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DanielLC comments on Can you recognize a random generator? - Less Wrong
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All of them will decay and all of them will not decay in different worlds.
Let's assume for the sake of argument that the Copenhagen interpretation is true. Before the particles decay, there is no way to tell which decay. It's random. After they decay, you can tell by looking. You know which ones decayed. It's not random at all.
Randomness is a state of the mind. All indeterminism tells you is that randomness must be a state of every mind that exists before the event.
Imagine a universe where it's always possible to tell the future from the past, but it's not always possible to tell what's on the right from what's on the left. This is deterministic, but if you rotate it 90 degrees, it isn't. A coordinate transformation can't be changing whether or not something is random, can it?
Time is not quite like space (or, in the PDE language, initial value problems are quite different from boundary value problems). There are QFT techniques that treat time as "imaginary space", but their applicability is quite limited and they certainly do not justify the view that "Randomness is a state of the mind", which is either untestable or manifestly false.
Could you be more specific, in what sense different?
In any case, final (or terminal?) value problems are the same as the initial value problems, PDE-wise.
Start with the obvious link, look for hyperbolic and elliptic PDEs and ways to solve them, especially numerically. The wave equation techniques are different from the Laplace equation techniques, though there is some overlap. Anyway, this is getting quite far from the original discussion.
Not quite. For example, the heat/diffusion equation cannot be run backwards, because it fails the uniqueness conditions with the t sign reversed. In a simpler language, you cannot reconstruct the shape of an ink drop once it is dissolved in a cup of water.
I was mainly asking what differences are, in your opinion, important in the context of the present debate.
1) You can run the diffusion equation backwards, only you encounter problems with precision when the solution exponentially grows.
2) Fundamental laws of nature are [second order in time and - edit:that's not true ] symmetric with respect to time reversal.
Well, we are quite a ways from the original context, but I was commenting on that treating time and space on the same footing and saying that future and past are basically interchangeable (sorry for paraphrasing) is often a bad assumption.
In other words, it is ill-posed and cannot be used to recover the initial conditions.
This is a whole other debate on what is fundamental and what is emergent. Clearly, the heat equation is pretty fundamental in many contexts, but its origins can be traced to the microscopic models of diffusion. There are other reasons why the apparent time reversal might not be there. For example, if you take the MWI seriously, the branching process is has a clear time arrow attached to it.
With precise measurement you can. Once started to be solved numerically different initial conditions (for the standard diffusion equation) are all going to yield constant function after some time due to rounding errors, so the information is lost and can't be recovered by time-reversed process. But as a mathematical problem the diffusion equation with reversed time is well defined and has unique solution nevertheless.
From what I recall, the reverse-time diffusion u_t=-u_xx is not well posed, i.e. for a given solution u(t), if we perturb u(0) by epsilon, there is no finite t such that the deviation of the new solution from the old one is bounded by epsilon*e^t. A quick Google search confirms it: (pdf, search inside for "well posed")
I didn't realise that "well-posed" is a term with a technical meaning. The definition of well-posedness I have found says that the solution must exist, be unique and continuously depend on the initial data, I am not sure whether this is equivalent to your definition.
Anyway, the problem with the reverse dissipation equation is that for some initial conditions, namely discontinuous ones, the solution doesn't exist. However, if a function u(x,t) satisfies the diffusion equation on the interval [t1,t2], we can recover it completely from knowledge of not only u(x,t1), but also from u(x,t0) with any fixed t0 lying between t1 and t2.
A small nitpick: The Schrodinger equation is not second order in time.
The Dirac one as well. Corrected.
Let me start over.
You could define "randomness" to mean indeterminism. If you do so, you would call the results of a fair, indeterministic coin toss random. Even so, if I tossed such a quantum coin, and you saw it land on heads, you would not be willing to bet even money that it landed on tails. P(coin landed on heads|your current information) ≈ 1. When you're finding expected value, this is all that matters.