Imagination and the Unreasonable Effectiveness of Mathematics by Armin Nikkhah Shirazi

Thank you for letting me know, I will leave a comment there shortly. Also, please note: My last name is Nikkhah Shirazi, not Shirazi.

Thanks,

Armin

Dear Harry,

Thank you for your comments. I am a bit surprised about your comments about special relativity because it was barely mentioned in my paper. In fact, the main result, the theorem on page 8 is expressly in the non-relativistic limit. You are correct that I did not define inconsistency; the reason was that I thought it would be reasonable to expect the reader to know this. If the reader doesn't know what an inconsistency is, then there is little hope they will understand the rest of my essay.

"It seems anyone with a high school math background can see that relativity is mathematically inconsistent while experts can not recognize this fact at all and insist that the theory is not inconsistent.

Well, why do you think it seems that way? Could it be because there is a grand conspiracy among physicists, mathematicians and nuclear engineers worldwide to impose on the rest an obviously false theory, presumably fudging all the technological innovations and experiments based on it, like nuclear reactors, atomic bombs, astrophysical observations, particle accelerators, GPS devices, not to mention all the physics labs so that each new generation of physicists gets brainwashed to join the grand conspiracy? Or could it be because if someone with a high school background misunderstands relativity and, in best Dunning-Kruger form, blames the theory and everyone who uses it for his misunderstanding?

I think I have a good idea which one you think is more likely.

Best,

Armin

Hi Jon,

"Am I the only one who thinks maybe HAL could be looked at in a positive light?"

No, I'm sure all the peers of HAL would also look at his actions in a positive light, presumably because they would be sufficiently advanced to have something like empathy for HAL, but the not the astronauts.

"But if the axioms are enumerable, couldn't a computer just systematically go through them, computing say the first 1000 "theorems" of each. If the goal of physics is to compress complex physical phenomenon into simple equations, then if you were to take this idea to the extreme, could you just systematically (without imagination) just looking through the space of all simple axiom systems (with simple updating rules/rules of inference) with some romantic hope that our universe might happen to be describable with a very simple system/program? Do you think that this is a worthwhile endeavor or just some pipe-dream?"

I think being able to answer these questions intelligently requires a background in both complexity theory and in computability theory, neither of which I posses, unfortunately. I would like to clarify that by using imagination in this sort of situation I had in mind something like the following: You pick some axiom system and think about what requirements it fails to satisfy, then use your imagination to add/subtract or modify the axioms and now you can take advantage of the ability to have computer system check the thousands of theorems that follow from the tweaked system.

"If equations didn't have closed-formed solutions that harnessed infinities and we saw things as a computation do you think the concept of time would stand out more in the equations/programs? Could time just be where you are in the calculation?"

I have a hard time understanding what it means for something physical to be a computation, so I am afraid I can't really answer that question. However, I suspect that computation conceptualized broadly in this way pre-supposes the existence of time, and if that it is true, then you cannot just say that time is defined by where you are in the calculation on risk of circularity.

"Do you think nature, evolution, or the universe is creating this way, with some high-level purpose in mind?"

No, I do not think there is such a high-level purpose. In my view, whatever adaptions you see are the result of environmental constraints, which though obeying regular patterns, do not serve to advance any particular purpose.

"How do I know that the truncated version of the one you provided is some particular noncomputable real and not another? How do you even know which one you had in mind that you are approximating?"

I would say that if you have to ask these questions, then you have not yet approximated the number to sufficient precision and need to approximate further. It seems doubtful to me that you would run into a realistic situation in which you need an arbitrary uncomputable number to infinite precision. What would you do with it? Any manipulation involving it to infinite precision is also going to be uncomputable, unless it happens to be expressible in simple form like e and pi.

"In the case of you flipping a coin, we could say that the outcome has already been logically determined, but we just don't know what the outcome is because the universe hasn't done the computation yet. In certain situations the calculation might be compressible, and these would be the cases in which we can make precise predictions. In the cases where the computation the universe is doing isn't compressible, we would resort to probabilities."

Again, because I have a hard time wrapping my mind around the idea of physical processes as computations, I'm afraid I can't say much. But if what you are saying is right, it seems that this would imply a novel interpretation of probability.

"...maybe some of Chaitin's youtube lectures might interest you if the Wikipedia page didn't offer too much insight on his Omega constant."

Thank you for the suggestion, I'll look into it.

I hope I was able to at least answer some of your questions, though I felt that they were somewhat outside of what I consider to be my area of knowledge.

Best,

Armin

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 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 Mark,

Thank you for your comments.

"My own view on the subject is that what distinguishes an "actual" (or "physical") mathematical structure from a "regular" (or "potential") one is whether it contains sub-structures that have the correct properties to be "self-aware" and can "feel" the actuality of the mathematical structure "from the inside"."

I'm afraid I do not quite follow. Can you give an example? Or, how would the thrown coin toss vs. the unthrown one differ according to your view?

"Not having enough of a background in your field of research, I have to confess I could not follow your presentation in detail,"

Well, that's ok because the details have not yet all be worked out, so if you were able to do that, you would have done my work for me;-)

"I hope you'll have the chance to continue your research and look forward to what it can teach us about what it means for a mathematical structure to be "actual"."

Well actually I don't think my work will be able to say anything about what it means for a mathematical structure to be "actual" because, remember from page 2 of my essay, my work is just concerned with mathematics as a representation of objects which in the real world exist as actualities or potentialities.

Thanks again, sorry I did not get any challenges from you:(

Best,

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