Unity, Set And The ‘Principia Mathematica’

If the early philosophical links with the West were Greek, the first mathematical links were Italian. The development of the Concept of Zero, its Western intellectual foundations, begin with Guiseppe Peano [1858-1932] and Fibonacci [1170-1240; ‘The Greatest Western Mathematician of the Middle Period’; see: ‘Venice’].

The latest subscribed research into Number Theory goes back less than 150 years with the formal conceptualization of the Symbol ‘0’ by Guiseppe Peano in his five famous postulates. [The first postulate reads: ‘Zero is a natural number’].

The Mathematician’s notion of Unity or Wholeness is the idea of the Mathematical Set. [In the wonder-world of the Self-Loop, the word with the most number of posted definitions the last time I checked seems to be the word ‘Set’, as in Mathematical Set, which is another word for ‘Definition’. See the Post.]

The problem becomes intractable once the investigators take the notion of ‘Unity’ to its necessary limit. And that required immersion into Set Theory [again, Peano pioneered its early application]. And in Set Theory they hit head-on into the Self-Loop.

The Self-Eating Expression comes alive from the shadows anytime something is used on itself, anytime we are wading, knowingly or otherwise, into the swamp of the Self-Loop.

The single tripper-upper? The notion of Self-Reference [recursion, repetition, reflection and numerous variants].

Bertrand Russell and Alfred Whitehead wanted to produce a founding Bible that built up the rules of Mathematics from the ground-up. So they began with Peano’s work and titled their opus Principia Mathematica. [1910; Latin titles are always dangerous, vide Wittgenstein and his Tractatus Logico Philosophicus.]

The celebrated 3-volume opus was among other much grander ambitions, an attempt to resolve the conflicts between the observing Subject and his inclusion or exclusion in a Set.

The issue never found a resolution [Russell and Whitehead had to use a variety of sequenced exceptions to hold the logic together]. Until Kurt Godel came along.

[No point in repeating here what is well in the public domain. The story of the Godel iceberg is common knowledge among the mathematically-inclined and there is a large readable literature out there on Godel, Russell’s Paradox, Principia Mathematica and the rest. I’ll limit myself to the outlines.]