Newly posted research paper:

**Abstract**

As opposed to the classical logic of true and false, viewed as an axiomatised theory, ordinary arithmetic conveys the three logical values: provable, negatable and logically independent. This research proposes the hypothesis that Axioms of Arithmetic are the fundamental foundation running arithmetical processes in Nature, upon which physical processes rest. And goes on to show, in detail, that under these axioms, quantum mathematics derives and initiates logical independence, agreeing with indeterminacy in quantum experiments. Supporting arguments begin by explaining logical independence in arithmetic, in particular, independence of the square root of minus one. The method traces all sources of information entering arithmetic, needed to write mathematics of the free particle. Wave packets, prior to measurement, are found to be the only part of theory logically independent of axioms; the rest of theory is logically dependent. Ingress of logical independence is via uncaused, unprevented self-reference, sustaining the wave packet, but implying unitarity. Quantum mathematics based on axiomatised arithmetic is established as foundation for the 3-valued logic of Hans Reichenbach, which reconciles quantum theory with experimental anomalies such as the Einstein, Podolsky & Rosen paradox.

**Conclusions**

This research set out to discover logical artefacts in mathematical physics that derive and initiate indeterminacy, agreeing with quantum experiment. The main finding tells how indeterminate information is constituted and where, in quantum theory, it is present. But in arriving at these deductions, a hypothesis is proposed concerning arithmetic’s place in Nature, demanding we regard arithmetic in physics as a formal, axiomatised theory.

The original question inspiring this research asked whether logical circularity, or self-reference, might possibly be present in Nature. And this speculation was reinforced, knowing self-reference is a feature in the proof of Kurt Gödel’s Incompleteness Theorems, which guarantee the existence of non-provable, but true statements in arithmetic. In the language of Mathematical Logic, these statements are logically independent of arithmetic’s axioms, being neither provable nor negatable. One well-known example is the statement asserting existence of the square root of minus one. And so, given the insistent presence of this number in quantum theory, this statement was taken as entry-point for investigation.

This is the thesis. A hypothesis is posed assuming the proposition:\quad

Axioms of arithmetic exist in Nature. This viewpoint is the reverse of the ordinary notion that fields of scalars are fundamental in Nature, with arithmetic being an abstraction, encoding their rules of combination. The difference is subtle but profound; instead, axioms of arithmetic collectively assert existence for fields of scalars. Arithmetic results, influencing physical processes, including logically independent statements. And these we interpret as logical anomalies in experiments. To gain logical isomorphism between quantum theory and experiment, quantum mathematics must view arithmetic as an axiomatised theory, also — as Nature views it.

Treating arithmetic as an axiomatised theory, this paper finds that formulae representing wave packets (prior to measurement) are logically distinct from all other formulae in quantum mathematics. Only these are essentially unitary; only these require existence of imaginary-i

; only these rely on self-reference and only these are logically independent of axioms.

A wave packet consists of a pair of mutually consistent superpositions of complimentary variables, such as wave-number and position. The wave packet is unitary, but the superpositions, as individuals, are not. Considering an individual superposition as unitary makes no sense because both must coexist as a pair. To be unitary, a superposition must feed off information offered by its complimentary partner. From each superposition in the pair, there is a flow of information, satisfying a void, deficient in the other. As a sustainable entity, existence of the whole wave packet is dependent on self-reference. At first, the exchanging information might be indefinite, but after repeated cycles of self-reference, information settles toward something definite with square-integrability guaranteed. The self-reference does not contradict axioms, but is consistent with them and therefore is not prevented. It is possible through coincident coexistence of the superpositions.

This artefact of self-reference divides quantum mathematics into two logical partitions: that part of theory, logically dependent on axioms, separate from wave packets which are logically independent:\quad

on the one side, theory provable from axioms, and on the other, theory consistent with axioms, but not provable — this partitioned theory being interpretable as a causeology that causes observables, but permits different spectral outcomes to result from identically prepared experiments.

This theory posits axioms of arithmetic as profound and fundamental foundation in Nature. It does not tell us the origins of these, but philosophical questions reduce neatly to them. Axioms assert a theory of existence. They explain: persistent, stable existence; caused, deterministic existence; uncaused, indeterminate existence. The theory tells us observables are always real because provable existence is always rational and it tells us that amplitudes are on the complex plane because wave packets exist unprovably. It dispenses with all possible existence of unbounded superpositions. And uncaused existence is suppressed in classical physics, because there, no scalar product is invoked in arithmetic.

Finally it acts as foundation for the 3-valued logic of Hans Reichenbach which he showed resolves the EPR paradox, the logic of complimentarity and the logic of states, prior to measurement.