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.
$(i i)$ If $x$ is a natural number, there is another natural number denoted by $x^{'}$ (and called the successor of $x$ ).
$(i i i)$ $0 \neq x^{'}$ for every natural number $x$ .
$(i v)$ 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
- $(v . I I)$ 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 $π$ . 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_{0} x^{n} + a_{1} x^{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 $π$ 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 $π$ 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.