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.]
In 1952, Harvard University honored the Logician Kurt Godel with an award that read: [For] ‘The discovery of the most significant mathematical truth of the century.’
The award was for Kurt Godel’s seminal 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 [There were other issues such as ‘Completeness’. But this will do for now.]
Can you sniff the Self-Loop?
A Theorem which itself drew upon the work of Jules Richard and others and went forward in seminal revelations that have hugely influenced modern computer, information and cognitive theories [as with the work of Alan Turing and numerous less famous mathematicians].
In a footnote to his paper Godel notes: ‘Any Epistemological Antinomy, such as the ‘Liar’s Paradox’ could be used for a similar proof’. Godel’s resolution relies on the form of 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. A full-blooded Self-Eating Expression.
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 ‘Liar’s Paradox’ is a paradox only because we don’t know if Epimenides is a liar or not. He is neither. 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.]
So 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.
Mathematician’s hurry to defend their work by drawing lines around terms like ‘Axiomatic’ and ‘Formal’. They are red-herrings. Godel’s Theorem is the tip of the iceberg. The issues with self-reference go deep. Any real foundation must begin by unwinding it all the way and laying it out for view in sunlight.
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 repeatedly exploding absurdities, contradictions and paradoxes in Logic and Mathematics, which among all subjects are the most carefully thought out, the most precisely expressed, will not end until the central issue of self-reference in all its forms is confronted head-on.
A Turing Machine is a device that uses a set of rules to work a list of symbols on a length of tape.
And it was the progenitor of the idea that became the modern computer. Or more accurately, what we call ‘Software’.
Kurt Godel’s paper was published in 1930. Within a decade, Alan Turing applied Godel’s work to solve issues fundamental in the birthing of Modern Software [‘Undecidability’; The’Halting Problem’, 1937]. And Academic Departments went about marveling at the quirkiness of ‘Strange Loops’.
Both Kurt Godel and Alan Turing reach for variants of the Self-Eating Expression to seal their respective proofs. So why not just start with the opening Self-Eating Expression: ‘I don’t exist!’ It’s a lot more fun to work with than the cryptic symbols of these two pioneers.
The famed ‘Enigma Machine’ had a host of creative mathematicians far less known from across Europe contributing to it. But Turing’s greater fame was matched by his tragedy.
What is it with brilliant Logicians who see too far? Why are the deities of Consistency and Rule so indifferent to their first-born? Godel died from self-imposed starvation, according to the coroner; Turing, from cyanide poisoning.