Classical and Quantum Probabilities as Truth Values

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http://arxiv.org/abs/1102.2213

Classical and Quantum Probabilities as Truth Values
Authors: Andreas Doering, Chris J. Isham
(Submitted on 10 Feb 2011)
Abstract: We show how probabilities can be treated as truth values in suitable sheaf topoi. The scheme developed in this paper is very general and applies to both classical and quantum physics. On the quantum side, the results are a natural extension of our existing work on a topos approach to quantum theory. Earlier results on the representation of arbitrary quantum states are complemented with a purely logical perspective.

Today i spotted this paper by Doering an Isham it's really interesting it reminded me somewhat of Careful's new book and also of Fra's ideas. Take a look !.
 

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  • #2
Fra
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Thanks for the link, I'll take a closer look later.

On the frist skimming of the first paper I think they correctly note some of the problems of the probability notion, but their escape to seek the neo-realist resolution doesn't seem like what I advocated.
About careful's and Noldus ideas I'm not sure. I still have not had time to read more of his paper yet. If you did read more of Noldus book that I did, please add your associations to this paper and this ideas, it could be interesting. For myself, things on my todo list are constantly beeing timeout as new things are added all the time.

About their choice of representation I'm not sure what the motivation is. I admit that I am no expert in category theory, but I am working with "sets of non-commutative sets" where there are flows defined in between. Here probability and quantum information are special cases.

I'll try to get around to read it later. I read so many things that I keep forgetting it. I know I read Isham before but I do not recall in depth what his reasoning goes like. I would assume it was not to my liking as it didn't stick to my memory.

/Fredrik
 
  • #3
Fra
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Since the category theoretic perspective is not something I'm fluent in I see the paper is somewhat "hard to read" for someone that first has to get familiar to all the symbolic notions. All that first, in order to read out the physics content.

If anyone is already familiar with Ishams research vision, it would be great if you can jump in.

Apart from "reformulating" existing models into a different language, what other physical conjecture or benefits does he add?

I'll try to skim later but I see now that it's written very densely - meaning it takes at least me even longer time to read.

/Fredrik
 
  • #4
Fra
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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
 
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  • #5
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I fully agree by giving this paper a second thought. By my humble opinion their taking the classical physical worldview much to serious. All fingers point to deepenining of quantum mechanics and a generalization of quantum mechanics. But they say nothing about that.

How this al will work out i don't know.
 
  • #6
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I think physics can laern a lot from Biology, Neuroscience and psychology, if they ever want to put more evolutionary ideas in physics and to put it in a more philosophical view. And don't lean only on mathematics that is to rigid !.
 
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  • #7
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Whenever I see this topos-based physics stuff that Isham&co have been doing it seems very interesting to me, and it looks valuable as pure math, but I always struggle with trying to connect what they're doing to "reality" or determine how, if at all, their construction practically differs from more conventional constructions of quantum physics. It may be this would be quite clear to the people actually in Isham's group but I get lost in the category theory vocabulary very quickly :(

It seems like we should not have to satisfy ourselves with considering what their approach teaches us about the conceptual/philosophical underpinnings of quantum physics, they're doing MATH, it ought to be possible to take their technique and try to calculate things. Has anyone ever attempted this?
 
  • #8
Fra
3,097
144
Whenever I see this topos-based physics stuff that Isham&co have been doing it seems very interesting to me, and it looks valuable as pure math, but I always struggle with trying to connect what they're doing to "reality" or determine how, if at all, their construction practically differs from more conventional constructions of quantum physics. It may be this would be quite clear to the people actually in Isham's group but I get lost in the category theory vocabulary very quickly :(

It seems like we should not have to satisfy ourselves with considering what their approach teaches us about the conceptual/philosophical underpinnings of quantum physics, they're doing MATH, it ought to be possible to take their technique and try to calculate things. Has anyone ever attempted this?
The main problem I see that they do not motivate their approach physically in the beginning (*). Instead it turns out like declaring a whole new mathematical formalism, that is tedious to adapt to. All that, without even knowing what increased fitness of physical predictions and modelling it gives in the end. This is what disturbes me the most. I find this typical for certain mathematical type mindsets. I think it's pretty obvious at least from a sales perspective that this motivation should come first in the paper, or how would you expect physicists to keep reading, except those interested in the mathematical formalism alone (those do exists, and they also need feeding I suppose).

(*) Or maybe the DID, but if that was it, then I'd say the motivational parts are extremely thin. I find their argument to search for the realist abstraction not even close to sufficient motivator for their new complex formalism. But I can imagine that those who for some reason share their motivation, it may make more sense.

My hunch admittedly based on very incomplete technical understanding of their ideas is that given their constructing principles it's unlikely to be of much help for me in the search for interacting inference models. The realist stance is even righly opposite in spirit to the inference view.

/Fredrik
 
  • #9
Hi all. I'm new to the forum. Just stumbled across the link to this thread. Anyway, I noticed that some of you were asking for an explanation of some of the motivations and reasons behind the topos theoretic approach to quantum mechanics, and since I'm currently doing a PhD partially focused on this area, I felt obligated to at least try to help in this regard.
Basically, the approach was invented in the 1990s by Chris Isham and Jeremy Butterfield (who is a philosopher rather than a physicist). There were several main motivations. Firstly, in the traditional Hilbert space formulation of quantum mechanics (qm), it is natural to interpret the algebra of propositions that can be made about a given quantum system as the lattice of projection operators on the Hilbert space associated with that system. Notoriously, this lattice lacks many of the crucial logical properties that hold with respect to Boolean algebras of propositions associated with classical systems. Firstly, they are non-distributive, and this means that it becomes very difficult to define a suitable notion of implication, and hence of deduction, with respect to an algebra of quantum propositions. Also, the Kochen Specker theorem tells us that it is impossible to simultaneously assign truth values to all of the propositions in such an algebra.
The first major result of the topos theoretic reformulation of qm is that it provides a new representation of the algebra of propositions of a quantum system that is both distributive and allows for the simultaneous assignment of truth values to all the propositions in the algebra, but of course, these truth values are generally non classical. In this framework, the Kochen Spekker theorem has an interesting, though rather technical interpretation, I.e. that the state space of a quantum system has no elements! So in this sense, the topos theoretic formulation can be seen in part as a response to problems originating in quantum logic.
I guess that what makes the idea interesting for physicists is not this aspect of the project, but rather the fact that it provides a general universal framework in which all physical theories could conceivably be formulated. For example, the universe of sets, in which classical physics is formulated, is a topos. The fact that we can find a topos in which to formulate quantum mechanics means that we are able to see the mathematical relationship between classical and quantum physics in a clearer light, since they both inhabit the same kind of mathematical structure. The hope is that this will also allow us to gain a better understanding of the relationship between Gr and Qm and thus will have implications for quantum gravity.
Anyway, I rushed many of the explanations because it's late where I am, but I hope this helps. Let me know if you want any more detail.
Ben
 

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