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GODS DICE H I G H l I G H t S

Vasil Penchev

Bulgarian Academy of Science: Institute for the Study of Societies and Knowledge: Dept. of Logical Systems and Models

vasildinev@gmail.com VIII COnfErEnCE Of tHE SpanISH SOCIEty fOr lOGIC, MEtHODOlOGy anD pHIlOSOpHy Of SCIEnCE, unIVErSIty Of BarCElOna, SpaIn, 7-10tH july 2015

tHE tHESIS: Gods dice is a qubit A qubit = |0 + |1, where , are two

complex numbers so that 2 + 2 = 1, and |0, |1 are two different subspaces of the complex Hilbert space

Gods dice need an infinite set of different symbols for all sides of them

PresenterPresentation NotesNonetheless that qubit is a single one, one should use dice (plural) rather than a die (singular) for any qubit means two independent choices and thus a pair of dice. Each of the orthogonal subspaces, | 0 and | 1, can be interpreted as a single die of that pair of dice, and the complex numbers and as the symbols (numerals), which are fallen in a given roll of Gods dice. This more accurate definition is fundamentally important as Section V will elucidate.

COntEntS: INTRODUCTION I A SKETCH OF THE PROOF OF THE THESIS II GLEASONS THEOREM (1957) AND THE THESIS III GODS DIE, GLEASONS THEOREM AND AN IDEA FOR A SHORT PROOF OF FERMATS LAST THEOREM IV INTERPRETATION OF THE THESIS V GODS DICE (A QUBIT) AS A LAW OF CONSERVATION AND TRANSFORMATION VI CONCLUSION

tHE StruCturE Of tHE prESEntatIOn

1 2

3 4 5

6 7

9

8

10 11

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The context of my paper

PresenterPresentation Notes

The thesis of the paper is fourfold: (1) Metaphor can be seen as the interaction of at least two frames in a sense of frame semantics. (2) Then representation can be interpreted as the particular case of zero interaction between the frames. (3) In turn, this allows of the frames to be interpreted formally as correspondingly reality and the image of reality, and language as an (even one-to-one) mapping between those two universal and formal frames of reality and its image. (4) Metaphor can be further represented formally as the entanglement of two or more frames and thus in terms of quantum information. Entanglement can be interpreted as a kind of interaction due to wholeness: If two or more entities constitute a common system, they can interact with each other by the whole of the system itself, i.e. holistically, rather than only by some deterministic and unambiguous mechanism.

IntrODuCtIOn

DEr nICHt wrfElt

Die Quantenmechanik ist sehr achtunggebietend. Aber eine innere Stimme sagt mir, da das noch nicht der wahre Jakob ist. Die Theorie liefert viel, aber dem Geheimnis des Alten bringt sie uns kaum nher. Jedenfalls bin ich berzeugt, da der nicht wrfelt (Einstein 1926)

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tHE quantuM DICElOGy An interdisciplinary field between: probability theory quantum mechanics information theory history and philosophy of science popular representations of quantum mechanics

and Einsteins polemic with it ontology epistemology, and even theology

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HIlBErtS SIxtH prOBlEM (1900)

Durch die Untersuchungen ber die Grundlagen der Geometrie wird uns die Aufgabe nahegelegt, nach diesem Vorbilde diejenigen physikalischen Disziplinen axiomatisch zu behandeln, in denen schon heute die Mathematik eine hervorragende Rolle spielt; dies sind in erster Linie die Wahrscheinlichkeitsrechnung und die Mechanik.

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tHE SEttInG Of Our prOBlEM Quantum mechanics is granted as the rules

generalizing all results of playing some imaginary Gods dice

How should Gods dice look like: What is the elementary choice determining any

single result unambiguously? What is the elementary event if the space of all

events consists of all possible experimental results, which can be obtained for that system, in the sense of probability theory and Kolmogorovs axioms (1933)?

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I a SKEtCH Of tHE prOOf Of tHE tHESIS:

waVE funCtIOn In tErMS Of quBItS

Any point of Hilbert space can be represented as an infinite series of qubits:

The rolls are independent of each other for any two qubits are orthogonal to each other

One can easily check out that Kolmogorovs axioms (1933: 2, 13) for probability are satisfied

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tHE SpaCE Of rOllInG quBItS anD KOlMOGOrOV S axIOMS

1. All sets of rolls constitute algebra of sets 2. That algebra of sets includes the set of all rolls 3. and 4. 2 is a real number, which is less or equal to 1 4. The probability of all rolls () is 1 for the wave function is normed: .* = 1 5. Indeed ( + ) = () + () as all rolls are independent of each other 6. The axiom of continuity is implied by the continuity of wave function

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II GlEaSOnS tHEOrEM (1957) anD tHE tHESIS

GlEaSOnS tHEOrEM anD ItS ExCEptIOn

A qubit falls just into the exception of two dimensions in Gleasons theorem not allowing of unambiguously defining the probability of any linear subspace of a qubit

The space consisting only of any single elementary event does not satisfy Kolmogorovs axioms in fact

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III GODS DIE, GlEaSOnS tHEOrEM anD an IDEa fOr a SHOrt prOOf Of fErMatS laSt tHEOrEM

fErMatS laSt tHEOrEM Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-quadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet (Nagel 1951: 252) Andrew Wiles (1995) managed to prove that theorem however on more than one hundred journal pages still recently and utilizing many of the last achievements of mathematics

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fErMatS laSt tHEOrEM anD GODS DICE

1. Any three Pythagorean numbers , , (consequently, any Pythagorean prime ) implies a different measure 2. Any three other real numbers share one single common measure for it is necessarily infinitesimal 3. The dimensions of one and two ( = 1, 2), which are under the exception of Gleasons theorem, allow of an infinite set of measures necessary for the equation + = to be able to be satisfied for some triples of natural numbers , , .

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fErMatS laSt tHEOrEM anD GODS DICE

4 Gleasons theorem excludes the existence of more than one single measure for any other number of dimensions ( > 2) 5. The utilized concept of measure should be much closer to the Pythagorean one implicitly suggested in the shocked the Pythagoreans incommensurability of the hypotenuse and cathetus of almost all right triangles rather than to the contemporary too generalized notion of measure:

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fErMatS laSt tHEOrEM anD GODS DICE

6. What remains to be proved is that concept of measure in Gleasons theorem is relevant and consistent to the above Pythagorean kind of measure: Whether a space of events is one and the same

where it is standalone as a set and where it is a subset of another

It seems that neither Gleason touches that question nor it in turn refers to his proof in any way

If no, Gleasons theorem seems to imply the above Pythagorean concept of measure, and if yes, it does not

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IV IntErprEtatIOn Of tHE tHESIS

tHE MaSS anD EnErGy Of InfOrMatIOn?

An analogy between two Einsteins famous formulas (1905), = 2, and = :

= This implies that the physical quantity of mass

corresponds to the flux of information passing through the present (i.e. per a unit of time)

It can be interpreted physically as the frequency of the associated de Broglie (1925) wave specifying any quantum system and corresponding to its mass

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V GODS DICE (a quBIt) aS a law Of COnSErVatIOn anD tranSfOrMatIOn

tHE SyMMEtrIES Of GODS DICE

A qubit as one is to wait for Gods dice

possesses many symmetries being as symmetric as possible at all

Any symmetry of them corresponds to some law of conservation

That of the physical quantity of action and probability (entropy and information) seems to be the most important one

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tHE unIty anD tranSItIOn BEtwEEn tHE MatErIal anD IDEal

The physical meaning should be that the physical quantity of action and the probability of an element or a state (respectively, entropy or information of the entire system) are conserved only transforming into each other

In particular, this implies the conservation of action rather than of energy

Gods dice demonstrate the unity and transition between the physical (i.e. material) action of a part and its disposition among the whole by meditation of probabilities (entropy or information), which are rather nonmaterial and mathematical

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VI COnCluSIOn

GODS DICE turn Out tO BE a quBIt

... and thus having the shape of a unit ball. Any item in the universe as well the universe

itself is both infinitely many rolls and a single roll of that dice for it has infinitely many sides

Thus both the smooth motion of classical physics and the discrete motion introduced in addition by quantum mechanics can be described uniformly correspondingly as an infinite series converges to some limit and as a quantum jump directly into that

MuCHaS GraCIaS pOr Su atEnCIn y ESpErO SuS COMEntarIOS y prEGuntaS

GODS DICE Vasil Penchev, Bulgarian Academy of Sciences: Institute for the Study of Societies and Knowledge: Dept. of Logical Systems and Models, vasildinev@gmail.com

INTRODUCTION Prehistory: Einstein wrote his famous sentences in a letter to