As noted earlier, 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.
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’].
Now, 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. [Latin titles are always dangerous, vide Wittgenstein and his Tractatus Logico Philosophicus.]
The wreck of this Titanic, the Principia Mathematica, on the Godel iceberg is common knowledge among pop. mathematicians and I’ll limit myself to the outlines.
Everything went along fine until they entered the zone of Set Theory [again founded by Peano]. Its most prolific user was the celebrated Professor Georg Cantor in his construction of various types of ‘Infinities’. And in Set Theory they hit head-on into the Self-Loop. [I chuckle, when I think how Cantor mocked Immanuel Kant, one remarkably alert to the Self-Loop, as a mathematical ignoramus.]
In 1952, Harvard University awarded the pioneering Logician Kurt Godel an Honorary Doctorate for: ‘ The discovery of the most significant mathematical truth of the century’ for his paper: ‘ On formally undecidable propositions of Principia Mathematica and related systems.’
Godel 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. [ Where have we seen that before?]
Godel discreetly states in a footnote to his paper: ‘Any Epistemological Antinomy, such as the ‘Liar’s Paradox’ could be used for a similar proof’. ‘The Liar’s Paradox’ is the famously confusing declaration by Epimenides the Cretan who pronounced: ‘All Cretans are liars!’
The proof of ‘The most significant mathematical truth of the century’-lay in a Self-Eating Expression. But Kurt Godel was shaking things up at a deeper level. What was the flower of Peano’s seed?
Following his discovery a professor of mathematics summarizing Godel’s work solemnly intoned: ‘[Godel’s Theorem] ‘requires that the ultimate foundations of Mathematics and all its derivative truths remain a mystery’. [In other words: ‘We don’t really know what we are doing, but we are doing it anyway’.]
Less kindly, it suggests that all Mathematical Modeling cannot be differentiated in any provable way from a manufactured reality in indeterminate Self-Loop.
Are Logico-Mathematical truths intrinsic, hard-wired into Nature? Or are they a man-made convenience, a modeled-understanding of Self and World?
Get to True Nothing and find out for yourself.
The ‘Liar’s Paradox’, venerated but winked away in Logician’s circles, is a paradox only because we don’t know if Epimenides is a liar or not. He is neither, for the phrase is a Self-Eating Expression. This might be more transparent if: ‘All Cretans are liars’ is replaced with: ‘All Words are false’ which denies you the ability to extract yourself out of the loop and pronounce on its truth or falsity.