# Pi Day (2023/03/14) Piaxid Monochromatic 1/4: Axiom

Axioms are the fundamental unit of mathematical logic that inductively enable creation.

Axioms are true statements that enable meaningful definitions, and any other kind of high-level statements like theorems.

Math is pure as it’s homogeneous: math creates math.

Math is the abstraction of abstractions.

One immediate example of this are the axioms for the formal creation of the natural numbers :

• $(i)$ $0$ is a natural number.
• $(ii)$ If $x$ is a natural number, there is another natural number denoted by $x'$ (and called the successor of $x$).
• $(iii)$ $0 \neq x'$ for every natural number $x$.
• $(iv)$ If $x' = y'$, then $x = y$.
• $(v)$ If $Q$ is a property that may or may not hold for any given natural number, and if
• $(v.I)$ $0$ has the property $Q$ and
• $(v.II)$ Whenever a natural number $x$ has the property $Q$, then $x'$ has the property $Q$ then all natural numbers have the property $Q$ (mathematical induction principle).

Another example is the creation of the Fibonacci sequence inductively:

The declarativeness of math enables creation with mathematical elegance unlike the imperativeness of those impure where every instance is to be addressed (i.e. defined or demonstrated) imperatively at a time.

Finally, the last example shows an immediate application of $\pi$. First, lets define algebraic numbers :

A real number $x$ is said to be an algebraic number is there is a natural number $n$ and integers $a_0, a_1, ..., a_n$ with $a_0 \neq 0$ such that

$a_0x^n + a_1x^{n-1} + ... + a_{n-1}x + a_n = 0$

A real number which is not and algebraic number is said to be a transcendental number.

It can be proved that $\pi$ is transcendental, and with this, it can be proved that it’s impossible to “square” the circle by a ruler-and-compass construction .

Math enables us to create from fundamentally abstract concepts like axioms and I devised the concept of mapping it $1:1$ to the virtual universe of software via mathematical software (engineering), and today (2023/03/14) is the glorious day of $\pi$ in which I take the opportunity to get inspired and celebrate it.

 Mendelson, E. (2015). Introduction to Mathematical Logic. CRC Press.

 Morris, S. A. (1989). Topology Without Tears.

 Liaw, C., & Ulfarsson, H. A. (2006, April 7). Transcendence of e and π. Williams College. Retrieved March 14, 2023, from Web | Transcendence of e and π | Williams College.

 The Fibonacci sequence - HaskellWiki. (n.d.). The Fibonacci sequence | Wiki | Haskell.