Finding Structure in Science and Mathematics by Noson S. Yanofsky

Dear Noson,

i now read your essay in detail. It is written in clear language, simple to understand and the lines of reasoning can be traced very easily. Good work.

You contrast order with disorder, structure with chaos. You seem to have a rather pessimistic view on things like goals and intentions. But nonetheless, you argue your case very well and stringently. Let me annotate some thoughts i had during the reading.

Firstly, if considering the contest's questions, i think one has to presuppose as an axiom that all that exists does all things the right way. If we drop that assumption, we end up at nihilism. So i presuppose that the universe and its possible causes and all the rest is a consistent whole.

Your take on mathematical number systems is fascinating, innovative and thought provoking to me. The progression in the dimensionalities of the mentioned number systems is clearly an expression of logic, it is a kind of algorithm and it has therefore structure. In contrast to this - as you annotated at the end of your essay - one may arrive with such a progression at a level of description of 'all there is' which seems to say that the universe, viewed objectively, is devoid of structure. The interesting question is (and i formulated similar questions at the essay page of Cristinel Stoica) how order and disorder, randomness and information are intertwined.

If the mentioned progression of the dimensionalities of number systems indeed leads to the conclusion that the universe is devoid of structure, this would pose serious questions. For example, does it make sense right from the start to extrapolate a mathematical algorithm like the one for building up ever higher dimensional number systems? Isn't this somewhat similar to Cantors cardinalities, building up ever higher infinities out of a simple algorithm? Surely, your progression should not be infinite but terminate at a certain level where no axioms are left. Although i am not a mathematician and certainly not an expert on octonions and so forth, i ask myself how can such a progression of number systems be able to drop one axiom after the other, until there are no axioms left? But taking it as given, what you arrive at is simply a tautology, namely a 'number system' (although without any axioms) saying that the universe and all the rest is just what it is (without specifying it further). Obviously this is plainly true, indeed the universe and all the rest is just what it is. One does not need a single axiom to conclude this!

The problem only arises when one wants to specify the whole thing due to a certain category. I assume it to be true that our universe has many phenomena which do not have goals and intentions. But this does neither imply that the universe's main characteristic must be termed as 'chaos', nor does it mean that the universe's existence and the many 'mindless' phenomena in it are senseless from a higher point of view. Maybe mindless phenomena serve a higher purpose; surely, this purpose then had to be determined via the construction and rules of the universe and also surely not by some physical mechanisms, but by an entity which has goals and intentions. I see no contradiction that randomness, chaos and disorder cannot support goals and intentions. This may be a hard to swallow statement, but i will explain it further.

As you know, a perfectly random sequence of 0's and 1's follows a certain mathematical law. The digits 0 and 1 should be evenly distributed over the whole pattern. Random in this case means that the occurrence of either of the two digits does not depend on the value of the preceding or following values of such digits. Every event should be totally independent of each other event. Have we catched 'chaos' and 'randomness' with this? In no way. We only catched the extreme case at one end of the continuum of order. 'Randomness' as a nihilistic ontological fact should have other features, i suspect. It should be not catched up with any mathematical description. For example, imagine that you are a kind of Boltzmann brain, but without all the physics needed for it. Just imagine you are aware of something. This something does not reveal any correlations. At one time you see a flash, then you see the flasch forming to a vague kind of bubbeling-up of some melted cheese-like thing, you see all sorts of visual impressions and they do not make any sense to you. 'Randomness' defined as this would be just like a nightmare.

Now, let us elevate the mathematical concept of randomness to perhaps meet what you intended to say in your essay's last paragraphs. Maybe the observed structure of our universe is a lucky fluke within a chain of random events (like the 0's and 1's, randomly encoding some kind of theory of everything). Would this be a convincing scenario to explain the order in our universe and its - assumed - dichotomy to the observed chaos in our universe? I would say no, because it does not answer where the randomness comes from, how and on what existencial features it operates on and why it can be mathematically explored to the point where it produces conscious beings which indeed then mathematically have explored it.

The whole point for me here is to say that behind the concepts of randomness and order, there must be a common fundamental origin of all of this. Alternatively one must conclude that reality is somewhat irrational. Because i am not the kind of Boltzmann brain described above, i conlude that behind the interplay between randomness and order, there is some deeper origin of all there is. As you envisioned with the progressions of number systems, mathematics seems to be able to transcend itself - in the sense that its very limits show us how the universe cannot be. It cannot be infinitely dividable, it cannot be overall deterministic and at the same time be overall consistent, it cannot be overall chaotic and nihilistic in a Boltzmann-brain sense and it cannot capture the whole of existence. All this cries out for an explanation that is metaphysical. It is no wonder that the progression of number systems tends to converge towards a non-axiomatic description of reality. Because mathematics cannot capture all there is due to the nature of it. Its nature is not overall compressible. But this does not mean that for explaining the nature of reality otherwise than with mathematics, one wouldn't need some axioms. My axioms are that there are realms beyond space and time from which our universe originated. It is limited in time, duration and space. At the outer borders (at the microscale) of this universe, quantum mechanics rules and spontaneous collapses occur. These collapses may not have an explanation in terms of physical causality, but i am convinced they have an explanation in terms of purpose and intention. For me, it is no problem to think of a creator that has the power to give some order to his creations to behave spontaneously (although here again, this spontanity is restricted by the rules of QM, also given by this creator). Surely, these considerations are all axioms, choosen by me, they do not arise necessarily from what i have written. But i see no alternatives between a creator and nihilism, the latter in the sense that reality is absurd and logics cannot conclude something logical from the fact that logics has its limits in deducing the ultimate layer of reality. But it surely can *induce* these layers on the basis that logics has realized its own deductive limits. Therefore i tend to say that logics, as an expression of a formal system, is as incomplete in the sense of Gödel's results as every other formal system subject to Gödels results. The main result for me is that the limits of deducability do not necessarily mean that the universe is absurd. It is open for the possibility being not so. This openness is installed even within the very structure of logics, since logics can discriminate between a possibility and a necessity.

I would be happy if you could write what you think about this all.

Best wishes

Stefan Weckbach

Hi, Noson, thanks for the good read, I specially appreciated the analogy with mathematics, which was original and (at least for me) instructive. I share also the view that "intention is in the eye of the beholder". You have focused in the role of symmetry, I chose to focus in predictability: humans design their seives in order to be able to predict the future. Regarding your question "What is it about human beings that renders us so good at being sieves?", I believe there are good evolutionary arguments to become good seives, which I mention briefly in my essay. Given the similiarity of our approaches, I would appreciate your comments - if you have any.

Thanks again!

inés.

Dear Yanofsky

A great introduction into the fascinating land of quaternions and octonions. I honestly had no idea that such mathematics existed and that they were promising candidates for future scientific developments. Having read your essay, I am now convinced that they have a role to play in future discoveries. The analogies that you pointed out from the past development in physics are just too powerful to be ignored. You may want to check out the essays of Dickau and van Leunen. They areboth talking in terms of the number system that you are advocating. All the best!

Warm Regards, Willy

Dear Noson, I like how you contrast selecting subsets with taking quotients. I rather focused on taking quotients in my contribution but I have to admit that selecting subsets may be equally important when discussing emergent phenomena. Cheers, Stefan

Greetings Noson,

Your essay complements mine well, in terms of telling the other side of the story I tell. I once wrote about the value of the octonions, and in the same paper said I thought the sedenions were unlikely to have uses in Physics. And then I learned geometrically the sedenions are truly aimless like a blank slate, having no preferred direction, but heir decompositions via fibration yields only the C, H, and O algebras. So they give us only the set of algebras useful to Physics.

I must find fault in your chosen sieve criterion, after more than 30 years of research into the possible applications for Physics of the Mandelbrot Set, which is maximally asymmetric. I had a few phone conversations with Ben Mandelbrot, and published a brief letter in the 80s, before setting it aside, but the theory of gravitation I presented last year at GR21 is an outgrowth of that work. Ergo; I have serious doubts about the hypothesis that symmetry is the feature that characterizes genuine Physics.

I will send a PDF of what I presented at GR21 by e-mail, if you like. But I had to grapple for many years with the subject of asymmetry in Physics, as a result of my finding parallels to Cosmology in M, or rather its family of related figures, years ago. My algorithms reveal the trends in iteration, where coloring in monotonically diminishing iterands shows basins of attraction near the Misiurewicz points.

Theories of entropic or emergent gravitation, like those of Jacobson, Verlinde, and Padmanabhan, are well modeled by M, but Mandelbrot gravitation most closely resembles DGP gravity, where the 5-d black hole into 4-d spacetime idea of Pourhasan, Afshordi, and Mann is exactly modeled at (-0.75, 0i, in M, when it is embedded in the octonions. This spot is also a precise replication of Cartan's rolling-ball model of G2 - which is what creates the bubble we inhabit- so symmetry does emerge victorious in the end.

More later,

Jonathan

By the way..

Seeing the value of this work, and especially seeing it is ranked well below that value, I gave it an honest rating of 8 out of 10, which should boost your score a bit. I am discouraged to be in the 90th %-ile myself, and be highly regarded, and yet still have such a low score (below the median of 5.5). It is even more tragic when an essay like yours gets pushed down in the pack so far where it can easily be lost.

Good luck. I may want to continue this conversation further.

All the Best,

Jonathan

I wanted to comment further..

What the Mandelbrot Set seems to teach us is that Physics is about how exact local symmetries are bounded by global asymmetry. So this is my proposal for a more realistic sieve condition. For the record; the Mandelbrot Set admits the Multiverse hypothesis but denies the possibility that the range is endless, and instead spells out specific spectral ranges where bubble universes can form.

All the Best,

Jonathan

Dear Prof. Yanofsky,

Your very interesting essay asks two important questions: Why are there structures, and why do we see structures?

I think that the answer to both of these questions lies in the biological concept of evolutionary adaptation. Particularly on a macro scale, only ordered structures can be maintained. Secondly, our tendency to see structure and agency all around us is itself a successful adaption to perceiving and acting in the real world.

I address the issue of adaptation in my own essay, "No Ghost in the Machine". I argue that recognition of self, other agents, and a causal narrative are built into specific evolved brain structures, based on neural networks, which create a sense of consciousness as part of a dynamic model of the environment. The reason that this is such a difficult problem is that we are being misled by the subjective perceptions of our own minds.

Alan Kadin

I also wanted to thank you..

Your bottom-up explanation and discussion of the octonions was especially lucid, and I will probably refer other contestants to your essay for its value in clarifying what I leave out. I think this contest is a learning experience for many of us, and is especially valuable for seeing the ways different ideas fit together or relate, to give us a better perspective on the whole truth of the matter we are examining.

All the Best,

Jonathan

Hello Noson,

I greatly enjoyed your essay. I think it actually would have been perfect for earlier FQXi contest "Trick or Truth" about the relationship between mathematics and the laws of physics. What I am less sure of is how you're addressing the question of the current contest which is the emergence of goals and intentions from mathematical laws.

What is your view?

Thanks,

Rick Searle

Thank you dear professor, for answering my post and favorable words on my work. This important for me as a opinion of one deeply thinker specialist. Unfortunately our approach on the role and significance of math are some different from opinions of many important bosses in present science. However, we can thinking as we see it correct.

Maybe I have not enough level to say this, but I think your clear approach to a relation between facts with math may induce a lot of perspectives, therefore I am going to rate your work!

Best regards

George K.

Dear Noson,

I can't agree with your argument quite as you put it, but I think you're on the right track. I like the premise "that the universe is chaotic and lacks structure," and that something "acts like a sieve" to pull out only the very small subset of phenomena we actually observe. But as you note, it hardly seems reasonable to make scientists the primary agency of the selection process. I would rather say that science involves the discovery of nature's own selection rules, which are complex and operate on many levels.

You say at the start - "These laws of nature are fine-tuned to bring about life, and in particular, intelligent life." Well, they are evidently fine-tuned, and they do support life, but I argue in my essay that the structure of the physical world makes the emergence of life exceeding difficult. And while biological evolution did eventually produce quite intelligent animals, the leap to our human kind of intelligence hardly looks to be preordained in the selective principles of biology, much less in the laws of physics. This entire history looks much more like a series of unlikely accidents than something built into the structure of the universe from the start.

My suggestion is that physics itself provides the "sieve" - specifically, in the complex system of recursive processes that scientists make use of when they observe and measure things. We humans aren't responsible for the fact that physics has a complicated set of symmetries that can make all its components empirically observable. Nor is this something the universe does just for the sake of intelligent observers.

I argue that in order for any kind of information to be meaningfully definable or communicable, there always needs to be a context consisting of other definable and communicable information. The physical world clearly provides such contexts... that is, the observable universe consists of the subset of structureless chaos that not only has structure, but succeeds in defining all its own structure and communicating it interactively. This is the source of the predictability of phenomena, and the many kinds of symmetry that make it work - which we observers take advantage of for our own ends.

In any case, I appreciate your imaginative line of thought here and the clarity of your writing.

Thanks - Conrad

Noson,

Things that need to be said: Rather than looking at the universe, we should look at the way we look at the universe.

In my essay I speculate about discovering dark matter in a dynamic galactic network of complex actions and interactions of normal matter with the various forces -- gravitational, EM, weak and strong interacting with orbits around SMBH. I propose that researchers wiggle free of labs and lab assumptions and static models.

As you suggest, static models are based on "static mathematical functions." and "phenomena with certain symmetry."

Your essay is instructive.

Jim Hoover

Dear Noson Yanofsky,

I enjoyed your essay, and found fascinating the idea of thinking of octonions as fundamental and all other number systems as subsets of the octonions. I very much like your statement: "All the axioms that one wants satisfied are found "sitting inside" the octonions."

You rightly focus on symmetry in physics. While much of particle physics is based on symmetry [such as SU(3)xSU(2)xU(1)] these are not 'exact' symmetries in that the masses of the particles are not equal. In fact, approximate symmetries are applied to cases where one mass is almost 100 times greater than another. Yet these approximate symmetries still yield results.

I believe your key point is that physicists act like a sieve and significantly constrain the class of problems they tackle, limiting themselves for the most part to predictable regularities. At the end of your first paragraph you ask "What exactly are these laws of nature and how do we find them?"

In my reference 5 (The Automatic Theory of Physics) I design a robot physicist to derive theories (models) of physics from observational data. The general approach, group the numbers via inter-set and intra-set distances to derive feature vectors, is summarized in my endnotes. Thirty years later Schmidt and Lipson applied this theory via pattern recognition algorithms to

"automatically search motion tracking data captured from various physical systems..."

Whereas I had treated little more complicated than trajectories of rocks, etc, Schmidt and Lipson treated complex systems such as weights on springs and the double pendulum. In other words, systems with predictable regularity as you note. Based on their pattern recognizing robot they found:

"Without any prior knowledge about physics, kinematics, or geometry, [the robot] discovered Hamiltonian's, Lagrangians, and other laws of geometric and momentum conservation."

This agreed with my theory. However what I found most fascinating was that they found the 'type' of law that the robot derived was determined by what variables were presented to the robot observer. They discovered:

"... if we only provide position coordinates, the algorithm is forced to converge on a manifold equation of the system's state space. If we provide velocities, the algorithm is biased to find energy laws. If we additionally supply accelerations the algorithm is biased to find force identities and equations of motion."

Especially with regard to your question, 'how do we find these laws' I hope you find this as interesting as I do.

My very best regards,

Edwin Eugene Klingman

Dear George,

Thank you for the nice rating.

All the best,

Noson

Professor Yanofsky,

That was an extremely interesting essay to read. The idea of us being sieves I felt was very insightful. I also enjoyed the journey you took us in a simple manner through the different hierarchies of the number system (I definitely have to look into quaternions in more detail), as well as drawing the corresponding parallels over the course of research in physics.

"One possible conclusion would be that if we look at the universe in

totality and not bracket any subset of phenomena, the mathematics we would need would have no axioms at all"--- I would be very pleasantly surprised if that truly happens to be the case.

While I agree that your essay provides an interesting new perspective, I personally am interested in why we are sieves in the first place? Can we only be sieves in this universe? I would be interested in your thoughts on my submission "Information is Physical", where I talk about the use of some thermodynamic constraints to explain the emergence of learning and intelligence in physical systems. Thanks.

Natesh