# 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 [1]:

- \((i)\) \(0\) 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 [2]:

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 [3].

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.

## References

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

[2] Morris, S. A. (1989). Topology Without Tears.

[3] 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.

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