Selected quotes from the paper:
"we and our collaborators have shown how quantum theory can be re-expressed as a type of ‘classical physics’ in the topos of presheaves on the partially-ordered..."
"All aspects of quantum theory—states and state space, physical quantities, the Born rule, etc.—find a new mathematical representation, which also provides the possibility of a novel conceptual understanding."
I want to know what motivates this choice of abstraction. (as a "new methamatical representation" in itlsef is not an answer) To look for clues I continue...
They first note some foundational problems of probability.
"By definition, the frequentist interpretation requires a large ensemble of similar systems on which an experiment is performed, or a large number of repetitions of the experiment on a single system."
"A particular challenge is posed by those physical situations in which a frequentist interpretation cannot apply, even in principle. For example, if the whole universe is regarded as a single entity, as in cosmology, then clearly there are no multiple copies of the system."
"Such an operational view does not readily extend to quantum cosmology."
"It is always interesting to reflect on what a weather forecaster really means when he or she says ”There is 80% chance of snow tomorrow”."
So they seem to acknowledge that the ensenble pictures simply doesn't make sense. So far I agree about the problem - but then they go on by revealing their intent.
"In particular, an observer-independent, non-instrumentalist interpretation becomes possible."
"As a side remark, we do not see quantum theory fundamentally as some kind of generalised probability theory. Such a viewpoint is almost invariably based on an operational view of physics and, worse, usually comes with a very unclear ontology of both probabilities themselves and the objects or processes to which they apply"
"Moreover, the instrumentalist concept of an ‘experiment’ performed on the entire universe is meaningless, since there is no external observer or agent who could perform such an experiment."
"Of course, in most of science there is a valid instrumentalist view in which the world is divided into a system, or ensemble of systems, and an observer. The system, or ensemble, shows probabilistic behaviour when an observer performs experiments on it. In the ensuing two-level ontology the system and the observer have very different conceptual status."
"Hence, it is desirable to have a (more) realist formulation of quantum theory—or, potentially, more general theories—that could apply meaningfully to the whole universe. This desire to avoid the two-level ontology of operational/ instrumentalist approaches is one of the motivations for the topos approach to the formulation of physical theories."
So it seems to me that their rational motivation for a new framework is,
Characterization of the problem: in the general case there is in fact no external observer, where the instrumentalist or ensemble view can be realized as the two-level ontology.
Preferred conceptual solution: to remove the two-level ontology and thus the observer. This is done by seeking a realist description of this entire game.
Presumably this is one of the motivators.
Now, while I fully agree with their description of the problem, I strongly disagree with their conceptual preference of the desired solution here. Their way of reasoning, and motivators here... in particular their conclusion that the described problem, suggests that we should seek a realist picture... is completely at face with the very spirit of the scientific process and heart of measurement theory. It seems to be to instead by a not so rational idea of finding a MATHEMATICAL picture which does away with the observer, but I sense this idea (while it may be doable to a certain point) will not help.
If I pretend not to see their IMHO misguided motivator and just look further, I see they still use uncountable states for the truth values. This details I have hare to see how it's motivated.
Maybe the motivation for the general topos formalism is one of the earlier papers, but I don't find it in this paper.
"It is always interesting to reflect on what a weather forecaster really means when he or she says ”There is 80% chance of snow tomorrow”."
Just to contrast: My resolution, is instead to take the subjective perspective even deeper. They choose the opposite way.
The difference in my view, between the "external observer" and the "internal observer" (which the seem to forget) is that the former is a descriptive problem, and the action of the observer is not really actively making a difference. It's in this persepective the current QM is formulated, and this is what also Smolin refers to as Newtonian scheme observing subsystems; and you described it as "timeless law"). Now the LATTER picture is not a descriptive problem, it's a decision problem! (at least in my view), this is not even mentioned in the analysis of Ishams paper. Meaning that the meaning of the staement of probability by an inside observer w/o ensenbles and repeatability is that the probability is then of course defined with respected to histories, and the statement is the solution to the decision problem how to "play your cards" rationally. So the external observer picture is a descriptive problem really, the inside observer picture is a GAME. And the corroboration is replaced by survival or continuation of the game. Game over means your'e falsified. This is as I see it completeley different from Ishams views.
"Moreover, the instrumentalist concept of an ‘experiment’ performed on the entire universe is meaningless, since there is no external observer or agent who could perform such an experiment."
Right. But there is an internal observer of course. To say there is no external observer, does not imply there is no observer at all.
The crucial and essential difference though (IMHO) is that the "picture" is not really "experiment" in the sense of ensembles or repetitive preparations etc... but it's rather a GAME. And instead of preparations, and corroborations, what we do is place our bets and play. Corroboration means the game goes on. Falsification corresponds to game over.
This is how evolution works. The goal is not to "predict the future per see" - the goal is to survive. This is NOT a descriptive problem, it contains elements of gaming and decision problems.
Edit: Almost needless to say but if anyone thinks so, there can of course not be an external DESCRIPTION of this game. because then we are back at the external view. That's the point. We only choose to enter the game and play. Still this main prove to be an important insight as it does change the attitude towards modelling.
/Fredrik
