Imagination and the Unreasonable Effectiveness of Mathematics by Armin Nikkhah Shirazi

Dear Adel,

I just left a comment at your blog with some suggestions on how you might make it easier for me (and possibly others) to follow your arguments. You say that I am trying to do what you have done with your simulations, but unfortunately this is far from obvious to me. So let me ask you the following questions (to which I have concrete answers based on the theory that I am pursuing) to see whether our answers match up:

1) You say that your simulation lead to the Schroedinger equation. Where does the imaginary factor i in that equation come from in your simulations?

2) The best I can tell, what you have built is a probabilistic model in which the states should be identified with probabilities, not probability amplitudes. If the states are in fact probability amplitudes, which part of your model leads to that conclusion?

3) How does the exponential phase factor associated with each quantum state arise in your simulation?

4) How do you obtain the Born Rule?

5) How do you formalize the distinction between actuality and potentiality? Note, contrary to your assertions above, my conception of actualizability is not the same as randomness. In particular, I talk about a specific kind of actualizability which I call pro-actuality, and it is as far from randomness as possible: As an example, consider a hypothetical coin which no matter how often you flip it, will alway land on heads. Using standard mathematics, I do not know of a way in which an experiment with such a coin in which you flip it can be distinguished from one in which you don't, because in both cases we would say that the probability of an outcome of heads is 1. Pro-actuality (and actualizability in general) is meant to bring in the distinction between cases in which the event has happened from those in which it has not.

Answering these will go some ways to help me better understand your model.

Best wishes,

Armin

Hi Marc,

I had been thinking about what might have led you to believe that I would consider abstract mathematical objects as potentialities, and it occurred to me that possibly it was something I said in the conclusion of my essay, namely "...the freedom to choose one's axioms

coupled with the requirement of consistency should naturally lead us to expect mathematics to be unreasonably effective in modeling reality, but that this unreasonable effectiveness only exists, as it were, as an actualizability..."

If this passage is responsible for that, I'd like to mention that I did not literally mean that undiscovered mathematics exists as an actualizability (as signaled by the words "as it were"). I just phrased it that way because it seemed pleasing to me that the central distinction in the case study was analogical to my overall argument.

Hope this cleared things up a little,

Best wishes,

Armin

PS. When you said that you could not follow my arguments in detail, I missed a chance to ask where you got lost, so that in future expositions I can take greater care to try to explain those parts better. I would still appreciate any constructive criticism in that respect.

Dear Adel,

Just a quick remark that your comment seems to confirm my suspicion that the probabilistic model at which you have arrived is mathematically inequivalent to QM. In and of itself, this is not necessarily a bad thing, if you can squeeze out testable predictions of experiments which have not been performed yet.

But if your goal is to re-derive QM from your model, it really has to be mathematically equivalent. That means that the central differential equation has to be one that is mathematically the same as the Schrodinger equation, including the imaginary factor i.

I understand that your concern is with getting to a more general framework that encompasses QFT, but if you don't pay attention to these intermediate steps, your chances of success will be greatly diminished.

Best,

Armin

Dear Michel,

Thank you for your comments. Yes, I still need to fill in many more details, which I hope will be forthcoming soon, but it is true that in my view all the "weirdness" associated with QM comes out the fact that the quantum states are spacetime manifestations of objects which are in the sense outlined in my paper incomplete.

I will shortly read your essay as well.

Thank you and best wishes,

Armin

Dear Mohammed,

Thank you for the kind words. Yes, I like to think that though people do mention the role of imagination in mathematics occasionally, they do not really appreciate how crucial it has been to all the advances in the field.

I will shortly leave a comment on your essay and wish you good luck in the contest as well.

Nest,

Armin

Dear Michel,

Given that your essay had a strong emphasis on phenomena related to contextuality and non-locality, I thought it would be a good idea to try to clarify one aspect of my paper that you might have come across which I already know others seem to have misunderstood, and which you might have perhaps also found not easy to understand when you read my paper.

I claimed that one of the analogies I gave permits one to understand the correlations which are analogous to those observed to obey QM as a form of pseudo-nonlocality. Applied to QM, this emphatically does not mean that I think that the usual arguments for non-locality are incorrect, but that they may not apply, because in order to be applicable they require "beables" between measurements.

You may have noticed that the object I defined as an incomplete spacetime vector is characterized by the absence of anything that could be described as a beable in between measurements; if there are no beables, then there is nothing that can receive an influence from the outcome of the measurement on another entangled quantum object. There is of course still "something" that is enforcing the correlations, my point is that if the mathematics I am trying to build is correct, the correlations cannot be enforced by any sort of superluminal influence, and the way I understand non-locality to be defined is in terms of some superluminal influence.

So how are the correlations enforced, then? I believe that the mechanism lies at pre-emergence level. But then, if the correlations are enforced before, say, an electron "comes into being" upon a "measurement" (i.e. the completion of the incomplete spacetime vector to a complete one), then it cannot be said to come from "within spacetime". I think so far Gisin is the only one who has been willing to stick his neck out to say something in this direction.

The bottom line is that I am pursuing a theory that is local at the expense of a radical form of non-realism, but then phenomena like contextuality would seem to support such a perspective.

I hope this helps make my perspective more understandable,

Armin

Dear Bill,

Thank you for your comments. I regret that you found the logical foundation part of my essay hard going. I could not omit it, because without it the novel mathematical structures that are characterized by incompleteness would seem like non-sense, and I could not describe it more fully, partly because of the length restrictions and partly because I have not yet worked out all the details.

As for my argument for pseudo-nonlocality, perhaps it helps if I explain it in this way: The incomplete objects I defined imply that in between measurements there are no "beables": The only spacetime events associated with the incomplete spacetime vector are those labeled as x_i and x_f, and both of those are measurement events.

By definition, non-locality implies the superluminal transmission of some influence as well as a receiver. In my example, if there is no beable before Bob's measurement but after Alice's measurement, then there is nothing that can "receive" such an influence. So, in essence, the scenario altogether side-steps the usual arguments for non-locality (I actually think the arguments are correct!!) by giving a description that might be called radically "non-realist" precisely in the sense that the path integral and wave function are actualizable manifestations of intrinsically "incomplete objects" which are are ontologically distinct from complete (and therefore actual) objects that satisfy the criteria for being "beables".

What I am saying so far is really perfectly in line with the orthodox understanding of QM, which says that unless you measure a quantum object it has no definite properties, clothed in novel concepts and terms that are meant to take the mystery out of these phenomena and promote deeper understanding as well as the ability to make novel predictions.

Where the "deeper understanding" and "novel predictions" come in is in the explanation in how the correlations are enforced. It is here that this framework leads to novel and very unfamiliar ideas (but note, every truly new idea was highly unfamiliar in the beginning) that have implications both for geometry and topology. I believe that the key concept here is what I call an "incomplete embedding".

I believe that a space A which is incompletely embedded in another space B is not a subspace of the latter, in the sense that you cannot arrive uniquely at metric relations within A by projecting from the metric of B (the problem, for instance, in 3D is which 2D plane to project to, since in my example, the plane simply has no z-position). If I am right, this means that the metric intervals of A and B are independent, in the sense that two objects could be far apart in B but right next to each other in A (if A were a subspace of B, then this could not happen: Two objects would have to be either next to each other in both spaces, or else be in different subspaces, say object 1 would be in subspace A and 2 would be in subspace A', where the two subspaces are separated along a direction not contained in either but contained in B). This is the intuition my Euclidean analogy was meant to bring out, but the proportionality of the metric interval to proper time in Minkowski space complicates this somewhat: Instead of a plane, we need to consider the boundaries of successive lightcones.

If the metric intervals of the two spaces (In Minkoswki they are the 2+1D space that is the boundary of each one of the successive the light cones and the 3+1D space that is spacetime) are independent (in fact, I believe they are "orthogonal"), then the correlation could be enforced at a pre-emergence level (i.e. before the underlying incomplete object is completed) without requiring any sort of superluminal influence (recall that entangled particles are always either within or at most at the boundary of the lightcone of the entanglement event). Once the emergence occurs, the actual object that emerged out of the superposition of actualizabilies by which the incomplete object manifested itself will have the correlated property, even though in the higher dimensional space it is "far away" from the other.

This is the qualitative picture of the correlated measurement phenomena that I believe lies behind all these Bell phenomena. Of, course a quantitative description will be necessary before anyone other than me believes this, but I really hope I could at least convey the rough picture. It boils down to the idea that the correlations reflect metric relations which are not those that characterize ordinary (meaning actual or complete) spacetime objects.

On the paper by Chiribella et.al. : I just read it, and that particular paper does not make it obvious how they recover standard QM from their list of axioms. I did take a look at the original paper, and, as it seems quite technical, it would require an investment of time to be able to truly understand their ideas and thereby judge how similar their ideas are to mine.

But let me at least say this: While I do not a priori think of actualizability in terms of information, I suppose one could try to make such an identification, in which case their ideas do, at least to some extent, become similar to mine. In fact, I wrote an essay in the essay contest 2 years ago which, among other things, made this point (titled "It, bit; Object, Background").

http://fqxi.org/community/forum/topic/1919

BTW, this was before I had any concrete ideas of formalizing my default specification principle.

Two major differences that I see are that just saying that "quantum states are information", even if it reproduces standard QM does not seem to point any deeper insight (though, I could be perceiving this because I have a geometric bias) and that they do not, as far as I can tell, seem to make an ontological distinction between immediate pre-and post-measurement states. That would seem to imply that everything is information. I have serious difficulties with thinking of, say, myself as "made out of" pure information.

So, in short, there may be a limited analogy, but I do not see much of a point yet in trying to pursue it further.

I hope you found my answer understandable and useful.

Best wishes,

Armin