Imagination and the Unreasonable Effectiveness of Mathematics by Armin Nikkhah Shirazi

Dear Akinbo,

"To form an all encompassing opinion, and to know where any conspiracy could be coming from..."

Are you seriously considering the possibility of such a worldwide conspiracy?!?

All those physicists working on the Manhattan project generations before GPS must have done an outstanding job not only maintaining the secrecy of the conspiracy but of giving terrifying fake evidence that the theory works.

Or wait...maybe you think Hiroshima and Nagasaki was not A-bombed, either?

My point is that if your really believe such things, then it seems that such beliefs, in order to be consistent with other beliefs, entail beliefs which become ever more absurd. And citing one person in one particular field is not going to help much.

I see that Ronald Hatch has written a lot of critical material on SR. I find it curious that, at least upon a cursory search, I was not able to find any critique of his work except for one particular blog.

"By any stretch of imagination going by his Bio above, Ron Hatch cannot be said to be someone with a high school background who misunderstands relativity. Why, my friend Armin would feel such a person should be disregarded along with other references like Prof. Herbert Dingle is what looks more like conspiracy, probably unintentional or well meaning, I can't really say."

This is called a straw-man, a fallacy which only further undermines your credibility. Anyone reading my comment knows that my "high school background" comment was a reference to Harry's claim that "anyone with a high school math background can see that relativity is mathematically inconsistent".

It is quite obvious to me that your reason for holding your physics beliefs is not because you are guided by evidence to your beliefs, but because you have already decided what you want to believe. This leads you to grasp every straw you can find, regardless of its merits, to support the position which you already had decided beforehand you wanted to hold. This became especially clear with the articles you cited on the purported invalidity of the Heisenberg uncertainty principle, all of which, except for the one in which the author himself suffered a misunderstanding, contradicted your position.

I have not investigated Hatch and Dingle's works, and so I cannot judge their merits. However, based on my understanding of special relativity and my own research in it, I think if their findings were so revolutionary, we would have heard of it by now. Until that happens, it is much more reasonable to work with the theory for which there are mountains of evidence.

Best,

Armin

Dear Marcel,

Thank you for your comments. As for scientific publication, yes that is a goal, but I omitted a lot of details that still need to be filled in.

Best wishes,

Armin

Dear Lloyd,

Thank you for your kind remarks.

Yes, as you can see, most essays here try to explain the relationship between physics and mathematics solely in terms of those fields (and perhaps some philosophy). But it is easy to ignore the fact that we do both activities in our minds, and therefore its role on the connection is less likely to be appreciated. My essay was in part an effort to bring attention to an important factor that might be easily overlooked.

Best wishes,

Armin

Dear Jon,

"Seth Lloyd has a view of the universe being a quantum computer, ..."

First, whenever you have someone proclaiming that the universe "is" a giant version of some contemporary piece of technology, you should always take that with (more than) a grain of salt . As you may know, a couple centuries ago, during the enlightenment, people proclaimed the universe to be a giant clock, and who knows what people will proclaim the universe is in a couple centuries. To me, the universe is just the universe.

I watched the short video and found Seth LLoyd to be a bit fast and loose with words (For example, he claimed that we already have quantum computers, but I am more skeptical (My view is informed by Scott Aaronson's blog posts on the subject matter).

"...which may be a little more appealing to you..."

No, you know (I think) enough about my ideas to understand that in my view our contemporary conception of the universe, which equates it with spacetime, is too undifferentiated.

If you want to talk just about spacetime, then I definitely disagree with him, because I conceive of quantum theory (well QM, QED, and EW interactions) as the physics of spacetime objects emerging out of areatime. If you already have spacetime to begin with (which is the reasonable interpretation of his use of "universe"), then the appropriate theory is Einstein's General Relativity, not quantum theory.

"Stephen Wolfram, Ed Fredkin, Jurgen Schmidhuber, and some other scientists talk about the universe being a (classical) computation, but these ideas aren't quite as accepted."

Do they give concrete examples for how some physical process can be reframed as a computation?

"From a purely theoretical perspective, no matter how far an uncomputable real number is specified, it will always be ambiguous, since there are an infinite number of real numbers that start with the specified sequence of numbers, and there is no way to refer to or conceive of one specific uncomputable real since they have an infinite amount of information that can't be compressed into a formula like those that represent/generate pi or e. If you don't think this point is relevant when it comes to physics and your ideas regarding ZFCD, then that's probably a slightly different discussion."

Yes, I interpreted you previous question as a physicist, from the point of view of mathematics I agree. Yes, I am uncertain about the relevance of AC. The only reason at this point that I chose ZFCD vs. ZFD (not to be confused with Zermelo-Fraenkel with axiom of Determinacy) is that ZFC is the standard set theory. This does not mean that subtle deep connections may not be there, in fact, more likely than not, I think there may well be.

"I don't think computation broadly defined pre-supposes time, although I do think it would imply a sequential relationship...which I think is slightly different."

You may be right. As mentioned, the notion of a physical process as a computation is not intuitive to me, so my intuitions are more likely to lead me astray than in other areas.

"I tried to raise the question that maybe some of the statistical laws of physics are actually modeling pseudorandom processes, as opposed to truly random processes. If the computation that the universe was doing was too complex, a statistical approach analogous to the PNT might be are only practical approach to making predictions. "

Can you tell me which statistical laws of physics you had in mind?

"I hope some of this made sense."

Rest assured that you did:)

Best,

Armin

Dear Colin,

Well, thank you for going through the trouble of researching modal logic in order to understand my theory better. I believe that one of the ways in which the mathematics of the future will be different from the mathematics of today is that it will have the power to formally express nuances that today most mathematicians would perhaps not even dream of expressing.

To reiterate a slightly modified analogy to the one I gave in my response to Vladimir Tamari's post, in my view the era of today's mathematics is like the era of black and white movies, and tomorrow's will be, l believe, like that of color films.

As for the subject of contextuality (and pseudo-nonlocality), these are meant merely as intuition building analogies. The hard work of matching the ideas to the known formalism still awaits. I believe the key is for me to understand how the Hilbert space is built up from orthomodular lattices, a subject I plan on learning this summer. Then I can hopefully take the step of showing how the absence of any "beables" in between measurements implies both.

"Anyway, I just wanted to say it is good to see your progress."

Thank you, it is going slower than I hoped, but it is going.

Best wishes,

Armin

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