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

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

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

It's those damn wave functions! They just seem to be everywhere (at least on this forum). No one mentions path integrals in QT. We can imagine waves, but not functional integrals.

They’re not actually waves but the distribution of probabilities. Just so happens they can be visualized as waves. N’est pas?

Let's let the master speak on that:

http://yclept.ucdavis.edu/course/215c.S17/TEX/GodelAndEndOfPhysics.pdf

All Godel showed in meta-language is that there are infinite things that cannot be prove in math and infinite things that could be proven (from sure foundations). What space IS cannot be precisely said by mathematics. Affirmation of a double negative is what infinity is, which translates to a double positive. The positives are abstract infinity (a single thing, undivided) and abstract finitude (the finite as idea). They merge to form infinite units, composing a single object that swallows itself. It curves back around like on a sphere. We experience objects in their finitude. In thinking abstractly, you have to go back and forth from the infinite as idea and the thought of "the finite" in dealings with uncountable infinity (in the form of points). I think this is how Leibniz understood it, but anyway it certainly is how Hegel understood Leibniz

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

He said some wanted an ultimate theory that can understand the world in finite ways with finite equations. He said he changed his mind when he reflected on Godel's theorems and realized that the world gives finite results when there is precision from us, who approach the universe as part of it. So there will always be a dialectic between the infinite and the finite in our dealings with the universe.
I am not allowed to quote that article

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

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 — Ryan O'Connor

The rankest whackdoodle nonsense.
And why should I not think you a complete fool at this point?

I believe that's a false equality. — Ryan O'Connor

Oh boy my mentions have piled up tonight. I wanted to respond to this before diving back into the maw of the slavering mob I triggered by daring to express a political opinion they don't like. But never mind all that.

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". — Ryan O'Connor

Before going on @Metaphysician Undercover pointed out a distinction we need to nail down. Please respond clearly to this question. Are you talking about

* Mathematics? or

* Some kind of notion of reality that goes beyond math?

If the former, you're just wrong. 1/3 = .3333... because the right hand side is just the sum of the infinite series 3/10 + 3/100 + 3/1000 + ... which equals 1/3 since it's a geometric series as taught in freshman calculus. It does you no good to demonstrate ignorance of the theory of limits. The sum is defined as exactly 1/3, not "approaching 1/3" or "infinitesimally less than 1/3" or any similar confused location that calculus students often have after slogging through the course. Mathematically you're wrong as a matter of fact.

On the other hand if the latter, I have no idea what you mean, because 1/3 = .333... as a matter of formal mathematics. I can prove it directly from the axioms of ZF set theory. There is no question about it. It's as certain as saying that a certain chess position follows from the rules of the game. You'd be better off self-studying real analysis than wasting your time telling me about your mathematical misunderstandings. And if you think it means something reality, you're likewise wasting your breath because I have no interest. I'm doing math, not metaphysics; and frankly I'm not sure 1/3 OR .333... OR the number 47 have any metaphysical meaning at all. That's a completely different subject.

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. — Ryan O'Connor

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.

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. — Ryan O'Connor

No not at all. First, video is nothing more than a series of stills, whether analog or digital frames. Have you ever seen a flip book? We see the illusion of motion from a sequence of still images because our eye-brain system retains a little afterimage for a fraction of a second. That's how movies work, I'm certain you must know that. Back in the old days I believe flip books were a common medium for the pr0n of the day but I found a really nice G-rated contemporary flip book artist. This video is worth checking out for its own sake and makes this point dramatically. Video is nothing more than a rapid sequence of stills. It depends on a quirk of your eye-brain system.

https://www.youtube.com/watch?v=ntD2qiGx-DY

Secondly, how about still photos? If you ever had a DSLR or film SLR you know that one of the important controls is the shutter speed. You hold the shutter open for a period of time, usually a fraction of a second, and during that time interval photons come in and hit the film or digital sensor. If an object is moving side to side in front of you, you can always determine its velocity by setting a long enough shutter speed that you get some motion blur; then you can work backwards from the blur to determine the velocity. You know, the same reason your low-light photos taken with your automatic camera are blurry. The camera compensates for the low light by choosing a slower shutter speed, and your shaky hands cause motion blur.

So still photos actually take time, and videos are just a sequence of stills.

We want points (photographs) — Ryan O'Connor

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.

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. — Ryan O'Connor

We're not going to solve Zeno's paradoxes here. Better you should try to understand some math, in particular the sum of a geometric series.

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

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. — Ryan O'Connor

Utter nonsense. If you put your webcam on yourself, record at 30 frames per second, and move around, then play it back frame-by-frame, you will most definitely see motion blur in each frame. Unless you are speaking metaphorically about an idealized camera, you are just wrong on the technology. And an idealized camera with infinitesimal shutter speed could not transmit any photons to the light-sensitive medium, so it wouldn't work.

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. — Ryan O'Connor

We're not going to resolve those ancient issues here.

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)? — Ryan O'Connor

What? The tangent line IS defined as the limit of the secant (for a function from the reals to the reals). That's how they motivate it. But in fact that is the DEFINITION of the tangent line. Without the definition, we don't even know what a tangent line is.

Only if you consider 0/0 a valid velocity. — Ryan O'Connor

You're confusing the formalism with reality.

The pre-Socratic ontology said that water was pure simplicity and with fire can make perfect movement. It bowls over into the reality of cars, etc (earth and wind) This was seen within the geometry of atomism and Zeno's thesis

It's the language of a culture

Imagine a dark room and a quantum sensor — Ryan O'Connor

So you're no longer talking about cameras but rather quantum sensors? It would be better if you dropped these labored analogies and just made your point some other way. Are you saying photons use quantum tunneling to show up on the sensor without the shutter being open? This is such an unproductive tangent to the discussion. To record an image on a photographic medium you have to expose the medium over a period of time. You're stepping all over your own point, we're not talking about cameras we're talking about something else and this isn't making sense.

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 — Gregory

Just an artifact of the decimal representation. Doesn't mean anything. Roman numerals had drawbacks, decimal representation has drawbacks. Every notation has some advantages and some drawbacks.

As to why a circle's circumference and radius are incommensurable, I think that's one of God's little jokes. God in Einstein's sense. Not necessarily religious, but a stand-in for all the ineffable mysteries of the world. With a sense of humor thrown in.

Your guess is as good as mine. There's no actual reason. But the diagonal of a square seems like an even simpler example. One unit to the right, one unit up, and the distance between your start and end points is incommensurable with 1. No reason at all, just a shocking surprise.

How can anyone deny the existence of the length of the diagonal? That's the part I don't get.

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. — Metaphysician Undercover

A moving body has an instantaneous velocity, even though our formalism requires two temporally separated measurements. But, I'm willing to concede that you've either made your point or at the very least caused me to doubt mine.

It's FALSE that motion through a point is the same as resting in a fixed point. There is energy that pushes right through. Imagine an arrow being pushed by the air rushing behind as it pushes forward. What's wrong with this picture (of the arrow)? It's that the motion forward is prior to air rushing behind so we can not saw the air pushes the arrow. The arrow moves through any point with forward velocity so it's never ever at rest

↪jgill
They’re not actually waves but the distribution of probabilities. Just so happens they can be visualized as waves. N’est pas? — Wayfarer

:cool:

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. — Ryan O'Connor

What is real and fundamental in quantum physics is the points where particles appear. The wave function is the mathematical apparatus which predicts where particles might appear. Yes, the wave function is fundamental to the model, but what is being modeled is the appearance of particles at specific points. This is why physicists understand light as photons, because the energy appears at, and causes an effect at a point.

You say that points only emerge from a measurement, but a measurement is an interaction between the energy, and the object which is the measuring device. So, such points exist wherever energy is interacting with objects. What this indicates is that energy, though it is modeled as existing in a continuum, (wave function) only interacts with the physical objects which we know, at discrete points. Therefore our only access to observe whatever substratum there is, which is modeled as wave functions, is through an understanding of these points where we can observe interactions.

Sure, you might say that the continuum, or substratum as I call it, is more fundamental, but from the point of view of the model, and this means the mathematics, the points must be fundamental. This is because we only find a route inward, toward understanding the substratum through a mapping of the points where it interacts with the spatial existence we know, observe, and understand. What must be fundamental, and basic to the model, is what we know the best, and this is the points. The substratum is modeled based on the existence of those points where we can observe it The better we know the points, the more reliable our speculation about the substratum will be.

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. — Ryan O'Connor

The substratum, which is represented by wave functions, may or may not be a continuum. That a wave function represents it as a continuum doesn't mean it is. Furthermore, a measurement is simply the substratum interacting with a physical object. So if this causes a "collapse", there are collapses occurring all the time, all over the place, as the substratum is interacting with physical objects. And, if measurements are only possible at particular points, then we ought to assume that other interactions between the substratum and physical objects are only possible at particular points, and this is most likely a feature of the substratum itself.

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). — Ryan O'Connor

I don't agree. A process which is never executed cannot be studied. It has no existence so it cannot be studied. Let's say that you write out a rule, an algorithm, but the algorithm is never implemented. You can study that rule, but you cannot study the process dictated by that rule, because it does not exist. The rule was never put to work, actualized, it exists only as the potential for the designated process Do you see the difference between the written rule, and the activity which is prescribed or described by that rule? To study one is not the same as studying the other.

A moving body has an instantaneous velocity,../quote]

Yes, because that is the convention, use some math, and figure out the "instantaneous velocity", just like the convention is to place a zero limit on the example of the op. But what these conventions really represent may not be what one would expect from the terms of usage. — fishfry

A process that is active is mental use of the four functions. This can be applied to reality but not perfectly

Whether 1+1 creates a new number (Kant) or a set (Frege I think) is a question for phenomenolog psychology, not math. And most of these questions are not useful for modern science.

Yes, because that is the convention, use some math, and figure out the "instantaneous velocity", just like the convention is to place a zero limit on the example of the op. But what these conventions really represent may not be what one would expect from the terms of usage. — Metaphysician Undercover

A car whose speedometer reads 40mph is going 40mph at that moment even though that's a mechanical approximation. But as I say I'll concede you've moved me off my certainty and I've run out of talking points. After all we don't know the ultimate nature of reality so who's to say if the notion of instantaneous velocity really makes sense. Based on that I'll concede the point.

I've been meaning to ask you: "Descartes great merit here was to have applied geometry to algebra; he was not the first to have applied geometry to geometry... And to Descartes we owe the first systematic classification of curves. After dividing 'geometric curves ' which can be precisely expressed in equations from 'mechanical curves' that cannot, he classified the former into three clases... This new geometry is more than a general theory of quantity: it led to the concept of continuity, from which was developed the theory of function and, in turn, the theory of limits... But he was mistaken in believing that equations of any order could be so resolved." ( Britannica Encyclopedia 1965)

I think applying numbers to geometry is how we apply Godel's numbering to physics

A car going at constant speed passes point A at stopwatch time=0, then passes point B, one mile further at stopwatch time=one minute. You ask, "What was the speed of the car back there at point A?" Your answer, "It was moving at 60 mph at point A".

Silly, but this entire discussion needs termination.

I consider this thread to be about the "geometry of Godel" but nobody uses that phrase and it does spin out of control. Peop!e start talking about space being outside itself and such.
I have two links saved somewhere on Leibniz' relationship to this question. I'll go find them

Ok I found the more relevant article:

https://www.humanities.mcmaster.ca/~rarthur/articles/lsi-final.pdf?fbclid=IwAR34tqTaNU4OY-YBbMA-ZpsGIxLFev1ZV8QRvHQF7UprFiXtdx9RgYoLHGc

Marx said Hegel was the best of mathematicians yet Hegel learned what little he knew of the subject from Leibniz, who could easily be the greatest of all mathematicians (so my intuition is saying).
This also relates:

https://en.wikipedia.org/wiki/Benacerraf%27s_identification_problem

This also relates: — Gregory

Thanks. Wasn't aware of that. Kind of anti-set theory regarding natural numbers.

I think applying numbers to geometry is how we apply Godel's numbering to physics — Gregory

I don't think physics in its present form is amenable to methods of formal logic because there are no axioms for physics. Axiomatizing physics is Hilbert's sixth problem and according to Wiki it's still open. Even so I doubt that the axioms of physics, whatever they may eventually turn out to be, satisfy the premises of Gödel''s incompleteness theorems.

But of course you could Gödel-number anything, Moby Dick or the contents of this post. You assign numbers to the upper and lower-case English letters and punctuation symbols, and you apply Gödel numbering. So if 'abc" is your string, and a = 1, b = 2, c = 3 is your encoding of the symbols, then 'abc' is encoded as $2^1 \times 3^2 \times 5^3 = 2 \times 9 \times 125 = 2250$.

The idea is to use the unique factorization of integers into prime powers. To go backwards from 2250, we note that it's $2^1 \times 3^2 \times 5^3$ and read off the exponents, 1, 2, 3 to get 'abc'. You could do this with any piece of text.

As you can see, if you believe that the positive integers exist (whatever existence means for you) then every finite-length document that could ever be written, already exists. It's already encoded as some positive integer. LIkewise if you pixellate a painting finely enough so that the pixellated version is indistinguishable from the original (to the naked eye, say) and encode its color and texture values, you can Gödel-number all possible paintings. Likewise songs, etc.

Or as the Pythagoreans said: All is number!

After all we don't know the ultimate nature of reality so who's to say if the notion of instantaneous velocity really makes sense. — fishfry

This is the point. When we use math to figure out things like instantaneous velocity, the volume of a supposed infinitely small tube, etc., it is implied that we know things about reality which we do not. This is a falsely supported certitude.

A car going at constant speed passes point A at stopwatch time=0, then passes point B, one mile further at stopwatch time=one minute. You ask, "What was the speed of the car back there at point A?" Your answer, "It was moving at 60 mph at point A". — jgill

There is a flaw with your example jgil. That the car was "going at a constant speed" is just an assumption, so it may not be the truth of the matter. And your answer as to how fast the car was going at point A requires that the assumption be true. So it needs to be proven.

You might lay out a series of such points, at equal distance, and do numerous similar measurements. If your measuring capacity is precise, you'll find that all the measurements will not be exactly the same. The assumption of "constant speed" cannot be validated. That's what we've found out about the nature of reality, motion consists of spurts and starts. So you'd have to establish trends, and the more measurements you took the better your graphing of the trends would be. But you'd be graphing averages which does not tell you the precise amount at any given point.

Yes. Goes back a hundred years if my memory serves. Sometimes it's very easy. For example, here is a linear fractional transformation written in terms of fixed points and multiplier.(I'm working on a theorem right now involving this). I think a more general case was dealt with in the discipline of functional equations. Can't recall the work offhand. — jgill

Thanks!

:smile:

I'm not looking for people to buy in, I'm looking for truth. — Metaphysician Undercover

A worthy goal. I am not going to ask you what truth is or what you think it is. But I am going to ask you to write something - anything - that is true. I suspect you will not be able to express anything (as) true that will also meet your criterium for truth. Because - aside from problems with truth itself - I'm thinking, from your many posts, that your own critical stances are themselves too variously destructive to allow you make such an expression. Or another way, when the AAF confronts himself, the result must be total destruction, as with matter, anti-matter.

My own solution to such difficulties is to remember that truth is always with respect to something, usually different somethings for different truths. And no one truth to bind them, except as a concept so abstract it refers only to itself, containing thereby only the shadow of truth, that being that the truth of true statements is truth because they are true.