The Square of Opposition & The Concept of Infinity: The shared information structure
6th WORLD CONGRESS ON THE SQUARE OF OPPOSITION Orthodox Academy of Crete, 1-5 November 2018 http://www.square-of-opposition.org/square2018. html
Vasil Penchev
Dept of Logical Systems and Models, Institute for the Study of Societies and knowledge, Bulgarian Academy of Sciences vasildinev@gmail.com
Project: Đ”Đ? 15/14 - 18.12.2017 to the "Scientific Research" Fund, Bulgaria: "Non-Classical Science and Non-Classical Logics. Philosophical and Methodological Analyses and Assessments"
Prehistory and background
“Two bits, two bits!� for Mr Finnegan! One may demonstrate that the square of opposition shares the least possible structure of the ordering which cannot be well-ordered directly This structure is equivalent to two bits of information, absolutely independent of each other
The alphabet of “[ ”יהוהYHWH] That minimal possible information structure is a kind of the minimal “alphabet” of the universe That “alphabet” is necessary for anything to be “written” unambiguously
The YHWH alphabet of four letters That alphabet consists of four letters corresponding one-to-one to two independent bits It can be found as the four “letters” of human genome as the four “colors” enough to be colored any map
“Four letters” vs “two letters” The minimal possible alphabet consists of only two letters This is a single bit of information That alphabet of two letters is used is our computers It is enough for any well-ordering to be written exactly
Why our computers do not think (yet) One may define ‘thinking’ as the ability of well-ordering of all (which is not ordered in general) The alphabet of two letters is insufficient to be written relevantly anything which is not well-ordered So, the alphabet of our computers and that of thinking are essentially different yet
A fundamental loss of information The record of anything which is not well-ordered by only two letters implies the loss of information necessarily For example, let that not well-ordered be written by four letters, i.e. as if “two bits� However, one needs furthermore the superposition of the order, in which the two bits are recorded:
One quantum bit (qubit) Their superposition
One bit
Another bit
An ontological example An exceptionally important exemplification of that loss of information is quantum measurement It reduces a coherent state before measurement unorderable in principle to a statistical and thus well-ordered ensemble after measurement
What the alphabet of two letters implies ... Any record of something not well-ordered by the alphabet of two letters implies the well-ordering principle (equivalent to) the axiom of choice One may mention Voevodsky’s univalence axiom (2009) able to link logic to topology on a similar basis
Adding the lost information... The sense to that application of the axiom of choice is to add outwardly, by external choices the information lost before Namely, that information lost after recording by that insufficient alphabet of two letters
Restoring the equation of information... The information equation before and after recording, respectively before and after measurement, need be restored What the axiom of choice does do actually...
‘Infinity’: the mathematical “God” The power of the square of opposition has been proved in logic during a few millennia It can be also utilized to be hinted the most general concept in the framework of logic, particularly the being of infinity
The alphabet of infinity... A reasonable conjecture in the same course of thought is: The concept of infinity admits to be written by the same alphabet of four letters or two independent bits consistently
‘Infinity’ by two bits? Or even a more strong hypothesis is possible: Infinity might be equated to te the same elementary information structure of two independent bits
The present talk’s intention The talk intends the discussion of that hypothesis: Infinity Four-letter Square of alphabet opposition Two bits of information
Thesis:
Four letters (or two bits) for infinity 1. The concept of infinity can be written relevantly by the alphabet of four letters Thus, infinity shares the universal alphabet enough for anything in the universe
All you need ‌ are four letters 2. Even more, the concept of infinity might be equivalent to that fundamental alphabet One means the alphabet consisting of four letters or two independent bits
The four-letter language of nature 3. Metaphorically, anything in the universe, or in other words, anything existing is written by the language of infinity However, it consists necessarily of only four letters...
An idea for the proof of the thesis
The two dimensions of two independent bits: The structure of two independent bits is two-dimensional necessarily Indeed, it implies a gap between the two bits due right to their independence
Infinity as .. a “gap” The gap between any two dimensions both requires and impies the concept of infinity to be described explicitly or “cataphatically”: Dimension 1
Dimension 2
The gap “of” (or “as” infinity)
The ‘gap’ of infinity and the “gap” of bits In other words, the gap between two dimensions (such as two independent bits) describes the same, however in an implicit, or “apophatic” way
The gap of infinity as the gap between qualities ‘Infinity’ means not any finite entity, but another quality different from any finiteness Quantity suggests homogeneity and thus, does not a gap Indeed, ‘two quantities’ means “two qualities”
Any single quality as a quantity Thus, infinity can be represented equivalently by two “finitenesses” Properly: by the gap between them symbolizing the different qualities of each of those “finitenesses”
The mathematical structure of two qualities Thus, the minimal structure of two independent bits or the alphabet of four letters, both can describe infinity That description of infinity coincides with infinity in a sense, even
A symmetry of four elements... Furthermore, the structure of four letters is symmetric to the transforms between elements and triples of elements And the triples of elements are all well-ordering available among four elements...
The structure of four letters as a tetrahedron That symmetry of four letters can be visualized as a “tetrahedron�: Correspondingly, by its symmetry to the transformation between vertices and faces
The structure of four elements as a qubit The structure of four elements is the minimal one able to represent equivalently a qubit It can be explained as the “superposition� of two bits, as minimal as possible
From a qubit to a string of qubits... Any entity in the physical world is necessarily a quantum one (according to the contemporary stage of cognition) Thus, it is representable as a linear string of qubits (words)
Conclusion
The power of the square of opposition The power of the square of opposition has been proved during millennia It supplies logic by the ontological language of infinity for describing anything...
A happy end! Thank you for your kind attention! Any questions? I would be very glad of many questions!