‘The Mathematical Discovery Of The Century’

The Gödel Iceberg is common knowledge among the mathematically-inclined and there is a large readable literature out there on Gödel, Russell’s Paradox and the rest. I’ll limit myself to the outlines.

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, the Titanic, 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 until Kurt Gödel came along [Russell and Whitehead had to use a variety of sequenced exceptions to hold the logic together].

In 1952, Harvard University honored the Logician Kurt Gödel with an award that read: [For] ‘The discovery of the most significant mathematical truth of the century.’

The award was for Gödel‘s seminal paper: ‘On formally undecidable propositions of Principia Mathematica and related systems.’

Gödel proved that the consistency of any Formal System cannot be proven using the methods of that System without simultaneously violating the basis of consistency of that System itself [There were other issues such as ‘Completeness’. But this will do for now.] Can you sniff the Self-Loop? 

In a footnote to his paper Gödel notes: ‘Any Epistemological Antinomy, such as the ‘Liar’s Paradox’ could be used for a similar proof’. Gödel is referring to a famously curious shout by Epimenides the Cretan who declared: ‘All Cretans are liars!’

‘All Cretans are liars’ is close. It would be closer if, unless you happen to be a Cretan yourself, it read: ‘All men and women are liars’, for then it would include you, the observer, in any interpretation of this claim. Better still: ‘All Words Lie!’ A full-blooded Self-Eating Expression.

[The most transparent Self-Eating Expressions tend to include words like: Always; Never; Everybody; Nobody; Nothing; Everything, and so on. Nice slippery words that can drown you if you don’t watch out.]

A Self-Eating Expression is meant to be unwound sequentially in the ‘Backward Step’. It is not to be stared at as a static logical complexity which is the way it has been treated in the professional literature. That just sinks you deeper into the swamp. The only way to get to its root is to buy yourself a Meditation Mat.

Are Logico-Mathematical Truths intrinsic, hard-wired into Nature? Or are they a man-made convenience, a modeled-understanding of Self and World?

Following Gödel‘s Theorem a professor of mathematics solemnly intoned: ‘[Gödel’s Theorem] requires that the ultimate foundations of Mathematics and all its derivative truths remain a mystery’.

Less kindly, it suggests that all Mathematical Modeling cannot be differentiated in any provable way from a manufactured reality in indeterminate Self-Loop.

Mathematician’s hurry to defend their work by drawing lines around terms like ‘Axiomatic’ and ‘Formal’. They are red-herrings. The issues with Self-Reference go deep; the repeatedly exploding contradictions and paradoxes in Logic and Mathematics cannot be band-aided.

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There is a 125 page book by Nagel and Newman [2008] called ‘Gödel’s Proof’ [dedicated to Bertrand Russell] which tries to explain it to the lay-reader. Like my shelved copy of Tolstoy’s ‘War and Peace’ [which ran 1225 pages in its first edition], I have yet to finish it.