Physics and the Integers by David Tong

Hello,

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.

Cheers LC

David,

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

David,

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?")

Tom

David

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

Peter

David,

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.

Jim Hoover

Dear David

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.

Regards,

J. Benavides

Dear David,

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.

httt://www.qsa.netne.net

David,

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.

David,

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.

Tom

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.

Tom

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...

Cheers,

Moshe

Dear David,

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?

Regards,

Eckard

Excellent essay, a breath of fresh air. Hope you will win a prize.

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.....

Steve

Dear David,

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?

Best wishes,

Alan