Prime numbers are the dwellings of the mystics

The Eternal Debate: What Came First?

In the Beginning

People have debated the origins of existence since the dawn of time.

The Chinese Perspective

In the beginning, according to the Chinese, there was Tao—formless chaos.

Tao

Jung’s Pleroma

Jung shared a similar idea but used a different term: "pleroma".

Pleroma

Chaos to Order?

How could order emerge from chaos? Philosophers often remain silent on this.

Coincidence as a Deity

Some attribute this transformation to Coincidence, elevating it to the status of a deity.

The Dual Hypothesis

Another intriguing hypothesis suggests that in the beginning, there was both Chaos and Order. Chaos occupied one half of Everything, Order the other.

The T=0 Moment

Chaos couldn't sustain itself alone. It grabbed something from Order, sparking the beginning: moment t=0.

Burkhard Heim's View

My favorite, Burkhard Heim, uses the Greek word "Apeiron" instead of Tao or Pleroma.

Apeiron

My Belief

I prefer to start from symmetry and let it break. You can call me an "aperionist".

Symmetry Over Chaos

Starting from chaos doesn't attract me. I often leave the question of order's origin unanswered.

Beauty and Love

At most, I suggest that order arose from beauty or love.

Prime Numbers

Pressed for an answer, I propose that in the beginning, there were prime numbers. What could be more fundamental than prime numbers?

The First Principle

In essence, prime numbers embody the essence of the beginning for me. Their purity and simplicity are unmatched.

Explore these ideas, ponder the origins, and join the eternal debate on what came first.

Prime numbers are the dwellings of the mystics

Definition: A positive integer p is prime if p has exactly two distinct positive divisors, namely 1 and p itself. If an integer n is not prime, it is said to be composite.

Behold, the numbers 2, 3, 5, 7, and 11, the smallest quintet of primes, like early stars twinkling in the infinite sky of mathematics. Notice how 2 stands alone, a solitary even prime among a sea of oddities, a beacon of uniqueness. Furthermore, under this poetic definition, the integer 1 is excluded from the prime pantheon, for it dances solo, bearing only itself as a partner.

In ancient times, mathematicians might have embraced 1 as prime, a simpler era's muse. Yet, as we delve into the elegant theory of unique factorization, the harmony and necessity of excluding 1 from prime status will unfurl, revealing the deeper symphony of numbers.

Definition: Given two positive integers a and b, we define the greatest common divisor of a and b to be the largest positive integer that divides them both. This number is denoted by gcd(a, b). We say that two positive integers are coprime, relatively prime or mutually prime if their greatest common divisor is 1, i.e., if the only divisor they have in common is 1.

Definition: Suppose b is an integer and a is a non-zero integer. We say that a divides b if there is an integer q so that b = aq. If there are such integers, we denote the fact that a divides b by using the notation a|b.

Theorem. (Unique Factorization of Integers)

(i) (Existence) Every positive integer n ≥ 2 may be written as a product of (not necessarily distinct) prime numbers; i.e., n may be written in the form

n = p₁ · · · p_r

where each integer p_i , i = 1, . . . , r, is a prime.

(ii) (Uniqueness) Moreover, this factorization is unique except for the order of the primes; i.e., if we also have n = q₁ · · · q_s where each q_i is a prime, then r = s and (if necessary) upon re-ordering, p_i = q_i , i = 1, . . . , r.

The horizon of prime numbers stretches infinitely,

Ladies and gentlemen, the moment of revelation has arrived. Behold, as we unveil the cornerstone of mathematical truth: the essence of prime numbers. Prepare to be awestruck, for we declare with unwavering certainty that the journey of primes is infinite! 

No matter how colossal a prime you conceive, there will forever exist an even grander prime beyond. 

The horizon of prime numbers stretches infinitely, a testament to the boundless nature of mathematics!

But this will be the subject of the next post...

References:

[1] Gove Effinger, Gary L. Mullen, Elementary Number Theory, CRC Press, 2022

[2] C.G. Jung, The Red Book, W.W. Norton & Company 2012

[3] K. Lange, A Connection Between the Apeiron to the Set of Supersingular Prime Numbers (2013)

It from Bit? Or Order out of Chaos?

 John Archibald Wheeler advocated the idea of "It from Bit". Everything in the universe, including mid, is made of information, and information consists of sequences of binary alternatives 0 and 1. More on this subject can be found here: "IT FROM BIT: WHAT DID JOHN ARCHIBALD WHEELER GET RIGHT—AND WRONG?". The article contains the view of an American mathematician, philosopher and theologian Willian Dembski. More about his views can be found in the article "HOW INFORMATIONAL REALISM SUBVERTS MATERIALISM". Let me quote just two paragraphs from this last article:

Informational realism does not deny the existence of things (i.e., entities or substances). But within informational realism, what defines things is their capacity for communicating or exchanging information with other things. Things are inferred from the information they communicate. Information, as the relational glue that holds reality together, thus assumes primacy in informational realism.

In informational realism, things make their reality felt by communicating or exchanging information. Thus, things that are not in immediate or mediate informational contact with other things might just as well not exist and so, in fact, don’t exist within informational realism. Informationally isolated or disconnected entities thereby become nonentities. 

This sounds to me as more reasonable approach. There is information and there are "things" that communicate this information  What is this "communication"? For bits of information to be considered as a "communication" they need to be "meaningful". And what that means? It is the role of the mind and of consciousness to decide what has a meaning and what doesn't. The main problem here is, as far as I can see, the ability to distinguish between order and chaos. These are two fundamental concepts: ORDER and CHAOS. Intelligent creation is the ability to crate order out of chaos. Thus not 0 and 1 are the fundamental elements of Being, but O and C. Both order and chaos must include in their description infinite sequences of bits, and infinitely complex algorithms for creating these bits. And there is no reason for supposing that Nature is scared of infinities. Exactly the converse might be true: Nature effortlessly operates with infinities, and of any order. With our finite means we can only approximate in our description the true workings of the Universe.

Concerning Order and Chaos, let me quote from the recent book I decided to study what a clinical psychologist Jordan B. Peterson has to say in his  "12 Rules for Life: An Antidote for Chaos"

Order is where the people around you act according to well- understood social norms, and remain predictable and cooperative. It’s the world of social structure, explored territory, and familiarity. The state of Order is typically portrayed, symbolically—imaginatively —as masculine. It’s the Wise King and the Tyrant, forever bound together, as society is simultaneously structure and oppression.

Chaos, by contrast, is where—or when—something unexpected happens. Chaos emerges, in trivial form, when you tell a joke at a party with people you think you know and a silent and embarrassing chill falls over the gathering. Chaos is what emerges more catastrophically when you suddenly find yourself without employment, or are betrayed by a lover. As the antithesis of symbolically masculine order, it’s presented imaginatively as feminine. It’s the new and unpredictable suddenly emerging in the midst of the commonplace familiar. It’s Creation and Destruction, the source of new things and the destination of the dead (as nature, as opposed to culture, is simultaneously birth and demise). 

Order and chaos are the yang and yin of the famous Taoist symbol: two serpents, head to tail.   Order is the white, masculine serpent; Chaos, its black, feminine counterpart. The black dot in the white—and the white in the black—indicate the possibility of transformation: just when things seem secure, the unknown can loom, unexpectedly and large. Conversely, just when everything seems lost, new order can emerge from catastrophe and chaos. For the Taoists, meaning is to be found on the border between the ever-entwined pair. To walk that border is to stay on the path of life, the divine Way.

From there I will particularly stress this observation:

meaning is to be found on the border between the ever-entwined pair.  

In the future posts I will deal with this "border" or "boundry" - using simple algebra.

P.S. Saturday's entertainment:

And something really really scary:

"People who pay strict attention to objective reality right and left, become the reality of the 'Future.'"