Right, so to actually be at that limit, as in having zero curvature, would be contradictory to having any degree of curvature at all. — Metaphysician Undercover

You have talked right past the point in your usual fashion. The uncurved line is what neatly separates the lines with positive curvature from those with negative curvature. Kind of like how zero separates the positive and negative integers.

So what is important is that it lacks curvature of both kinds.

Where that presents difficulties, is that there is no provision in most people's minds for things to exist in different ways — Wayfarer

If things were not bizarre enough: Rydberg polaritons

I'm trying to stick with traditional and modern philosophy. I'm interested in the question of the nature of abstract objects. Actually I've bought a textbook on it which I'm making some headway with. But I know I'll never understand quantum computers.

You have talked right past the point in your usual fashion. The uncurved line is what neatly separates the lines with positive curvature from those with negative curvature. Kind of like how zero separates the positive and negative integers.

So what is important is that it lacks curvature of both kinds. — apokrisis

Right, but "positive" and "negative" curvature is an arbitrary convention of measurement, just like the number of degrees in a circle. And as I said, to attribute both curved and uncurved to space, is contradiction, unless you can show how space changes from being curved to being uncurved or vise versa. Or perhaps you can show how space changes from being positively curved, to being negatively curved, to justify this convention as other than arbitrary.

Right, but "positive" and "negative" curvature is an arbitrary convention of measurement, — Metaphysician Undercover

What’s arbitrary about it? Parallel lines converge in the one and diverge in the other. An ant traversing a sphere sees a different world from an ant exploring a hyperbolic space. There are concrete differences.

What’s arbitrary about it? Parallel lines converge in the one and diverge in the other. — apokrisis

There's no such thing as parallel lines if space is curved. As I keep saying, that would be contradictory. So your reference, parallel lines, has no place here in a curved space. And your supposed concrete differences are just a product of contradictory premises. I suggest that you look at the differences you allude to, as the difference between internal and external, but the boundary between the two cannot be a straight line. What would constitute the difference between inner space and outer space?

There's no such thing as parallel lines if space is curved. — Metaphysician Undercover

Crikey. And yet two lines - as x = 1 and x = 2 - can start off as points in parallel.

So your reference, parallel lines, has no place here in a curved space. — Metaphysician Undercover

Cripes. You mean non-Euclidean geometry is legit?

And your supposed concrete differences are just a product of contradictory premises. — Metaphysician Undercover

Jeepers. You mean that one of Euclid's axioms just got violated? And hence all straight lines are really just an especially constrained instance of a curve?

I suggest that you look at the differences you allude to, as the difference between internal and external, but the boundary between the two cannot be a straight line. — Metaphysician Undercover

I suggest you read up on intrinsic curvature and stop making a fool of yourself.

The relation between positive and negative curvature is not about a contradiction but our old friend, the dichotomy - the reciprocal relation, the (inverse) unity of opposites.

With respect to LaPlace's Daemon - the accepted wisdom is that Heisenberg's Uncertainty Principle forecloses the possibility of absolute determinism, because there's an inbuilt degree of uncertainty at a foundational level of atomic physics. Banno posted an academic paper challenging the accepted wisdom somewhere upthread, but I confess I haven't had time to read it. — Wayfarer

Hidden variables make everything determined. The electron in an orbital always has a well-defined position and velocity like this. For example, an electron in an s-orbital always has zero velocity, and a position somewhere in the confines of the wavefunction.

Hidden variables make everything determined. The electron in an orbital always has a well-defined position and velocity like this — Haglund

As I'm not a physicist, I can't disprove that, but suffice to say that hidden variable theories are held by only a minority of scientists, to my knowledge. I think the majority opinion is that 'the electron' has no position until it is measured. All you have until then is the equation which describes the distribution of possibilities for where it might be when it is measured but it can't be deduced from that, that it is in an unknown position. This implies that 'the electron' is not an objectively real until it is measured (as explained on the third page of this article under the heading 'Phenomenon'.) Because that undermines scientific realism it is rejected by a lot of people, arguably, this is why the 'many-worlds formulation' is popular, because it avoids this anti-realist implication, but at the cost of introducing many worlds.

Yes, but that minority is because in Copenhagen the standard was set. Einstein didn't agree. But it's what the books teach today. A majority is no proof of being right.

Suppose my random number comes from an observation of unpredictable minute changes in atmospheric pressure?
— jgill

Those changes shouldn't be unpredictable to Laplace's demon — frank

That doesn't make them non-random. You can only predict the gas pressure variations if you know the initial state of the gas particles. You can't predict these. The initial momentum distribution is random.

A majority is no proof of being right. — Haglund

No proof is available in this respect, otherwise there would be no scope for interpretation.

No proof is available in this respect, otherwise there would be no scope for interpretation — Wayfarer

Precisely. There are plans for experiments to discern. Costly and difficult though. But even to imagine this could be discerned 100 years ago!

From my understanding the Copenhagen interpretation simply observes that we can't know what sub-atomic entities are apart from how they appear to us when they are registered by instruments. That's what makes it congruent with Kant's transcendental idealism: the wave function is analogous to the noumenal reality, while the registered particle is the observed phenomenon. We don't know what it is in itself, but only as it appears to us. It is that inability to know exactly which concerns scientific realists. But maybe it's just an acknowledgement of the limits of knowledge. It's an epistemically humble attitude.

It could be though that a particle is the noumenon and the wavefunction the phenomenon. The wavefunction being the observable jacket the particle wears. A wavefunction only reveals it's nature by repeated local interactions of the particles in it (say the flashes on a screen behind the two apertures).

If the sequence is random, no such function exists. Each outcome (B or R) is not determined by a function. Isn't that the definition of a sequence of random choices? — Haglund

Yes, according to the classical understanding of randomness, or rather we should say unlawfulness - in mathematical logic "randomness" tends to refer to algorithmic randomness that refers to the compressibility of a definable sequence such as Chaitin's Constant , whereas we are referring to the common understanding of "randomness" that indicates that the values of a sequence are being produced in an algorithmically unspecified manner -something that the intuitionists call unlawfulness, which includes sequences generated step-wise by repeatedly tossing a coin, or through the 'free willed' choices of the mathematician. For intuitionists, the distinction between lawfulness and lawlessness is practical rather than metaphysical or ontological - if a sequence is generated by using an algorithm, then it is can be said to be "lawful" in pretty much the same way that a good citizen might be said to be "law abiding" - neither of these examples appeals to the metaphysical notions of logical or causal necessity.

What you describe is the classical way of thinking that upholds the traditional philosophical dichotomy between lawfulness versus lawlessness . The classical ontological distinction between a lawless sequence versus a lawful sequence begs the existence of absolute infinity in order to conclude that a 'completed' infinite extension is possible that can be subsequently tested as to whether it corresponds to a definable function. But if the empirically meaningless notion of absolute infinity is rejected for the empirically meaningful weaker notion of potential infinity in which infinite sequences are understood to refer to unfinished sequences of a priori unknown finite length, then the previous conclusion can no longer be considered as meaningful and consequently there is no longer an ontological distinction between unlawful versus lawless processes; all we we can only speak of are similarities between finite observed portions of two or more processes that haven't thus far finished.

That every choice is based on pure chance? If you assess a finite sequence, BRRBRBRBRRBBRRBRB... (which probably ain't random since I typed it right now) and you find a program leading to this sequence, but can this be done with every sequence? Say that I base my choice on the throwing of a coin. Taking the non-ideal character of the dice into consideration and throwing it randomly (by making random movements). Will there always be a function a pattern, beneath the sequence? Is there non-randomness involved? If the underlying mechanism is deterministic, and we're able in principle, to predict an R or a B, can't we say the initial states of the throws are random? — Haglund

When repeatedly tossing a coin ad infinitum, at any given time one has only generated a finite number of outcomes that is always identical to the prefix of some definable function. A classicist is tempted to speculate " this infinite process if continued ad infinitum might eventually contradict every definable binary total function, and hence be lawless", but such speculation isn't testable as it again begs the transcendental idea of a completed infinity of throws only with respect to which lawfulness and lawlessness become ontologically distinguishable.

How about the genetic code? That determines outcomes, does it not? — Wayfarer

When we say that genotypes 'determine' phenotypes we are implictly referring to a class of situations that we recognize as bringing about this determination via the empirical contingencies of nature that we are unable to fully describe, control or predict. And so we are not appealing to causal or logical necessity when we recognise this determination, and are only appealing to our expectations of nature with respect to this recognisable class of situations. We could have alternatively said that genotypes 'miraculously' produce phenotypes with respect to such situations that bring about the magic of nature.

The ontological distinction between miracles and mechanics begs the principle of sufficient reason, which is but another form of absolute infinity in disguise.

So only in the context of infinite experiments we could say something is truly random? But what, say, about the distribution of momenta of the particles in a gas? If the particles move independently (apart from collisions), isn't the distribution random? Or do they still carry the imprint of someone previous interactions where they had functional interactions or common causes?

It could be though that a particle is the noumenon and the wavefunction the phenomenon. — Haglund

But the particle is what appears. The wavefunction never appears but can only be inferred. 'Phenomenon' means 'what appears'.

And so we are not appealing to causal or logical necessity when we recognise this determination, and are only appealing to our expectations of nature with respect to this recognisable class of situations. — sime

Fair enough.

The ontological distinction between miracles and mechanics begs the principle of sufficient reason, which is but another form of absolute infinity in disguise. — sime

:chin:

But the particle is what appears. The wavefunction never appears but can only be inferred. 'Phenomenon' means 'what appears'. — Wayfarer

In the double slit experiment, a 2D cross section of the wavefunction seems to show itself directly on the screen. You could even say that the observation of a single flash of light in this experiment is the wavefunction collapsed to one of the eigenstates of the electron (though a dirac delta is not an eigenfunction, but you get the picture).

The ontological distinction between miracles and mechanics begs the principle of sufficient reason, which is but another form of absolute infinity in disguise. — sime

Now that's philosophy!

That doesn't make them non-random. You can only predict the gas pressure variations if you know the initial state of the gas particles. You can't predict these. The initial momentum distribution is random. — Haglund

Laplace's demon knows the initial states. Obviously the answer to Beanhead's question regarding randomness is: your conclusion will follow from your assumptions.

But the question is, is the initial state random? The initial particle states of the universe seem to be in a low entropy state. Does that make them non-random? Dunno. There doesn't appear to be any patterns in their distribution yet. But time started.

I suggest you read up on intrinsic curvature and stop making a fool of yourself.

The relation between positive and negative curvature is not about a contradiction but our old friend, the dichotomy - the reciprocal relation, the (inverse) unity of opposites. — apokrisis

The mathematics is irrelevant, because as I've explained to you numerous times, and you persistently ignore, the measuring system employed (i.e. the mathematics) is arbitrary. The fact that the measuring system is arbitrary allows you to apply contradictory measuring systems, the system for a flat surface together with the system for a curved surface, then you speak of "space" as if it has contradictory properties.

The reality is that the contradictory systems are really incommensurable, and this incommensurability produces the illusion which you suffer from, the illusion that there is a real, substantial difference between negative curvature and positive curvature. That difference only manifests as a result of applying the premise that "zero curvature" is a real possibility, for curved space. Of course it is not, because curved and not curved are contradictory.

Can't parallel lines on a sphere intersect?

I don't think a line has positive or negative curvature. A 2D surface can have positive or negative curvature, like the sphere and saddle. A torus has both, but can be defined to have zero curvature, like a cylinder.

If you are on a sphere you can only walk at the same distance from someone if you walk with different speeds.

Can't parallel lines on a sphere intersect? — Haglund

I see this as incoherent. A sphere is a 3d curved surface. It requires 3D. Parallel lines require a flat 2D surface, a plane. The two are incompatible, and there cannot in any way, be parallel lines on the surface of a sphere. A two dimensional object is incompatible with a three dimensional object because the one has a premise which the other is lacking, the third dimension.

What is at issue in apokrisis' representation is the claim that the premise of the third dimension might be present within the two dimensional representation as 'the flat plane has "zero" extension in the third dimension'. But that's nothing more than the blatant contradiction of saying that the object is both three dimensional, and not three dimensional (the plane is curved and not curved) at the same time.

A 2D surface can have positive or negative curvature, like the sphere and saddle. — Haglund

This again is incoherent. A 2d surface is a flat plane. To give that plane any type of curvature requires a third dimension. You could give a line (1D) curvature, with a second dimension, but then what you get is a circular plane.

But the question is, is the initial state random? — Haglund

Randomness is just a matter of how a thing is determined, not whether it is.

Randomness is just a matter of how a thing is determined, not whether it is. — frank

Yes, but how is randomly determined different from non-randomly determined? How do initial conditions are different in a random and determined process? How can we see the determined motion of gas particles is random?

Yes, but how is randomly determined different from non-randomly determined? — Haglund

"Random" usually just means a thing wasn't intentionally selected.

The thesis:

.....laws..... — frank

......rational constructs derived from the principles of universality and absolute necessity.....

.....natural laws...... — frank

......rational constructs that act as explanatory devices for occurrences of a specific kind in Nature.....

.....happening by natural laws...... — frank

......that in Nature determinable by that rational construct.....

X is (...) happening by natural laws..... — frank

.....that as an occurrence of a specific kind in Nature determined by that rational construct....

X is logically necessary if it's happening by natural laws. — frank

....X is an occurrence of a specific kind determined by the principles of universality and absolute necessity, therefore because X occurred, it is necessary that it occurred, iff such occurrence is determinable by law......

Given the above, is not mistaken.
————-

The antithesis:

That isn't true, (X is logically necessary if it is happening by natural law), because we can imagine the counterfactual: our universe with different laws. — frank

.....therefore is mistaken.
————

The theorem:
Even if we imagine different laws, they are still laws, by definition. Otherwise, something must be constructed that doesn’t adhere to universality and absolute necessity, in order to permit happenings that are not necessary merely because they happened, as natural law demands. In which case, it isn’t a law that is constructed, which leaves the truth of the original proposition is unaffected.
(Propositions regulated without universality and absolute necessity shall be deemed as rules, and depending on which predicates are assigned, deemed only convictions, and of ever lesser power, mere persuasions)

The proof:
Counterfactual indicates fact that negates established fact. You’re imagining our universe factually different. Regardless, facts are predicated on law, law is predicated on principles, but imagination is predicated on mere inclination. It is classically irrational to exchange the legislative power of principle, for the indiscriminate power of inclination, which are conditions of conviction or persuasion. It is therefore permissible to imagine anything to which one is inclined, but he has no business immediately addressing it as lawful. It follows that even if one images our universe as explainable by different laws, the universe in itself cannot be explained as being different in itself merely because our explanations relative to it, are.

In effect, there is no epistemologically legislative profit in imagining counterfactuals in opposition to established law, absent the exchange of imagination for law. Our universe as it is but explained by different laws is an empty conjecture until we actually have the different laws with which to explain it, to determine that it is possible to still understand our universe as well as or better than we do now, however different such understanding may be.

The conclusion:
That the universe may be explainable by different laws is not sufficient to falsify the truth of the proposition that X is logically necessary if it happens by natural law.
————-

Epistemic possibility has nothing to do with that. — frank

How could it not? It is our human epistemology alone, which immediately makes any epistemic relation inescapable. We create the doctrine, we subject ourselves to it, therefore it is us. Nature, on the other hand, has nothing to do with our epistemic possibilities, but is only the occasion for its exercise.

To select implies intent already. I guarantee you that you can't randomly select 10 marbles out of a vase with red and white marbles. Approximately random, maybe... Quasi-random. Even if you take one gas atom at a time out of a gas mixture of oxygen and helium gas (equal amounts of both atoms) it's still tricky.