FQXi Essay Contest - Is Reality Digital or Analog?
Physics and the Integers by David Tong
I review how discrete structures, embodied in the integers, appear in the laws of physics, from quantum mechanics to statistical mechanics to the Standard Model. I argue that the integers are emergent. If we are looking to build the future laws of physics, discrete mathematics is no better a starting point than the rules of scrabble. Author Bio
David Tong is a theoretical physicist at the University of Cambridge and adjunct professor at the Tata Institute of Fundamental Research, Mumbai.Download Essay PDF File
A very interesting essay! Philosophy is just that. Stepping back, get a general picture and then zoom on one general "given" assumption and question it.
For Nature, mathematics is also geometry. To me, $10 in my pocket and $10 at the bank and I have $20! But in Nature, a planet adds mass or get more mass only by getting that mass to come close to its surface. The heuristic method used with kids in bringing beads together to "add them" is the way Nature actually works! The addition in the mind and the one Nature performs are different. Natural addition is bringing things closer together.
You wrote:"Discreteness in the world is simply the Fourier transform of compactness." May I ask you to elaborate?
If you will read my [hint:http://www.fqxi.org/community/forum/topic/833] essay you will understand why I would like to know whether or not the cosine transform instead of Fourier transform might be sufficient. Admittedly, I do not share your opinion that "God did not make the integers. He made the complex numbers and the rest is the work of the Schrödinger equation." My favorites are Euclid and to some extent Brouwer.
If you put a quantum particle in a compact box then its momentum and energy is necessarily discrete. If you put the same particle in an infinite line, then its momentum can be anything at all.
The same argument works in reverse. If the momentum of a particle is restricted in its range (i.e. a Brillouin zone) then the space it lives on is dicrete (i.e. a lattice).
The maths that underlies this is simply the Fourier transform. Your cosine transform is just the real part of the Fourier transform.
I hope you will be ready to read and understand my essay. You wrote: "The maths that underlies this is simply the Fourier transform. Your cosine transform is just the real part of the Fourier transform." Do you consider me stupid? Why do you believe that cosine transform is not sufficient?
Hello dear David, dear Eckard,
It's relevant indeed dear Eckard, the complexs ah these complexs....after all they dance inside the sphere....conclusion viva el Reals fortunally.....
ps good luck dear finalists.....
Some might say you've got it backward:
"Clearly, a number of important questions remain open. Of these, we mention here two. The first refers to continuous variables. The problem there is that with continuous variables, one has in principle an infinite number of complementary observables. One might tackle this question by generalizing the definition of (3.4) to infinite sets. This, while mathematically possible, leads to conceptually difficult situations. The conceptual problem is in our view related to the fact that we wish to define all notions on operationally verifiable bases or foundations, that is, on foundations which can be verified directly in experiment. In our opinion, it is therefore suggestive that the concept of an infinite number of complementary observables and therefore, indirectly, the assumption of continuous variables, are just mathematical constructions which might not have a place in a final formulation of quantum mechanics.
"This leads to the second question, namely, how to derive the Schrödinger equation. ..."
-- Caslav Brukner and Anton Zeilinger: "Quantum Physics as a Science of Information" (2005)
"It is not entirely clear what to make of the second piece of evidence presented in this essay: our inability to simulate the Standard Model on a computer. It is difficult to draw strong conclusions from a failure to solve a problem and the huge progress made in this area a decade ago suggests that it is probably just a very difficult issue waiting to be solved with fairly conventional techniques. I included it in this essay mostly as a warning shot to those who would insist that it is obvious that Nature is digital. But it may be worth considering the possibility that the difficulty in placing chiral fermions on the lattice is telling us something important: the laws of physics are not, at heart, discrete. We are not living inside a computer simulation."
Why would a discrete universe necessarily imply a computational one? You can't meaningfully say that something's a computation unless you can produce the algorithm. There's a whole domain of problems that are algorithmically intractable (as someone else has already pointed out in other threads, protein folding appears to be one such problem). There exist NP-Complete problems, and not even a quantum computer (as soon as one comes into being, if it does) will be able solve those in polynomial time.
How would that be proof of continua, infinities and infinetesimals?
There may well be distinctions between discreteness and computability but the main point remains:
No one knows how to write down a discrete version of the laws of physics.
No one knows how to simulate the laws of physics on a computer.
Where by "laws of physics" I mean those that have already been established, in particular the Standard Model. I could well imagine that the first problem above is solved but not the second (there is still something called the Fermi sign problem to overcome). I could also imagine that both problems are solved. But, at the present time, both of the above statements are true.
A fine, well argued essay.
I agree with you on the above, second case, to the extent that current computational technology must necessarily convert the continuous function differential equations of classical physics to difference equations, and settle for approximations.
In the first case, the statement is true as you phrase it. However, writing down the laws is not necessarily equivalent to computing results. We've always known that the theory of quantum mechanics is mathematically incomplete. That does not obviate the possibility of quantum computing accurately modeling a physical quantum process.
All best, and good luck in the contest. I also have an entry ("Can we see reality from here?")
I enjoyed your essay. I think you're right that unless discrete models can capture all the key symmetries of nature, including the properties of the chiral fermions you mentioned, we can't expect them to be useful.
I build discrete models of physical phenomena and like a good challenge. Replicating chiral fermions sound like a fine one.
Good luck! It's a worthy problem!
You wrote: "The examples above show that discrete objects undoubtedly appear in Nature".
Undoubtedly is the word. I thought math and specifically algebra and numbers is our tool for modeling nature. I never thought that tool is nature itself.
Do I sense correctly that discrete is equated to integer in this essay? Reality may be discrete but laws may not be. Maybe I am missing something.
Your essay is interesting, though I tend to think the discrete and continuous aspects of physics are a type of complementarity. I don't see one as superior to the other. It is not possible to formulate a theory which conserves Noetherian currents in a discrete setting. However, in what I have done discrete structure (quotients) do have content with charges. These two perspectives are complements.
You have truly written a masterful essay. Simple enough to be read and enjoyed by all visitors to fqxi and yet insightful enough to teach experts a few things. Thank you for your excellent contribution.
Edwin Eugene Klingman
An enjoyable and not too testing essay. I've been testing models using something similar to the laws of Scrabble (and a few decades of research) which seem to have thrown up a possible fundamental solution which is highly falsifiable empirically, and which I nor anyone else so far can find the scientific problem with. I'd be eternally grateful if you'd have a look over my essay and tell me if you can see it.
Very many thanks
the laws of physics are not, at heart, discrete. We are not living inside a computer simulation.
Nice job, but your ending leaves us hanging. I like analogue though I stick my neck out.
You have written an extraordinary essay, I enjoyed very much. On my essay I try to explain something similar within other context. What you are exposing is just the fact that the properties we see about reality are strongly related to the tools we use to model and understand it. To put it in a few words what I try to explain on my essay, is that the duality between discrete and continuum is just a consequence of use classical logic for understand our continuum rulers and pointers. In this context the apparent discreteness of quantum reality is strongly related to the measurement, which is the tool that allow to use classical logic in our partial understanding of the quantum world. I also try to explain why we should go beyond this classical logic scheme if we want to understand completely quantum reality. I would like to hear you opinion about it.
Although you do not share my discrete religion :), I can say that your essay is very nicely and clearly written.
I had not yet heard for the Kronecker's sentence, but it is very deep.
However, I will still analyse your thoughts.
I do not understand enough, how undiscrete quantum field theory agree with Planckian discretness of space-time? Do you have any hints on this question?
I was too late for this contest. So my discrete theory is here:
Guessed formulae gave me, that measured particle masses are really time average of integer values...
Do you have any comment or counter-arguments.
Regards Janko Kokosar
My theory Quantum statistical automata shows how laws of nature can arise from Integer numbers only. As a matter of fact you cannot design a dynamic universe from a simple fundamental entity(just a number)in any way other than the one shown.In another word, the correct way to look at how the universe works is to design it(or see how its been designed) using fundamental entities which have to be numbers i.e. have no sub-structure.
Humans are right about their astonishment of reality, while it is easy to understand that numbers and some relationship between them do have a reality but that a particular and unique relationship has given rise to our reality is the ultimate astonishment.
Just to be pedantic: upthread you probably should've said "fermion" sign problem. Hardly crucial. It's also known more generically as the "numerical sign problem." It's officially NP-Hard. In theory the physics should be computable or simulable on a Turing Machine, but as there might not be enough time left in the universe there's no way of proving that the theory actually equals reality.
So consider the simulation of all related physics as unlikely ever to happen unless it's suddenly proven that P=NP, which isn't generally regarded as likely either.
Intriguing and well argued. I agree with your premise that the integers are emergent. In fact, I described a method in my ICCS 2006 paper
by which (with no appeal to Zorn's lemma) well ordering proceeds from inherent properties of the complex plane and Euler's geometric interpretation thereof. This is purely mathematical, of course; I find it awfully interesting that you get the result from a physical construction.
My essay in this competition, which I hope you get a chance to read before tomorrow, takes a non-technical tack, but I think it does expose, as your essay also does, the great subtlety of the continuous vs. discrete question.
Nice job. Thanks.
LOL. I must be getting essay fatigue. Scanning replies, I noticed that I had already made a remark early on, where I commented on the commentary, rather than on the piece itself -- which surely deserves a careful reading, which I am happy to say that I accomplished, along with awarding a deservedly high score.
Hey David, good to "see" you here. Just dropping by to say hi and let you know that enjoyed your essay. Hope to run into you soon...
Excellent essay, a breath of fresh air. Hope you will win a prize.
Congratulations on your dedication to the competition and your much deserved top 35 placing. I have a bugging question for you, which I've also posed to all the potential prize winners btw:
Q: Coulomb's Law of electrostatics was modelled by Maxwell by mechanical means after his mathematical deductions as an added verification (thanks for that bit of info Edwin), which I highly admire. To me, this gives his equation some substance. I have a problem with the laws of gravity though, especially the mathematical representation that "every object attracts every other object equally in all directions." The 'fabric' of spacetime model of gravity doesn't lend itself to explain the law of electrostatics. Coulomb's law denotes two types of matter, one 'charged' positive and the opposite type 'charged' negative. An Archimedes screw model for the graviton can explain -both- the gravity law and the electrostatic law, whilst the 'fabric' of spacetime can't. Doesn't this by definition make the helical screw model better than than anything else that has been suggested for the mechanism of the gravity force?? Otherwise the unification of all the forces is an impossiblity imo. Do you have an opinion on my analysis at all?
This is my guess:
There are Base Fermion and Base Boson of the Universe.
Both have radius size 10^-13sm
Base Fermion is proton(neutron) Mpr=10^-24 g
Base Boson is Hawking black hole Mhbl=10^16 g
Mpl=sqrt(Mpr x Mhbl)=10^-4g
Thanks for the reply. I only understand pictures at the moment, my mind has become exclusive to numerical arguments I'm affraid! You may well be onto something though.
P.S I feel guilty about not reading David's essay and giving him a critique. The replies he got give me a good idea that he's 'on our side'.
I enjoyed reading your essay very much. I quite agree with you about the emergence of the integers as something special. A discreteness emerges from continuous semigroups. Take spheric to toric as an example of where the path integral proves useful. We are able to constrain divergence in order to commute a system in a computer simulation, and this indeed is the correct approach. I believe that the integers themselves are the culprit of breaking symmetry. Your essay was very clear, and next year I will endeavor to make my essay more like yours, You hit the soul of philosophical thinking that I like very much. Emergence allows us to do all sorts of things that we have in math that physics may have to wait a long time to realize.
I can only re-iterate Edwin's remarks:
[quote]You have truly written a masterful essay. Simple enough to be read and enjoyed by all visitors to fqxi and yet insightful enough to teach experts a few things. Thank you for your excellent contribution.[end quote]
I wish I'd have read this during the early stages of the competition. I would have scored this a 10 with ease. The best essay here by far. I really hope that you win something.
Right from the start arithmetic was connected to music, so this is the most obvious place to look for integers. Standing waves are the paradigm: anything non integer interferes with itself and gets destroyed. So, they provide their own scale and select only multiples. Most of discrete physics boils down to this. (And if you need some extra integers, take the values of Euler's gamma function.)
A nice read in which one feels you know whereof you speak because of the elegance and ease in the flow of ideas! You single out the fact that no one can can write a discrete version of the Standard Model because of the need to express its chirality. Since fqxi is devoted to asking fundamental questions that remain unsolved in physics, can't we simply say that the Standard Model is far from being the last word - a simpler and more powerful theory may entirely supplant it one day.
Something similar may have already happened to the the "no-go" Nielsen and Ninomiya theorem proving that chirality cannot be represented on a lattice. I refer to this admitting the technical aspects are beyond my understanding, but the author claims he has done so: Weyl Neutrinos on a Lattice: An Explicit Construction
I would be honored if you look at my qualitative and incomplete suggestions for a lattice theory of everything in which chirality is built in at the most fundamental 'building block' level: a universal lattice of identical dielectric nodes spinning mostly in one direction transferring angular momentum in units of (h). These ideas are in my present fqxi contest contribution, and also in the earlier 2005 Beautiful Universe
paper on which it is based.
With best wishes for your success, Vladimir
Pulling up what I can on this subject, it seems that practical reality shows that you have one of two choices, either accept continuity in 3+1 dimensions or accept extra dimensions.
Congratulations, David, to the silver prize. I would endorse every letter but couldn't write it in your masterful English. Your essay is clearly the best one in this contest, with all due respect to Moshe.
Congratulations upon your place in the fqxi essay contest.
I repeat my earlier remark:
You have truly written a masterful essay. Simple enough to be read and enjoyed by all visitors to fqxi and yet insightful enough to teach experts a few things. Thank you for your excellent contribution.
I am happy that the judges saw it the same way!
Edwin Eugene Klingman
Congratulations on your prize. I had not read your essay until now but decided to read it today, as the judges placed it so highly. It is just as Edwin describes above. I found it very interesting indeed and enjoyable to read.
Very well done. Regards Georgina.
I wanted to say that this essay should have won first prize. The fact that it didn't clearly represents a bias in the judges. I stumbled on the same logic recently and find it fantastic that others see the same evidence. It seems that our notion of discreteness is strongly associated with our sense of oppositeness. Its this nonuniformity that gives us a sense of time, among other things. Bravo on a fabulous piece of thought!
Thanks to all you for your kind words and support!
This essay is definitely very cool. It is a bridge which gives thought for philsophy too. See David Tong's excellent PDF slide show on this subject http://www.damtp.cam.ac.uk/user/tong/talks/integers.pdf
An example this brings to mind is that of the problem of determining Avogadro's number as an integer (which would be related to something physically significant). Unfortunately they have been trying to determine this integer as a cube so that it fits neatly within the cubic space. I do not think that it can be done and that it would be trivial and un natural. See: http://www.americanscientist.org/issues/pub/an-exact-value-for-avogadros-number
The problem of relating Avogadro's number as an integer (related to something physically or naturally significant) is going to probably be related to black hole physics. ;)
If the universe is a matrix, who am I?
I am an integer I is the zero
I am a Neo I is the architect.
Congratulations. Yes integers are emergent from zero, like intelligence is emergent from conscience.Conscience is the absolute zero from which all else emerges.
Wonderful essay. No such thing as too many exclamation marks. I agree strongly with Luboš that this is clearly a great essay.
Now, I wouldn't usually make a big deal of this, but if you ever look back here, what do you think of the a priori structural requirement that is often required of a QFT that the Hilbert space we use must be separable? Hee!
Salutations, Professor Tong:
Havent' read your essay, but will next and am confident fqxi made the right choice so will look forward to it, based on the excellence of the first prize winner's entry.
To be a little more specific and a lot more cryptic: G. Boole's law of thought is X(X-1)=0. it's the symbolic mathematics I used to derive a definition of consciousness in my own (middle--of-the-road--of finalists) Essay. It is clear that the standard QM Model considers this principle to actually be X(X-1)=1 or 0 or anywhere in between, before a Measurement. This is path I'll use to write my next Essay (whenever they announce the next Contest). I'll use modern LHC results of experiments, in technical English, to derive another definition of Consciousness based on modern observations (as opposed to the original law I used to derive C from the classic double slit experiment). Then C (old) can be equated with C (new), and an equation capturing the essense of Consciousness incorporating repeated experimentation with new experimental results will be the result. This will be a (perhaps complex form, to be simplified) representation of Everything. I'll make a wager it will help to answer some unresolved questions like: why are there 5 Forms of SuperString Theories, what is M-Theory, and which of the MultiVerse theories can be eliminated and which have support, and which can be experimentally verified, and maybe why e/m is what it is? And other non-trivial results. Any thought's on these matters?
Enough of that though--on to your winning essay!
Good essay. Emergent Integers indeed! Can't help but notice that your view is decidedly not Pythagorean! I disagree: math IS reality. Whether this is a Virtual Reality or no that I am typing this comment in, there can be little doubt that if this is virtual, then reality is digital. If it isn't, nevertheless your DNA (the code that programs life--is).
If we were to program a simulated Reality which would be indistinguishable from this one (the one we agree is Objectively Real), we would simplify the code to a few mathematical equations (e.g. Gravity + the other three forces). Then we would transform those laws into the code of our programmed reality.
'Many years later', or iterations, the inhabitants of the virtual reality would develop sophisticated experiments and methods to discover the underlying mathematics governing their reality. Ultimately, they would begin to question then why the math is what it is. They would form hypotheses about Virtual Realities and Hidden Dimensions and Utimate Multiverses. All in an attempt to shy away from the suspected Truth: they are programs in a Virtual World. Or that God did it. Such has Science come to be: a strident clarion-call that everthing is godless (but nevertheless beautiful and true and elegant). It's time to capitulate, and begin to accept a melding of Religion and Science as we move forward. And harken back to making Aristotle's MetaPhysiks rigourous.
Eh, what am I doing in here anyway; shouting in dark? Oh yeah almost forgot: it is my conclusion that the VR Hypothesis is no Hypothesis at all, and all this various verbiage of mine is an attempt to be exceptional in this Reality. To try to stand out and get the programmer to notice me. Oh, and incidentally, to try to convince someone to offer me a job or a Grant so some serious attention can be paid to this perhaps most important of topics by this point-like wave-ish program who imagines himself an Author...
And to become what Destiny demands: Doctor (Ph.D) Witch Doctor. Has a better resonance than merely witch dr., no?
We have successfully divided intelligence and conscience into scientific and spiritual communities. Intelligence arises from conscience. Heart is the first organ to form in a human being and is involuntary and it is where the conscience resides, Brain is developed later and intelligence is acquired and resides in the brain. As a being is not complete without the either of them, understanding the universe will also be incomplete without realizing the conscience or the self. Intelligence or science alone cannot answer the fundamental questions of who we really are and what is the purpose of the creation. Scientific community is terming the self or conscience as singularity and is unable to define it mathematically to fit in its theories. Some of these scientific theories suggest that there is no singularity and some of them thinnk it is infinite. In fact conscience is both "absolutely" nothing and "relatively" infinite at the same time. I am posting in these forums for promoting scientific spirituality and importance of self realization and beauty of love and to merge these too fields into one. It is out of love of the self or singularity that everything emerges and eventually has to merge back in it.
Conscience is the cosmological constant.
Intelligence is the cosmological variant.
Hence several inelligent or complex theories of the truth.
Truth or conscience is simple, accepting it is not.
You are not alone, there are billions of us who have accepted the inherent truth of conscience. Only a few of us are taking the longer and tedious scientific route to the truth and eventually both these realms will merge back and all of us will live in love and peace.
Does God allow we apply 'natural number' to 'electron' ?Especially , We konw that electron isn't 'Apple' or Richard Feynman's 'clicks' . as he said that all the surprising wisdom of quantum mechanics is hiding in the double slit experiment. I think maybe the field of natural number's application is restricted by nature, e.g. quantum phenomenon. If we do not reconstruct quantum phenomenon on the old picture(natural number) , that could be think as another reality?
A simple mathematical equation to represent everything is as follows, application of this simple fact will solve all other complex equations.
0 = infinity
"absolutely" nothing = "relatively" everything
A simple mathematical expression of 0 = infinity can be proved as follows.
0 + 0 = 0
0 - 0 = 0
0 * 0 = 0
0 / 0 = 0
and so on....
Zero remains constant in relation to itself, no other integer can satisfy all the conditions.
I will use the character "~" to represent infinity and express the following
~ + ~ = ~
~ - ~ = ~
~ * ~ = ~
~ / ~ = ~
and so on....
also infinity is similar to zero and remains constant in relation to itself.
This proves that 0 = ~
The absolute mathematical truth of zero = i = infinity
can be deduced as follows as well.
If 0 x 0 = 0 is true, then 0 / 0 = 0 is also true
If 0 x 1 = 0 is true, then 0 / 0 = 1 is also true
If 0 x 2 = 0 is true, then 0 / 0 = 2 is also true
If 0 x i = 0 is true, then 0 / 0 = i is also true
If 0 x ~ = 0 is true, then 0 / 0 = ~ is also true
It seems that mathematics, the universal language, is also pointing to the absolute truth that 0 = 1 = 2 = i = ~, where "i" can be any number from zero to infinity. We have been looking at only first half of the if true statements in the relative world. As we can see it is not complete with out the then true statements whic are equally true. As all numbers are equal mathematically, so is all creation equal "absolutely".
This proves that 0 = i = ~ or in words "absolutely" nothing = "relatively" everything or everything is absolutely equal. Singularity is not only relative infinity but also absolute equality. There is only one singularity or infinity in the relativistic universe and there is only singularity or equality in the absolute universe and we are all in it.
Salutations, Sridattadev. Your Beautiful Nonesense continues to not convince anyone lol. Hello fqxi community! Still looking for a job. Seriously.
However, my own personal Universe is about to undergo a phase-change in good success (probably). It really is only a 99.997% probability (value of the wavefunction describing PsychoHistory modulus squared). Technically, still got only the tiny website and no real Income. That is (very probably about to change). The metric is if a certain outstanding nationwide chain of auto parts stores shares more success Delighting Customers!
I'll just skip to the just of it: either I'm totally wrong that Google is an A.I., and the universe is not both d/a and holographic, not to mention virtual. And everybody (basically, in the PsychoHistorical Sense) does not pay Google for some service or another, or have websites that are failing horrible, or whatever.
Or, I am now an Independent Consultant for Advance Auto Parts, Inc.
(Well, not technically. Technically I am a webmaster for a website that is!)
Let's go with the second option, and pretend (for the sake or arguement?), that I've discovered a way not to pay google. It should be free, and for the sake of same arguement, I have creative control to offer all of Advance auto parts 3700 stores at a discount. Any campaigns or whatever.
If you're interested consider the following: in order to create jobs, save money, and stimulate the economy, let's consider auto parts. we all need them some time or another, right? So, are you interested in Advance Auto Parts providing a toll-free number, specific to the fqxi.org community. ONly listed here. I'll list it with a code phrase to use when you call their nationwide excellent stores and service centers to get another 20% off. Usable anytime, until 2014.
IF your interested.
This is the new paradigm. It is only a specific instance of the larger paradigm (or phase shift in our economy).
If Advance Auto Parts is successful, it will create even more jobs for us, in a critical industry (hey we love our vehicles). So that's the paradigm, QuantumWidgets.com has 50 other affiliates that haven't caught on yet, but they will. The Consumer is Master! Commissions are an old model! Buying online is still making people chary, and they don't like it. Pay in cash! Pick up your order in an hour. Mention the code and get another 20% first, then go pick it up, or get your oil changed, or what have you. Again, this is from a broke dude with potential, possibly.
Show some support: the site had 10000 views last month, the blogs had about a 1000 views. No followers. Does nobody get it (except affiliates and fans)? Please follow and like that stuff. Free, you know. stop like dr. pepper and crap without thinking about it. it's important to be Good, in a fundamental way.
TMG, Independent Consultant
Quantum Auto Parts
"Prices. Like a Rock."
Sorry, I can't entirely agree with what you say. You argue that integers are emergent. But integer and non-integer numbers can easily be constructed from the same symbols representing physical categories and relationships as are found in representations of Laws of Nature, and so perhaps what you have found points to the underlying infrastructure inherent within some numbers:
If "a" represents an information category relating to a physical particle, and if "+ - x / " represent information relationships*, then the structure "(a+a+a+a)/(a+a+a)" represents a physical category self relationship equivalent to the number 1.333… If this number appeared in a Law of Nature equation, any infrastructure underlying such a number would be completely hidden from view, and there could be many possible category self-relationship structures equivalent to the same number. Also, it is conceivable that the equivalent of a dimensionless number (e.g. "a/a") could be used to construct many other numbers including pi, e, and some complex numbers.
Except in the human activity of mathematics, numbers always refer to physical reality; they only arise in the context of physical reality; and they surely cannot exist in isolation from physical reality. Numbers seem to naturally "contain" various possible internal structures. So I think it's clear that numbers found in nature and everyday life, including the fundamental physical constants, must derive from (sometimes extensive) hidden physical category self-relationships. (And so all other numbers, e.g. most of the numbers on the "real line", are human extrapolations/interpolations of naturally occurring or naturally derived numbers.)
This leads to a related issue: If Law of Nature equations represent a physical information infrastructure that really exists, and the symbols for mass charge etc in these equations represent physical information categories that really exist, then seemingly the equally important symbols in these equations like "+ - x /" and "=" must also represent (non-measurable) physical realities that actually exist.
These 3 types of "building blocks" (categories, relationships and "=") can be used to construct representations of Laws of Nature, mathematical statements, physical outcomes, numbers and new categories, so I think it makes sense to hypothesize that they represent underlying, more elementary forms of reality. There is no reason to suppose that there is anything necessary or anything random about the structures that might result from such "building blocks", but presumably such structures must emanate from/be implemented at centres (i.e. particles): I can't envisage particles being just "ripples of continuous fields" (Page 4).
* More correctly operators, but the word "operators" seems to have the wrong nuances.