Finding Structure in Science and Mathematics by Noson S. Yanofsky

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

Dear Noson,

That was a somewhat deja-vu experience. A brilliant essay, and great shame we seemed not to read each others last year. We have parallels, and mine was scored highest & yours won a prize. I hope you'll do so this year as I'd value your response to what may be a big advancement in understanding, leading to a real physical sequence of interactions 'classically' reproducing QM's predictions (and rather more besides). Last year I analysed 'brackets' in terms of quantum or 'propositional dynamic' logic (PDL) which hierarchical architecture I employ this year, but describing real physical phenomena rather than just the abstract descriptions of them.

I agree with just about all you wrote. OK it may be a touch off topic and incomplete, but all essays are, and it's fundamental insight surpasses almost all. I certainly agree we; "do not take into account all phenomena", and indeed suggest we miss much, including consistent application of things which may reveal certain more complex or fundamental symmetries.

I'm not a mathematician (maybe why I missed yours last year!), so you did loose me a little for a while (though I knew what you meant) but I've consciously refined, over decades, a more physical (and geometrically dynamic) way of looking at the universe.

At the end you suggest; the universe in totality is devoid of structure and needs no axioms. There are just plain sets without structure. Have you thought a hierachy may have a larger 'elephant in the room' structure? or that your concept may be very close to Einsteins final 1953 inertial systems as; "spaces in motion within spaces", with only the same local rules, but 'transformable' (physically!) in a fundamental Lorentzian way?

Very best

Peter

Dear Noson,

I struggled a bit with your essay at first, then came up with examples.

Consider a differential equation commonly used to model mechanical or electrical resonance. The model takes as input random noise and amplifies a narrow band of frequencies. If the bandwidth is narrow enough, the output is practically indistinguishable from a pure sine wave. We notice the (nearly) predictable sine wave, but it is the noise doing the actual physical work. To describe a realization of the process, the noise is an essential part.

Quantum mechanics seems to have a noisy aspect. For example, the position of the next photon (or particle) to show up in a diffraction pattern is unpredictable. As your essay proposes, the necessity to account exactly for unpredictable events leads to the conclusion that, when used to describe the physical universe, mathematics becomes a collection of sets without structure.

On the other hand, it is the differential equation which models resonance that seems to belong in "Plato's little treasure chest of exact ideals". Because the noise itself can be idealized as having a uniform amplitude spectrum, it belongs as an archetype even if it is not, strictly speaking, exact. Or perhaps there is another chest with inexact ideals.

I think you would be interested in my essay, "Seeking the Analytic Quaternion". Shared symmetry plays a major role in guiding the selection of quaternion derivatives involved in determining the analyticity of a function of a quaternion variable. I find two varieties of analytic functions, and two anti-analytic. My speculation is that these are related to complementarity in quantum mechanics.

Best regards,

Colin

Dear Professor,

Your essay is very educative to say the least. However, as a follow-up to your logic,the geometric increase in mathematical ( dimensional) structure and the decreasing axiomatic scaffolding can only reach zero axiom ,and therefore lack structure,aims and intentions in an infinite universe. Am i right?

Noson -

Thanks for an interesting essay. It's a highly challenging notion to conceptualize mathematics without axioms and a physics of perfect, unbroken symmetry. The unity of the indistinguishable void - timeless, motionless, and yet recursively related to the infinity of all potential and all time and place. Great stuff! I put some thought into these question in my last FQXi essay The Hole at The Center of Creation.

I am left wth a question - how does it all get started? I know first causes are a problematic issue - the responses ranging between nothing and God, but I do think it is relevant to the contest. If the beginning is pure symmetry and no distinctions - what gets the ball rolling? My sense is there is of necessity some form of intentionality and direction (whether from divine agency or otherwise).

Sincere Regards - George Gantz (The How and The Why of Emergence and Intention).

I tried to do that in my last FQXi essay

Dear Noson S. Yanofsky,

Excellent informative essays about the temporal and spatial symmetry, about complex numbers, quaternions, etc., written in good academic style. It would be nice if you would consider tensors, which Einstein coded their theories from prying eyes.

I inform all the participants that use the online translator, therefore, my essay is written badly. I participate in the contest to familiarize English-speaking scientists with New Cartesian Physic, the basis of which the principle of identity of space and matter. Combining space and matter into a single essence, the New Cartesian Physic is able to integrate modern physics into a single theory.

Don't let the New Cartesian Physic disappear! Do not ask for himself, but for Descartes.

New Cartesian Physic has great potential in understanding the world. To show potential in this essay I risked give "The way of The materialist explanation of the paranormal and the supernatural" - Is the name of my essay.

Visit my essay and you will find something in it about New Cartesian Physic. After you give a post in my topic, I shall do the same in your theme.

Sincerely,

Dizhechko Boris

Noson,

You didn't respond to my post above and haven't read my essay. Is there a reason or just pressure of time. I'd hoped we may discuss, including some consistencies, also with my last years (top scored) offering perhaps helping shed some light on; "This idea that we only see structure because we are focusing on a subset of phenomena is novel and hard to wrap one's head around"

Very best

Peter