Physics and the Integers by David Tong

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.

David,

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

Dear David,

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!

David

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.

http://translate.google.com/translate?hl=en&sl=de&u=http://de.wikipedia.org/wiki/Nielsen-Ninomiya-Theorem&ei=wlvvTeesFcm3twfotMmhCQ&sa=X&oi=translate&ct=result&resnum=9&ved=0CG0Q7gEwCA&prev=/search%3Fq%3Dnielsen%2Bninomiya%2Btheorem%26hl%3Den%26client%3Dsafari%26rls%3Den%26prmd%3Divns

http://translate.googleusercontent.com/translate_c?hl=en&prev=/search%3Fq%3Dnielsen%2Bninomiya%2Btheorem%26hl%3Den%26client%3Dsafari%26rls%3Den%26prmd%3Divns&rurl=translate.google.com&sl=de&u=http://de.wikipedia.org/wiki/Domain-Wall-Fermion&usg=ALkJrhgLwUg8PtxZIK4j6WuASB7n1a-5pw

http://cerncourier.com/cws/article/cern/28271

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. ;)

Dear David,

Congratulations. Yes integers are emergent from zero, like intelligence is emergent from conscience.

Conscience is the absolute zero from which all else emerges.

Love,

Sridattadev.

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!

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?

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?

Dear David,

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

Love,

Sridattadev.

Dear Tommy,

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.

Love,

Sridattadev.

Dear All,

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 = ~

Love,

Sridattadev.

Dear All,

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.

Love,

Sridattadev.

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."

David,

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).

Lorraine Ford

* More correctly operators, but the word "operators" seems to have the wrong nuances.

People with not enough training in mathematics stumble on the continuous vs. discrete dichotomy much too often and I'm afraid David Tong shows some innocent immaturity here. Real analysis is a degenerate case of discrete analysis and he is entertaining a false dichotomy. The clearer dichotomy concerns the parts vs. the whole. Continuity leads to serious self-contradictions and I encourage Tong to read Solomon Feferman's work on Cantorian mathematics and its relation to physics (note: countable infinity is as guilty as uncountable).

Tong is simply engaging in a circular form of reasoning by assuming the illogical notion of completed infinity and accusing computers of not understanding it (note: Wen has done work on chiral fermions but the gauge community is neglectful with refereeing). The calculus really uses finite arguments in disguise (my computer can do it with ease) and it's actually still debated in the mathematical community whether the Cauchy regime consists of nothing but trite non sequiturs (it's not based on classical logic). We only see integers (multiplied decimals) in any empirical setting and NOT ONE purely real number has ever been recorded or seen.

We do have "buy one, get one free" Banach-Tarski paradoxes and genuine contradictions debated by philosophers but I suggest Norman Wildberger's article "Real fish, real numbers, real jobs" in The Mathematical Intelligencer March 1999, Volume 21, Issue 2, pp 4-7. Tong shows little knowledge of the various views in the mathematical community and I suggest consulting Jean Paul Van Bendegem to consider a directly contrasting perspective. He is right to see that something is amiss but he has the argument precisely backwards.

There is nothing particularly profound about the technical problems in mathematics but a great deal of popular confusion stems from amateur accounts. Example: people making obsessive and crackpot arguments about Cantor, Gödel, and Turing diagonalization. We have self-referential sentence proofs: Gödel, Rosser, Kleene, Post, Church, Turing, Smullyan, Jech, Woodin, etc... Epsilon-naught induction proofs: Kripke, Paris-Harrington, Goodstein, Hydra, etc... Kolmogorov complexity proofs: Chaitin, Boolos, etc...

It's really quite simple:

Proof: Given the program "THEOREMS" which outputs theorems (it could be doing deductions in Peano Arithmetic, for example), write the computer program SPITE to do this:

SPITE prints its own code into a variable R

SPITE runs THEOREMS, and scans the output looking for the theorem "R does not halt"

If it finds this theorem, it halts.

"Gödel's theorem is a limitation on understanding the eventual behavior of a computer program, in the limit of infinite running time." (Ron Maimon)

The same way you can't even list the infinity of reals, you can't even complete the entire list of computer programs. The initial assumptions are the problem. The Gödel sentence is basically a gibberish freak that has no provable relation to meaningful mathematical problems. The real take away: "No compelling evidence has yet been presented that G1 affects, or future refinements of it will affect, mainstream mathematics." (doi:10.1007/s11787-014-0107-3) If you think undecidability is profound, I can comically ask you if you stop beating your wife.

Cantor makes a more obvious appearance in the non-standard model, which requires infinite non-natural numbers after the natural numbers. Incompleteness can be done away with using an infinitely axiomatizable system but that very fact reveals what it's really about. Poincare knew it was trivial all along and he also knew set theory was just backpedaling. There are genuine concrete coloring problems in the plane that give different answers depending on continuous assumptions in set theory. Set theory ends up limited by the analog of an inescapable chaotic initial value problem.

The limitative results of Gödel and Turing are very poorly understood. The ability to answer Goldbach-like questions is not necessarily dependent on infinite output. Programs can also examine the symbolic structure and content of other programs. Gödel's result has nothing to do with meaningful mathematical problems but nonsensical ones that only look like well-formed formulas. Self-referencing formulas and impredicative sets are not about mathematics proper. I can't stress how badly people don't understand this! A computer program may very well be able to answer all meaningful mathematical statements. Nothing from incompleteness stops this.

In terms of consistency, logical explosion can prove anything. If I ask a system whether it's consistent, it can say it is by logical explosion. Hilbert was simply confused to ask such a question. Gödel is that trivial and simple.

And then we have people so confused they think incompleteness has some bearing on the continuum hypothesis. This is an embarrassing conflation that fails to understand structural generation of the hierarchy of freak statements. I agree with Feferman on this question.

The point of the above is to say that it was the introduction of completed infinity that got people so confused that they are even turning to the logic of Priest and "true contradiction" to find a way out. Why not throw out the nonsense rather than change the rules of logic? Continuous (infinite) assumptions are baseless and break down when pushed. If you throw them out the problems go away. Nature abhors a nonsense.