Are natural numbers sacred in the universe?

[nobox]

There is a profound principal in the universe that says there is no central entity or notion anywhere, and that everything has no special significance than any other things in physics terms. This principal dispelled ‘earth-centric’ idea and later the Newtonian absolute time-space concept. It is a universally accepted principal in modern science. If math and physics are intertwined inextricably then it seems natural numbers ought to have an equal standing as any other numbers, irrational, complex, or even numbers yet to be invented.

Is there any physical underlying reason for natural numbers’ special status? Or are the natural numbers just a convenient way for people to count and were invented by macro intelligent beings like us?

Since all natural numbers are mere derivatives of the number ‘1’, so let’s look closely at what this number one really means. There are two broad meaning of the number one. First it registers a definitive state of some physical attribute, such as ‘presence’ or ‘non-presence’. We can find its application in information theory, statistical physics, counting and etc. The second interpretation of number one is that it denotes the ‘wholeness’ of an entity.

In physics, natural numbers virtually have no sacred places prior to the establishment of quantum mechanics. After all, we don’t need any natural numbers in our gravity functions or the Maxwell electro-magnetic wave functions. Some sharp observers would argue that the ‘R squared’ contains a natural number 2. However on close examination the number 2 is merely a mathematical notation for a number multiplying by itself, and it has no actual physical corresponding object or attribute. The fact that there is no natural number in the formulae represents the idea that time-space is fundamentally smooth. For instance there is no such law in physics that requires 7 bodies (non quantum mechanical) to form a system in equilibrium.

Had we obtained calculus capability before we can count our fingers, we probably would have been more familiar with the number e than 1-2-3. We might have used e/2.718 to represent the mundane singletons. There is no logical requirement that we couldn’t or shouldn’t do it. It is all due to the accident that people happened to need to count their fingers earlier than the invention of calculus. There is no physical evidence that the number ‘2’ is more significant than the any other numbers in the natural world.

However with the standard model of quantum mechanics, energy is quantized, that is, it can only take natural numbers. This idea profoundly altered the status of natural numbers in physics and is a direct contradictory of the notion of ‘no center in the universe’ principal. In this sense it is far more unorthodox than the two relativity theories combined because the latter in fact enhance the ‘no center in the universe’ law. Why does the quantum have to be integer times of a certain energy level, and not an irrational number like square root of 17, or the quantity e? Does it really mean there are aristocrats in the number world, where some are nobler than others? Were the ancient Greek mathematicians right after all, who worshiped the sacredness of natural numbers and even threw the irrational number discoverer into the sea?

From this standpoint we can almost say that quantum theory has some bad taste among all branches of natural science.

Before the quantum theory got its germination, actually people should have noticed the unusual role natural numbers play in rudimentary chemistry. For instance, why two hydrogen atoms and not five, are supposed to combine with one oxygen atom to form a water molecule? If scientists are sharp enough back then they ought to be able to be alarmed by the oddity underlying the strange status of natural numbers. It could almost be an indirect way to deduce the quantized nature of electrons.

Fundamentally if natural numbers indeed play a very unusual role in nature, then nature resembles a codebook not just from a coarse analogy standpoint. It is the ultimate codebook filled with rules for a limited number of building block codes. The DNA code is an excellent example.

If it’s a codebook, inevitably it takes us to surmise if information itself is the ultimate being in the universe. It is probably not electrons, strings, quarks or whatever ‘entities’ people have claimed. It is the information that is the only tangible and verifiable entity out there. Everything else is a mirage or manifestation of some underlying information, the codebook.

In this sense physics has somewhat gone awry by focusing on the wrong things, the ‘attributes’ such as momentum, position and etc. Instead, information is what contemporary physicists talk about and experiment with. Otherwise, the physicists would have no right to laugh at the medieval scholars who based their intellectual work on the measurement of the distance between a subject and God’s throne.
The nature has revealed her latest hand of cards to us. It looks like it’s the final hand but no one can be sure of course.

[Strange]

There is a profound principal in the universe that says there is no central entity or notion anywhere, and that everything has no special significance than any other things in physics terms.

 

I don't believe that is a principal (profound or otherwise).

 

 

Or are the natural numbers just a convenient way for people to count

 

That is pretty much the definition of natural numbers. Hence the name.

 

 

Since all natural numbers are mere derivatives of the number ‘1’

 

Are they? The derivations I have seen start from zero.

 

 

However with the standard model of quantum mechanics, energy is quantized, that is, it can only take natural numbers.

 

That is not true. There is no simple integer relationship between the energy levels of an electron in an atom, for example. And electromagnetic radiation is quantized but can have any energy/frequency.

 

 

For instance, why two hydrogen atoms and not five, are supposed to combine with one oxygen atom to form a water molecule?

 

Quantum theory explains that.

[John Cuthber]

"actually people should have noticed the unusual role natural numbers play in rudimentary chemistry. For instance, why two hydrogen atoms and not five, are supposed to combine with one oxygen atom to form a water molecule?"

Well, for a start, five is also a natural number.

For a middle, people did notice this- it's called stoichiometry and

For a finish it's not always true. plenty of chemicals are not stoichiometric.

 

On the whole, as far as I can tell, the answer to "Are natural numbers sacred in the universe?" is simply no.

[nobox]

Ancient Greeks had pondered exactly the same questions for long. People like Pythagoras adamantly asserted 'yes' to this question. We are back to square one after 2 or 3 millenniums. Natural numbers is a set of math objects and it happened to have a very well defined physical correspondence with it. The question is whether this set is fundamental in math? If so, it is the fundamental set for physics? I inclined to think it is NOT a fundamental set in math. In this physical world we occupy, maybe natural numbers got some fundamental role to play, or maybe not. Quantum mechanics seems to give us some clue to this question.

I don’t know the universes that humans do not occupy and the relevant status of natural numbers there.

[Strange]

Ancient Greeks had pondered exactly the same questions for long. People like Pythagoras adamantly asserted 'yes' to this question.

 

And they had to give that belief up when they found that even things as simple as the diagonal of a square or the circumference of a unit circle are irrational.

 

 

Quantum mechanics seems to give us some clue to this question.

 

Apparently not.

[nobox]

Mapping is a fundamental idea in math. A large part of knowledge body in math is about mapping. In a domain A, the relationship among its members a1, a2… an, can be mapped to another domain B, based on a set of rules R, and vice versa. The benefits of doing this mapping are multitude. It certainly can reduce computation complexity; it gives rise to a difference viewpoint at the old relationships; the mapping can be a daisy chain without ending therefore the mathematicians can stay employed.

I am not suggesting mapping is the basic characteristic of math although I admitted being tempted to do so.

If we consider the physical world as a domain where we can abstract all the elements within as its members. Apparently there exist innumerable relations among the members. It can be further thought as a math domain. So far we have very few problems mapping the physical domain to the broader math domains. It is fascinating to see further that some of the logic constructs in physical domains are even shared by the math domains. (Whether or not it has to be that way is a debatable subject)

Obviously the math domains are inexhaustible, which is hardly a feature of the physical domain. If we think out of the box, we can invent many different new math domains that are quite different from our existing math domains. Non Euclidean geometries are great examples of that but the attempt had fell short in other math branches. For instance, probability is such a fertile field for a paradigm change in designing a new math domain and new mappings, where 1+1 does not equal 2. The traditional math domains can still handle probability but it is not very beautiful.

I'd further argue that if we change our current math representation (domains and rules) we may invent a type of math that can express quantum mechanics or high physics in far simpler terms. I don't see the reason why there shouldn't exist a math domain where 2+2=5, for instance.

The math as we know now is built on a number of self-evident axioms. The problem with these axioms is that they are seemingly correct with our everyday experience. However, that does not grant these axioms any immunity or validity. If we are as small as an electron, I’d bet we will be accustomed to a completely new type of math, a math where calculus is the heredity.

Further the notion that the whole universe runs on the math we have invented/discovered is nothing more than hubris, a distinctive characteristic of mammals.

[Bignose]

The traditional math domains can still handle probability but it is not very beautiful.

Beauty is in the eye of the beholder, I guess then, eh?

 

I'd further argue that if we change our current math representation (domains and rules) we may invent a type of math that can express quantum mechanics or high physics in far simpler terms.

Ok. Argue it all you want. Can you actually demonstrate it in any way? If no, then I guess I don't see what the point in 'arguing' it is...

 

The problem with these axioms is that they are seemingly correct with our everyday experience. However, that does not grant these axioms any immunity or validity. If we are as small as an electron, I’d bet we will be accustomed to a completely new type of math, a math where calculus is the heredity.

More opinions. Here on the science forums, we like to deal with science. Any evidence you can provide to support anything here?

Edited January 12, 2015 by Bignose

[nobox]

Sequence is assumed in natural numbers, i.e., one is the next math construct after 0. Sequence is also exclusive, i.e., there is only single symbol to denote what’s after 0. However, it doesn’t have to be so. In the quantum world, there is no definition of sequence, hence no concept of natural numbers as we understand.

In the realm of speed of light world, we do not have addition either; and we may see c+c=c.

Modern physics calls for new type of math ever since the early 20^th century, and so far no mathematicians dare to risk careers to venture outside the conventional wisdom.

Of course the details are much complicated and I am no expert to explain. Everything is just a conjecture here. I found it far easier and gratifying to point out others’ insufficiencies, than to improve one’s own understanding. This is human nature, just like why people prefer being fed than being hungry.

[Bignose]

I found it far easier and gratifying to point out others’ insufficiencies, than to improve one’s own understanding.

I guess I should be glad that you willingly admit this. It is easier to tear down than build up.

 

...

 

Kind of useless, though, no?

 

I guess I am ultimately a practicalist. If you don't have something to improve the situation, just pointing you what you perceive to be problems isn't all that interesting, because it actually isn't doing anything.

Edited January 12, 2015 by Bignose

[Sato]

I don't see the reason why there shouldn't exist a math domain where 2+2=5, for instance.

 

Someone posed a similar question earlier, and I tried to elaborate here; maybe that will be of use. If you redefine the addition operation or the symbols '2' and '5' and so on you might arrive at the result "2+2=5", but in our system of natural numbers base-10 and their arithmetic, applying 2 and 2 to the addition operation, 2+2, or +(2,2), will result in 4. This is because we define 2 as the second successor of 0, s(s(0)), and applied to itself is s(s(s(s(0)))), which is symbolized short-hand as '4'. That's the basic structure of what we call the natural numbers, however you define the symbols and all.

 

The math as we know now is built on a number of self-evident axioms. The problem with these axioms is that they are seemingly correct with our everyday experience. However, that does not grant these axioms any immunity or validity. If we are as small as an electron, I’d bet we will be accustomed to a completely new type of math, a math where calculus is the heredity.

 

Our abstractions (of which the mathematical objects that we study are a subset) are extrapolations from our perceptions, so how we present or interpret the different facets of mathematics is deeply influenced by our vantage as humans. And that's alright, because mathematics is the generalization, formalization, and study of our abstractions, and so we don't need to be concerned with what some other sentiences in some other circumstances would conceive of; unless for some reason it would be fruitful for our own use and application. Nonetheless, we do have different schools of thought and proposals for foundations of mathematics; from some who deprecate of the real numbers (Wildberger) to some who think/thought all mathematics could be derived from formal systems (Hilbert) to those who support types as a foundation of mathematics and do away with sets in that respect (HoTT movement). An excerpt from a blog post I read recently may interest you:

 

 

...Now imagine that this hypothetical industrial society also skipped the hunter-gather phase of development. That’s the period that gave birth to counting and natural numbers. I know it’s a stretch of imagination worthy a nerdy science fiction novel, but think of a society that would evolve from industrial robots if they were abandoned by humanity in a distant star system. Such a society could discover natural numbers by studying the topology of manifolds that are solutions to n-dimensional equations. The number of holes in a manifold is always a natural number. You can’t have half a hole!

 

Instead of counting apples (or metal bolts) they would consider the homotopy of the two-apple space: Not all points in that space can be connected by continuous paths

,,,

 

 

P.S. Your idea that mathematics can exist as and be founded on other forms is sound, but sometimes you throw out hokum like "...nothing more than hubris, a distinctive characteristic of mammals.", "It certainly can reduce computation complexity", "In the realm of speed of light world, we do not have addition either"; remarks that are either incoherent, unsubstantiated, or just wrong. Unnecessary stuff like that plagues your posts and dilutes the discussion. I think it would be best for you to look into and develop your ideas further first, but I hope this post assists you in someway anyway.

[Mike Smith Cosmos]

Remembering ( without the detail ) , one of the memorable moments in the presence of a maths lecturer. ( in university) , He eat , drunk and and lived mathematics . I asked him how he knew where to go in a proof. He replied " take the simpler more beautiful route "

 

He also said " there is a transform ( like s/ 1- s squared or something ,) not sure . He went on to say will transform to 'anywhere' . Such that when you transform back from the s domain ( ' anywhere ' in the s domain ) , you will come back to this exact spot" .

 

This sounded fascinating to me at the time , and to this day. As it means mathematically , one could nip off to 'anywhere' , and yet get back home to exactly the right spot. Yet life in the other domain would be quite quite different.

 

I think this has some bearing on what is being discussed here, ! I think !

 

Or was it , " you transformed to an exact spot , but could reverse transform to ' anywhere ' . In which case you could travel to anywhere in the universe! From this spot ? " Not sure how you get home again ? Probably reverse transform (better get the sums right , or you could end up anywhere ) ! Hey ! Where am I ? I suppose this is the difference between maths and physics. Maths is a construct ! Physics is reality ?

 

If the maths is not, or can not ever be anywhere near , explicit enough , then it can never attain reality , whereas physics by definition IS the reality ? Possibly ? Or possibly not ! Eek !

 

Mike

 

I need some sleep !

Edited January 12, 2015 by Mike Smith Cosmos

[nobox]

Prime numbers have some mysterious properties. Are they more fundamental than the natural numbers? Natural numbers can be mapped through a set of rules from the set of prime numbers. Is there a math system that utilizes prime numbers as the base set? Does prime number set have any physical mapping that has not be discovered in the past?

The universe that we now live in has some remarkable feature: Seemingly complexity stems from a relatively few math/physics principals. Observation selection looks attractive to me. But unless there is proof of possibility that the other unknown universe can have any effect on our universe, this conjecture is meaningless. The same guideline is also relevant in this thread about math systems.

Another interesting idea is that math never implies causality but physics does. This is because the time element in physics. There is no place or treatment for time in math. What does that tell us about time? My guess is that time is the culprit of a lot of misunderstanding in physics. Time essentially is a psychology quality in living organisms. It’s not a physical entity at all. Is there is physics that exorcizes time vetor?

[Greg H.]

 

Are natural numbers sacred in the universe?

 

No. The idea that something is sacred is a human construct, and varies depending on which human you ask. The universe, by and large, doesn't really care.

[nobox]

 

No. The idea that something is sacred is a human construct, and varies depending on which human you ask. The universe, by and large, doesn't really care.

Agree. Maybe I shouldn't use 'sacred', but 'special' here.

[Bignose]

Natural numbers can be mapped through a set of rules from the set of prime numbers.

So can any other series. See https://oeis.org/wiki/Welcome for a pretty large list. Are all of these 'sacred', too?

 

Is there is physics that exorcizes time vetor?

Define time vector. The time variable is involved in a lot of vectors, but time itself isn't typically taken to be a vector.

There is no place or treatment for time in math.

What?!? This statement seems totally unjustified to me.

Edited January 12, 2015 by Bignose

[nobox]

Math is able to stay self-consistent, partly is because the domains are constructed arbitrarily and the mappings amongst domains are constructed arbitrarily. The only thing that are rigorously constructed and tested is the relationship within one particular domain. Use human languages are an example. A domain corresponds to a language, say English. Mapping is translation, and the elements in a domain are words, sentences, meanings and etc. The relationship of domain elements is the relationship of the words in a language. The grammar of a language is equivalent to one relationship within a domain. You will need coherent and correct grammar to have meaningful language. However, you can construct any arbitrary language, such as sign language and so forth, as long as there is some order and coherence in that language. Further you have mapping that can completely or incompletely translate among different languages. Math is always ‘correct’ because it keeps inventing new domains and new mappings. There is no requirement of what the mappings should be as long as it fits the purpose. For the new domains themselves, the only requirement is to have a set of relationship rules that are coherent and logical. You are able to achieve any types of proof by continuously mapping from domains to domains.

One requirement of a good mapping is to avoid contradictory results in mapping operations. Another requirement is to use as few as possible rules in mapping. Unlike human language mapping which requires a tome of dictionary, math mapping prefers general rules that can be summed up in a few lines.

[ajb]

Mapping is a fundamental idea in math. ....

I am not suggesting mapping is the basic characteristic of math although I admitted being tempted to do so.

 

Category theory kind of takes that attitude and places morphisms between objects as the 'fundamental objects'. You then have functors which take us from one category to another, again the ideas of 'mappings' is fundamental. And then we have natural transformations which are 'mappings' of functors.

 

So maybe the idea of a 'mapping' really is the basic characterisation of mathematics.

[nobox]

We may consider the current world as a domain A and the world in the next moment as another domain B. Time is the operating mapping rule to map each element in A to B. But unfortunately there exists none reverse mapping, hence time travel is a meaningless phrase. In every domain there is no time element. Time itself is an illusion for organisms to incorporate the mapping operation into innate experience.

The physics law can be re-interpreted under this new paradigm. Instead treating time as a presumed independent physical variable, it is a function T. For every element (positions and masses) of domain A, we have a time function T(.) that operates on it.

2^nd Thermal dynamic law:

T(a) = b, where a is the entropy of element a in domain A and b is the entropy b in domain B, and the rule is b>a;

Speed of light:

T(a) = b/c, where a and b are relative position coordinates in domain A and B, and c is a constant, i.e. speed of light.

…

If my scheme works, then it will be a profound breakthrough. Of course the probability is admittedly low. But any critique is welcome, as long as it is not personal attack.

Fundamentally we cannot measure time, and time is not an absolute variable like position. Making time-space a 4 dimension doesn't help, except more confusion. It caused crisis after crisis in physics because of the confusion of time. If time is treated as an operator things will be much clearer.

Quantum mechanics will be instantly making sense the moment we take out the time as a variable.

Edited January 13, 2015 by nobox

[Bignose]

If my scheme works, then it will be a profound breakthrough. Of course the probability is admittedly low. But any critique is welcome, as long as it is not personal attack.

Show me any equation with time in it replaced by your method and demonstrate it works.

 

If you need an example, how about the wave equation:

 

[math]\frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u[/math]

 

There is a clear dependence on time in there. And, in fact, this equation works perfectly fine whether you go forward or backward in time. And it is verified accurate a very large number of times.

 

Show me how your idea is better than what we have now.

[Strange]

Fundamentally we cannot measure time

 

Yes we can. We have these things called "clocks". (I hope this isn't going to turn into yet another "time doesn't exist it is only change/movement/bat-poo/etc")

 

and time is not an absolute variable like position

 

Postion is not absolute either. It can only be defined relative to something else.

 

Making time-space a 4 dimension doesn't help

 

It seems to work extremely well. In everyday life, if you want to meet someone, you need to specify both the spatial location and the temporal coordinate. And, of course, GR is an extremely succesful theory.

 

It caused crisis after crisis in physics

 

Can you give an example of these crises?

 

Quantum mechanics will be instantly making sense the moment we take out the time as a variable.

 

That is a bold claim. Can you back it up?

[nobox]

Someone is figuring out what proposed here: inventing a new math and do away time and etc for the quantum world. Searh "amplituhedron".

[Strange]

Someone is figuring out what proposed here: inventing a new math and do away time and etc for the quantum world. Searh "amplituhedron".

 

Or provide a link:

https://www.quantamagazine.org/20130917-a-jewel-at-the-heart-of-quantum-physics/

https://en.wikipedia.org/wiki/Amplituhedron

http://www.math.columbia.edu/~woit/wordpress/?p=6607

http://www.preposterousuniverse.com/blog/2014/03/31/guest-post-jaroslav-trnka-on-the-amplituhedron/

 

This isn't "new math" though. And it isn't clear yet if it is actually useful.

[nobox]

My intuition told me that the common physical variables such as location, time, velocity, and etc are very inappropriate in the quantum world, which created vast amount of confusion. The confusion can be dispelled after removing these concepts/variables, and instead using geometric interpretations. After all, nothing is real in this world. We are fooled by nature all along.

“They are very powerful calculational techniques, but they are also incredibly suggestive,” Skinner said. “They suggest that thinking in terms of space-time was not the right way of going about this.”

Read more: https://www.quantamagazine.org/20130917-a-jewel-at-the-heart-of-quantum-physics/#ixzz3Oq1kq1uK

At its core, space or time do not exist. Relativity and quantum mechanics have to twist these constructs to fit our understanding of space and time. This is equivalent to defective mapping in math. Why bother with the concept of space or time, just because we have everyday experience and think we know what they are?

Use languages as an analogy, we speak them with speed, but does that mean that English has a property called 'velocity'?

I predict that a revolution in physics may have started this year. I suggest physicists picking up geometric algebra at once.

Obviously we have to have physical forms to use languages, which involve pitch, speed, paper, digital code and etc. Without some physical form languages cannot 'exist'. But none of them is the innate property of language itself. I tend to think nature is similar to the language. Alas, I am closer to Plato's idealism that I want to.

Edited January 14, 2015 by nobox

[Bignose]

My intuition told me that the common physical variables such as location, time, velocity, and etc are very inappropriate in the quantum world, which created vast amount of confusion.

Hmmm. It used to be intuitive to think that the moon was made of green cheese, the earth was flat, and heat was a fluid called phlogistan. If only there was some way of checking intuition...

 

The confusion can be dispelled after removing these concepts/variables, and instead using geometric interpretations. After all, nothing is real in this world. We are fooled by nature all along.

So, quit telling us it is wrong, and I don't know, actually demonstrate something. Like maybe answering the question I asked above. Show me how to remove time from the wave equation.

[Strange]

and heat was a fluid called phlogistan.

 

I hate to be picky (*) but: caloric. Phlogiston was something else.

 

(*) No, you're right.