I gathered that the quote I referenced means that the state of a brain at time t is not sufficient to reconstruct memories or other meaningful information. The fundamental point of contention between you and others criticizing the article appears to be that you all believe that there is a storage mechanism in the brain in a similar (analogous?) fashion as a computer. I gather the author claims this is not so. Information is not stored in neurons in such a way that one “retrieves” it by accessing a storage location.
I don’t know enough about this stuff to intelligently comment on the veracity of it. I just know that someone far more knowledgeable than me and just about everyone else commenting says that our intuition about how this stuff works is wrong. That alone is worth causing me to reconsider my intuition on this stuff.
The author is saying our brains do not function like our digital computers, something I think we do all agree on. It is not so clear how the author thinks our brains do work, but he apparently wants us to stop using computer metaphors when discussing their function.
He would have this prohibition extend to the notion that our brains store and retrieve information, which is absurd; one might as well argue that a computer is not a Turing-equivalent device because RAM is not a tape. The author says that scientists will never find copies of words or grammatical rules in the brain, and if, by copies, he means coded in something like UTF-8, then that is, of course, true, but beside the point: if his brain did not have some mechanism that supports the storage and retrieval of this information in some manner, how was he able to write the article in the first place? He claims you won't find copies of Beethoven's 5th. symphony in a brain, but I suspect that at least Beethoven himself, and many conductors of the piece, have had just that - and the soloists who play his piano concertos are not reading from a score, so where does that come from?
I think the author may have ended up making these absurd claims because he is trying to use the trivial brain-does-not-function-like-a-computer argument to prove something that is just an unargued-for intuition: he doesn't seem to think RAM (or perhaps any form of physical information store) could possibly be the foundation for something that works like human memory. He is apparently unaware of the extent that software such as neural networks (or even relational databases) have already extended the concept of information storage and retrieval beyond the simple model of randomly-addressable bytes (which does not, of course, make the point that human memory is like a computer's; what it does show is that the author's low-level comparisons are insufficient to make the larger point he is trying to squeeze out of them.)
I don’t know enough to understand how that is possible or why someone knowledgeable about this stuff thinks this. I have basically the same conception of the brain and how it works as you do. But I’m confronted with the fact that a person far more knowledgeable than me thinks otherwise. It is that fact that causes me to persist in my view with caution. The author may be a crank. I don’t know.
This is not the equivocation that it may appear to be, as it establishes a sort of asymptotic boundary between what is possible and what is not (the more memory we have, the closer we can get to it.) It also means, for example, that we don't have to wonder if there is one computer instruction set or architecture that can perform computations that are impossible by another (again, up to having sufficient memory to complete it.)
The author of the claim you are questioning has not, so far, returned to explain what he means, but I think he is saying that the brain is Turing-equivalent in the informal sense given above: we can compute like a Turing machine, up to the available tape/memory (though with a very limited tape, if we are not writing things down...)
If that is so then I (one of the people here criticizing the article) must say that I don't think it is relevant. An alternative interpretation of the statement, that it says it has been shown that that there is a Turing machine equivalent to the human brain, would seem to depend on believing (as I happen to) that the brain's functioning is a matter of electro-biochemistry that could, in principle, be simulated by a computer, but no-one, so far, has given a demonstration, or even a convincing explanation, of how that works at a Turing-machine level of abstraction.
With regard to the quote you offer: I think it is a simple case of rhetorical overreach -- one might need to know the entire history of that brain to fully understand everything there is to know about its current state, but that does not mean that, absent that full history, the state is meaningless. In understanding what a person is thinking, what they remember (which is an aspect of their brain's state) is more important than what actually happened.