Imagination and the Unreasonable Effectiveness of Mathematics by Armin Nikkhah Shirazi

Dear Marc,

Thank you for engaging with me even after my relatively heavy criticism.

"For you (correct me if I'm wrong), the "real world" is the observable physical universe and mathematics is a way for us humans to represent it: naturally, you find it is most important to study how mathematics does this, and hopefully improve the definition of mathematics to help it do its job better."

Yes, that is largely correct. I would only add that the effort "to improve the definition of mathematics" is for me a means to an end, which is to understand reality at the deepest level. I did not start out with the foundations of mathematics but was in some sense "forced into it" by the realization that some of the ideas I thought would explain how the universe works simply could not be expressed using the language of contemporary mathematics. As you know, this is one of the ways in which new mathematics comes about.

"For me, what is most important is to find a satisfying answer to the question "Why is there something?". It seems to me that the only answer that does not create more questions has to be something like "all abstract structures simply ARE, and one of these IS our observable universe". I believe all of mathematics (being abstract) simply IS, but it doesn't mean of course that we, human mathematicians or physicists, automatically have access to it all."

Yes, I understand this point of view because I have entertained it myself. My criticism in the last post was meant primarily to

1) compel a self-examination of what appeared to me an instance of "moving the goal post" in response to one of my challenges

2) compel an examination of what, absent further specification, appears as contradictory evidence. Note, contradictory evidence is not contradictory proof. Perhaps there is a way of overcoming the difficulty I referenced, but before we know this it has to be acknowledged as such.

Perhaps it helps if I lay out an analogous difficulty that I face in my project (i.e. I am throwing a challenge at myself in your place). As you know, ZFC is regarded currently as the foundation of mathematics because starting from it, one can define ever more complicated kinds of sets which either serve as mathematical structures e.g. groups, rings, fields etc, and as numbers. Most of these begin with the concept of an ordered pair, which is usually defined in terms of sets in a manner first given by Kuratowski. It turns out that Kuratowski's definition of an ordered pair fails for an incomplete ordered pair. It is possible to come up with a more complicated kinds of sets which satisfy the definition, but then I have to make sure that it does not unintentionally collide with other well-established set theoretic structures (or if it does, I have to make sure that this difficulty can be resolved). I have not yet solved this problem, which is to show unequivocally that there is a set which both satisfies the definition of an incomplete (and complete) ordered pair and which agrees with all the well-established structures with which the Kuratowski definition agrees, and until I solve this problem, my framework has no chance (Incidentally, the function of the Kuratwoski definition is only to make sure that ordered pairs are well-defined, beyond what I just mentioned, the definition of an ordered pair is completely arbitrary and usually forgotten about by mathematicians) .This is my version of the problem that I pointed out to you about inconsistent mathematics, in the sense that it lies at the very core of the undertaking. I acknowledge the difficulty and all the while I am working on developing the overall framework, I attempt to overcome it as well.

On the other hand, I perceived in your response to my bringing to your attention the possibility that inconsistent mathematical structures might render the maxiverse as a whole inconsistent a refusal to acknowledge that there is a problem that needs to be addressed. Since I explain the perceived refusal to myself in terms of psychology, I thought I share what I consider to be the explanation with you as well.

"You asked me to clarify my view of actuality vs potentiality, ideally with an example. I will take Tegmark's example of a dodecahedron: it is a mathematical structure, but it is not complex enough to be a physical universe --- it just exists abstractly in the space of all mathematical structures. Perhaps you would say that something that only "exists abstractly" is a "potentiality" --- fine, that is a valid way to define potentiality. "

No, I would not say that. If it has no chance of becoming an object in the real world, then, I agree, it would in some sense "exist abstractly" but it would not exist as a "potentiality". To me, the essence of the concept of potentiality is the possibility of the "coming into being" as an object in our real world.

"The way I see things, my mind also is a mathematical structure, but I know from direct experience that it does not merely have an abstract existence. It has (at least) a "mental" existence, so we could say it is an "actuality". Moreover, I perceive myself as a physical being in a physical universe: this "physical universe" is also a mathematical structure, but it is precisely because my conscious states are part of it that it makes sense to say that it exists as a physical "actuality". "

This is quite metaphysical, and I am not quite sure in what sense you are referring to your mind and your perception. There are certainly neural correlates, very small changes in the electromagnetic fields in the brain etc. that correspond to these, but I have the impression you are not talking about them. If you are talking about your mind in the sense of, say, a consciousness, or your perception in the sense of qualia, then I think it would be much more convincing to give some examples, or at least analogies, to how they can be mathematical structures instead of just positing that they are.

"As you can see, I'm looking at the philosophical roots of the concept of actuality vs potentiality, while you seem to take these concepts for granted. "

Well, based on the above discussion, I am not sure we are talking about the same thing. I have not yet tried to check it, but I suspect that even a child could tell that there is a difference between, say, the outcome of an experiment in which a fair coin which has been flipped, and the (lack of) an outcome of an experiment in which it hasn't been flipped yet. This sort of distinction, which I use, seems to me not to require deep metaphysical thoughts.

"What is potential vs what is actual is purely contextual: from the point of view of the year 2014, the year 2015 is potential, but from the point of view of the year 2016, it is actual."

Yes, that is an excellent point, and I think the reason why the operators I have defined can also be conceptualized in terms of temporal modal operators.

"Ultimately, I don't think that the distinction between actual and potential is very useful when you try to understand reality at the most fundamental level. But of course, since I believe that this most fundamental level is a spaceless and timeless abstract realm where everything simply is, that I hold such a view should not be too surprising."

Well yes, we each start out with our own intuitions, biases, and prejudices and try to work our way towards building something concrete, which in this field is still ultimately a framework that is consistent with what we already know and which makes new testable predictions. That will be the ultimate arbiter of whether our intuitions had merit or not.

I hope you found my responses useful.

Best,

Armin

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 Luca,

Thank you for your comments, it is always gratifying to see when others recognize the greater context of the project one is pursuing. In fact, I would say all my previous FQXi essays with the exception of last year's (which was my one and only foray into more political matters) have explored different aspects of the grand scheme, but I do not expect this to be obvious until most of the pieces of the puzzle are in place.

I am glad that you found my essay It, Bit, Object, Background useful. Yes, in my mind, the question about the essence of quantum states and their possible conception as information is inseparable from the question of the relationship of quantum states to a background. Although it is only implicit, the connection is there even in the current work. The path integrals associated with harmonic oscillator and other quantum states with potential states are still more fundamentally describable as in section 6 in terms of an incomplete spacetime vector, so in that sense all quantum states are the same. What gives rise to the different manifestations is the difference in the spacetime background (together with the difference in the phase factor), and I have the impression that you have understood this point.

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