At least two and even rather different scientific areas met the problem of completeness as being fundamental: the foundation of mathematics and quantum mechanics.
Hilbert’s program tried to ground mathematics on arithmetic in a finite way (finitism). However Gödel (1931) proved that any theory containing Peano arithmetic including itself is either consistent or complete. Nevertheless, Gentzen (1936) adding transfinite induction or Heyting arithmetic suspending the “excluded third” as to infinite sets of natural numbers managed satisfyingly enough to overcome that kind of incompleteness.
The famous triple article of Einstein, Podolsky, and Rosen (1935) questioned whether quantum mechanics was complete answering negatively. The proof was based on postulating “elements of reality” therefore rejecting any “spooky action at a distance” in Einstein’s words. Bell (1964) suggested an experimental test for the existence of that “spooky action”. Clauser and Horne (1974), Aspect, Grangier, and Roger (1981; 1982) and many others after them realizing those experiments demonstrated categorically entanglement as a real physical phenomenon, thus confirming the “spooky action” and rejecting the alleged incompleteness of quantum mechanics. The concept of quantum information was introduced to explain all phenomena of entanglement, and quantum mechanics even turns out to be reformulated entirely as a theory of quantum information:
The complex Hilbert space underlying quantum mechanics as the fundamental mathematical formalism is interpretable as the free variable of the quantity of quantum information. Any wave function being both a point in that Hilbert space and describing exhaustedly the state of some quantum system is a value of that quantity of quantum information. If “quantum computer” is defined as a device processing quantum information as a computer processes information standardly defined, the universe can be seen as a huge quantum computer, and any system or change in it should share the same nature. So, quantum information has claimed to be a fundamental metaphysical conception.
Shannon’s articles (since 1948) introduced absolutely independently the concept of information as well as its quantity to describe coding and calculating by artificial devices. Winner (1948) suggested cybernetics as a general theory of control based on processing information and comprising both artificial and natural systems. Shannon’s information turns out to share a common or at least similar mathematical formula with the thermodynamic quantity of quantity. Kolmogorov, Martin-Löf, Chaitin and many others after them interpreted information algorithmically, i.e. as the minimal length of bits in a set of algorithms able to calculate some given problem. Thus information was seen in a second way, namely as an ordinal number whether of a finite or an infinite set.
Computer science and technics have thrived since the 60th of the last century and now it is the main determiner of all human progress. Their base is the processing of information. Many other sciences especially those exploring human brain and capabilities, and life as well as even chemistry share the same information paradigm. Philosophy of information is a new but accelerated developing area.
The conception of information demonstrates a huge metaphysical potential, which is not less forthcoming to extend itself.
The thesis in that background:
The conceptual and particularly metaphysical power of information grounds on being the solution of the problem of completeness. Seen as such, it should be interpreted as ordering and the quantity of choices necessary for a given well-ordering.
Arguments:
1. Information defined in Shannon (1950) and in all other alternative ways shares the “bit” as its unit. A bit represents the elementary and simplest case of choice between two equally probable alternatives. Any finite choice or series of those contain a certain number of bits and thus a value of the quantity of information.
2, Quantum information can be seen as that generalization of ‘information’, which refers to infinite series or sets. The Kolmogorov representation of information as the minimally possible algorithm, which can construct a message equivalent to a given object, can be also generalized as algorithms of transfinite length indiscernable of randomness and processed by quantum computers.
3. Then a quantum bit (qubit) usually defined as the normed superposition of two orthogonal subspaces of the complex Hilbert space can be equivalently interpreted as the elementary choice among an infinite set of alternatives. Quantum information is measured as a number of qubits just as the classical information is a value of bits.
4. Any state of any quantum system being a “point” in the complex Hilbert space is a value of quantum information. Then any physical process being always quantum in a fundamental level is informational. The concept of information unifies technics and nature as finite in relation to the former and infinite (quantum) as to the latter.
5. Even the Schrödinger equation fundamental to quantum mechanics can be interpreted in terms of quantum information as equating the energy of a transfinite bit (in Kolmogorov) to that of a quantum bit standardly defined. That understanding of the Schrödinger equation reveals new horizons to general relativity and the Standard model from the viewpoint of quantum mechanics.
6. The theorem about the absence of “hidden variables” in quantum mechanics (Neumann 1932; Kochen, Specker 1968) can be interpreted as a proof of completeness of quantum mechanics based on the complex Hilbert space, which represents the variable of quantum information. Thus the property of completeness can be transferred to and deduced from the quantity of quantum information. It includes even the coincidence of model and reality in that completeness.
7. Complete in that sense, quantum information turns out to be not only a fundamental physical, but furthermore metaphysical quantity. It is that quantity, to which any quality can be related in final analysis.
8. Even arithmetic and all mathematics after that can be underlain by both quantum information and complex Hilbert space interpreting the Gentzen or intuitionistic completeness of arithmetic as reducible to the completeness of two independent Peano arithmetics (or Turing machines in the Nash equilibrium as to a quantum computer). Indeed the complex Hilbert space can be represented as a set of subsets of two independent (complementary) Peano arithmetics.
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