Dear Ian,
I am very glad that you were able to submit an essay to this contest.
I was delighted to read it and, as expected, it was interesting and thought-provoking, which I think are the two most important qualities.
That being said, you can correctly infer that it has me thinking, which means I have a lengthy response.
In the essay, you focus on the topic of questions, which is rather dear to my heart. It is, interestingly, the reason that I got into quantum foundations in the first place. When I was at NASA Ames, I was interested in designing intelligent instruments so that our probes could be more autonomous and function more effectively at greater distances from Earth, such as Jupiter and Saturn (which have round-trip light travel times of anywhere from something like 2 to 4 hours). Being expert in Bayesian data analysis, and familiar with the various foundations and derivations of probability theory, I wondered if there was some way to consistently quantify the relevance of questions, so that our machines could compute with questions and thus decide which experiments were most relevant to the mission. This was a way of automating experimental design.
You won't be surprised to learn that you can derive a calculus for relevancy by considering the underlying symmetries among questions. And, it was while discussing this over dinner, that I casually mentioned to Philip Goyal that we ought to be able to derive the Feynman rules similarly. As you know, we succeeded at that.
Back to questions.
In my work, I settled on defining questions in the way suggested by Richard T. Cox (1974) where a question is defined as the set of all statements that answer it. Read that previous sentence several times until you understand it, as this idea is critical to much of the commentary that follows. In terms of lattices, questions are then downsets of logical statements on the Boolean statement lattice (if that helps).
Since questions are sets of statements, set union and set intersection result in a logical OR and AND for questions. However, one can show that there is no negation, so that algebra is NOT Boolean. (Sadly, Cox got this wrong, despite the fact that he had published a proof that the algeba could not be Boolean. He was just TOO familiar with Boolean logic.) I was able to show that, based on Cox's definition of a question, the resulting algebra is the Free Distributive Algebra. There is an OR and AND operation, but no negation (https://arxiv.org/abs/physics/0403089).
The symmetries of this algebra result in a quantification (think, measure), which I called relevance, which has a Sum Rule and a Product Rule, and a Bayes theorem. Relating this relevance to the probabilities of the statements that define the question results in relevances being entropy (https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20040081076.pdf). Beware there are a few conceptual mistakes in this paper, since the question algebra is kind of upside-down from what you would expect thinking Boolean. These are most corrected in (https://www.semanticscholar.org/paper/15-Valuations-on-Lattices-%3A-Fuzzification-and-its-Knuth/a92e2780f218dc1b177f5f6778c42162465fad58). I am now working on an updated paper that fixed these problems.
Whew! That is a lot of background, and this is now more of an essay than it is a commentary!
You really get into questions when you discuss Comprehensibility.
You note that the answer to "What is the color of my hair?" will never be "narrow" or "dog". Logically, it couldn't be, because none of those answers would be members of the set that define the question, nor would they, by themselves, imply any statements that answer that question. But you are not as interested in the underlying logical, as you are the physicality as in the outcome of an experiment.
This is where your essay relates more to the relevance of experiments (which I would call experimental questions) to the issue that you want to resolve. An experiment that could result in an answer "narrow" or "dog" would have a very low relevance to the issue "What color is my hair?" And you would be foolish to perform such an experiment, and certainly would not expect the experimental result to be relevant to the question. For this reason, I would not say that experimental tests do not tend to remain within the physical context because of a principle. Instead, I would say that the experiments are selected, or rather, designed, so that their results are expected to be most relevant to the issue to be resolved. This is designed to work.
When discussing Comprehensibility, you say "By simply asking a question, we immediately establish a context which limits the scope of the inquiry." This is exactly right! I would turn that around and be more pragmatic saying, "the context, or the scope of the question, defines the question."
There were also many side points that you made which resonated with me.
At one point you discuss computing the digits of pi and how one might decide how accurate a certain computation was. This is an interesting question, especially since you discuss computations in different number bases. I wondered if you were aware that in base-16, the Nth digit of pi can be computed independently from the others. there exists a formula, called the Bailey-Borwein-Plouffe (BBP) formula, which can be used in base-16 to compute the nth digit of pi independent of the other digits (Bailey, Borwein, and Plouffe, 1997). In base-16, the digits of pi are absolutely predictable. (Perhaps the lesson is that we should be working in base-16 rather than a base system determined by the number of digits possessed by the common ancestor to the modern tetrapods.)
I like your example of calculating 1/10 exactly on a computer. I have used this as a demonstration on Computational Physics that computers do indeed make errors and that (physicist) programmers need to be aware of how those errors happen and propagate.
I also liked your comment that "anyone who has spent any time in the laboratory will attest to the fact that the real world is far messier than theory would have us believe". I think of this fact every time that I see Stern-Gerlach experiments described in theory papers. There are always two neatly separated beams of sliver ions... yeah, right!!! I took an experimental course on quantum photonics last year so that I could incorporate those experiments into our Advanced Physics Lab course. It was a joy to do those experiments myself: delayed choice, Bell inequality, etc. It was also astonishing to see that some of what theorists say happens in an experiment really doesn't. It made me appreciate that one problem with theoretical QM is that the theorists thought experiments are often just wrong. QM theorists should really do QM experiments to build up an intuition because some of the intuition we have been taught is simply wrong. But that is another commentary altogether!
Your comment about John Wheeler's stance that the answers to binary questions were the basis of all that exists is interesting in this context. I need to give this some careful thought. But at first sight I feel that this may be a truism in that questions are sets of statements. So what Wheeler is saying could be rephrased as saying that the statements that we consider as possibilities form the basis of our knowledge of what exists. My response to such a rephrasing, "Well, I should hope so! Otherwise, you have missed something!"
Further along in that paragraph, you mention that some of physics might be unknowable. This is Shcumacher and Westmoreland's Information Isolation. We can't measure quantum phase, only phase differences. Yes! My thought about this is that the phase does not matter as much as the phase difference. Again, a matter for more careful thought!
Later you talk about truth and non-commuting observables. However, I think of these observables as incommensurate descriptions (think the FT). If two descriptions are incommensurate, then you cannot expect to use both descriptions simultaneously. How relevant is this to Godel? I am not sure. Again, some careful thought is called for. But it might come down to it not being appropriate to assign truth values to descriptions.
Thank you again, for an impressive and thought-provoking essay. I think that re-reading your essay with the ideas of question-statement duality in mind, would be helpful. There is much to ponder here!
Thank you!!!
Kevin
PS Sorry to respond to your essay with an equally long essay!
