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.
Jung’s Pleroma
Jung shared a similar idea but used a different term: "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.
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.
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 formn = p1 · · · pr
where each integer pi , 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 · · · qs where each qi is a prime, then r = s and (if necessary) upon re-ordering, pi = qi , 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)
Base 60 as mentioned earlier can't be the mother to base 10 since both are just how you write numbers not what the numbers themselves are. As you say here, it's symmetry in the beginning as in Clifford Algebra, the mother symmetry, as you've said in a paper.
ReplyDeleteAlso as you say here, you can get smaller useful structures via breaking the mother symmetry. An infinite Clifford Algebra tensor product is kind of just the natural numbers with patterns like Bott periodicity, the Pascal triangle, etc and prime numbers are kind of the most enigmatic "pattern".
Prime numbers can be interpreted in many ways, one of them being stability. Since primes are only divisible by 1 and themselves, we can argue that they are "tougher to crack" compared to composite numbers, which can be divided into more parts. In the context of symmetry, this perspective could shed more light on crystallographic arrangements.
ReplyDeletehttps://www.semanticscholar.org/paper/Primes,-Geometry-and-Condensed-Matter-Rabeh-Riadh/a066dd666ef47e969dbc59266c729e93471ed919
"We next suggest that a non-prime can be considered geometrically as a symmetric collection that is separable (factorable) into similar parts- six is two threes or three twos for example. A collection that has no such symmetry is a prime. As a result, a physical prime aggregate is more difficult to split symmetrically resulting in an inherent stability. This “number/physical” stability idea applies to bigger collections made from smaller (prime) units leading to larger stable prime structures in a limitless scaling up process."
"A prime is formed every time we have highly symmetric combination with one to be added or subtracted to it to break the symmetry and produce a prime."