• Bulgarian Academy of Science: Institute for the Study of Societies and Knowledge: Dept. of Logical Systems and Models vasildinev@gmail.com
10:15 - 10:45, June 29th , University of Istanbul, Room “C” 5th Congress in Universal Logic, University of Istanbul, Turkey, 25-30 June 2015
The Gรถdel incompleteness can be modeled on the alleged incompleteness of quantum mechanics
Then the proved and even experimentally confirmed completeness of quantum mechanics can be reversely interpreted as a strategy of completeness as to the foundation of mathematics
The one supposes that the GĂśdel incompleteness originates from the deficiency of the mathematical structure, on which mathematics should be grounded However that deficiency can imply two alternative and maybe equivalent ways for the cherished completeness: ďƒ˜ Qualitative deficiency: some other mathematical structure rather than arithmetic (e.g. geometry) ďƒ˜ Quantitative deficiency: arithmetic but more than one (e.g. two ones)
Which is the mathematical structure, on which completeness can be proved?
In tradition originating from Hilbert and Gรถdel, that should be arithmetic, but what are the reasons for that choice?
Indeed arithmetic seems to be the simplest one, but whether not too simple in order to be able to be sufficient for grounding completeness?
In fact, the Gรถdel incompleteness theorems means only that it is insufficient, but nothing about some other one eventually ...
Set theory and arithmetic were what was put as the base of mathematics
However set theory seemed to be controversial allowing of paradoxes such as that of Russell (1902) unlike arithmetic
So, the Hilbert idea (1928) for grounding set theory on arithmetic appeared
That idea has not ever been more than one hypothesis and still less its refusing can mean anything about the foundation of mathematics at all
In fact, there is a well-known result, that of Gentzen (1936) It claims the self-foundation of arithmetic and thus of mathematics at all merely substituting induction with transfinite induction (and even only to đ?œ”đ?œ” is what is necessary) One can distinguish the Peano arithmetic from the newly Gentzen one only by the axiom of induction Then the Gentzen arithmetic would be sufficient for the self-foundation of mathematics
However the transfinite induction seems to involve implicitly and in advance infinity , that controversial concept of set theory just which is what should be to be grounded
Thus (along with his real or alleged complicity in Nazism unlike Gödel who was a refugee from it) Gentzen’s result has tended to be neglected in favor of Gödel’s
In fact the real problem should be: What is transfinite induction in comparison with the standard, “finite” one?
Induction is the only “interesting” axiom among the Peano ones in turn abstracted from Dedekind’s (1888), which grounded arithmetic on set theory, and therefore breaking the vicious circle
Transfinite induction has used to be thought as a kind of super-induction in infinity rather than to (or until) infinity and thus containing the usual one as a true subset
However it can be not less well defined as a second induction therefore a second and independent Peano arithmetic along with the “first”, standard one
Transfinite induction can be (e.g.) defined as a second and independent induction thus: ďƒ˜ Merely postulating it as such: After that the first and second induction can be ordered (not idempotent) or not (i.e. idempotent) ďƒ˜ By distinguishing the successor function as follows:
ďƒź No one-to-one mapping of sets of đ?‘› and đ?‘› + 1
elements for the first induction (always � ≠� + 1)
ďƒź There is at least a one-to-one mapping of sets of đ?œ” and đ?œ” + 1 elements for the second induction (not
Arithmetic and furthermore mathematics can be self-founded consistently
This is able not to involve infinity either explicitly or implicitly (which is an interpretation of Gentzen’s finitism)
Infinity can be equally well defined as both continuation of finiteness (continuity) and a leap to a new dimension of finiteness (discreteness)
The concept of (quantum) information as the quantity of choices underlies the foundation of mathematics in fact:
Indeed the unit of information (a bit) is the choice between two equally probable alternatives and thus describes the mapping between a single arithmetic (finiteness) and two ones (infinity)
The unit of quantum information (a qubit) is the choice among an infinite set of alternatives and describes the mapping between the “finite” arithmetic and the “infinite” set theory
Quantum mechanics being a physical and thus experimental science can be nevertheless thoroughly reformulated in terms of (quantum) information
Then quantum mechanics should be interpreted as an empirical doctrine about infinity after (quantum) information can describe the relation between infinity and finiteness quantitatively
Quantum mechanics is inherently dualistic theory for it rests on the system of two fundamentally different elements:
o The studied quantum entity, and o The macroscopic apparatus measuring it Of course both are finite, but two too different kinds of finiteness: microscopic and macroscopic
If quantum mechanics studies eventually infinity in an experimental way, this turns out to be possible just by reducing infinity to a second and independent finiteness
If the first lesson repeated Gentzen’s, the second one is unique and furthermore allows of building a link between it and Gentzen’s
It consists in involving Hilbert space, a properly geometric structure in its foundation and thus in the foundation of mathematics
Indeed mathematics turns out to be able to found itself as both two arithmetics and geometry implicitly including arithmetic
Anyway why the arithmetic? This turns out to be a random historical fact appealing to intuition or to intellectual authorities such as Cantor, Frege, Russell, Hilbert, “Nicolas Bourbaki�, etc. rather than to any mathematical proof
However arithmetic keeps its place in the foundation of mathematics but forced to share it whether with still one and independent arithmetic or with geometry generalizing it in a sense
The so-called Gรถdel incompleteness theorems (1931) demonstrated that set theory reducible to a single arithmetic is irrelevant as the ground of mathematics
However they said nothing about some other mathematical structures relevant for self-grounding of mathematics
The quantum strategy allows of at least two direction for researching those structures relevant to completeness and still one corresponding to their unification in terms of information as well
One can utilize an analogy to the so-called fundamental theorem of algebra:
It needs a more general structure than the real numbers, within which it can be proved
Analogically, the self-foundation of mathematics needs some more general structure than the positive integers in order to be provable
Still one key is Einstein’s failure (however nevertheless exceptionally fruitful) to show that quantum mechanics is incomplete
The triple article (1935) designated merely “EPR” as well Schrödinger’s study (also 1935) forecast the phenomena of entanglement on the base of Hilbert space
The incompleteness of set theory and arithmetic and the alleged incompleteness of quantum mechanics can be linked to each other inherently ...
The close friendship of the Princeton refugees Gödel and Einstein (Yourgrau 2006) might address that fact
However, Gödel came to Princeton in 1940 The famous triple article of Einstein, Podolsky, and Rosen “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?” was published in 1935
So, there should exist a common mathematical structure underlying both “completeness and incompleteness”
The mathematical formalism of quantum mechanics is based on the complex Hilbert space featuring by a few important properties relevant to that structure capable to underlie mathematics:
It is a generalization of positive integers It is both discrete and continuous (even smooth) It is invariant to the axiom of choice
Hilbert space is a generalization of positive integers: Thus it involves countable infinity Indeed it can be considered as a countable series of “empty” qubits equivalent to 3D unit balls
If one “shrinks” these unit balls to 3D points (balls with zero radius), Hilbert space will degenerate to Peano arithmetic
Hilbert space is both discrete and continuous (even smooth) in a sense:
It is that mathematical structure, in which the main problem of quantum mechanics about uniformly describing both discrete and smooth (continuous) motion can be resolved
Furthermore, it is discrete between any two qubits but smooth (continuous) within each of them
Thus it can unify arithmetic and geometry
Hilbert space is invariant to the axiom of choice in a sense:
Indeed any point in it (a wave function in quantum mechanics) can be interpreted both as:
o The characteristic function of a certain probability distribution of a single coherent state before measurement, i.e. before choice (the Born interpretation of quantum mechanics)
o The smooth space-time trajectory of a “world�
after measurement, i.e. after choice (the manyworlds interpretation of quantum mechanics)
This would mean the unification of: • The externality and internality of any infinite set • Model and reality in principle • The probabilistic and deterministic consideration of the modeled reality
• Along with that property of it to allow of uniformly describing both discrete and smooth motion for resolving the main problem of quantum mechanics
One can say that the crucial concept of all those unifications is that of choice and thus (quantum) information as the quantity of choices
Indeed it allows of reducing o Two arithmetics to only one single (as bits of information)
o Geometry to arithmetic (as qubits of quantum information)
o And even much, much more: qubits of quantum information to bits of information
The essence of set theory is the concept of infinity and its link to arithmetic
Even more, that essence of set theory allows of it to ground all mathematics though in a way yet not consistent enough
Right the concept of information is what can capture that core consistently
The Schrรถdinger equation is the most fundamental equation in quantum mechanics By the concept of (quantum) information, it can be interpreted in terms of the foundation of mathematics Then its sense would merely be that both ways for infinity to be represented are equivalent two each other. That is: oA bit and a qubit can be equated energetically, i.e. per a unit of time oInfinity is quantitatively equivalent to a second finiteness
One can describe that simple way for the Gödel undecidable statements to be resolvable in two arithmetics (besides Gentzen’s proof by transfinite induction):
Any statement of that kind can be interpreted as if its Gödel number coincide with that of its negation
The second dimension (for the second arithmetic) allows the Gödel numbers of the statement and its negation to be different always, i.e. for any statement
Then once the Gรถdel incompleteness can be anyway sidestepped, mathematics can found itself consistently at a certain and rather surprising cost:
Mathematics turns out to be equivalent to the being itself rather than to some true and thus limited part of it: Of course, this might be called quantum Pythagoreanism
Mathematics can self-ground only at the cost of identifying with the world
Infinity is equivalent to a second and independent finiteness
Two independent Peano arithmetics as well as one single Hilbert space as an unification of geometry and arithmetic are sufficient to the self-foundation of mathematics
Quantum mechanics is inseparable from the foundation of mathematics and thus from set theory particularly
Dedekind, R. (1888) “Was sind und was sollen die Zahlen?“ Einstein, A., B. Podolsky, N. Rosen (1935) “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?”
Gentzen, G. (1936) “Die Widerspruchfreiheit der reinen Zahlentheorie“ Gödel, K. (1931) “Über formal unentscheidbare Sätze der Principia mathematica und verwandter Systeme I”
Hilbert, D. (1928) “Die Grundlagen Der Elementaren Zahlentheorie“ Russell, B. (1902) “Letter to Frege” Schrödinger, E. (1935) “Die gegenwärtige Situation in der Quantenmechanik”
Yourgrau, P. (2006) A World Without Time: The Forgotten Legacy of Gödel and Einstein. New York: Perseus Books Group
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