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. — apokrisis
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. — apokrisis
There's no such thing as parallel lines if space is curved. — Metaphysician Undercover
So your reference, parallel lines, has no place here in a curved space. — Metaphysician Undercover
And your supposed concrete differences are just a product of contradictory premises. — Metaphysician Undercover
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
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 — Haglund
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
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
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
How about the genetic code? That determines outcomes, does it not? — Wayfarer
It could be though that a particle is the noumenon and the wavefunction the phenomenon. — Haglund
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
The ontological distinction between miracles and mechanics begs the principle of sufficient reason, which is but another form of absolute infinity in disguise. — sime
But the particle is what appears. The wavefunction never appears but can only be inferred. 'Phenomenon' means 'what appears'. — Wayfarer
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
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
Can't parallel lines on a sphere intersect? — Haglund
A 2D surface can have positive or negative curvature, like the sphere and saddle. — Haglund
Randomness is just a matter of how a thing is determined, not whether it is. — frank
.....laws..... — frank
.....natural laws...... — frank
.....happening by natural laws...... — frank
X is (...) happening by natural laws..... — frank
X is logically necessary if it's happening by natural laws. — frank
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
Epistemic possibility has nothing to do with that. — frank
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