Zeno, the favorite of Parmenides [‘Venerable and Awful’], a pioneer of the logico-mathematical paradox, describes his new treatise to Socrates:
‘It is…a defense of Parmenides against those who make fun of his ideas…this book is a retort against those who assert a Plurality…pays them back in the same coin with something to spare. For it shows that on a thorough examination, their own supposition that there is a Plurality leads to even more absurd consequences than the Hypothesis of ‘The One’.’
The Parmenides is considered the most difficult of the Platonic Dialogues. That is because Parmenides [and a few others; see Aristotle’s Metaphysics excerpts in the Posts] was alert to the Self-Loop, and to which his modern interpreters are conspicuously innocent.
[This and all other excerpts from Plato’s Dialogues are from the Hamilton and Cairns, Princeton, ’61 Edition.]
In a recent article reflecting the Bodleian Press Release, a Writer on Mathematics offered the above sharp observation on the origin of the symbol. [I forget the name; drop me a note if you recognize this piece.]
‘The concept of zero seems intuitive, but that’s because we’re already familiar with it. There’s a big conceptual leap between saying ‘there are no apples on this tree’ to saying ‘this tree has zero apples on it.’
Both ‘No Apples’ and ‘Zero Apples’ are conceptual extracts, neither being Shūnyam [True Nothing]. But the above is a helpful distinction and it can be made more visible.
Around 600 CE, Chandrakirti, an articulate spokesman for the Mādhyamika school which claimed itself a dialectic, itself holding ‘No views of its own’, gave this illustration of one who retains a ‘View’ of Absence [i.e. an Idea, a conceptual elaboration].
‘It is as if I ask a shopkeeper:’What do you have to sell?’. And he replies: ‘I have nothing to sell’. And I say: ‘Oh, fine! That will do. Sell me this nothing then.‘
The claim: ‘We have no views!’, is problematic and ultimately undermines the Mādhyamaka school itself. It is an unwound cousin of the primal Self-Eating Expression Shūnyam, a stopping short of the terminus [see ‘The Heart Sūtra’ and later posts]. But that does not take away from the insight of Chandrakirti’s illustration.
I’d like to briefly set forth the two most significant entries of close variants of the Self-Eating Expression, the vehicle of the acknowledgment of the Self-Loop, in Logic and Mathematics. The first, less than a 100 years old; the second over 2,000.
‘Logic is to Philosophy what Mathematics is to Nature’. So goes the line. For me the distinction has always been fuzzy. But I will use both headings.
The problem-zone is the point of intersection between Mathematics and Philosophy. Or more precisely, the Mathematician’s application of philosophical notions which have remained unresolved for millennia in Philosophy itself.
First, the Mathematician’s interpretation of the idea of ‘Unity’ and secondly, his application of the idea of ‘Truth’. [And underlying it all, a blithe take on the notions of ‘Unboundedness’ and ‘Isness’ or Ontological Presence: ‘There is an X such that…’. Its discussion, along with other related issues, can be found in the later posts.]
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’].
And it has been repeatedly tripping up on one single idea: the notion of Self-Reference [recursion, repetition, reflection and numerous variants].
The Mathematician;s notion of Unity or Wholeness is the idea of the Mathematical Set. [Of interest, the word ‘Set’ has been marked as the word with the most number of definitions in the Dictionary, the last time I checked.]
The celebrated 3-volume opus of Russell and Whitehead, the Principia Mathematica was, among other things, an attempt to resolve the conflicts between the observing Subject and his inclusion or exclusion in a Set.
The issue never found a resolution [R and W had to use a variety of sequenced exceptions to hold the logic together] until Kurt Godel’s celebrated 1931 paper. [‘The most significant mathematical discovery of the century’ cooed Harvard in presenting him an award in 1952.]
A result which 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].
Godel’s resolution relies on the form of a famously curious shout by Epimenides the Cretan who declared: ‘All Cretans are liars!’ [‘Any epistemological antinomy of this form’..noted Godel in a footnote to his paper.] Note the similarity of its logical form to that of a full-blooded Self-Eating Expression.
‘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 [see: ‘Vishnu’s Dream’] Else, as has happened in the academic investigations of the proof, it simply leads to more convoluted and strained results.
Following Godel’s discovery a professor of mathematics 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. The issues with self-reference in Mathematics go far deeper than that.
Just as the notion of ‘Set’ expressed the Mathematician’s notion of Wholeness, the idea of ‘Proof’ marked his interpretation of Truth.
‘Proof’, a word that makes every kid wet his pants in high-school, is largely one of demonstrating the consistency of a set of logico-mathematical assertions based on a set of agreed upon foundational axioms.
There are various levels of ‘Proof’ and numerous interpretations of what exactly the word means [it was this very need that ultimately lead to Godel’s work]. But what we do know is that they all take life upon a central principle, the principle of non-contradiction, in delightful irony also called the principle of contradiction.
It is in a sense the founding principle of Logic and Mathematics. Once you have made the first cut between ‘Self’ and ‘World’, the Contradiction Principle follows as its immediate progeny.
The pioneering Aristotle, the founder of Classical Logic, understood this. In his Metaphysics he called it: ‘The First Principle of Rational Knowledge‘, that ‘Which one must have to understand anything whatsoever‘.
Two thousand later, his faithful disciple Immanuel Kant, who defined the domain of Academic Philosophy for two hundred years, in his Critique of Pure Reason called it the: ‘Principle Sine Qua Non-the universal and fully sufficient principle of all analytic cognition‘.
It is Aristotle’s Principle for it was he who had the courage of conviction to place it on center stage and defend it vigorously.
The Contradiction Principle demands that all derivations meet its condition of internal-consistency. But what then is the defending criterion for this celebrated Principle itself?
Aristotle called it the ‘Self-Destroying Argument’ as in [his example]: ‘All things are false’ [for the full story, see the later posts.] From his Metaphysics:
‘The possibility of a middle between contraries is excluded: for it is necessary to assert or deny one thing or another. This is clear from the definition of Truth and Falsity: either what is, is affirmed or denied, or else what is not, is affirmed or denied, there can be no middle ground..
Similarly, every thought and concept is expressed as an affirmation or a negation, this is clear from the definition of Truth and Falsity.
Hence also, the frequent saying befalls all such arguments, that they destroy themselves. For he who says that all things are true presents even the statement contrary to his own as true, and therefore his own as not true: whereas he who says that all things are false presents also himself as false.’
Aristotle’s ‘Self-Destroying Argument’ is as close a cousin as you will find to the primal, paradigmatic expression of which is the graphic visual Symbol ‘0’.
So what are we saying? The Principle of Contradiction is wrong? No. As with the notion of’Self’, of ‘Substance’ [see: ‘Nothing and Everything‘], we need to extract ourselves in imagining some universe of stable ‘Inner Entities’ that are at war when in violation of the Principle.
Red is not blue; nor is an orange an apple. They are not so, not because the apple has an ‘Inner Apple Center’ in conflict with an ‘Inner Orange Center’. They are not so because we have defined them that way in the sign-based referential system of Language. Coherence would increase substantially if we displayed a little more modesty.
The Self-Loop will continue to infiltrate every aspect of Logic and Mathematics until there is an investigation of the opening assumption, that of a Separated ‘Self’. [itself an astonishingly amorphous idea which Logicians and Mathematicians would reject as loose talk if they themselves were’t deeply ensconced in it].
The posts carry many more examples of the ‘Blink and Wink’, and not just in Logic and Mathematics.