Randomness

[FrediFizzx]

It is a classic and simplest possible example of reducible randomness. It exhibits epistemic randomness, not an ontological one. It embodies the fundamental determinism of nature.

***

Ok, but what does it reduce to? The phrase "reducible randomness" means that it must reduce to something non-random, i.e. deterministic. So what is the deterministic mechanism in your model? There are no explanations in your paper for this except a reference to a coin toss, but we all know that God does not toss coins. (S)he throws dice.

[Joy Christian]

It is a classic and simplest possible example of reducible randomness. It exhibits epistemic randomness, not an ontological one. It embodies the fundamental determinism of nature.

Ok, but what does it reduce to? The phrase "reducible randomness" means that it must reduce to something non-random, i.e. deterministic. So what is the deterministic mechanism in your model? There is no explanations in your paper except a reference to a coin toss, but we all know that God does not toss coins. (S)he throws dice.

The word "reducible" in this context simply means that the said randomness is epistemic in origin, like the randomness in the outcome of a coin toss. The randomness in the outcome of a coin toss is not intrinsic to the coin or to Nature, but stems from our subjective lack of knowledge of the initial state of the system; namely, of the coin. Likewise, if we know the initial state λ in my model (which specifies an orientation of a 3-sphere) for a given run of the experiment, then the outcome A(a, λ) for a given parameter a is completely and deterministically known. This is in sharp contrast to the irreducible randomness presumed within orthodox quantum mechanics. In the case of quantum mechanics, no matter how much information about the initial state of the system is known, the precise outcome, say, of a spin detection, would remain unpredictable. Therefore quantum mechanical randomness is not reducible in general.

***

[Heinera]

The randomness in the outcome of a coin toss is not intrinsic to the coin or to Nature, but stems from our subjective lack of knowledge of the initial state of the system; namely, of the coin. Likewise, if we know the initial state λ in my model (which specifies an orientation of a 3-sphere) for a given run of the experiment, then the outcome A(a, λ) for a given parameter a is completely and deterministically known.
***

The initial state of the coin, unknown to us, is a continuous variable that ends up in a binary outcome, but still assumed to be the result of a completely deterministic dynamical evolution (for which we know the laws: Classical dynamics). What is the equivalent continuous initial state in your model? In other words, how does lambda get its binary outcome? By what process?

[Joy Christian]

The randomness in the outcome of a coin toss is not intrinsic to the coin or to Nature, but stems from our subjective lack of knowledge of the initial state of the system; namely, of the coin. Likewise, if we know the initial state λ in my model (which specifies an orientation of a 3-sphere) for a given run of the experiment, then the outcome A(a, λ) for a given parameter a is completely and deterministically known.

The initial state of the coin, unknown to us, is a continuous variable that ends up in a binary outcome, but still assumed to be the result of a completely deterministic dynamical evolution (for which we know the laws: Classical dynamics). What is the equivalent continuous initial state in your model?

The orientation, geometry, and topology of the 3-sphere.

***

[FrediFizzx]

The randomness in the outcome of a coin toss is not intrinsic to the coin or to Nature, but stems from our subjective lack of knowledge of the initial state of the system; namely, of the coin. Likewise, if we know the initial state λ in my model (which specifies an orientation of a 3-sphere) for a given run of the experiment, then the outcome A(a, λ) for a given parameter a is completely and deterministically known.

The initial state of the coin, unknown to us, is a continuous variable that ends up in a binary outcome, but still assumed to be the result of a completely deterministic dynamical evolution (for which we know the laws: Classical dynamics). What is the equivalent continuous initial state in your model?

The orientation, geometry, and topology of the 3-sphere.

***

I think it actually has to go back to the creation of the 3-sphere.
.

[Heinera]

The initial state of the coin, unknown to us, is a continuous variable that ends up in a binary outcome, but still assumed to be the result of a completely deterministic dynamical evolution (for which we know the laws: Classical dynamics). What is the equivalent continuous initial state in your model?

The orientation, geometry, and topology of the 3-sphere.

***

But the orientation, which is a random binary variable in your model, must obviously be the result of something, and not the cause, if the randomness is supposed to be reducible to something else. So what causes the orientation to be either +1 or -1?

[FrediFizzx]

The initial state of the coin, unknown to us, is a continuous variable that ends up in a binary outcome, but still assumed to be the result of a completely deterministic dynamical evolution (for which we know the laws: Classical dynamics). What is the equivalent continuous initial state in your model?

The orientation, geometry, and topology of the 3-sphere.

***

But the orientation, which is a random binary variable in your model, must obviously be the result of something, and not the cause, if the randomness is supposed to be reducible to something else. So what causes the orientation to be either +1 or -1?

I think you end up with a "turtles all the way down" scenario. Whatever created the 3-sphere in the first place was either left or right handed. Then whatever that was was either left or right handed, etc. So what was the original cause for left or right handedness? Things are just left or right handed in Nature. It is a fact of having a 3D space.
.

[Heinera]

The initial state of the coin, unknown to us, is a continuous variable that ends up in a binary outcome, but still assumed to be the result of a completely deterministic dynamical evolution (for which we know the laws: Classical dynamics). What is the equivalent continuous initial state in your model?

The orientation, geometry, and topology of the 3-sphere.

***

But the orientation, which is a random binary variable in your model, must obviously be the result of something, and not the cause, if the randomness is supposed to be reducible to something else. So what causes the orientation to be either +1 or -1?

I think you end up with a "turtles all the way down" scenario. Whatever created the 3-sphere in the first place was either left or right handed. Then whatever that was was either left or right handed, etc. So what was the original cause for left or right handedness? Things are just left or right handed in Nature. It is a fact of having a 3D space.
.

That sounds very irreducible.

[Joy Christian]

The initial state of the coin, unknown to us, is a continuous variable that ends up in a binary outcome, but still assumed to be the result of a completely deterministic dynamical evolution (for which we know the laws: Classical dynamics). What is the equivalent continuous initial state in your model?

The orientation, geometry, and topology of the 3-sphere.

But the orientation, which is a random binary variable in your model, must obviously be the result of something, and not the cause, if the randomness is supposed to be reducible to something else. So what causes the orientation to be either +1 or -1?

Mathematical necessity (see the definition below) dictates that there are only two possible orientations of a given vector space or a manifold: λ = +1 and λ = -1. Therefore It would be presumptuous to assume that a chance of the initial state form these two possibilities could be anything other than 50/50.


***

[FrediFizzx]

The orientation, geometry, and topology of the 3-sphere.

***

But the orientation, which is a random binary variable in your model, must obviously be the result of something, and not the cause, if the randomness is supposed to be reducible to something else. So what causes the orientation to be either +1 or -1?

I think you end up with a "turtles all the way down" scenario. Whatever created the 3-sphere in the first place was either left or right handed. Then whatever that was was either left or right handed, etc. So what was the original cause for left or right handedness? Things are just left or right handed in Nature. It is a fact of having a 3D space.
.

That sounds very irreducible.

It may sound irreducible but we just don't know do we? Handedness disappears in a 2D space. So perhaps the "cause" is from going from 2D to 3D.
.

[Heinera]

Mathematical necessity (see the definition below) dictates that there are only two possible orientations of a given vector space or a manifold: λ = +1 and λ = -1. Therefore It would be presumptuous to assume that a chance of the initial state form these two possibilities could be anything other than 50/50.
***

But that didn't answer my question. So you have no fundamental theory of why this 50/50 possibilities could be reduced to something deeper?

[Joy Christian]

Mathematical necessity (see the definition below) dictates that there are only two possible orientations of a given vector space or a manifold: λ = +1 and λ = -1. Therefore It would be presumptuous to assume that a chance of the initial state form these two possibilities could be anything other than 50/50.
***

But that didn't answer my question. So you have no fundamental theory of why this 50/50 possibilities could be reduced to something deeper?

I have answered your question.

What could be deeper than the depth itself? The physical space, S^3, is what it is. By mathematical necessity, its orientation can only be either λ = +1 or λ = -1, with 50/50 chance.

You are grasping at straws.

***

[Heinera]

What could be deeper than the depth itself? The physical space, S^3, is what it is. By mathematical necessity, its orientation can only be either λ = +1 or λ = -1, with 50/50 chance.

You are grasping at straws.

***

Ok, so the 50/50 is the depth itself. Irreducible. Why this is less irreducible then QM probabilities is only for Humpty Dumpty to decide.

[FrediFizzx]

Mathematical necessity (see the definition below) dictates that there are only two possible orientations of a given vector space or a manifold: λ = +1 and λ = -1. Therefore It would be presumptuous to assume that a chance of the initial state form these two possibilities could be anything other than 50/50.
***

But that didn't answer my question. So you have no fundamental theory of why this 50/50 possibilities could be reduced to something deeper?

I have answered your question.

What could be deeper than the depth itself? The physical space, S^3, is what it is. By mathematical necessity, its orientation can only be either λ = +1 or λ = -1, with 50/50 chance.

You are grasping at straws.

***

I think what Heine is looking for is a "cause". While what you are saying is good logical physical sense, I don't see any problem in saying that the "cause" is 3D space itself.
.

[Joy Christian]

What could be deeper than the depth itself? The physical space, S^3, is what it is. By mathematical necessity, its orientation can only be either λ = +1 or λ = -1, with 50/50 chance.

You are grasping at straws.

Ok, so the 50/50 is the depth itself. Irreducible. Why this is less irreducible then QM probabilities is only for Humpty Dumpty to decide.

Actually, John Bell explains it very well why any randomness in terms of a hidden variable like my λ is reducible while quantum mechanical randomness is not. Just read the first chapter of his book, or read the less technical chapter about Bertlmann's socks. The fundamental difference between a reducible explanation and irreducible explanation is not difficult to understand.

***

[gill1109]

What could be deeper than the depth itself? The physical space, S^3, is what it is. By mathematical necessity, its orientation can only be either λ = +1 or λ = -1, with 50/50 chance.

You are grasping at straws.

Ok, so the 50/50 is the depth itself. Irreducible. Why this is less irreducible then QM probabilities is only for Humpty Dumpty to decide.

Actually, John Bell explains it very well why any randomness in terms of a hidden variable like my λ is reducible while quantum mechanical randomness is not. Just read the first chapter of his book, or read the less technical chapter about Bertlmann's socks. The fundamental difference between a reducible explanation and irreducible explanation is not difficult to understand.

Trouble is, as you go down, randomness *has* to become more fine-grained. More precisely, it can’t ever be *less* fine-grained. In the lab, when we measure spins of two particles, there are four *joint* outcomes, not two. And they typically have probabilities different from zero, one half, or one. The deeper level has to have at least as many “atoms” of probability, and their sizes have to add up in groups to the sizes of atoms at the top level.

Now I’m happy to believe that at some deep level the handed-ness of space is responsible for the randomness of spin measurements. But I cannot believe it works through a single binary fair coin toss λ.

A real fair coin toss provides an excellent example. An idealised coin is launched vertically into the air with a rotation speed around a horizontal axis X > 0 and a vertical velocity Y > 0. It rises and then falls and hits a flat surface which instantly absorbs all energy and leaves the coin lying flat. In suitable units this could result in the binary outcome Parity(IntegerPart(XY)). If X and Y have a joint smooth probability distribution which extends in both directions far enough, the outcome is almost a fair coin toss. This has been studied long ago by Persi Diaconis. Who is both a great mathematician and a great conjuror. He can toss a coin and have it land heads 95% of the time.

I’ll add some references and pictures later.

Added later:
Preprint:
https://statweb.stanford.edu/~susan/papers/headswithJ.pdf

Picture (Figure 6)
https://www.instagram.com/p/BzPX7h1HuqK

Persi Diaconis, Susan Holmes, and Richard Montgomery
SIAM Review, 2007
https://doi.org/10.1137/S0036144504446436

We analyze the natural process of flipping a coin which is caught in the hand. We show that vigorously flipped coins tend to come up the same way they started. The limiting chance of coming up this way depends on a single parameter, the angle between the normal to the coin and the angular momentum vector. Measurements of this parameter based on high‐speed photography are reported. For natural flips, the chance of coming up as started is about 0.51

[FrediFizzx]

Trouble is, as you go down, randomness *has* to become more fine-grained. More precisely, it can’t ever be *less* fine-grained. In the lab, when we measure spins of two particles, there are four *joint* outcomes, not two. And they typically have probabilities different from zero, one half, or one. The deeper level has to have at least as many “atoms” of probability, and their sizes have to add up in groups to the sizes of atoms at the top level.

Now I’m happy to believe that at some deep level the handed-ness of space is responsible for the randomness of spin measurements. But I cannot believe it works through a single binary fair coin toss λ.
…

It is not the "handed-ness of space". It is 3D space "causes" handedness. Now, why wouldn't the handedness that is caused by the 3D space be exactly 50-50? Of course it would.
.

[gill1109]

Trouble is, as you go down, randomness *has* to become more fine-grained. More precisely, it can’t ever be *less* fine-grained. In the lab, when we measure spins of two particles, there are four *joint* outcomes, not two. And they typically have probabilities different from zero, one half, or one. The deeper level has to have at least as many “atoms” of probability, and their sizes have to add up in groups to the sizes of atoms at the top level.

Now I’m happy to believe that at some deep level the handed-ness of space is responsible for the randomness of spin measurements. But I cannot believe it works through a single binary fair coin toss λ.
…

It is not the "handed-ness of space". It is 3D space "causes" handedness. Now, why wouldn't the handedness that is caused by the 3D space be exactly 50-50? Of course it would.
.

Obviously, if anything, it would have to be 50-50, by symmetry.

But aren’t we saying the same? Aren’t you saying: 3D space *has* “handedness”?

I agree that a mathematical representation of 3D space has a handedness.

Anyway, the words are just a guide to understanding the formulas, as long as we are talking about mathematical models. If we are talking about physical reality ... well that’s the whole question. Can we know anything about it? Probably not, without making some assumptions.

I’m not a professional philosopher, nor a physicist. I think you should tend to trust me on mathematics. But it’s up to you, of course.

[FrediFizzx]

Trouble is, as you go down, randomness *has* to become more fine-grained. More precisely, it can’t ever be *less* fine-grained. In the lab, when we measure spins of two particles, there are four *joint* outcomes, not two. And they typically have probabilities different from zero, one half, or one. The deeper level has to have at least as many “atoms” of probability, and their sizes have to add up in groups to the sizes of atoms at the top level.

Now I’m happy to believe that at some deep level the handed-ness of space is responsible for the randomness of spin measurements. But I cannot believe it works through a single binary fair coin toss λ.
…

It is not the "handed-ness of space". It is 3D space "causes" handedness. Now, why wouldn't the handedness that is caused by the 3D space be exactly 50-50? Of course it would.
.

Obviously, if anything, it would have to be 50-50, by symmetry.

But aren’t we saying the same? Aren’t you saying: 3D space *has* “handedness”?

I agree that a mathematical representation of 3D space has a handedness.
...

I doubt very much that 3D space itself has a handedness. It is matter or forms of energy transport that can have a handedness. But sure, you can have left or right handed basis definitions for defining cartesian coordinates. Usually that is for defining objects via vectors that exist in the 3D space. Right hand rule or left hand rule. Most people are "locked" into the right hand rule and don't think about the left hand rule much. But anyways, yeah we have that 3D space is the cause of the 50-50 "fair" coin toss definition. No problem there.
.

[gill1109]

Trouble is, as you go down, randomness can only become more (it cannot become less) fine-grained. In the lab, when we measure spins of two particles, there are four *joint* outcomes, not two. And they typically have probabilities different from zero, one half, or one. The deeper level has to have at least as many “atoms” of probability, and their sizes have to add up in groups (over the elements of a partition) to the sizes of atoms at the top level.

Now I’m happy to believe that at some deep level the handed-ness of space is responsible for the randomness of spin measurements. But I cannot believe it works through a single binary fair coin toss λ.
…

It is not the "handed-ness of space". It is 3D space "causes" handedness. Now, why wouldn't the handedness that is caused by the 3D space be exactly 50-50? Of course it would.
.

Obviously, if anything, it would have to be 50-50, by symmetry.

But aren’t we saying the same? Aren’t you saying: 3D space *has* “handedness”?

I agree that a mathematical representation of 3D space has a handedness.
...

I doubt very much that 3D space itself has a handedness. It is matter or forms of energy transport that can have a handedness. But sure, you can have left or right handed basis definitions for defining cartesian coordinates. Usually that is for defining objects via vectors that exist in the 3D space. Right hand rule or left hand rule. Most people are "locked" into the right hand rule and don't think about the left hand rule much. But anyways, yeah we have that 3D space is the cause of the 50-50 "fair" coin toss definition. No problem there.
.

OK

But you still haven’t responded to this, Fred:

Trouble is, as you go down, randomness *has* to become more fine-grained. More precisely, it can’t ever be *less* fine-grained. In the lab, when we measure spins of two particles, there are four *joint* outcomes, not two. And they typically have probabilities different from zero, one half, or one. The deeper level has to have at least as many “atoms” of probability, and their sizes have to add up in groups to the sizes of atoms at the top level.

Now I’m happy to believe that at some deep level the handed-ness of space is responsible for the randomness of spin measurements. But I cannot believe it works through a single binary fair coin toss λ.

A real fair coin toss provides an excellent example. An idealised coin is launched vertically into the air with a rotation speed around a horizontal axis X > 0 and a vertical velocity Y > 0. It rises and then falls and hits a flat surface which instantly absorbs all energy and leaves the coin lying flat. In suitable units this could result in the binary outcome Parity(IntegerPart(XY)). If X and Y have a joint smooth probability distribution which extends in both directions far enough, the outcome is almost a fair coin toss. This has been studied long ago by Persi Diaconis. Who is both a great mathematician and a great conjuror. He can toss a coin and have it land heads 95% of the time.