Physics Letters, which ceased publication after 2006. :cool:
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The problem with the declaration that points in time are unreal is that the photoelectric effect demonstrates that there must be very real points in time. The way that electrons react to electromagnetic radiation indicates that there must be a point in time when an electron is emitted. The emission occurs as a "quantum" of energy at a determinate point in time, rather than as a continuous flow of energy.

The fact that we have not been able to understand or properly identify these real points in time manifests as the misunderstanding of the wave function collapse. The collapse must be in some sense real because it is observable, but since the conventional employment of points in time is done through arbitrariness rather than correspondence, as points are apprehended as unreal, the result is many worlds, according to many possible points. On the other hand, complete denial of the reality of points in time produces a continuous wave function with no possibility for any real points of collapse.
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As I read these, there's a failure to distinguish between what we might call a map co-ordinate and a dimension/duration. Of course there are "points in time." 2:00 can stand as an example. Any time before 2 is earlier, and later, later. And at 2 precisely, it is 2. But what is the duration of the point of time? Of course it is no duration at all. And I would observe, as I did above, that if time is duration, and you specify no duration, then you have a no-time time, which seems whatever that is, it is not time. Similarly with points on a line. If a line implies length, and a point is dimensionless, then a point on a line is a lengthless-line, and thus itself no line at all.

And most folks seem to understand that a point on a map is just a co-ordinate, descriptive but in itself no-thing. Same with points in time. The o'clock is a descriptive and useful coordinate, but in itself, nothing. Reifying the concepts, making them what they are not, seems a ready source of trouble, avoided by including/acknowledging qualifications as needed and appropriate.

Consider the intervals [0,1] or (0,1). Each requires end points, one includes its end points and the other does not.
Kindly correct me as needed, but I'm thinking both include their endpoints; in the one case the endpoints are known and identified, and in the other, unknown and unidentifiable. But whatever the status of their endpoints, both intervals.

And finally, just for the heck of it, what is a "metaphysical object"? And what exactly is "the wave function collapse"? As an informal descriptive term, I (think I) get it. But if it's more than that, if it's a something, then what is it?
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Consider the intervals [0,1] or (0,1). Each requires end points, one includes its end points and the other does not. — jgill

Kindly correct me as needed, but I'm thinking both include their endpoints; in the one case the endpoints are known and identified, and in the other, unknown and unidentifiable. But whatever the status of their endpoints, both intervals.

[0,1]={t:0<=t<=1} and (0,1)={t:0<t<1}

And finally, just for the heck of it, what is a "metaphysical object"? And what exactly is "the wave function collapse"? As an informal descriptive term, I (think I) get it. But if it's more than that, if it's a something, then what is it?

A dimensionless point, not a pencil dot on a map. Or an infinitesimal in non-standard analysis. An object of the mind, not something that has a physical presence. IMO.

"the wave function collapse": Differential equations can have more than one solution, and a linear equation thus has all linear combinations of these solutions. Upon measurement, one discovers which of these is correct. For example, dy/dt=1 implies y=t+c, c is an arbitrary constant. Upon measurement I find that c=-.5, e.g. But this is overly simplistic and there does appear to be some weird stuff going on. In my opinion the wave function is not ontological. In fact, the Schrödinger equation in its simplest form is
dy/dt=Ky
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As I read these, there's a failure to distinguish between what we might call a map co-ordinate and a dimension/duration.

The problem is that there is no specific real thing, in physics, which corresponds with "duration". Duration is simply a relation between one activity and another, and as a relation it is a feature of the coordinate system employed. That is why there is serious ontological discussion as to whether time is real or not, and the general consensus is that the principles employed in physics assume that time is not real.
In physics, time is defined by its measurement: time is what a clock reads. — Wikipedia

Notice, a clock does not read time, people read time with the use of a clock. So it's just like any other measurement. If I measure between the house and the car, the reading, 50 meters for example, is part of the map. Likewise with time and also "duration". In physics duration is part of the map, as a measurement.

Since the measurement of time is just a number of 'units' produced by relating one activity to another, physicists have no idea of what is actually being measured. The 'units' are completely artificial. The tendency is to claim that there is nothing real, and time is an illusion. This effectively avoids the issue. But any physicist (more correctly metaphysician), who wants to understand the reality of what is being measured, is confronted with the question of what constitutes a real unit of time.
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The problem is that there is no specific real thing, in physics, which corresponds with "duration".
Sure there is, it's called time. If you'll read your own post, your comments are on the measurement of time, not time itself. As to units of measurements, when did the thing measured ever care about how it was measured? And to say that there is no specific real thing in physics that corresponds with duration, how about a physics lecture?
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:up: IMO.
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Sure there is, it's called time. If you'll read your own post, your comments are on the measurement of time, not time itself.

The point though, in physics, time is defined as the measurement, as the Wikipedia quote indicates. There is no "time itself" in physics, because "the measurement" is simply a product of the application principles. Therefore time is part of the map, not the territory, just like space.

This is the same issue which we discussed with the aether earlier. Without the medium, (the substance within which the waves exist), "space" is just a feature of the measurement system, the map. This means that what is referred to by terms like "space-time", as well as "the wave function", are features of the map's measurement principles or rules, with conventions for application in physics. but without any corresponding reality in the physical universe.

Think of the way that a coordinate system is a feature of the map, a product of the measurement principles. There are conventions for application in various situations, the proper technique for applying a coordinate system in practise, but there is no part of the physical universe which corresponds with the coordinate system. That is the reality of both "space" and "time". in the practise of physics, they are a system of measurement principles, application techniques, with nothing in the physical universe which corresponds.
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