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I was emphasizing the possibility that what we see is a self-unfolding process (an algorithm in process of writing itself, as it were), wishing to set that over against the idea that there was a pre-existing program that was then 'run'. I'm not sure that's a distinction without a difference (although it could be).

I'm not sure if I'm following this, but an algorithm cannot create create information. An algorithm writing another algorithm only reduces the order of the expression of that algorithm but adds no information to the system (i.e. creates no new algorithms) that was not already there. The reason this is not obvious is that algorithmic induction (discerning the high-order expression of an algorithm from a lower-order expression) is grotesquely intractable such that we prove it in mathematics but we will never have the computational capacity to experience it in anything but a vaguely probabilistic sense. In fact, cryptography relies heavily on the fact that algorithmic induction is extremely expensive while algorithmic deduction (how most software works) is extremely cheap, effectively creating a one-way function.

In short, an algorithm never generates a new algorithm, only a lower order expression of the same algorithm. But if we observe that lower order expression, it is nearly impossible to discern the higher order algorithm that generated it even in toy cases where very little information is involved. As a point of reference, an algorithm with 256 bytes of information can easily generate a low order expression with an apparent complexity the size of our universe -- even if we observed that universe for a million years, we would never be able to divine the 256 bytes of state that generated what we were observing as a mathematical fact.

In fact, I tend to the view that all information is embodied information, that there was never, is not now nor will there ever be any information that is not physically carried or mediated.

Sure. Information is existence, and at any level of abstraction one cares to use -- that we can conceptualize some bit of information means it exists physically somewhere. We like to think of the substrate being independent of the information, but that is a bit of an artificial distinction (though very convenient for engineering purposes).

In short, an algorithm never generates a new algorithm, only a lower order expression of the same algorithm. But if we observe that lower order expression, it is nearly impossible to discern the higher order algorithm that generated it even in toy cases where very little information is involved. As a point of reference, an algorithm with 256 bytes of information can easily generate a low order expression with an apparent complexity the size of our universe -- even if we observed that universe for a million years, we would never be able to divine the 256 bytes of state that generated what we were observing as a mathematical fact.

One of the most intriguing posts I've seen in a long time. Can you recommend any websites (for the math novice, if possible) that explain this in greater detail?

But if we observe that lower order expression, it is nearly impossible to discern the higher order algorithm that generated it even in toy cases where very little information is involved.

As a "practical" example, trying to discern the hyperspatial expression for the interaction potential for two molecules, solely from the experiemental scattering information, can be a pain in the ass to say the least...

...even aside from the difficulties of fitting a functional form that behaves in the way you want.

Cheers!