Re: [PrimeNumbers] Distributed as regularly as possible

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• ... It is in fact a tautology to define primes by the way you find them. Whatever you find by the Sieve is prime. If the Sieve way is regular, then prime is
Message 1 of 7 , Feb 7, 2007
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--- "Werner D. Sand" <Theo.3.1415@...> wrote:

> Demonstration of the statement "the prime numbers
> are distributed as
> regularly as possible":
>
> Regula 1: All numbers > 1 are prime.
> Try: 2,3,4 oops, 4 is not prime. 4 is equal, hence
>
> Regula 2: All numbers >1 which are not multiple of 2
> are prime.
> Try: 2,3,5,7,9 oops, 9 is not prime. 9 is multiple
> of 3, hence
>
> Regula 3: All numbers >1 which are not multiple of 2
> or 3 are prime.
> Try: 2,3,5,7,11,13,17,19,23,25 oops, 25 is not
> prime. 25 is multiple
> of 5, hence
>
> Regula 4: All numbers >1 which are not multiple of 2
> or 3 or 5 are
> prime.
> Try: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,49
> oops
>
> and so on. The regulas become more and more exact.
> The prime numbers
> are distributed "as regularly as possible".
> It's simply the Sieve of Eratosthenes.
>
> Werner
>

It is in fact a tautology to define primes by the way
you find them. Whatever you find by the Sieve is
prime. If the Sieve way is regular, then prime is of
course regular. but it is a tautology. The fact that
the Sieve is regular cannot prove itself to be
regular. If regularity is already defined in to
primes, that same regularity cannot be used to prove
that prime is regular. the same kind of regularitys
cannot prove the same regularity. If prime is defined
by regularity or the regular way of finding primes,
those way of finding primes cannot be used to prove
that primes are regular. What needs to be done is to
define primes by a kind of regularity that can then be
used to prove another kind of prime regularity.
Regularity A proves regularity B (B could be the sieve
or the non-divisibility). But regularity B cannot
prove itself. If prime is defined by regularity B,
then B cannot be used to prove prime regularity. It
is to use B to prove B, a tautology. You have to use
A to prove B. To prove prime is regular is to prove
that the essence of prime is regular. From that
regular essence, we deduce other manifest regularities
such as the Sieve or the non-divisibility. But the
regularity manifestation of a prime does not
necessarily mean or prove that the essence of prime is
regular. The essence could be random but still
produce a manifestation of regularity. The manifest
regularity of nature does not prove that the essence
of the universe, the quanta, are regular. In fact,
many believe that the regularity nature is caused by a
random essence. Thus, the regularity of primes (the
sieve or non-divisibility) could be caused by a random
essence and does not prove that prime has the true
property of being as regular as possible.

To prove or explain the hardness and rareness
properties of diamonds, one must start from its
essence carbon and its way of creation. To explain
particles and its way of creation. To prove or
explain the regularity of a non-prime number, one must
start from its essence which is the prime number and
its way of creation.

To prove or explain the regularity of primes, one also
must or may have to start from its essence and its way
of creation. What is the essence is unknown at
present, which is most likely the reason why the RH
cannot be proven. Non-divisibility is not the essence
of primes, just like divisibility is not the essence
of non-primes (the essence of non-prime is the prime).
Essence here simply means the building block. If the
essence of non-prime is the prime, it is only fair and
logical to go down the hierarchy to ask what is the
essence of prime. That essence logically cannot be a
number since prime number is the lowest level a number
can be.

Assuming we know the essence of prime, we can imagine
two ways to create a prime. A random way of
manipulating the essence or a non-random and highly
regular way. It is of course possible with near zero
probability for the random way to create a population
of primes that would show regular pattern. But it is
far more likely that the random way will not produce
any regular pattern of primes. On the other hand, the
non-random and regular way of creating primes would
exclude any irregularity as a viable outcome. It
would demand or predict that primes are distributed as
regular as possible. In this case, the RH would be a
logical deduction of the prime essence and its lawful
way of creation, just like hardness and rareness of a
diamond is a logical deduction of the carbon essence
and its way of creation. What canbe logically
deducted from an essence or axiom must be true. To
prove something is to deduce it from some basic truth,
essence, or axioms. To prove that human is made by
God using quantum particles is to deduce humans from
the particles essence and God in a step by step and
lawful way. Without knowing the law of God or the way
of creation using particles, we can never prove or
disprove the hypothesis that God created humans.
Analagously, without knowing the way of creating
primes from its essence, we can never prove or
disprove the hypothesis that prime is created in a way
that is as regular as possible, which automatically
predicts the RH.

The regularity of primes is what we observe. The
regularity includes many forms, such as
non-divisibility, prime number theorem, and the RH.
But it leaves open the question of what causes the
regularity, is it randomly caused or not. To prove
the RH means we can exclude any randomness in primes
including their causes. If the RH is true, the cause
for the regularity of prime cannot be random. The
phyical universe provides a good analagy. The
regularity and lawfulness of matter above the quantum
level is what scientists observe and is why scientists
have a job. The regularity includes many forms, such
as the various physics laws and the fact that life is
built by DNA. But it leaves open the question of what
causes the regularity, is it randomly caused or not.
Scientists believe, without any proof, that it is
randomly caused because of lucky accidents. Religions
say it is caused by God. The position of science can
never be proven since randomness can cause many
outcomes, including randomness itself and the near
zero probability event of regularity. Since both
randomness and regularity can be logically deduced
from a random cause, neither is a certainty and
neither can be proven by invoking a random cause. But
the God position has a chance to be proven true. If
we give God a creation law and that law predicts only
one outcome which is regularity, then we would have
proven that the regularity of nature is caused in a
non-random way which is a way of law that predicts
only one specific outcome of regularity. To prove
that an outcome is caused by the essence is to show
that all other outcomes are non-viable. The random
position cannot exclude randomness as a viable outcome
and cannot therefore prove that the regularity outcome
is caused by the random essence.

So to prove the RH, we need a creation law for the
primes that predicts only one outcome regularity.
Irregularity is not allowed by that law and is not a
viable outcome. By finding that law, we prove that
the cause for the regularity of primes is not random.
If the cause is not random, then there is no chance
for the primes to display any aspects of randomness,
which is the same as saying that primes are as regular
as possible, which thus proves the RH.

So it is my assertion that the only way to prove the
RH is to find the essence of primes and the law of
creation. It is comparable to prove the hypothessis
that the regularity of nature is caused by God and his
law rather than randomness. But the only way to prove
God is to find his law of creation using particles.
Until one finds a creation law that uses
essence/particles to create primes/nature, one can
always say that the regularity of primes/nature is
caused by chance, no matter how small a chance. One
can never say that primes/nature are as regular as
possible or as non-random as possible. If a thing is
caused randomly, it surely cannot qualify as the most
regular possible or as regular as possible, because
another equivalent thing that is caused non-randomly
would have more regular properties and would better
qualify as the most regular possible.

To say primes are as regular as possible means
everything about primes, including its way of
creation, are non-random. The RH makes a big claim
that is inherently or in principle unprovable by
anything that deals only with behaviors of primes.
The RH does not deal with how prime is created from
its essence. Thus, even if proven true, it cannot
support its claim that primes are as regular as
possible. The RH is logically in capable of proving
its own claim. This is perhaps the reason why it
remains unprovable. Its claim is bigger than it can
possbily deliever. It simply cannot deliver its
claim. Proving the RH will not prove its big claim,
which is the same as saying proving the RH will not
prove the RH. The claim of RH is logically in capable
of being proven by proving the RH. The RH is simply
unprovable from its own perspective. Its claim has to
be proven from another perspective.

To use an analagy. The laws of physics above the
quantum level says that nature is as regular as it can
be, limited only by the random noise of atoms. But
such a claim imply that the cause of nature is also
regular, since a nature that is regularly caused would
be more regular than a nature that is randomly caused.
Proving the laws of nature that describe behaviors of
nature cannot prove that nature is caused/created
non-randomly. So they can never justify the claim
that nature is as regular as possible. The claim can
only be proven or justified from finding the
non-random law that is used to create nature.

So, if the RH really claims, in layman terms as the
Wikepedia says, that primes are as regular as
possible, proving RH will not justify the claim. This
simply means that the RH is inherently unprovable from
the same level of thinking that leads to the RH. It
means that the RH cannot be proved by any number
theory. It can only be proven by finding the essence
of primes and by finding the non-random law of prime
creation. To prove the RH is the same as to prove
that the regularity of nature is caused by a
non-random creation law and a law giver. In this
sense, it is the same as to prove God. This is the
reason for the RH to be regarded as the most
fundamental problem of mankind. It is interesting
that another equally fundamental problem with
relevance to God, the Darwinian theory of evolution,
came out the same year as the RH in 1859. Darwinists
are people who look at the primes and say that they
are caused randomly. The RH implies that everything
about primes, including its essence and its way of
creation, is not random. To prove the RH is to
disprove Darwinism and is to prove God. The two
opposite theories have the same birthday and most
likely will have the same judgement day, which
hopefully will be very soon.

Shi

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• ... Werner, You certainly have no jusitification or proof to call primes mathematical constants. If everyone agrees that the essence of numbers or non-prime
Message 2 of 7 , Feb 8, 2007
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--- "Werner D. Sand" <Theo.3.1415@...> wrote:

> Hello Shi,
>
> I do not define the primes by the E-Sieve, they are
> defined as they
> are. The Sieve is only a method to develop them. I
> don't prove a
> regularity of the primes by a regularity of the
> Sieve, but I show
> that the Sieve IS a regularity and thus the primes
> regularity. A regularity is not necessarily a
> formula like f(n) but
> may also be a calculation instruction like Euclid's
> algorithm or the
> Fibonacci numbers or the prime numbers. The prime
> numbers are a
> natural constant just as pi or e or sqrt(2). You
> cannot predict the
> 678th digit of pi, but pi/4 = 1 - 1/3 + 1/5 - 1/7 +
> Random or not?
> Do you call sqrt(2) random because you cannot
> predict the 101th
> digit? Probably you say "sqrt(2) is as it is". And
> such are the
> primes.
>
> Tell me something about the "essence" of
> mathematical constants: are
> they "God"? Are they "Tao"? Are they "Nothing"?
>
> Werner
>

Werner,
You certainly have no jusitification or proof to call
primes mathematical constants. If everyone agrees
that the essence of numbers or non-prime numbers is
the prime, then tell me what is the essence of prime?
Dont tell me that prime has no essence. That simply
shows that you dont know the answer.

Just like from the essence of non-primes, we can prove
or deduce that non-primes are always regular and
predictable, we should be able to deduce or prove from
the essence of primes whether primes are always
regular but unpredictable. Whthout knowing the
essence of primes, certain things can never be proven,
such as the predictability or the RH. Without knowing
the essence of diamond to be carbon, how can you be
sure or prove that diamond cannot be rock made of
silicon?

Primes are touted as the building blocks or atoms of
numbers. Well, you should ask what is the building
block of primes, is it quantum particles? Isnt is odd
that no one has a clue about such fundamental
questions?

Shi

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• Shi, first of all: what do you mean by essence ? I love doing philosophy instead of mathematics, but I must know about what we are speaking. Maybe I tell you
Message 3 of 7 , Feb 8, 2007
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Shi,

first of all: what do you mean by "essence"? I love doing philosophy
instead of mathematics, but I must know about what we are speaking.
Maybe I tell you that the "essence" of the primes are the primes. You
didn't answer my question about pi or e or sqrt(2). What do you
consider to be the essence of sqrt(2)? Sqrt(2) has a definition, but
no essence. What difference is there between the prime numbers and
pi or e? pi=3.1415926, e=2.718281828, p=2.35711131719 That's my
justification to call the primes a natural or mathematical
constant.You may as well call the set of the prime numbers
P={2,3,5,7} or each single prime number a constant.

In a sense you can say the primes are the atoms of the numbers. Then
you mean multiplication. In another sense you can call One the atom
of the numbers. Then you mean addition. What do you think about
the "essence" of One? Perhaps the "essence" of the prime numbers is
the combination of addition and multiplication. If there is only
either addition or multiplication, there are no prime numbers. If you
have the number 2 and only multiplication, you get only numbers 2^k.
If you have the number 1 and only addition, you don't know what
division is. It is somewhat like Yin and Yang: each of them alone is
nothing, both of them together are the reality. Is that a bit nearer
to what you call "essence"?

Werner
• snip ... I have heard and read that the primes are the building blocks of numbers and I don t appreciate that at all. I was going to say get that but that
Message 4 of 7 , Feb 9, 2007
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snip>

> Primes are touted as the building blocks or atoms of
> numbers. Well, you should ask what is the building
> block of primes, is it quantum particles? Isnt is odd
> that no one has a clue about such fundamental
> questions?
>
> Shi

I have heard and read that the primes are the building blocks of
numbers and I don't appreciate that at all. I was going to say "get
that" but that isn't true, I do get it, I just don't appreciate it
well. "1" is the building block of numbers. If you build numbers
from 1, you get a finitely generated abelian group with group
operation addition. If you build the numbers multiliplicatively,
then you get an infinitely generated monoid with no zero divisors
("cancellative monoid"?). I vote for the abelain group over the
monoid. It is a stretch to even understand the monoid without
addition-I have got this big multiplication table and I "happen" to
observe certain entries that are minimums in the POSet ordered by
divisibility. That is really stretching the layman's understanding
of "number". I think it is even stretching the mathematician usual
view of "number".

• ... Has this thread not suffocated from the morass of ill- or un-defined terms yet? It certainly looks moribund. ... What do you mean by tautology here? ...
Message 5 of 7 , Feb 10, 2007
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--- Shi Huang <shuangtheman@...> wrote:
> --- "Werner D. Sand" <Theo.3.1415@...> wrote:
> > Demonstration of the statement "the prime numbers
> > are distributed as
> > regularly as possible":
> >
> > Regula 1: All numbers > 1 are prime.
> > Try: 2,3,4 oops, 4 is not prime. 4 is equal, hence
> >
> > Regula 2: All numbers >1 which are not multiple of 2
> > are prime.
> > Try: 2,3,5,7,9 oops, 9 is not prime. 9 is multiple
> > of 3, hence
> >
> > Regula 3: All numbers >1 which are not multiple of 2
> > or 3 are prime.
> > Try: 2,3,5,7,11,13,17,19,23,25 oops, 25 is not
> > prime. 25 is multiple
> > of 5, hence
> >
> > Regula 4: All numbers >1 which are not multiple of 2
> > or 3 or 5 are
> > prime.
> > Try: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,49
> > oops
> >
> > and so on. The regulas become more and more exact.
> > The prime numbers
> > are distributed "as regularly as possible".
> > It's simply the Sieve of Eratosthenes.
> >
> > Werner

Has this thread not suffocated from the morass of ill- or un-defined terms yet?
It certainly looks moribund.

> It is in fact a tautology to define primes by the way
> you find them.

What do you mean by 'tautology' here?

> Whatever you find by the Sieve is
> prime. If the Sieve way is regular,

What do you mean by 'regular' here?

> then prime is of
> course regular.

What do you mean by 'regular' here?

> but it is a tautology.

grrrr.

> The fact that
> the Sieve is regular

grrrr.

> cannot prove itself to be
> regular.

What do you mean by a sieve 'proving itself to be [something]'?

> If regularity is already defined in to
> primes,

grrrr.

> that same regularity

grrrr.

> cannot be used to prove

grrrr.

> that prime is regular.

grrrr.

> the same kind of regularitys

grrrr.

> cannot prove

grrrr.

> the same regularity.

grrrr.

> If prime is defined
> by regularity

grrrr.

> or the regular

grrrr.

> way of finding primes,
> those way of finding primes cannot be used to prove
> that primes are regular.

grrrr.

> What needs to be done is to
> define primes by a kind of regularity

grrrr.

> that can then be
> used to prove another kind of prime regularity.

grrrr.

> Regularity A proves regularity B (B could be the sieve
> or the non-divisibility). But regularity B cannot
> prove itself. [...]

I give up.

Phil

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