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AlNg
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But there are many statements of infinite length. For example, one third is 3/10 + 3/100 + 3/1000 + 3/10000 + …
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rom
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If reality is chaotic and fuzzy, and I know it to be chaotic and fuzzy, then I have a true knowledge of reality. What is the problem? Or did I miss your point?
Things exist according to the way that God knows them. Therefore, things are always ontologically true (even if there is no human mind to know them) because they correspond perfectly to God’s Knowledge. But that is ontological truth, or “the truth of things.” There is another kind of truth, and that is philosophical truth, which is the correspondence of our mind to reality.
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StephieNorthCo
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Since you’re measuring in 3 space with time as the 4th dimension, of course you can both think of the system and measure the spherical components at different times, allowing you to discover anisotropies in the propagation field. That’s how many so called green lasers operate… doubling frequency of the ir emitter. So, for that matter do you measure and guide light in a fiber optic cable.StephieNorthCo:
Granted. Does that mean the electromagnetic wave is a 4D sphere and that, therefore, the 4D sphere is real? Or does it merely mean that it is OK to think of the electromagnetic wave as a 4D sphere for the purpose of mathematically predicting its propagation into space? It is a mathematical model, useful perhaps, but it is only in the mind. I have known electrical engineers who used a spring-mass system to set up the differential equations that describe the behavior of alternating current circuits. Would you therefore equate an alternating electric circuit to a spring-mass system?
As far as electrical / mechanical dual, yes, of course. One can model a system as an electrical LCR circuit or spring/mass/dash pot system.
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Dovekin
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I believe the material world is a 4d world. Einstein’s Relativity theories established that time and space are aspects of 4d spacetime. Time dilation, the different measures of time depending on perspective, shows that space and time are related as comparable realities. By equating mass and energy, we can understand our 4d universe as shaped by gravitational fields.
I see a 3d universe, but the array of evidence uncovers a 4d material world. I am not sure what you mean by saying it is 3d. You accept a 4 dimensional descriptive system (x,y,z,t). What else do you want?
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Dovekin
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That is equivalent to a statement of finite length, one third is 1/3.
Your phrasing suggests why including statements of infinite length would make the set uncountable, because the real numbers are uncountable.
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rom
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The material world is really just a 3D world. Although physicists have spoken of time as a fourth dimension, they realize that it is not a dimension in the same sense that the other three are dimensions of a body in the real world. A 4D sphere is a mathematical “model” that exists only in the mind because one dimension (the time dimension) is imaginary. If you have a line in the x-axis defined by the end-points P1 and P2, you can see that all the points between P1 and P2 really exist. You have a real line. But if you have a timeline in the t-axis defined by T1 and T2, you will notice that there is always only one point between T1 and T2 that actually exists (the “now”), because the points before it are no more (since they are already past), and the points after it are not yet (because they are still future). Therefore, the timeline of the physicist does NOT really exist except in his mind. The same is true of the 4D space-time continuum. It is just a mathematical model useful for setting up the equations that will describe the interaction of bodies in the world, the movement of and bending of light, etc. But the model itself is not real, but only a “tool for calculation.” That the model works does not prove that the model is real. It proves that the model is useful.
You don’t see evidence of a 4D world. You see plenty of evidence that your 4D model works. It is a big leap to think that a model is real just because it works. Think of the electron, for example. There are many physical phenomena that can be described properly by thinking of the electron as a particle. But there are other phenomena that can be better described by thinking of the electron as a wave. Now, is the electron a particle or a wave? I say neither. The particle concept is only a model of the electron that works in some cases. The wave concept is also just a model of the electron that works in other cases. I am not saying that the electron is not real. It is. It is the model that is not real. The model is only a tool for setting up equations and for predicting the outcome of certain experiments. But it would be “jumping to conclusions” to think that the electron is what the model happens to be. The fact that the model works is not evidence that the model is “true” or real. It is evidence that the model is useful.
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rom
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I guess you and I do not disagree on that. I just want to warn people that a 4D sphere does not necessarily exist just because there is a physical application of that model that works. Please see my comments to Dovekin’s post just above this one.
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rom
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If God knows more about the chaotic world than I do, it would mean that my knowledge is incomplete, but not that it is false. Philosophical truth is not the correspondence of the human mind to the Mind of God, but the correspondence of the mind with reality. If philosophical truth were to be defined as matching God’s mind, then no one, absolutely no one, will be in possession of philosophical truth. Even if the world were not as chaotic as it is now, God will still know infinitely more about a blade of grass than I could ever learn in a lifetime.
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StephieNorthCo
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As soon as you introduce chaotic nature, you also have to deal with the concept of fractional dimensions. You can end up making sense of “chaos” by modeling the chaos as a fractional dimension. Note the Mandelbrot /Julia sets and the beautiful chaotic pictures they produce. They also reside between 2 and 3 dimensions, that’s why they are both quite periodic and quite chaotic. I think it was J H Conway that had that insight.
In my field (obviously electronics) chaotic oscillations can relatively easily be demonstrated by properly constructing a feedback system. Those chaotic oscillations, with careful manipulation, can be precisely controlled. Presto. Chaos gone.
The first, that I know of, very practical example of clever manipulation of the chaos to produce clean , non chaotic output was your touch tone phones. Two sine waves of different but precise frequencies non integer related concurrently produced with exactly one amplifying/oscillating device (transistor). Not magic, but clever.
Incidentally, that also where the electrical LCR and spring/mass/dash pot become difficult if not impossible to mesh. (And that may only be because I don’t understand mechanical systems nearly as well as I do electrical systems).
In my field (obviously electronics) chaotic oscillations can relatively easily be demonstrated by properly constructing a feedback system. Those chaotic oscillations, with careful manipulation, can be precisely controlled. Presto. Chaos gone.
The first, that I know of, very practical example of clever manipulation of the chaos to produce clean , non chaotic output was your touch tone phones. Two sine waves of different but precise frequencies non integer related concurrently produced with exactly one amplifying/oscillating device (transistor). Not magic, but clever.
Incidentally, that also where the electrical LCR and spring/mass/dash pot become difficult if not impossible to mesh. (And that may only be because I don’t understand mechanical systems nearly as well as I do electrical systems).
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rom
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That is very interesting. So, perhaps there is really nothing chaotic in nature except our chaotic understanding of it. To God, who knows all the initial conditions of dynamic processes, even what we consider as stochastic processes could really be determinable and orderly. Even Brownian movement, which appears to us as random motion, will not appear to Him as random.
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DignumEtJustum
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Math is something understood by us as it correlates to our frames of references (numbers, letters, etc.) and works in the box we put it into but I think it is ultimately an imperfect system as we understand it that could perhaps one day no longer be useful for what we need it to do for us, or at least have to be reformed or some new mathematical realization made in order to really help us advance. That said I am no mathematician and my friend who is has disagreed with me on this subject before. The thing is we are not technologically where we could be and it’s because of this imperfect system, or at least our imperfect understanding of math along with our imperfect understanding of science in general, that this is the case. I was thinking the other night, what would be the mathematical equation to explain me standing on a patio having a cigarette? Surely there is some code of some sort that could explain it mathematically, but it’s not something we have much of a semblance of an understanding of or would be so intricate and complicated that it couldn’t likely be fathomed by the human mind.
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Dovekin
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Granted, if you model the universe in a way that excludes time as a dimension, you have a 3d universe. It will be hard defining a 4d sphere with that stricture. I would have to start talking about imaginary numbers.
All numbers and models are images of reality in some sense, abstractions based on real objects. The imaginary number t is an abstraction based on the behavior of real things. The truth of a model depends on how well it conforms to reality.
A 3d model does not conform to reality as well as a 4d model. In the model based on relativity, space and time measurements are part of a single imaginary system that is constrained by things like the speed of light. Even though time is imaginary, reality is better comprehended by including it than by excluding it. This shows up mostly at extremities, like near light speeds, electrical interactions, gravitational influences.
If we define reality as you have, 4d spheres cannot be real. But a 4d model can more closely conform to reality than any 3d model could, that is, it is closer to the truth. Our naive perception of reality as 3d is not closer to the truth than a 4d model, which raises questions about your exclusion of time as imaginary.
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rom
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We are getting closer to agreement. 
Our world is really a 3D world. But a 3D model is not what I use to describe and predict phenomena in my 3D world. Like you, I also use an imaginary 4D model (which takes time into account) to describe and predict phenomena in my real 3D world. But I don’t jump into the conclusion that I live in a 4D world just because the imaginary 4D model works great in describing phenomena in my 3D world. I am ok with thinking of a “warped spacetime continuum” to describe how bodies interact. But I find it nonsense to say that real space or real time is warped just because the imaginary spacetime model can flex and bend. The 4D spacetime model is really just that – an imaginary model. And it can be replaced by a different model in the future, if a better one is found.
At one time people thought that the behavior of bodies can be described by the gravitational force. Was gravity real, or was it just a conceptual model? I think it was just a model, and it worked great in many cases. However, we now have a better model – spacetime. Now the same question comes up: is it real or is it just a model? At one time the Rutherford model of the atom was very popular. Was the atom really just a miniature solar system or was that just a model? It worked great, and is still being used today. But the real atom is far more complicated than Rutherford envisioned. We have to be careful about the concepts we use in physics. Many of them are mathematical models and concepts, not real entities. Think of work, for example. I don’t deny that there is such a thing as real work. But work in physics is not what real work is. In physics work is the mathematical product of force and distance. It is a concept. That means, a man holding a heavy barbel for 1 minute but not moving it, has done zero work. He might perspire, but he has done less work than an ant that walked one inch. And it is correct, because “work” in physics has a different meaning than real work. The same can be said of “mass,” “energy,” the “quarks,” the “anti-particles,” etc. They are all concepts and models of modern physics. It is useful to think of reality in terms of those concepts. But it is an unpardonable mistake in the philosophy of nature to think that they are real entities just because they work well in describing the real world. They are describing something real, of course, but they themselves are just models that can be replaced in the future depending on the progress of our scientific knowledge. Medieval philosophers have a term for concepts like these. They are called entia rationis cum fundamento in re, or “beings of reason with a foundation in reality.”
Our world is really a 3D world. But a 3D model is not what I use to describe and predict phenomena in my 3D world. Like you, I also use an imaginary 4D model (which takes time into account) to describe and predict phenomena in my real 3D world. But I don’t jump into the conclusion that I live in a 4D world just because the imaginary 4D model works great in describing phenomena in my 3D world. I am ok with thinking of a “warped spacetime continuum” to describe how bodies interact. But I find it nonsense to say that real space or real time is warped just because the imaginary spacetime model can flex and bend. The 4D spacetime model is really just that – an imaginary model. And it can be replaced by a different model in the future, if a better one is found.
At one time people thought that the behavior of bodies can be described by the gravitational force. Was gravity real, or was it just a conceptual model? I think it was just a model, and it worked great in many cases. However, we now have a better model – spacetime. Now the same question comes up: is it real or is it just a model? At one time the Rutherford model of the atom was very popular. Was the atom really just a miniature solar system or was that just a model? It worked great, and is still being used today. But the real atom is far more complicated than Rutherford envisioned. We have to be careful about the concepts we use in physics. Many of them are mathematical models and concepts, not real entities. Think of work, for example. I don’t deny that there is such a thing as real work. But work in physics is not what real work is. In physics work is the mathematical product of force and distance. It is a concept. That means, a man holding a heavy barbel for 1 minute but not moving it, has done zero work. He might perspire, but he has done less work than an ant that walked one inch. And it is correct, because “work” in physics has a different meaning than real work. The same can be said of “mass,” “energy,” the “quarks,” the “anti-particles,” etc. They are all concepts and models of modern physics. It is useful to think of reality in terms of those concepts. But it is an unpardonable mistake in the philosophy of nature to think that they are real entities just because they work well in describing the real world. They are describing something real, of course, but they themselves are just models that can be replaced in the future depending on the progress of our scientific knowledge. Medieval philosophers have a term for concepts like these. They are called entia rationis cum fundamento in re, or “beings of reason with a foundation in reality.”
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Dovekin
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Really? How do you know?
The argument you offered earlier, that there is only one real point on any time interval, describes the local situation. But what is the basis for generalizing to the whole world? It is like saying “it is flat here, we must live on a flat earth.”
“The Present” means something different depending on time, place, relative speed, gravity, etc. Since time and space are linked in spacetime, you cannot eliminate time without distortions in space.
Our world is really a 4d world. Or more.
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rom
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Then why can’t you make a Klein bottle without an intersection?
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Dovekin
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I think the problem is with twisting in the time dimension. I only know how to travel forward in time, not backwards or sideways or even twisty, though that should be a couple of dimensions.
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rom
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OK then. I will wait until you succeed in making one, and I will celebrate that with you in 4D. Make sure the Klein bottle is filled with good beer. (I am excited to know how beer taste like in 4D). For the meantime, until you succeed, we will just have to agree to disagree. Deal?
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Dovekin
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You don’t taste beer in 4 dimensions? It is much better when you have time, to savor it.
I do not think a Klein bottle could improve the taste!
I am off to take a run around the Möbius track nearby.
I do not think a Klein bottle could improve the taste!
I am off to take a run around the Möbius track nearby.
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