Finding Structure in Science and Mathematics by Noson S. Yanofsky

Noson Yanofsky

Essay Abstract

One can view the laws of nature as having goals and intentions to produce the complex structures that we see. But there is another, deeper, way of seeing our world. The universe is full of many chaotic phenomena devoid of any goals and intents. The structure that we see comes from the amazing ability that scientists have to act like a sieve and isolate those phenomena that have certain regularities. By examining such phenomena, scientists formulate laws of nature. There is an analogous situation in mathematics in which researchers choose a subset of structures that satisfy certain axioms. In this paper, we examine the way these two processes work in tandem and show how science and mathematics progress in this way. The paper ends with a speculative note on what might be the logical conclusion of these ideas.

Author Bio

Noson S. Yanofsky has a PhD in mathematics (category theory). He is a professor of computer science in Brooklyn College and The Graduate Center of CUNY. In addition to writing research papers he also co-authored "Quantum Computing for Computer Scientists"(Cambridge University Press, 2008) and is the author of "The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us" (MIT Press 2013). The second book is a popular science book that has been received very well both critically and popularly. He lives in Brooklyn with his wife and four children.

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Steve Dufourny

Hello Mr Yanofsky,

I loved your general papper.It is one of my favorite.Because your have well generalised about maths.Noether I like also ,she was very relevant.I like the determionism and the objectivity of methods.

I liked also your interpretation of chaos and order.Especialy how you show the harmony in its generality and order by these mathema and symmetries.I like also these geometrical algebras, I try to formalise the 3D sphères and the spherical geom alg that I invented.The convergences with the spherical volumes could be analysed instead of points.The associativity, the commutativity or not the domains, the scalars the vectors.It is possible to find the correct universal partition of 3D sphères.The récurrences when we follow this order of numbers like you said appear with these maths like a quiet harmonical road.But there are still many convergences, the Mtheory especially to find to harmonise all thjis puzzle.They turn these 3D spherical volumes Professor Yanofsky.I see relevances so in your reasonings about the convergences between the mathematical singularities and their main codes of buildinbg if In can say and the gravitational physical codes.That becomes so very relevant there when we consider the maths and the singularities.The main primordial in 1D of Mr Witten with strings seems relevant ,but If I can the aether doe not seem to be luminiferous but gravitational.That implies an other reasoning.These spherical volumes Professor,They turn so they are .....Inside this 3D sphere.Now with the multiverse and mathematical singularities and the spherical volumes considering these singularities, that becomes very intriguing considering the real universal singularity because if a multispheres exist so we must also consider a central sphere even for this multiverse multispheres for me.That is why that becomes very very relavnt about our constants for each universe ,in logic the main central cosm singularity and its volume implies these aethers, gravitational so implying our constants like c ,G,h,alpha fine struct cst or this or that.That intrigues me a lot at this moment these volumes and these singularities if Mr Tegmark is right about the fact that we have several universes, 3d sphères for me.These sphericalgeometrical algebras with these volumes and the good operators ,vectors and scalars,domains, groups....can explain many things in our universe and others if they exist.In all case an interesting thing is that we retrun always at a main primordial singularity,cosmological even for multispheres.Dirac helps us :)

Congratulations for your papper and good luck in this contest.

Best

Jochen Szangolies

Dear Noson,

glad to see you entering this contest!

I like your idea that finding (simple) mathematical laws in nature is, in part, due to a certain selection bias. Ultimately, to connect with the contest's topic, one might then speculate that it's not nature working according to mathematical laws that gives rise to goal-directed behavior, but rather, that it's the other way around: human beings, following their goals, investigate precisely those systems simple enough as to be easily mathematized, in order to further these goals (which, one may presume, include some measure of predicting the future of such mathematizable systems).

Where most would see physics as an increased specification, a sort of chipping-away process in order to get at the underlying capital-T Truth of it all, you see rather a broadening process, being forced to accept ever-more general structures into the formulation. I think that's an interesting take on the matter (and in some conceptual sense related to investigating the world not by means of uncovering some set of positive truths, but rather, by pointing out the photonegative limitations that eventually serve to define the boundaries of this set, as you do in your popular-level book).

In the end, it seems to me you come close to a formulation of 'law without law', as envisioned by Wheeler. Something like that might offer a way forward on one of the problems for your view you note: namely, that it seems that in particular, the fundamental dynamics of the world are the most exactly mathematizable. One might view this description however as merely effective, originating in some essentially chaotic underlying dynamics.

Take a process operating on a given bit string: even if bits are flipped at random, without any law at all, there are certain predictions that can be made for the gross properties of the string---such as that it will tend towards an equilibrium of roughly half as many 1s and 0s, e.g. This also eliminates the worry of 'who fixes the laws': in that sense, the laws are merely descriptive, rather than proscribing the way the universe unfolds.

Good luck with your essay!

Noson Yanofsky

Dear Jochan,

Thank you for the kind words.

Yes, Wheeler's law without law does come in here. However I always got the impression that he used that as a way of introducing his participatory anthropic principle. While the PAP might be true, it seems to bring in some quantum magic which makes me nervous. I am trying to point to a more general way of picking out laws.

Your point about any stream of bits automatically having structure is very true. It is probably the simplest version of Ramsey theory. This says that as chaotic as you can get, there is always some order that has to show up. This is what I am aiming at. What is needed is some way of quantifying the complexities of observed physical phenomena and show that although we focus on the structured phenomena that we see it is only a small part of all the phenomena that exists.

I look forward to reading your essay today.

All the best,

Noson

Francis Moore

Hello Noson, Very nice representation of symmetry structure subsets. If you are interested In the upper and lower limit numbers of a quantum field with subsets described by plane immersion,read my essay "Proton Three Plane Immersion Connection theory" Thanks francis Duane Moore

Wilhelmus de Wilde

Dear Professot Yanovsky

Thank you for a very clear explanation of your view points. You are finishing with:

qoute

If the structure that we see is only an illusion, then why do we see this illusion? Instead of looking at the laws of nature that are formulated by scientists, we have to look at scientists and the way they pick out (subsets of phenomena and their concomitant) laws of nature.

unquote

In my essay I called "Illusion" EMERGENT PHENOMENON and I tried to explain my perception on the question you are proposing. It is quite different of course, but the totality of perceptions gives us all the colours of the rainbow.

You can link to my essay here and I hope to hear your opinion :

[link:fqxi.org/community/forum/topic/2770]The Purpose of Life[link]

best regards

Wilhelmus de Wilde

Joe Fisher

Dear Professor Yanofsky,

Please excuse me for I have no intention of disparaging in any way any part of your essay.

I merely wish to point out that "Everything should be made as simple as possible, but not simpler. Albert Einstein (1879 - 1955) Physicist & Nobel Laureate."

Only nature could produce a reality so simple, a single cell amoeba could deal with it.

The real Universe must consist only of one unified visible infinite physical surface occurring in one infinite dimension, that am always illuminated by infinite non-surface light.

A more detailed explanation of natural reality can be found in my essay, SCORE ONE FOR SIMPLICITY. I do hope that you will read my essay and perhaps comment on its merit.

Joe Fisher, Realist

Noson Yanofsky

Dear Joe,

What if Einstein was wrong and the universe is not simple?

Also, I am not a surface. "If you prick us, do we not bleed?"

I will look at your essay.

All the best,

Noson

Satyavarapu Gupta

Dear Prof Yanofsky,

Good essay on Structures of the mathematics and number systems required for explaining this Universe or Multiverse...

Your observations like...

1. "Since we have no contact with possible other universes, the question of the existence of the multiverse is essentially metaphysics." And "Rather than saying that the universe is very structured, say that the universe is chaotic and lacks structure. The reason why we see so much structure is that scientists act like a sieve and pull out only those phenomena that are predictable. They do not take into account all phenomena; rather, they select those phenomena they can deal with."

2. "science predicts predictable phenomena"

3. "Despite these failings of our explanation for the structure, we believe it is the best candidate for being the solution."

4. "in the middle of the 19th century, physicists started using complex numbers to discuss waves. In the 20th century, complex numbers became fundamental for the study of quantum mechanics."

5. "the octonions as fundamental and all the other number systems as just special subsets of octonions. The only number system that really exists is the octonions. To paraphrase Leopold Kronecker, "God made the octonions, all else is the work of man." The octonions contain every number that we will ever need"

6. "This is similar to what we are doing in physics. We do not look at all phenomena. Rather, we pick out those phenomena that satisfy the requirements of symmetry and predictability. In mathematics, we describe the subset with the axiom that describes it. In physics, we describe the selected subset of phenomena with a law of nature."

7. "As physics progresses and we become aware of more and more physical phenomena, larger and larger classes of mathematical structures are needed and we get them by looking at fewer and fewer axioms."

8. With the final concluding sentence... "Rather than looking at the universe, we should look at the way we look at the universe."

Are really excellent.

It leads to a question, while describing some portions of Universe, if some mathematical structure is used, later we find some undefined area or singularities.... We will be left with no alternative except to find those singularities, is it not? Or can we change the structure of mathematics to find a better solution...

Here in Dynamic universe model an attempt was made to find a singularity free tensor mathematics, which provides a solution to many of the present day Physics problems.

Many papers and books were published by the author on unsolved problems of present day Physics, for example 'Absolute Rest frame of reference is not necessary' (1994) , 'Multiple bending of light ray can create many images for one Galaxy: in our dynamic universe', About "SITA" simulations, 'Missing mass in Galaxy is NOT required', "New mathematics tensors without Differential and Integral equations", "Information, Reality and Relics of Cosmic Microwave Background", "Dynamic Universe Model explains the Discrepancies of Very-Long-Baseline Interferometry Observations.", in 2015 'Explaining Formation of Astronomical Jets Using Dynamic Universe Model, 'Explaining Pioneer anomaly', 'Explaining Near luminal velocities in Astronomical jets', 'Observation of super luminal neutrinos', 'Process of quenching in Galaxies due to formation of hole at the center of Galaxy, as its central densemass dries up', "Dynamic Universe Model Predicts the Trajectory of New Horizons Satellite Going to Pluto" etc., are some more papers from the Dynamic Universe model. Four Books also were published. Book1 shows Dynamic Universe Model is singularity free and body to collision free, Book 2, and Book 3 are explanation of equations of Dynamic Universe model. Book 4 deals about prediction and finding of Blue shifted Galaxies in the universe.

With axioms like... No Isotropy; No Homogeneity; No Space-time continuum; Non-uniform density of matter(Universe is lumpy); No singularities; No collisions between bodies; No Blackholes; No warm holes; No Bigbang; No repulsion between distant Galaxies; Non-empty Universe; No imaginary or negative time axis; No imaginary X, Y, Z axes; No differential and Integral Equations mathematically; No General Relativity and Model does not reduce to General Relativity on any condition; No Creation of matter like Bigbang or steady-state models; No many mini Bigbangs; No Missing Mass; No Dark matter; No Dark energy; No Bigbang generated CMB detected; No Multi-verses etc.

Dynamic Universe Model gave many results otherwise difficult to explain

Have a look at my essay on Dynamic Universe Model and I request your esteemed opinion...

Best wishes................

=snp. gupta

Noson Yanofsky

Thank you for looking at my essay. Thank you for the summary of the best lines. I will look at your paper.

All the best,

Noson

Jeff Yee

Dr. Yanofsky,

Your essay is on point with topic and you've brilliantly summarized the link between physics and mathematics, with relevant examples from history. Another example would have been Newton creating calculus for his works.

If you haven't seen Gary Simpon's essay, he is another fan of quaternions. Thought you might have an interest in it. Or, you may have an interest in our essay too: (The Relation of Particles Numbers to Atomic Numbers). It's not as similar to yours, but my co-authors and I would certainly appreciate your feedback.

Regards,

Jeff

Noson Yanofsky

Hi,

Thank you for the kind words. I will look at those other essays.

All the best,

Noson

David Brown

"Since we have no contact with possible other universes, the question of the existence of the multiverse is essentially metaphysics." The preceding statement is an interesting hypothesis which might, or might not, be true. Does string theory with the finite nature hypothesis imply MOND and no supersymmetry?

Consider 3 conjectures: (1) Milgrom is the Kepler of contemporary cosmology, and the empirical validity of Milgrom's MOdified Newtonian Dynamics (MOND) requires a modification of Einstein's field equations. (2) The Koide formula suggests that there might be a modification of Einstein's field equations. (3) Lestone's heuristic string theory suggests that there might be a modification of Einstein's field equations. Are (2) and (3) sure bets? No. Is (1) a sure bet? I say yes. I suggest that there might be 3 possible modifications of Einstein's field equations. Consider Einstein's field equations: R(mu,nu) + (-1/2) * g(mu,nu) * R = - κ * T(mu,nu) - Λ * g(mu,nu) -- what might be wrong? Consider the possible correction R(mu,nu) + (-1/2 + dark-matter-compensation-constant) * g(mu,nu) * R * (1 - (R(min) / R)^2)^(1/2) = - κ * (T(mu,nu) / equivalence-principle-failure-factor) - Λ * g(mu,nu), where equivalence-principle-failure-factor = (1 - (T(mu,nu)/T(max))^2)^(1/2) -- if dark-matter-compensation-constant = 0, R(min) = 0, and T(max) = +∞ then Einstein's field equations are recovered. Can gravitons escape from the boundary of the multiverse into the interior of the multiverse? Does Lestone's theory of virtual cross sections suggest a theory of the multiverse in which virtual energy is shared among many different universes and is indirectly measured in every alternate universe in the multiverse?

Alexander Ilyanok

Dear Professor Noson Yanofsky

You essay is very interesting. You clearly see the problem in modern physics. I also think on the problem "where is the boundary between science and non-science?" If we consider that it is the metaphysics shape public opinion through the media it is a real danger that an adequate conception of science, its methods and ways of existence in the public mind will be replaced by substitute of abnormal knowledge.

Today humanity is in deep crisis associated with the transition to a new techno-economic paradigm - the knowledge economy. It uses the knowledge to generate tangible and intangible values. This requires abandoning the biblical paradigm based on myths. This paradigm has lost its relevance for the management of large masses of people, as it is based on mythology and distortion of objective knowledge. For example, the relativistic world view does not contradict the Old Testament in which God created the world not from the finished material but created the matter itself. Therefore, all the experimental data that contradict the relativistic worldview are falsified or silenced. Today, the struggle for resources in the modern world of physics is reduced to competition of hypotheses and as a consequence to fight without rules on the basis of clericalism and postpositivism.

I am in more agreement with you, that the job of physics is to describe a function from the collection of observed physical phenomena to mathematical structure. The basis of any science is the experiment that is why the science was called natural philosophy. In mathematics the experiments are not used. It is based on axioms and theorems. So formally it should not be considered as science. Of course, the mathematics is very important for science as well as it allows you to see patterns and to predict new effects, i.e. it is a kind of glasses to the experimenter. So the question is - Are the glasses we choose proper?

So the "elegant" mathematical equations, which are attributed to some physical phenomena, often do not allow us to see the fallacy of the original physical models that are used in mathematical physics, and are essentially metaphysical.

More detailed information on the subject can be found in Galactic Internet and Femtotechnologies

Femtotechnologies Presentation

Quantum_Astronomy_Part_II

My best wishes

Eckard Blumschein

Dear Noson Yanowsky,

I am looking for someone with whom I may largely agree on some rather uncommon views while I nonetheless intend to defend my criticism of seemingly mandatory tenets.

Someone who rated my essay 1 did not reveal his reason. I guess he judged me a moron because I am arguing against symmetry as a pillar of reality. You correctly explained symmetry as invariance against shift, rotation, and so on. While I am not familiar with S. Lee I vaguely recall the notion continuous symmetry.

To me, perfect symmetry is rarely a property of nature. I see it rather indicating an artificial mathematical ideal. Don't get me wrong, I don't question the essence of your essay. We are in agreement on that reality needs a sieve. I merely distinguish between what I defined to be reality and what the sieve has been abstracted from it. The symmetry you have in mind belongs to the level of abstracted laws of nature. Nature is not invariant against shift or reversal of time. The invariance is artificial.

I cannot hide that my criterion non-arbitrariness has unwelcome consequences.

Respectfully,

Eckard Blumschein

George Kirakosyan

Dear Noson,

I have read your work with big interest as I find some judgments which has excited also my mind with time. Particularly, the matter concerns to your assertion on a priority of representation the calculus with complex numbers as more capable - powerful tool than the ordinary numerical (which may be represented as the trivial case of the first). I am fully agree with you. I can say even that specialists have used complex representations in many important areas that mainly are joined, especially, with the harmonical (and non harmonical) oscillations. But, one amazing thing may be derived from this (from your assertion). It is the formal possibility to interpretation the quantum relations as the derivative from harmonical movement (i.e. from causal relations).

So, I see main merit of your formulation in what I am saying. Moreover, I try even to realize this opportunity in my works that I hope may serve to your attention (see in refs). So, I can only welcome your essay!

I hope to see some your comment on this in my page

Best wishes

Noson Yanofsky

Hi,

Thank you.

I will look at your paper.

All the best,

Noson

Gary Simpson

Noson,

This is a very good explanation of the relationships between the Division Algebras ... well done.

The observation that scientists act as sieves is also very appropriate. There is an old saying ... "If you are a carpenter then every problem is a nail". Essentially, people use the tools that they know how to use on everything ... even if it is not the correct tool.

I will offer one small criticism though ... truly ground breaking science is not simply sifting through data and finding order or symmetry. The ground breaking stuff predicts what the order and symmetry will be. That was the case when Paul Dirac predicted the existence of anti-matter as a consequence of his solution to the relativistic wave equation.

Best Regards and Good Luck,

Gary Simpson

Noson Yanofsky

Dear Gary,

Thank you for taking an interest in my paper.

As to your example about predicting symmetry: many people make such predictions. The ones that are true are recorded. The ones that fail are not recorded. Dirac was one of the best sieves around. : )

All the best,

Noson

Stefan Weckbach

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

Noson Yanofsky

Thank you for the long post. I agree with most of what you say. But I am not a nihilist... : )

I commented on your nice essay.

All the best,

Noson