Introduction:

It is not clear to me how Pythagoras developed the intuition that everything is a number, but in retrospect it appears that he was correct. In fact, his intuition may be made precise using the method of Gödel numberings. According to this system developed by Gödel, any mathematical proposition may be mapped to a unique integer.

The key idea is to use the Fundamental Theorem of Arithmetic.

Assigning Gödel numbers to logical propositions:

Given a formal system \(F\) with \(N\) symbols, each symbol may be assigned a unique integer from \(A \subset \mathbb{N}\) where \(|A|=N\). In particular, to encode any logical proposition such as:

\begin{equation} a = (a_1,a_2,…,a_n) \in A^n \end{equation}

we may use the prime factorisation:

\begin{equation} G(a) = \prod_{i=1}^n p_i^{a_i} = 2^{a_1} \cdot 3^{a_2} … \cdot p_n^{a_n} \end{equation}

and by the fundamental theorem of arithmetic, i.e. unique prime factorisation, it is possible to recover the original logical proposition from its Gödel number.

Discussion:

Upon closer inspection, this theorem appears to suggest that all the algorithmic information in the Universe is stored in the distribution of primes. However, I shall save this analysis for another day.

What I do wonder about meanwhile is whether Pythagoras might have devised a system similar to Gödel’s approach to Gödel numberings.