Pi Day (2023/03/14)

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:

fib 0 = 0
fib 1 = 1
fib n = fib (n-1) + fib (n-2)
Also Known as "the 'Hello, world!' of Haskell programming"
The Fibonacci Sequence [4]

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.