*6.5. Interference*

Let us go on with comments as to involving the *physics*-related argumentation to explicate the quantal behavior. We have already mentioned above that for this purpose there is simply no language of physics (Sections 2.1 and 6.4) and of mathematics ye<sup>t</sup> (Sections 2.3 and 5). That is why analogies of this sort are not only deceptive but must be prohibited for exactly the same reasons that accompanied boxes (5). The typical examples in this connection are the simultaneous measurability mentioned above and the two-slit interference [5,17].

First and foremost, the two cases—whether one or two slits are open—are utterly "different experimental arrangements" [153] (p. 236), [64] (p. 58):

$$
\langle \mathcal{S}, \mathbf{M}, \dots \rangle' \neq \langle \mathcal{S}, \mathbf{M}, \dots \rangle'' \dots
$$

There is nowhere to seek a means of their comparison or the transference of one into another ([153], p. 236). Nonetheless, the classical approach, when opening another slit S, **<sup>M</sup>**,... 2 together with the first one S, **<sup>M</sup>**,... 1, does literally envision properties for S, **<sup>M</sup>**,... (see Section 6.4). In doing so, the transference method itself—"addition of the two 1-slit S, **<sup>M</sup>**,... -physicae" by the rule of arithmetical addition of statistics (60)—is meanwhile considered self-apparent. Thus, natural questions arise, such as "why/where are the zeroes coming from, they should not be there". In accordance with the aforesaid, everything here is erroneous, including the "natural" questions. There are no rules at the outset whether (non)classical and even quantum, just as there is no addition per se. An a priori assumption that stem from the obvious images for S, **<sup>M</sup>**,... 1 and S, **<sup>M</sup>**,... 2 is actually a declaration of the physical properties for S, **<sup>M</sup>**,... , but they do not follow from anywhere [17], [64] (p. 55), [162]. The (illegal) assumption of the "negligible effect of which-slit detectors" were mentioned on p. 28 is identical with a declaration of a physical property, as well as a solenoid's switch-on/off in the Aharonov–Bohm effect.

Taken alone, the -distributions—separate for S, **<sup>M</sup>**,... 1 and S, **<sup>M</sup>**,... 2—are entirely correct observational pictures, but introducing the rule (60) is indistinguishable from "invention" of physics—a logically prohibited operation. As Slavnov had put it, "to invent the physical exegesis of a . . . mathematical scheme" ([76], p. 304). "Our custom of seeing classical mechanics as a no-nonsense description of 'reality as it is' does not seem to be justified. This custom is actually based on a confusion of categories . . . " (W. de Muynck ([4], p. 89)). In other words, the mere fact of non-adherence to this rule means that the grammatical conjunction of the verbs "to understand/deduce" with the noun "micro-phenomena" is unacceptable even linguistically. It is the point T that prohibits predefined (classical) semantics, and this was faithfully summarized by C. Fuchs: "badly calibrated linguistics is the predominant reason for quantum foundations continuing to exist as a field of research" ([2], p. xxxix). To figure out or deduce (from mathematics) that quantal phenomena are unfeasible ([5], p. 111) and are "*absolutely* impossible, to explain in any classical way" (quotation by Feynman). Just as with the elucidation of the nature of the quantum state on p. 23, any (circum-)classical justification or even motivation are guaranteed to fail here since they are based on significant and implicit assumptions.

The classical theory is a theory of *observational* objects with *observational* properties expressed by *observational* numbers. We possess none of the three items required to create the quantum (= correct) description (Section 2). The adjective "observational" itself is a linguistic notion of the classical vocabulary (Section 2.2). Accordingly, the description can only be changed "to describe in newly created terms". A. Leggett notes [95] that which is understood as common-sense should also be changed (see also [12] (p. 10)). The reason is clear.

• Common-sense operates—and that is perfectly normal—with observational categories rather than with structureless "microscopy" (9) and ∪-abstractions of Section 5.1;

cf. Bohr's correspondence principle [78]. In effect, we have dealt with a "fundamental chasm" between the right description—"what is really going on?" ([12], p. 12)—and our ability to give a (naturally speaking) explanation in terms of these categories:

"All our intuition, all our sense of what constitutes concreteness are based upon our everyday experience, and the terms used to describe a phenomenon concretely are necessarily drawn from that experience. There is no indication that such a language could be used without contradictions for phenomena which are as far removed from it as those of microscopic physics" (A. Messiah. *Quantum mechanics*).

The total dismissal of this has to be at the heart of quantum reconstructing.

#### 6.5.1. Detector Micro-Events

For similar reasons, we may not think or envision that a particle in an interferometer "flies through the slit", "has (not) arrived", "is located somewhere in the region of space" ([26], p. 7)], "here, not there", "now/later", that "the choice of a detector has been delayed" ([27], Wheeler), [62], or that a "photon . . . interferes . . . with itself" ([26], p. 9), and that, generally, "something is flying along a trajectory", and "something" is a particle at an intuitive understanding. Cf. Dirac's description of "the translational states of a photon" in Section 3 of [26].

•"Photons are just clicks in photon detectors; nothing real is traveling from the source to the detector" (ascribed to A. Zeilinger),

and this point is supported by all the known varieties of interferometers. There has to be an amendment here.

The clicks themselves are not the clicks *of photons/particles*, just "merely clicks". "[T]he click is no . . . produced by a particle. . . . nothing takes place in the source that could be a cause of the click . . . , the genuinely fortuitous click comes without a cause and has no precursor" ([126], pp. 758, 765). Nothing really interferes inside interferometers, nor is anything superposed/reinforced. For example, the fact that the path of "photons" is not represented by trajectories was impressively demonstrated with the nested Mach–Zehnder experimental setup in the work [163]. Asking "where the photons have been" [163] is also the matter of a certain *α*-distinguishability. An interferometer—the entire installation— should be perceived as nothing more than a black box S, **<sup>M</sup>**,... —the box (5)—outside of space and time. This is a kind of irreducible element that produces the only entity— distinguishable *α*-events, and no other. The box contains no "flying particles". Exempli gratia, none of the words in the typical sentence "photon propagates a definite path" are well-defined. Any assessment of the screen flashes observed within the interferometer, e.g., "is zero statistics possible in any spot?", lacks meaning until the theory's numerical apparatus is presented.

**Remark 13.** *Thus, Young's interference of the light beams (1803) is* inherently *the quantum not the classical effect: a micro-events' accumulation is usually termed as the light intensity. The classical electromagnetism and optics, in an exact sense, do not explain, only describe, the phenomenon quantitatively with the use of the numerical concepts of the positive and, which is important, the negative values of observable strength-fields* - *E and* - *H. (The negative numbers are specifically discussed further in Section 9.2.) Accordingly, operations of their addition/subtraction "rephrase" the effect in words "superposing, suppressing, waves, intensities"', and we call this "the explanation". In a quantum way of looking at it, all of these concepts are not yet available, and the phenomenon per se is no more than statistics of the "positively accumulative" quantal clicks:*

• *There are no particles, waves, or subtractions there.*

*The same macroscopic effect, which is visible with the naked eye and "explainable by waves", would take place if we had a "laser" of, say, mono-energetic very slow electrons (a proposal for*

*experimentalists). To put it more precisely: a gun or emitter of something we envision as the "tiny bodily formations" the electrons, molecules, microbes, and the like. It is self-evident that we would have seen the wave-like manifestation even from a single slit.*

Criticism of the typical (a common event-space) examination of the two-slit experiment [164] is already abundant in the literature. See, for example, the works [17] (Sections V.1, VI.1–2), [64] (pp. 55–58), [66] (Ch. 2), and [121] (p. 93), [162] (!), [165].

By way of continuing the last sentence in Remark 3, we add the following. To force an electron-click to happen each time at the same (or predictable) place is no different from "completely describing *everything* that we have", i.e., from the precise setting of "the same" and of macro-context S, **<sup>M</sup>**,... . It is amply evident that this is a manifest absurdity. Hence, it immediately follows that *the unpredictability of microscopic events must exist in principle* and macro-determinism may be only an idealization through a (math) model: the model description of the S, **<sup>M</sup>**,... itself.

Summing up, it is not the quantum interference that requires interpretative comprehension but its classic "roughening". In other words, a scheme that latently presumes the rule (60) of extrapolation of what is observed in macro and micro ([160] (!), last sentence on p. 101). It is this scheme and not the quantum approach that contradicts the logic and experience. Pauli characterizes this as habits "known as 'ontology' or 'realism'". More than that, the chief component of constructs—,observation state-—is cast out and replaced with (19) under such a transformation. The DataSource object (p. 31) begins to be identified with *observational* and *numerical* characteristics (see a paragraph preceding Remark 4), while the logic of the micro-world requires precisely distancing these two concepts, with no need for the characteristics themselves.

Thus, we should not be deriving the physics of one phenomenon from another ([58], p. 92) and making (super)generalizations, as soon as the incorrectness of the previous derivation method was established.

• Quantum-mathematics is not a physical theory—and that is its distinguishing feature— but rather a single syntactical (meta)*principle of forming the mathematical models being subsequently turned into* (the physical) *theories*. This principle is not subject to any physical validation.

Scott Aaronson was likely the first to advance the line of thought about non-physicality. On page 110 of the book [20], he writes that "it's *not* a physical theory in the same sense as electromagnetism or general relativity . . . quantum mechanics sits at a level *between* math and physics . . . *is the operating system* . . . ". Fuchs–Peres provoke: "quantum theory does *not* describe physical reality" ([43], p. 70).

To create the models, we already have a good deal of *latitude*: the toolkit O = {A , B, ...}, the parameter D, the families {TA , T<sup>B</sup>, . . . }, numbers {*s*} of mixtures (23), and—thanks to the notion of covariance III—spectra, a structure of a group, and the concept of (different) representations of a mathematical structure. This liberty will be subsequently augmented with the key notions of a mean and of time *t* and also with the composite systems, the classical Lagrangians/Hamiltonians, their symmetries, gauge fields, and phenomenological constants. This is what is currently termed the quantum phenomenology or a *quantization procedure* of the classical models: the path-integrals, *S*-matrix, etc.

All that remains is to examine the numerical constituent of quantum mathematics. The further strategy (Sections 7–9) lies in the fact that the numbers need to be created at first as a theoretical concept—arithmetic—and then as the "numerical values for observable quantities"—the observations numbers. Sections 9.1 and 9.2 contain some more explanations along these lines.

## **7. Numeri**

By number we understand not so much a multitude of unities, as the abstracted ratio of any quantity to another quantity of the same kind, which we take for unity—I. Newton (1707)

#### *7.1. Replications of Ensembles*

In connection with the emergence of a group, the numerical representation of brace also undergoes a change since the "doubling" of a semigroup into a group through adjoining the inversions deprives coordinate a of its distinction in comparison with the inversion <sup>−</sup>a. Given the involution

$$-\left(-\mathfrak{a}\right) = \mathfrak{a}\,,\tag{61}$$

it makes no difference what to call an element and what to call its inversion in the pair {<sup>a</sup>, <sup>−</sup><sup>a</sup>}. This doubling is formally known as a symmetrization of the commutative associative law (monoid) [166]. Curiously, under commutativity and associativity ([167], Section 1.10), the solution to the problem of embedding is unique ([166], pp. 15–17), and otherwise, no solution, in general, exists. There exist the classes (Mal'cev (1936)), which are not axiomatized by finitely many ∀-formulas ([168], pp. 216–217).

The aforesaid is best demonstrated by another way of "numeralizing" the empiricism, which is realized as the infinite replication of finite ensembles

$$\left\{ \{\overline{\mathbb{A}}\}^{\boldsymbol{\uppi}}\{\overline{\mathbb{A}}\}^{\boldsymbol{\uppi}}\cdots\right\} = \left\{ \{\overline{\mathbb{A}}\}^{\boldsymbol{\uppi}}\right\}^{\boldsymbol{\uppi}} =: \{\overline{\mathbb{A}}\}^{\boldsymbol{\uppi}}\,.\tag{62}$$

That is, empirically, any infinite ensemble is thought of as created by repetitions (copies) of the finite objects {Ψ}n. It is in this sense, and in this sense alone, that one should read the writing Σ ∞ for the infinity postulate (14) because, at the moment, we possess neither the mathematics nor the topological concepts, such as a passage to the limit lim Σ→∞ . For example, the expression Σ × ∞ can be viewed as a conjunction of the actual and potential infinity [105,169]. Simply put, the case in point is not an *axiomatic* act—an imposition of the *math*-existence condition for numbers {<sup>f</sup>*j*} in (14). The latter has been typically criticized as an idea of the stable limiting frequencies in QM [8] (pp. 15, 183, 211, . . . ), [23], [170] (pp. 97–99), [171]. Rather, we claim that the only way to consistently incorporate the language notions of infinity and of the finite (observational) numbers in theory—"to cross an abyss" (Poincaré)—is the above semantics and correspondence between symbols {*nj*, f*j*; Σ,, <sup>×</sup>, <sup>∞</sup>}. See also subsection "StatLength and infinity" in the work [6].

In turn, the above-mentioned copies {Ψ}n are replications of the atomic primitive {Ψ}1. Replication is thus an operation of the same significance as ∪ and 0. With this point, the ⎧⎩Ξ⎫⎭-brace is characterized by the "numerical" combination

$$(\text{25}) \quad \leftrightharpoons \quad \left\{ \left[ \mathfrak{n}\_1 \mathfrak{so}, \mathfrak{m}\_1 \mathfrak{so} \right], \left[ \mathfrak{n}\_2 \mathfrak{so}, \mathfrak{m}\_2 \mathfrak{so} \right], \dots \right\} \right\} \implies (\text{25})$$

(indices label the *<sup>α</sup>s*-primitives), which has been created from the unitary brace by the scheme

$$(34) = \left\{ \begin{array}{c} \{\frac{\langle\!\{\overline{\Psi}\}^{\alpha\_{\prime}}\{\underline{\Phi}\}^{\alpha\_{\prime}}\} \\ \downarrow \end{array} \right\}\_{m} \longrightarrow \left\{ \begin{array}{c} \{\frac{\langle\!\{\overline{\Psi}\}^{\alpha\_{\prime}}\{\underline{\Phi}\}^{\alpha\_{\prime}}\} \\ \downarrow \end{array} \right\}\_{m} \longrightarrow \left\{ \begin{array}{c} \{\frac{\langle\!\{\overline{\Psi}\}^{\alpha\_{m}}\{\underline{\Phi}\}^{\alpha\_{m}}\} \\ \downarrow \end{array} \right\}\_{m} \longrightarrow [n\infty, m\infty]\_{\mathfrak{K}}.\tag{63}$$

The semigroup union ⎧⎩Ξ ⎫⎭ 0 ⎧⎩Ξ ⎫⎭ is then conformed with the writing

$$\begin{aligned} \left\{ \left[ \mathfrak{n}\_{1}^{\prime} \infty, \mathfrak{m}\_{1}^{\prime} \infty \right], \left[ \mathfrak{n}\_{2}^{\prime} \infty, \mathfrak{m}\_{2}^{\prime} \infty \right], \dots \right\} & \left\{ \left[ \mathfrak{n}\_{1}^{\prime} \infty, \mathfrak{m}\_{1}^{\prime} \infty \right], \left[ \mathfrak{n}\_{2}^{\prime} \infty, \mathfrak{m}\_{2}^{\prime} \infty \right], \dots \right\} = \\ &= \left\{ \left[ \left( \mathfrak{n}\_{1}^{\prime} + \mathfrak{n}\_{1}^{\prime} \right) \infty, \left( \mathfrak{m}\_{1}^{\prime} + \mathfrak{m}\_{1}^{\prime} \right) \infty \right], \left[ \left( \mathfrak{n}\_{2}^{\prime} + \mathfrak{n}\_{2}^{\prime} \right) \infty, \left( \mathfrak{m}\_{2}^{\prime} + \mathfrak{m}\_{2}^{\prime} \right) \infty \right], \dots \right\}. \end{aligned} \right\}. \end{aligned}$$

Moreover, the n-, m-quantities may be freely thought of as real ones due to the R2- continual infinity of ensembles proven above (Section 4). The empirical rationale of this is apparent; namely, fractions of the arbitrarily large ensembles {Ψ Ψ ···}.

This way of matching the infinity with <sup>Σ</sup>-postulate automatically inherits translation of associativity/commutativity because the "percentages", such as *s* and *w*, just as the rules (44) and (45) themselves, do not even emerge. There, these numbers originated from <sup>Σ</sup>-postulate, but it, in turn, was *demolishing the pair* (<sup>κ</sup>, S) *itself* in (43): S → ∞. It is clear that, according to (64), the semigroup structure G is also inherited, turning into the addition of the numerical pairs

$$(\mathfrak{u}', \mathfrak{w}') \circ (\mathfrak{u}', \mathfrak{w}') = (\mathfrak{u}' + \mathfrak{u}'', \mathfrak{w}' + \mathfrak{w}'') \,. \tag{65}$$

Returning to the group, we observe that the "negative symbols" (−n, −<sup>m</sup>) might be initially taken as the semigroup G being duplicated, with equal success and with the same arithmetical addition <sup>⊕</sup>, while the positive (<sup>n</sup>, m) could be thought of as inversions thereof. Summing up, let us specify the rules of passing to the numerical representations

$$(\exists 4) \qquad \Longleftrightarrow \quad \{\pm \not p, \pm \not q\} \not \mathfrak{k}, \qquad (\mathfrak{p}, \mathfrak{q}) \in \mathbb{R}^2 \tag{66}$$

.

and, to avoid ambiguity, replace the binary-composition symbols {0, <sup>+</sup>} with a new symbol + for objects (66):

$$
\underline{\hat{\mathbb{P}}\{\underline{\mathbf{u}},\underline{\mathbf{u}}\}} + \underline{\mathbb{P}}\{\underline{\mathbf{p}},\underline{\mathbf{q}}\}
$$

The previously dropped primitives Ψ, Φ have been restored here since they will be further needed for the theory's covariance (Sections 7.4 and 7.5), although they are still unnecessary at the moment.

It is not accidental that we spoke of "numerally labeling" the brace (p. 22) since the question of arithmetic on them had not ye<sup>t</sup> arisen. Although f-statistics—the real R-numbers—are already involved, their use was based on an accustomed perception of the number. In accordance with pr. II, the numerical formalization of ensemble empiricism should be considered in greater detail.

#### *7.2. The Number as an Operator*

Let us take up the "process of manufacturing" the numbers (pr. II). We begin with the classical simplification

$$\mathfrak{X} = \{ \{\mathfrak{X}\}, \{\mathfrak{X}\widetilde{\mathfrak{X}}\}, \{\mathfrak{X}\widetilde{\mathfrak{X}}\}, \{\mathfrak{X}\widetilde{\mathfrak{X}}\widetilde{\mathfrak{X}}\}, \dots \}, \tag{67}$$

.

and the notion of the number does not ye<sup>t</sup> appear in any form.

The mathematical abstracting the observation micro-acts is an employment of the operation ∪ and of its closedness (see Section 5.1). For example, {Ψ}∪{ΨΨΨ} = {ΨΨΨΨ}. All the symbols in (67), as well as the character ∪, is of course merely a convention, and they may be changed. By writing (67) in symbols such as {*<sup>a</sup>*, *b*, *c*, *d*, ...} and +, this set should be supplemented with identities as *a* + *b* = *c*, *b* + *b* = *d*, . . . , i.e., with a binary construction +. Then, (semi)group and commutative superpositions arise. Though note that introducing the numbers at this point—even if only as symbols—is not necessary. It would reduce to re-notating the set's elements, to be precise. However, the empirical description calls for their unification, as manifested in the numerical notation as {Ψ} =: <sup>1</sup>{Ψ}, {ΨΨ} =: <sup>2</sup>{Ψ}, . . . . It is precisely this pattern that was implicitly kept in mind in procedures (34), (35), (62), and (63), i.e., when introducing the numbers n by means of replication of finite or infinite ensembles:

$$\{\overline{\mathbb{A}}\cdots\mathbb{A}\}^{\iota} \iff \iota\{\overline{\mathbb{A}}\}^{\iota} \qquad \{\{\overline{\mathbb{A}}\}^{\iota\iota}\cdots\iota\{\overline{\mathbb{A}}\}^{\iota\iota}\}^{\iota} \iff \mathfrak{u}\{\overline{\mathbb{A}}\}^{\iota}$$

The symbol ⇐⇒ should read here as "the same thing as". Clearly, the very idea of the conjunction of the two entities—empirical brace (34) and the concept of a (quantitative and ordinal) number (Sections 5.1 and 5.2)—is not otherwise implementable. That is to say:

• We have no any means of translating the aggregates of micro-acts A (i.e., macroobservations M) into the numerical language other than through the counting of things [172], i.e., through the natural-language notion of the "quantity of something":

··· A -transitions ··· ⇓ ⇓ ,quantity of- ,something- (replication) ⇓ ⇓ ,numbers- , Ψ-primitives, ensembles-4 5 n{Ψ} . (68)

Heisenberg stresses an obligatory relationship with "the natural language because it is only there that we can be certain to touch reality" ([98], pp. 201–202). Otherwise, the quantitative theory would have nowhere to originate even at the level of calculating the natural entities by the N-number tokens. It may be added that arising the numbers is a permanently present (innate) process of creating the thought objects by an abstraction in the human brain: the mental suppressing/neglecting of the inessential and identifying the distinguishable entities—perceptual objects—irrespective of their nature ([84], "forming collections, . . . putting objects together"; pp. 99, 251). It is something that humans do all the time without even realizing they are doing it. This process, say,

$$\{\text{langugage, words}\} \cdot \cdot \cdot \cdot \longrightarrow \{\text{sheep, \underline{\forall}} , \text{verb}, \underline{\text{\forall}} , \text{theory}, \dots\} \cdot \longrightarrow$$

$$\{\text{a sheep, a \underline{\forall}} , \text{a } \underline{\text{\forall}} , \text{a } \text{verb}, \dots\} \cdot \cdot \cdot \longrightarrow \{\text{something}/\text{thing}/\dots\}/\text{Stickle} \} \cdot \cdot \cdot \cdot \longrightarrow$$

$$\{\text{\bullet Stück}, \bullet \text{Stück}, \bullet \text{Stück}, \bullet \text{Stück}, \bullet \text{Stück}, \dots\} \cdot \longrightarrow \{\text{\bullet}, \bullet, \bullet, \bullet, \bullet, \bullet \text{Stück}\} \leftarrow$$

$$\{1, 1, 1, 1, 1, \text{\textbullet} \text{ücke}\} \cdot \leftarrow \\$\\$ \text{tiick} \longrightarrow \\$ \text{\bullet}\\$ \text{üek} \longrightarrow \\$ \longrightarrow \lceil \text{abtraction} \text{\textquotesingle} \rceil \} \, \, \, \}$$

is akin to Cantor's concept of a Menge ([136], Ch. 1, Section 1.1) and has *no* the mathematical (math-logic) nature. Rather, the math of numbers does originate from it [84] (Ch. 3); see also [173] (Section 2.4.5.1 ARITHMETIC).

Incidentally, the "inessential and identifying" just mentioned have the nature just like the "the same" in Section 5.4. It is with these notions—a key feature of the natural/physical language and of speech—that any abstracting begins: the "abstracting from . . . ".

On the other hand, the numerical tokens are "affixed" not only to the "atom" {Ψ} but also to other objects, *any* at that; for more details, see Remark 16 further below. Therein lies the primary meaning of this still proto-mathematical concept [174] ("Psychologie du nombre"). One might even say, a definition according to which this notion has been conceived ("20 Stück", "half an hour" . . . ) and is being used universally. Here are a few examples:

$$\{\underline{\Phi}\underline{\mathbf{X}}\} \stackrel{\scriptstyle \!=}{\longrightarrow} \{\underline{\Phi}\underline{\mathbf{X}}\Phi\underline{\mathbf{X}}\}, \qquad \{\underline{\Phi}\underline{\Phi}\underline{\Phi}\underline{\Phi}\underline{\Phi}\} \equiv 2\{\{\underline{\Phi}\underline{\}} \cup \{\underline{\Phi}\}\}$$

$$\{\underline{\Phi}\underline{\mathbf{X}}\} \stackrel{\scriptstyle \!=}{\longrightarrow} \{\underline{\Phi}\underline{\Phi}\underline{\Phi}\underline{\mathbf{X}}\}, \qquad a \stackrel{\scriptstyle \!=}{\longrightarrow} \{3a, \qquad c \stackrel{\scriptstyle \!=}{\longrightarrow} \mathbf{1}\}$$

Accordingly, in between the elements, there arise identities such as 2*b* ≡ 4*a*, 3*a* ≡ *c*, 1*c* ≡ *c*. In other words, as we complete Simplification (67),

• While abstracting the empirical contents of the number entities into math-symbols, they should be defined as unary operations { 1, 2, ..., 3 /4, ..., *π* , . . . } that take action at A-set (67) as automorphisms: { 2*b* = 4*a*, 1*c* = *<sup>c</sup>*,...}.

That said, replication is formalized as an operator *n* with its numerical symbol n:

$$
\Psi \stackrel{\hat{\pi}}{\longmapsto} \mathfrak{n} \psi, \qquad \Psi, \mathfrak{n} \psi \in \mathfrak{A}, \qquad \mathfrak{n} \in \mathbb{R} \tag{70}
$$

where *ψ* is understood to be any (sub)ensemble/(sub)set. In the language of the ZF-theory, n*ψ* would be formally organized as an ordered pair (<sup>n</sup>, *ψ*) := {{n}, {<sup>n</sup>, *ψ*}} [134], where n

is a cardinality of a set consisting of copies of the object/set *ψ*. We will refer to these facts as the implementation of a replication operator by numbers.

Attention is drawn to the fact that the case in point at the moment *is not* a math-logical *definitio/formalization* of the concept of a number, such as (106), but is an introduction of what is understood by number in the empirical/physical theory (II). For example, Chomsky says with regard to this point: "When multiplying numbers in our heads, we depend on many factors beyond our intrinsic knowledge of arithmetic" ([132], p. 3).

#### *7.3. QM and Arithmetica*

We immediately observe the following properties.

The operators are applicable to each other. Being a family {*n* , *m* , *p* , ...}, they are closed with respect to their composition *n* (*m ψ*)=(*n* ◦ *m* )*ψ* = *pψ*, and among them, there is an identical operator 1*ψ* = *ψ*. The empirical meaning of the concept indicates a fractional portion of the ensemble (see (62)) requires that for each *n* there exists its inversion *n* −1. Hence, the composition of replications *n* ◦ *n* −1 must return the former "quantity": (*n* ◦ *n* −<sup>1</sup>)*ψ* = 1*ψ*. Therein lies "the actual meaning of division. . . . this [operator] construction really corresponds to division" ([85], p. 37). By virtue of the fact that family {*n* , *m* , *p* , ...} provides automorphisms of the A-set, these operators entail the associative identities ((*n* ◦ *m* ) ◦ *p*)*ψ* = (*n* ◦ (*m* ◦ *p*))*ψ*. This point is a property, and it has a proof [175] (Section I.1.2). The common nature of the replication and of the ∪-union also signifies that there are relations in place that mix the actions of the unary *n*'s and the binary union of ensembles. At a minimum, suffice it to define the action of the replicator on a "∪-sum" of replications. Clearly, the case in point is the distributive coordination of ◦ and ∪:

$$
\hat{p}\left(\hat{n}\,\psi\cup\hat{m}\,\psi\right) = \left(\hat{p}\circ\hat{n}\right)\psi\cup\left(\hat{p}\circ\hat{m}\right)\psi\,\,.
$$

We now observe that the indication of *ψ* everywhere in the identities above loses the necessity, and the *ψ*-label becomes a semblance of a dummy index or the unit symbol (kg), which can be changed. As we omit it, the theory is freed of *ψ* as a "calculation unit". Then, the last relation, as an example, acquires the form of a property between the operator n-symbols (70), if {∪, ◦} are replaced with the symbols of binary operations {<sup>+</sup>, ×}:

$$
\mathfrak{p} \times (\mathfrak{n} + \mathfrak{m}) = (\mathfrak{p} \times \mathfrak{n}) + (\mathfrak{p} \times \mathfrak{m})\,. \tag{71}
$$

Supplementing this relation with other empirically determining properties, one infers that the *unary* operationality of *n*-replications (70) is *indistinguishable* from the *binary* operationality on their n-symbols. The latter, in turn, acquires the multiplicative structure of a commutative group

$$
\mathfrak{n} \times \mathfrak{m} = \mathfrak{m} \times \mathfrak{n}, \qquad \left(\mathfrak{n} \times \mathfrak{m}\right) \times \mathfrak{p} = \mathfrak{n} \times \left(\mathfrak{m} \times \mathfrak{p}\right), \qquad \mathfrak{n} \times \mathfrak{1} = \mathfrak{n}, \qquad \mathfrak{n} \times \mathfrak{n}^{-1} = \mathfrak{1}, \tag{72}
$$

and, as for the addition +, it is already binary and commutative due to properties of ∪ (Section 5.1):

$$
\mathfrak{n} + \mathfrak{m} = \mathfrak{m} + \mathfrak{n}, \qquad (\mathfrak{n} + \mathfrak{m}) + \mathfrak{p} = \mathfrak{n} + (\mathfrak{m} + \mathfrak{p}), \qquad \mathfrak{n} + 0 = \mathfrak{n} \,. \tag{73}
$$

Incidentally, the three-term multiplicative associativity relation in (72) has the same operatorial nature and origin as operations {∪, *∪*} do in (39). We have already commented on the additive analog in this situation—a determinative structure of the binary operation— after Formula (57).

It is also clear that Rules (71)–(73) must be supplemented with the concept of a negative number

$$
\mathfrak{l} + (-\mathfrak{v}) = \mathfrak{0} \text{ , \tag{74}}
$$

 for such numbers have been fully justified in the superposition principle.

n

After having acquired Properties (71)–(74)—call them *arithmetica*—symbols {<sup>n</sup>, m, ...} turn into abstract numbers, although their operator genesis does not go away and is ye<sup>t</sup> to be involved. This is where a full list of requirements for the concept of a real number should be added, and which have to do with ordering <, completeness/continuality, *and* their relations with Rules (71)–(74). We will assume that this is conducted axiomatically ([176], pp. 35–38), although the algebraic constituent of this "axiomatics", as we have seen, is not axiomatical but deducible from empiricism. Multiplication <sup>×</sup>, and also the subsequent \*-multiplication of C-numbers (82), is a most nontrivial part in deriving the structure from "the arithmetic".

As an outcome, we reveal an essential asymmetry in the genesis of the standard binary structures + and × (cf. [84] (p. 60)), and thereby a greater primacy of QM-consideration even over the (seemingly self-evident) arithmetic. Indeed, binarity may come only from operation ∪, which is primordially unique and, thereby, is inherited only to the one natural prototype—addition.

• Multiplication is not featured in the superposition principle, nor does it arise directly as a binary structure. The absence of a multiplication symbol in (58) and (59) is no accident.

The multiplication originates in the closedness of replications *n* ◦ *m* , and they are required according to the <sup>M</sup>-paradigm (12). In effect, any non-operatorial way of introducing the n-numbers is not self-evidence for empiricism. An operator nature of the number is precisely that which gives rise to the second binary operation. Moreover, without such a comprehension of the number, the "linear nature" of QM (Section 8.1) will remain axiomatic at all times, and, as will be seen below, quantum foundations will be doomed to neverending interpreting the mathematical symbols. However, the pure axiomatic declaration of Arithmetic (71)–(74) will, in one way or another, require a (reciprocal to (68)) treatment of the number in a context of "the quantity of what?", while its empirical pre-image always appears in the pair ,the quantity of- + ,something-. Another way to put it is:

•In the foundations of theory, there arises a predecessor/analog to the notion of a physical unit,

though the ultimate description is a description in terms of binary structures in Arithmetic (71)–(74). It is carried out by dropping/attaching the symbols such as *ψ*, which is a quantum generalization to the independence of a physical theory, from the measurement units.

Certainly, when formalized, the *n*-replication and its binary n-twin become universally abstract. For example, the *n*-operator (70) may be applied to the quantum case in which the object *ψ* has already an internal structure associated with the presence of Ψ, Φ-primitives. This changes no the essence of the matter. Another example is when numbers n give birth to really observable quantities. See also Section 5.3, Remark 16, and additional discussion in Section 9. Let us now proceed from the fact that the comprehension/relation of the number and its operator has been formalized as described above. This is "Axioms" (71)–(74).

As concerns the philosophical literature, the issue of numbers was likely discussed [177–179] (see also [172] and non-philosophical book [174]), and it would be appropriate to quote T. Maudlin: ". . . numbers: they can be added to one another, *perhaps* multiplied by one another, .... However, it is typically obscure what sort of *physical* relation these mathematical operations could possibly represent" ([151], p. 138; first emphasis ours, second in original). Cf. Einstein's remarks regarding the "concepts and propositions" and "the series of integers" on p. 287 in [180].

#### *7.4. Two-Dimensional Numbers*

A number in and of itself, as a replication operator, may be applied to any ensemble and to anything at all. However, in the quantum case, the "upper" primitives are attached to every "lower" *α*-event. These primitives, as was noted above, have to be discarded. At the same time, the minimal structure associated with the homogeneous array {*<sup>α</sup>s* ··· *<sup>α</sup>s*}

as a whole is a unitary brace {nΨ, mΦ }*<sup>α</sup>s* containing two "upper" primitives Ψ, Φ. Their order, however, is arbitrary there. That is to say, given (<sup>n</sup>, <sup>m</sup>)*<sup>α</sup>*, there are two quite equal objects {nΨ, mΦ }*α* and {nΦ, mΨ }*α* that are subjected to a replication. Each of them should be in a relationship (see Section 5.1) to any other brace (63), which is already apparent in the example of "one-dimensional" versions (<sup>n</sup>, <sup>0</sup>)*α* and (n, <sup>0</sup>)*<sup>α</sup>*. We mean that for each pair {(<sup>n</sup>, <sup>0</sup>)*<sup>α</sup>*,(n, <sup>0</sup>)*α*}, there always exists the number m such that *m* (<sup>n</sup>, <sup>0</sup>)*α* = (n, <sup>0</sup>)*<sup>α</sup>*, i.e., m × n = <sup>n</sup>.

As in the classical case (69), the sought-for generalizations of replicators are the transitive automorphisms on *unitary α*-brace (66), but they *are not* abstract and *not* arbitrary. They are strictly bound to the declared meaning of the number: *N*-operation of creating the copies. Therefore, by virtue of the equal rights of Ψ and Φ, it is imperative to bring the two one-fold copying acts *<sup>N</sup>*{n<sup>Ψ</sup> , mΦ }*α* and *M* {nΦ , mΨ }*α* into play, which differ in the permutation of primitives Ψ Φ. This point will determine a quantum extension of the replication.

As a result, since we have nothing but the copying *N* and "union" + , the most general transformation of the brace {nΨ, mΦ }*α* into (any) brace {nΨ , mΦ }*<sup>α</sup>*, which has been in a quantum-replication relation with it, is determined by the rule

$$\{\mathfrak{a}',\mathfrak{a}'\}\mathfrak{d} \quad \xrightarrow{\{\overline{\mathfrak{a}},\overline{\mathfrak{a}}\}} \quad \{\mathfrak{a}',\mathfrak{a}'\}\mathfrak{d} \quad \implies \qquad \{\widehat{\mathfrak{a}}',\widehat{\mathfrak{m}}'\}\mathfrak{d} = \bar{N}\{\widehat{\mathfrak{a}},\widehat{\mathfrak{m}}\}\mathfrak{d} \quad \langle \mathfrak{a}',\mathfrak{m} \rangle \mathfrak{d} \quad . \tag{75}$$

This is the quantum version of Operators (69) and (70), and the foregoing ideology of *N*-operators and of liberation from the Ψ-symbols remains in force and entails the following. The numeral implementation of replicating the unitary brace (66), along with the (<sup>n</sup>, m)-representation of itself, is also determined by a certain pair (<sup>N</sup>, M) ∈ R2, i.e., by an operator symbol (*N* , *<sup>M</sup>*).

The aforesaid means that the numerical form (<sup>n</sup>, m) (*<sup>N</sup>* ,*<sup>M</sup>*) (n, m) of Transformation (75) is indistinguishable from a composition of pairs

$$(\mathfrak{N}, \mathfrak{M}) \circ (\mathfrak{n}, \mathfrak{m}) = (\mathfrak{n}', \mathfrak{m}') \text{ .}$$

where \* is a designation for the new binary operation. Its resultant structure is derived from the arithmetical nature (72) of the one-dimensional replication (69) described above, i.e., from the rules

$$
\hat{N}\{\hat{\boldsymbol{\hbar}},\hat{\boldsymbol{\mu}}\}\underline{\boldsymbol{\varepsilon}}=\{\mathcal{N}^{\mathtt{Z}}\boldsymbol{\eta},\ \mathcal{N}^{\mathtt{Z}}\boldsymbol{\varepsilon}\}\underline{\boldsymbol{\varepsilon}},\qquad \hat{M}\{\hat{\boldsymbol{\hbar}},\hat{\boldsymbol{\mu}}\}\underline{\boldsymbol{\varepsilon}}=\{\mathcal{N}^{\mathtt{Z}}\boldsymbol{\eta},\ \mathcal{M}^{\mathtt{Z}}\boldsymbol{\varepsilon}\}\underline{\boldsymbol{\varepsilon}}\,.\tag{76}$$

Here, a positivity/negativity of symbols (n, m) in (66) should also be taken into account. Having regard to the foregoing, Rules (75) and (76) generate the Ansatz

$$(\mathsf{N}, \mathsf{M}) \circ (\mathsf{n}, \mathsf{m}) = \left( \pm \mathsf{N} \mathsf{n} \pm \mathsf{M} \mathsf{m}, \,\,\pm \mathsf{N} \mathsf{m} \pm \mathsf{M} \mathsf{n} \right), \tag{77}$$

wherein all four signs ± are independent of each other, and the (×)-multiplication of onedimensional numbers in (72) and (76) have been re-denoted by the habitual standard Nm := N × m. What should the pair-composition rule (77) be?

As was the case previously, the just-emerged binarity for \* should inherit—due to its operator origin—associativity, the existence of unity 1, and inversions. Namely, if the (<sup>n</sup>, m)-pairs are identified with the notation (57) according to the convention

$$(\mathfrak{n}, \mathfrak{m}) = :\mathfrak{a}\,,\tag{78}$$

then the following properties should be declared:

$$(\mathfrak{a}\circ\mathfrak{b})\circ\mathfrak{c}=\mathfrak{a}\circ(\mathfrak{b}\circ\mathfrak{c}), \qquad \mathfrak{a}\circ\mathbb{1}=\mathfrak{a}, \qquad \mathfrak{a}\circ\mathfrak{a}^{-1}=\mathbb{1}\,. \tag{79}$$

From (75) and (76), it is not difficult to see that the combining (79) with (57) leads to a distributive coordination of operations ⊕ and \*:

$$\mathfrak{c}\circ(\mathfrak{a}\circ\mathfrak{b})=(\mathfrak{c}\circ\mathfrak{a})\circ(\mathfrak{c}\circ\mathfrak{b})\,.\tag{80}$$

However, the direct examination of this property shows that Ansatz (77) satisfies it automatically. More than that, we can even consider Ansatz (77) with parameters {*<sup>α</sup>*, *β*, *γ*, *δ*} instead of (±)-signs:

$$\mathfrak{a} \circ \mathfrak{b} = (\mathfrak{N}, \mathfrak{M}) \circ (\mathfrak{n}, \mathfrak{m}) = (\mathfrak{a}\mathfrak{N}\mathfrak{n} + \beta \mathfrak{M}\mathfrak{m}, \,\gamma \mathfrak{N}\mathfrak{m} + \delta \mathfrak{M}\mathfrak{n})\dots$$

Then, the straightforward calculation shows that Distributivity (80) holds under the arbitrary {*<sup>α</sup>*, *β*, *γ*, *<sup>δ</sup>*}.

In turn, the examination of associativity—the first equality in (79)—under the same meaning for {*<sup>α</sup>*, *β*, *γ*, *δ*} yields *α* = *γ* = *δ* and free *β*. Returning to the (±)-values of these parameters, this associativity particularizes Ansatz (77) into the expression

$$(\mathsf{N}, \mathsf{M}) \circ (\mathsf{n}, \mathsf{m}) = \pm (\mathsf{N} \mathsf{n} \pm \mathsf{M} \mathsf{m}, \mathsf{N} \mathsf{m} + \mathsf{M} \mathsf{n});$$

now, with two independent signs ±. Moreover, in passing, we reveal the commutativity

a\* b = b \* a , (81)

though it was not presumed prior to that.

The search for unity 1 and subsequent finding of an inversion of the element (<sup>n</sup>, m) yield:

$$1 = (\pm 1, 0), \qquad (\mathfrak{n}, \mathfrak{m})^{-1} = \left(\frac{\mathfrak{n}}{\Delta'}, -\frac{\mathfrak{m}}{\Delta}\right), \qquad \Delta := \mathfrak{n}^2 \pm \mathfrak{m}^2 \dots$$

Both the (±)-symbols continue to be independent here. The choice Δ = n2 − m<sup>2</sup> results in the absence of inversions (<sup>n</sup>, n)−1. This is in conflict with the group property (79) and also causes the unmotivated exclusivity of the unitary brace {n Ψ , n Φ }*<sup>α</sup>*. There remains the case Δ = n2 + m2, and it reduces the scheme to the form

$$\mathbf{1} = \pm (\mathbf{1}, 0), \qquad (\mathsf{N}, \mathsf{M}) \circ (\mathsf{n}, \mathsf{m}) = \pm (\mathsf{N}\mathsf{n} - \mathsf{M}\mathsf{m}, \mathsf{N}\mathsf{m} + \mathsf{M}\mathsf{n})$$

with a single symbol ±. It is a simple matter to see that the choice of sign + or − leads to the models that are isomorphic in regard to which of representatives (+1, 0) or (−1, 0) should be assigned for the identical replication I. By virtue of (61), it does not matter, and we declare

$$\mathbb{1} := (1, 0), \qquad (\mathbb{N}, \mathbb{M}) \circ (\mathbb{n}, \mathbb{m}) = (\mathbb{N}\mathbb{n} - \mathbb{M}\mathbb{m}, \mathbb{N}\mathbb{m} + \mathbb{M}\mathbb{n})\ . \tag{82}$$

This is nothing more nor less than the canonical multiplication of complex numbers n + i · m = a ∈ C, if the following identifications are performed:

$$(1,0)\rightleftharpoons \mathbb{1}, \qquad (0,1)\rightleftharpoons \text{i}, \qquad \{\circ,\circ\} \rightleftharpoons \{+,\cdot\}, \qquad (\mathfrak{n},\mathfrak{m}) \rightleftharpoons \left(\mathfrak{n}+\mathrm{i}\cdot\mathfrak{m}\right). \tag{83}$$

Notice that the known fully matrix (over R) equivalent to (82)

$$(\mathfrak{n} + \mathfrak{i} \cdot \mathfrak{m}) \mapsto (\mathfrak{n}, \mathfrak{m}) \mapsto \begin{pmatrix} \mathfrak{n} \\ \mathfrak{m} \end{pmatrix} \mapsto \begin{pmatrix} \mathfrak{n} - \mathfrak{m} \\ \mathfrak{m} \end{pmatrix}, \qquad \begin{pmatrix} \mathfrak{n}' - \mathfrak{m}' \\ \mathfrak{m}' \end{pmatrix} = \begin{pmatrix} \mathfrak{N} - \mathfrak{M} \\ \mathfrak{N} & \mathfrak{N} \end{pmatrix} \circ \begin{pmatrix} \mathfrak{n} - \mathfrak{m} \\ \mathfrak{m} \end{pmatrix}$$

does directly reflect the above ascertained operator essence

$$(\widehat{\vec{n'}\vec{m'}}) = (\widehat{\vec{N}\_{\prime}\vec{M}}) \circ (\widehat{\vec{n'}\vec{m}})$$

of both the number multiplication \* and the C-number itself.

In view of the paramount importance of the C-number field in QT [96,138,142], let us provide additional substantiations to the rigidity of the emergence of this specific number structure, i.e., of the axiom collection (57), (79)–(82). Among other things, the transpositions Ψ Φ used above fit more general reasoning.

#### *7.5. Involutions and* C˜ ∗*-Algebra*

Apart from a freedom in ordering the primitives Ψ Φ in brace {n Ψ , m Φ }*<sup>α</sup>*, there is one more arbitrariness: reappointing them ( Ψ Θ, . . . ) as elements of the set T. However, no physics predetermines any of these degrees of freedom. For, if other ingoing T-elements Θ Ω were present in (32) instead of Ψ, Φ, then the theory of semigroup G, strictly, should be declared the segregated theories GΨΦ, GΘΩ, etc. It is clear that the labeling the theories, or a family thereof, is a manifest absurdity, and they should be thus factorized with respect to all kinds of ways to label them by T-primitives. The liberation from the Ψ, Φ-icons and reconciliation of the result with pt. **R+** (p. 25) are then performed by the scheme ,primitive has changed- ,a number character is changing-.

Inasmuch as declaring the {<sup>Ψ</sup>, Φ, Θ, ...} to be ongoing primitives in (32) is a replacement of one to another, any such an appointment boils down to permutations of no more than *pairs* with two types (inner/outer):

$$
\hat{\mathfrak{T}}^{\boldsymbol{\alpha}\circ} \colon \quad (\overline{\mathbb{K}}^{\boldsymbol{\alpha}} \overline{\mathbb{B}}) \xrightarrow[\mathbb{L}^{\boldsymbol{\Delta}}]{} (\overline{\mathbb{B}}^{\boldsymbol{\alpha}} \overline{\mathbb{K}}) , \qquad \qquad \hat{\mathfrak{Y}}^{\boldsymbol{\alpha}} \colon \quad (\overline{\mathbb{K}}^{\boldsymbol{\alpha}} \overline{\mathbb{B}}) \xrightarrow[\mathbb{L}^{\boldsymbol{\Delta}}]{} (\overline{\mathbb{K}}^{\boldsymbol{\alpha}} \overline{\mathbb{B}}) \ \mathrm{.} \tag{84}$$

However, it is immediately obvious that these reappointments change nothing in the ∪-relationships between (32) and are defined by the structural relations ג 2 ΨΦ = I, *<sup>ℵ</sup>*<sup>2</sup>ΦΘ = I. Then, the need to indicate the primitives themselves, as required, is eliminated, and their symbols may be thrown away if semigroup G is properly furnished with the two abstract involutions and ג *ℵ*. The G itself, of course, also possesses involution (61) that turns it into the group H, but this involution has already had a numerical representation (66) by signs ±. To be precise, it suffices to identify here the term "numerica" with the group arithmetic of the ⊕-addition (57) coming from the superposition principle realized on pairs (65) and (66). Therefore, the operators' actions (84) should be carried over onto objects defined in precisely this manner; nothing more needs to be assumed.

Operator גΨΦ is immediately translated into a numerical form independently of the property that the objects {n Ψ , m Φ }*α* form a (semi)group. Indeed, since the swap Ψ Φ in the unordered pair

$$\begin{array}{ccc} \mathfrak{T}\_{\mathsf{r}\diamond} : & \{\mathfrak{h}, \mathfrak{m}\} \longrightarrow \{\mathfrak{h}, \mathfrak{m}\} = \{\mathfrak{m}, \mathfrak{h}\} & \cdots \end{array}$$

(the *α*-label is dropped here as superfluous) is indistinguishable from the permutation of numbers n m, the symbols Ψ and Φ may be thrown away, organizing the numbers themselves into ordered pairs

$$
\begin{array}{ccc}
\begin{array}{c}
\begin{array}{c}
\begin{array}{c}
\end{array}
\end{array}
\end{array}
\end{array}
\begin{array}{c}
\begin{array}{c}
\begin{array}{c}
\begin{array}{c}
\end{array}
\end{array}
\end{array}
\begin{array}{c}
\begin{array}{c}
\begin{array}{c}
\end{array}
\end{array}
\end{array}
\begin{array}{c}
\begin{array}{c}
\end{array}
\end{array}
\begin{array}{c}
\begin{array}{c}
\end{array}
\end{array}
\begin{array}{c}
\begin{array}{c}
\end{array}
\end{array}
\begin{array}{c}
\begin{array}{c}
\end{array}
\end{array}
\dots
\begin{array}{c}
\begin{array}{c}
\begin{array}{c}
\end{array}
\end{array}
\begin{array}{c}
\end{array}
\end{array}
\dots
$$

When required, the *α*-symbol returns hereinafter. Let us now proceed to the outer involution Φ Θ in (84):

$$\begin{array}{rcl} \mathfrak{R}\_{\lnot\flat} : & \{\mathfrak{h}, \mathfrak{m}\} \longrightarrow \{\mathfrak{h}, \mathfrak{m}\} \dots \\ \end{array}$$

It is indifferent to the (first) Ψ-element of the pair, and, by extracting it by the rule

$$\{\stackrel{\mathfrak{a}}{\mathfrak{m}}, \stackrel{\mathfrak{a}}{\mathfrak{m}}\} = \{\stackrel{\mathfrak{a}}{\mathfrak{a}}, \stackrel{\mathfrak{a}}{\mathfrak{b}}\} + \{\stackrel{\mathfrak{a}}{\mathfrak{b}}, \stackrel{\mathfrak{a}}{\mathfrak{m}}\}$$

,

the question boils down to finding a representation of the transformations

$$(\{\emptyset,\emptyset\} \mp \{\emptyset,\not m\}) \rightarrow (\{\emptyset,\not m\} \mp \{\emptyset,\not m\})$$

The component {n Ψ , 0 Φ } must go into itself since the symbol Ψ attached to it has not changed. It means that n = n, and one is left with the task

$$\{\emptyset, \stackrel{\mathfrak{a}}{m}\} \stackrel{\stackrel{\mathfrak{r}}{\rightrightarrows} \{\emptyset, \stackrel{\mathfrak{a}}{m'}\} $$

.

However, operation *ℵ*ΦΘ recognizes only the primitive's symbols rather than their numbers. That is, replications *m* {0 Ψ , ±1 Φ } = {0 Ψ , ±m Φ } do formally commute with *ℵ*ΦΘ. Hence, by omitting the letters {<sup>Ψ</sup>, Φ, <sup>Θ</sup>}, it will suffice to look for the representation of *ℵ* by numerical pairs (0, <sup>±</sup>m) factorized with respect to replications *m* , i.e., by the set {(0, <sup>1</sup>),(0, <sup>−</sup><sup>1</sup>)}. It, for its part, remains to be transformed into itself, and the replication operators *n*, *m* will recreate the generic case. The identical transformation (0, ±1) (0, ±1) is ruled out since *ℵ*ΦΘ = I; therefore, (0, ±1) *<sup>ℵ</sup>* (0, <sup>∓</sup><sup>1</sup>). Restoring all the symbols that were dropped, the effect of *ℵ* reduces to the sign change for the second element of the coordinate pair:

$$(\mathfrak{n}, \mathfrak{m}) \stackrel{\mathfrak{N}}{\longrightarrow} (\mathfrak{n}, -\mathfrak{m})\,. \tag{85}$$

There is no need to change sign for the first element, as this change is the operator − I ◦ *ℵ*. Furthermore, one observes that the already existing group inversion − I coincides with composition

$$(\hat{\mathfrak{K}} \circ \hat{\mathfrak{I}})^2 = -\hat{\mathbb{I}}\,,\tag{86}$$

( and we may even "forget" about (the "old") subtraction, leaving the equipment

$$\{\circ,\mathbb{L},\bar{m},\bar{\mathfrak{K}},\bar{\mathfrak{I}}\}\tag{87}$$

of semigroup G as an irreducible set of mathematical structures over it.

In this connection, ye<sup>t</sup> another—more formal—motivation of the passage ,semigroup group- and thus of the superposition principle does arise. Indeed, the derivation of *ℵ* above engaged the inversion (61), but the reappointment of primitives Φ Θ in (84) is a fully independent act. Therefore, if we forget about "(−)-copies of the positive pairs" (0, <sup>m</sup>), the involutory nature of automorphism *ℵ*ΦΘ would still reproduce the semigroup G in numbers by "duplication" m ±<sup>m</sup>, i.e., create the negative pairs (0, <sup>−</sup><sup>m</sup>), thus turning G into a group H. An analogous reasoning on the symbol "−" could be cited even earlier, when the C-field was being derived.

Now, remembering the above-described move to the binarity of \*-multiplication on the (<sup>n</sup>, m)-pairs, we arrive at the problem of matching it with structures (87). Clearly, one needs only to ascertain the functionality of operators and ג *ℵ* that were not available yet. 

Relation (86) immediately gives us the correspondence *ℵ* ◦ ג i since i2 = −1. Hence, one of these operators, say ג, manifests itself in the imaginary unit i. The origin of this operator—permutation גΨΦ in (84)—is the very same permutation Ψ Φ that generated the i-object in algebra (82) and (83). The second operator, i.e., (85), as is directly seen, is also not related to the binary ⊕ and \* but determines the change i −i. This means that the QM-consideration does not just give birth to the field C but to a division C ˜ ∗-algebra, which is equipped with two *non-binary* operations

$$\mathfrak{a} \stackrel{\mathfrak{s}}{\longrightarrow} \mathfrak{a}^\*, \qquad \mathfrak{a} \stackrel{\mathfrak{s}}{\longrightarrow} \mathfrak{a} \dots$$

Informally, it defines all the basic actions as "complex quantities" and thereby determines a QM-extension/generalization to the intuitive and habitual arithmetical manipulations (68)–(74) with real things. Consequently, the four binary arithmetical operations— addition/subtraction/multiplication/division—should be supplemented with the two unary ones: conjugation *ℵ* and swap .ג

**Remark 14.** *A curious observation for formal complex-number mathematics is appropriate here. None of these operators boil down to involution* − I*. We mean that each of the pairs* ( *ℵ*, − I) *or*

( ג, −<sup>I</sup>) *is expressible through* ( ג, *ℵ*) *and not the reverse; see* (86)*. To put it plainly, the self-suggested going from the natural sign-change (i.e.,* − 1 *over* R*) to the inversion of the two-dimensional* ⊕*- addition (i.e.,* − I *over* C*) deprives the involution* − I *of its primary character, as it has taken in the one-dimensional domain* R*. Furthermore, the second operation* ג *is, in a sense, more "primitive" than the complex conjugation ℵ, as this operation has had to conduct it with a formal pair* (n, <sup>m</sup>)*— merely transposes it—and does not invoke an arithmetic action, as does ℵ when changing the sign* m %→ −m *in* (85)*.*

*The relationship between the operators is by binary multiplication:* a˜ = i \* a<sup>∗</sup>*. By virtue of this relation, it makes no odds which one of these unary operators is left for* C*-algebra.*

We note—and this is important [6]—that the observational statistics f*j* are unchanged upon both operations and ג *ℵ*.

#### *7.6. Naturalness of* C*-Numbers*

Thus, the T-set primitives have been entirely banished from the theory, with the exception of the eigen-state *<sup>α</sup>s*-markers, which are needed only for distinguishability (Section 2.1) in A -observations. These markers may be interchanged, but permutability *αj αk* is already reflected by the superposition's commutativity. Taking now into account the fact that reassigning the *α*-labels does not touch on the concept of the number, one infers: the covariance attained above is exhaustive. As a result, we draw the following conclusion.

•The coordinate representatives {<sup>a</sup>, b,c, ...} of states and superpositions thereof (58) form a complex number field C ˜ ∗ equipped with the structures of conjugation and swap:

$$(\mathfrak{n} + \mathrm{im}) \stackrel{\mathfrak{\*}}{\mapsto} (\mathfrak{n} - \mathrm{im}), \qquad (\mathfrak{n} + \mathrm{im}) \stackrel{\curvearrowleft}{\mapsto} (\mathfrak{m} + \mathrm{in})\ . \tag{88}$$

Statistical weights f*j* in object (35) are invariant with respect to both the involutions <sup>f</sup>*j*(a<sup>∗</sup>) = <sup>f</sup>*j*(a) = <sup>f</sup>*j*(a˜) for each component a*s* independently.

What is more, the commentary on the primacy of QM over the abstract arithmetic (see p. 46) has a logical continuation.

• Quantum-theoretic description invokes no C-numbers, nor does it introduce them. It *does create* them together with the C ˜ ∗-algebra. The C-numbers are in and of themselves the *quantum* numbers.

This fact is remarkable in its own right because the "two-dimensional" numbers arise at the lowest empirical level, not from the need for solving any mathematical problems. Mathematics is still lacking. Therefore, pt. R<sup>+</sup> (p. 25) could have even been weakened by replacing ,homomorphism onto numbers-, roughly, with the ,homomorphism onto continuum-. Our minimal points of departure are replications and the ingoing/outgoing structure of brace (32). The imaginary part of the complex number—as a supplement to the real one—comes, as a rough guide, from the left-hand side of the conception Ψ *α*. The theory does not depend on the meanings that will be later attached to the physical concepts—observables, measurement, spectra, means, etc.—their interpretations, or rigorous definitions. At the same time, the interferential "effects of subtraction and of zeroes" are intrinsically present within the construct's foundation itself.

Let us add, in conclusion, two more formal vindications of rigidity of the emerging C-structure. In doing so, one assumes that we have already had the R-numbers.

Unitary brace contain pairs of the form {n Ψ , 0 Φ }. The binary operations {⊕, \*} on their numerical representatives (<sup>n</sup>, 0) are closed and, as easily seen, form a commutative field, which is isomorphic to R. It is a subset of the generic pair set (<sup>n</sup>, <sup>m</sup>). From the operator nature of \*, it follows that these pairs form a certain distributive ring with general-group properties (79) and (80). The presence of the field R contained in it tells us that these pairs can be realized by the elements n + m*x* of, at most, associative algebra *A* over R. Here, n, m ∈ R, *x* is a generator of any ring's element beyond R, and the habitual + replaces the sign ⊕. The multiplication of two such elements

$$\mathbf{x} \cdot (\mathbf{n} + \mathbf{m}\mathbf{x}) \circ (\mathbf{n}' + \mathbf{m}'\mathbf{x}) = \mathbf{m}' + (\mathbf{m}\mathbf{n}' + \mathbf{n}\mathbf{m}')\mathbf{x} + \mathbf{m}\mathbf{m}'\mathbf{x}^2 = \dots = \mathbf{x}'$$

immediately shows that the result does not depend on the order of factors, i.e.,

$$\cdot \cdot = (\mathfrak{n}' + \mathfrak{m}'\mathfrak{x}) \circ (\mathfrak{n} + \mathfrak{m}\mathfrak{x})\ ,$$

due to the permutability of {<sup>n</sup>, m, <sup>n</sup>, m} between each other and of any *x* with itself. This is a direct consequence of the two-dimensionality of the algebra *A*; it must be commutative. Invoking now the well-known Frobenius theorem on associative and commutative structures containing the field R [181], we arrive once again at a multiplication of the form (82). Körner puts this point as follows: "The complex numbers constitute the largest system of objects that most people are content to call numbers" ([172], p. 230).

#### Topologies on Numbers

Yet another reasoning about exclusivity of C-numbers follows from matching the topological and algebraic properties of the general number systems ([182], Section 27). The case in hand is the uniqueness and non-arbitrariness in the emergence of the topological field C; Pontryagin (1932). In our case, we have two continuums: the numerical symbols n and m, each of which, by the very method of constructing the ⎧⎩Ξ⎫⎭-objects (34), is equipped only with the natural ordering <. Since we do not have any more math-structures yet, the topology, continuity, and limits on each of the continuums can already be introduced with respect to this relation. For example, there is no need to introduce the topology a priori by creating the arithmetical operation of multiplication/divisibility of rationals (and a concept of the prime integer), as is conducted in the *p*-adic approaches to QM [66,183,184]. The "non-naturalness" of multiplication as compared with addition was already noted above. Moreover, in the *p*-adic versions for a numerical domain, the topologically and physically required matching between the natural ordering, connection, and continuity ([182], Ch. 4) is destroyed, and the approaches themselves stipulate the existence of the *observations* numbers with a comprehensive arithmetic. At the same time, questions about the "structure" of the physical *x*-space at Planck's scale and about measurements by rationals (see motivation in [183,184]) have not ye<sup>t</sup> emerged because we are not relying on physical conceptions and are not ye<sup>t</sup> introducing these notions as numerical. The *x*-space itself is as of ye<sup>t</sup> absent, and D. Mermin [3] overtly claims along these lines that "when I hear that spacetime becomes a foam at the Planck scale, I do not reach for my gun". From the low-level empiricism standpoint, any objects, apart from the R2-continuality and frequencies f, call for independent axioms. In turn, the primary nature of the R-continuality itself follows directly from the boolean 2<sup>T</sup> (p. 19) and <sup>Σ</sup>-postulate of infinity.

#### **8. State Space**

Quantum states . . . cannot be "found out"—W. Zurek ([8], p. 428)

. . . quantum theory refuses to offer any picture of what is actually going on out there—D. Mermin (1988)

#### *8.1. Linear Vector Space*

Once the replication (*N* , *M*) of brace {nΨ, mΦ }*α* has acquired a binary character

$$(\widehat{\mathcal{N},\mathcal{M}})\left(\{\widehat{\mathfrak{n}},\widehat{\mathfrak{m}}\}\underline{\mathfrak{a}}\right) \quad \Longleftrightarrow \quad \left((\mathbb{N},\mathbb{M})\circ(\mathfrak{n},\mathfrak{m})\right)|\mathfrak{a}\rangle = \mathfrak{a}|\mathfrak{a}\rangle \,. \tag{89}$$

the difference between "what is replicated" and "how many times" disappears. A symbol |*α*⎫⎭ of the eigen-state has been attached to the abstract C-number a. Construing this point as a quantum analog of re-choosing (liberation of) the measurement units (p. 46), we

obtain that the two formal states a|*α*⎫⎭ and <sup>b</sup>|*α*⎫⎭ are always connected by a certain number operator p:

$$\mathfrak{b}\left|\mathfrak{a}\right\rangle = \hat{\mathfrak{p}}\left(\mathfrak{a}\left|\mathfrak{a}\right\rangle\right), \qquad \hat{\mathfrak{p}} \rightleftharpoons \mathfrak{b} \circ \mathfrak{a}^{-1}$$

Manipulating the numbers becomes independent of symbols |*α*⎫⎭. The way to formalize this is to think of generic states a|Ψ⎫⎭ ∈ H as the "solid characters"

<sup>a</sup>|Ψ⎫⎭ |**Ξ** ∈ H , (90)

.

i.e., as the |**Ξ** -elements of a new set H, which is equipped with the p-replication images represented by the p-family (p ∈ C) of maps

$$\mathsf{H} \times \mathsf{H} \xleftarrow{\star} \mathsf{H} : \qquad \mathsf{p} \cdot |\mathsf{E}\rangle = |\mathsf{Φ}\rangle \in \mathsf{H} \,\,\,\,\tag{91}$$

and which is obliged to inherit the structure (89). This inheritance says that the coordination of \*-multiplication in (89) with the replication's p-realization is performed by a new operation *·* of the unary kind on H, i.e., (91), which should be subordinated to the rule

$$\mathfrak{p}\cdot(\mathfrak{a}\cdot|\Psi\rangle)=(\mathfrak{p}\circ\mathfrak{a})\cdot|\Psi\rangle \qquad (\mathfrak{p},\mathfrak{a}\in\mathbb{C},\quad|\Psi\rangle\in\mathbb{H})\,.\tag{92}$$

Due to this connection between operations \* and *·*, the latter is usually referred to as "multiplication" as well; however, such an intuition with dropping the word "unary" may have implications ([130], Section 6.2). An analogous rule had already occurred in the relationship (59) between the ⊕-number C-structure and the ( +)-group superposition, i.e., when the multiplicative structures {\*, *·*} were not available yet.

Among replication operators p, there exists an identical transformation

$$
\mathfrak{h} = \mathfrak{I} : \qquad \mathfrak{a}|\Psi\rangle \stackrel{\mathfrak{l}}{\longrightarrow} \mathfrak{a}|\Psi\rangle\ ,
$$

to which a symbol of the numerical unity p = 1 corresponds. From this, in accordance with (90) and (91), there follows the rule

$$|1 \cdot |\Xi\rangle = |\Xi\rangle' \qquad \forall |\Xi\rangle \in \mathbb{H} \dots$$

It is clear that the (*·*)-multiplication needs to be agreed with the 0-union. Let us make use of the fact that an object of (quantum) replication may be not only the unitary brace {nΨ, mΦ }*<sup>α</sup>*, which is equivalent to the eigen-element <sup>a</sup>|*α*⎫⎭, but a (+ )-sum of the like objects and, in general, any constituents of quantum ensembles (see p. 44). Therefore, the p-replication

$$
\mathfrak{b}\left(\mathfrak{a}|\mathfrak{a}\rangle \oplus \mathfrak{b}|\mathfrak{b}\rangle\right) = \cdots \tag{93}
$$

is known to have its twin-sum

$$\cdots \cdots = \mathfrak{a}'|\mathfrak{a}\rangle \oplus \mathfrak{b}'|\mathfrak{b}\rangle = \cdots \tag{94}$$

with certain coefficients <sup>a</sup>, b.

Let us, for the moment, give back (93) to the initial language of operators/brace according to the scheme

$$\begin{array}{cc} \{\overbrace{\underline{\mathfrak{h}}', \overbrace{\underline{\mathfrak{h}}'}}^{\mathfrak{g}}\}, & \{\overbrace{\underline{\mathfrak{h}}', \overbrace{\underline{\mathfrak{g}}'}}^{\mathfrak{g}}\} \mathfrak{g} + \{\overbrace{\underline{\mathfrak{h}}', \overbrace{\underline{\mathfrak{g}}'}}^{\mathfrak{g}}\} \mathfrak{F} \end{array} . \tag{95}$$

Take into account a pre-image of operation + on objects (93), i.e., the +. Then, (95) and the content of Sections 7.4 and 7.5 certainly show that the expression (94) must be of the form

$$\cdots \cdot = (\mathfrak{p} \circ \mathfrak{a}) | \mathfrak{a} \rangle \oplus (\mathfrak{p} \circ \mathfrak{b}) | \mathfrak{b} \rangle = \cdots \cdot |$$

Reconverting, by (92), expressions such as (p \* a)|*α*⎫⎭ into the operatorial <sup>p</sup>(a|*α*⎫⎭), we complete the ellipsis

$$\cdots \cdot = \mathfrak{p}\left(\mathfrak{a}\left|\mathfrak{a}\right\rangle\right) \oplus \mathfrak{p}\left(\mathfrak{b}\left|\mathcal{S}\right\rangle\right) \dots$$

Passing now to the p-number and to the |**Ξ** -objects (90), i.e., replacing a|*α*⎫⎭ |**Ψ** and <sup>b</sup>|*β*⎫⎭ |**Φ** , one derives an additivity of operation *·* when acting on a sum:

$$\mathfrak{p} \cdot (|\Psi\rangle \hat{+} |\Phi\rangle) = \mathfrak{p} \cdot |\Psi\rangle \hat{+} \mathfrak{p} \cdot |\Phi\rangle \dots$$

Here, the H-addition + has been carried over to the group H as a new symbol +ˆ . This is nothing but a distributive coordination of the (*·*)-multiplication with the group addition +ˆ .

In a similar way, through a number operator, one establishes ye<sup>t</sup> another relation

$$(\Xi \mid \cdot (\mathfrak{a} \circ \mathfrak{b}) = \langle \Xi \mid \cdot \mathfrak{d} \downarrow \langle \Xi \rangle \cdot \mathfrak{b})$$

between *·* and +ˆ . Its origin is equivalent to (59). From the constructs above, it is not difficult to see that we have examined all the possibilities of C-replicating the superpositions (58) or their constituents, which is why we have exhausted all the compatibility rules that stem from the two fundamental operations—replication and union ( +).

Thus, having considered the passage (90) and (91) as a final homomorphism of the H-group elements a|Ψ⎫⎭ onto the objects |**Ξ** ∈ H, i.e., adjusting the previous concept of a state and of DataSource (p. 31), we infer the following.

• The minimal and mathematically invariant bearer of the observation's empiricism is an abstract space H of states |**Ψ** of the system S. The structural properties

H := |**Ψ** , |**Φ** ,... ,commutative group under operation +ˆ - , (96) C := {<sup>a</sup>, <sup>b</sup>,...} ,field of complex numbers (57), (78)–(82)- , a |**Ψ** =: a *·* |**Ψ** ∈ H ,closedness under operation *·* ⇐⇒ operator automorphism a |**Ψ** - , (97)

$$\begin{aligned} \mathbf{a} \cdot (\mathfrak{b} \cdot |\Psi\rangle) &= (\mathfrak{a} \circ \mathfrak{b}) \cdot |\Psi\rangle, & \mathbf{a} \cdot |\Psi\rangle \mkern-1.i & \mathfrak{b} \cdot |\Psi\rangle \mkern-1.i & \mathfrak{b} \cdot |\Psi\rangle \mkern-1.i & \mathfrak{b} \cdot |\Psi\rangle \mkern-1.i & \mathfrak{b} \cdot |\Psi\rangle \mkern-1.i \\ \mathbb{1} \cdot |\Psi\rangle &= |\Psi\rangle, & \mathfrak{a} \cdot (|\Psi\rangle \mkern-1.i & \mathfrak{a} \cdot |\Psi\rangle) &= \mathfrak{a} \cdot |\Psi\rangle \mkern-1.i & \mathfrak{b} \Phi\rangle \end{aligned} \tag{98}$$

of the space coincide with the axioms of a linear vector space (LVS) over the field C.

Attention is drawn to the fact that this is the first place in our construct where the word "*linear*" has appeared, and even the superposition principle, page 35, was formulated without using this term. The "axiom list" (96)–(98) should also be complemented with a declaration of the global D-number value (53) established above.

In a nutshell, the nature of the quantum state space is two-fold: group superposition (58) and operator nature of the "a-numbering" the elements of the group. It admits the C-field scalars as operators. Relations (98) describe the rules of "interplay" between all the objects. It is known that such formations, while being implemented by a binary algebra of numbers, turn into the vector spaces and modules [175] (Ch. 5), [181] (Sections I(7.1–2), II(13.4)). Concerning the consistency of these rules—say, of numerical distributivity (80)— with relations (98), see the work [185].

**Remark 15.** *A certain oddity is in place. QM-empiricism is such that the standard definition of LVS by the all-too-familiar axioms* (96)*–*(98) *is more "*non*physical" by its nature than the "generalistic" abstraction of a group with operator automorphisms of the group* H*-structure itself [166] (Section I(4.2)), [175]. A point like this might be expected, though. This is because, as noted in Section 1.2, meaning all of the tokens in* (1) *and their origin are entirely unknown, and the linearity of QM is radically different from other "linearities" in physics.*

*All told, the appearance sequence of the mathematical structures is as follows:* ,*sets, union* ∪*,...* - ,*semigroup (Section 5.2)*-

> ,*group* H *(Section 6.3)*- ,*numbers & arithmetics (Section 7)*-

> > .


*This sequence is rigid, such as the box-diagram in Section 1.3, so the structure of LVS cannot be weakened because we have the two fundamental principia (*II *and* III*) in between the semigroup, the group, and the vector space.*

#### *8.2. Bases, Countability, and Infinities*

From a ban on transitions *αs αn* under *s* = *n*, it follows that unitary *<sup>α</sup>s*-brace (34) corresponds to vectors a*s ·* |*<sup>α</sup>s* that are linearly inexpressible through each other. Aside from the general ensemble brace (32), no other elements exist, and all of them are in oneto-one correspondence with the vector representations a1 *·* |*<sup>α</sup>*1 +ˆ a2 *·* |*<sup>α</sup>*2 +ˆ ···. Each such vector has a statistical pre-image (32), and vice versa; there are no gaps. This means that the system of vectors {|*<sup>α</sup>*1 , |*<sup>α</sup>*2 , ...} forms a basis of H as the basis of LVS—*basis of an observable* A *or* A *-basis*—and the number of symbols |*<sup>α</sup>s* is its dimension: dim H = D. The D = ∞ case, just like anything associated with infinity, cannot be formalized without topology, and its presence is presumed, but this discussion is dropped. We just remark that even earlier, when arising the two-dimensional continuum, we have silently assumed the (R × <sup>R</sup>)-product topology on it. This supposition is natural, inasmuch as it does not involve additional constructions/requirements. Thus, if Properties (96)–(98) are directly accepted as empirical, then the mathematical rigors augmen<sup>t</sup> them axiomatically on the outside because one constructs the mathematical theory.

The micro-transition <sup>A</sup> in Section 2.1 is a solitary entity. This means that the number of eigen *<sup>α</sup>s*-primitives for an actual instrument may be either finite or discretely unbounded. We base this on the fact that continual formations are products of mathematics rather than empiricism (see also [58] (p. 35)). The T-set, as an example, is also non-continual, but that premise may even be given up because only a discrete portion of this set is present at arguments (transitions <sup>A</sup> ). Notice incidentally that continuum, along with the number, does not feature in the ZF-axioms [134] but is also created, just as "an infinity is actually not given to us at all, but is . . . extrapolated through an intellectual process." [105] (p. 55; Hilbert–Bernays); see also the book [84] for the conceptualization of infinity. One obtains a countability of the A -basis. Hence, it follows a completeness of H and countability of dimension (53), as of the number LVS-invariant:

$$\mathbb{D} = \mathbb{2}, \mathbb{3}, \dots, \mathbb{N}\_{\mathfrak{a}} \qquad \left( = \dim \mathbb{H} \right) \,. \tag{99}$$

Finally, let us mention the following. The basis is a term that in no way is present in the abstract axiomatics (96)–(98), and LVS on its own account does not contain a motive for introducing that concept. However, empirically, the H-space is arising entirely and ab initio in all possible linear combinations over |*<sup>α</sup>j* , i.e., through A -bases. Because of this, in order for an abstract LVS to become the quantum state space, the LVS should be considered as being accompanied by the concepts bases and changes thereof. Conforming to such a requirement and the formal existence of a basis is given by a nontrivial math theorem invoking the axiom of choice ([166], Section **II**(7.1)).

#### *8.3. The Theorem*

The states |**Ψ** and sums thereof, at the moment, form a formal family of different elements. Recall that symbols {≈, ≈} in pt. R, as from the end of Section 5.4, have been replaced with the standard ones {<sup>=</sup>, <sup>=</sup>}. The physical aspects of S, **<sup>M</sup>**,... were being left aside so far, and, for example, |**Ψ** and c *·* |**Ψ** were the different vectors of the H-space. However,

• empiricism (deals with and) yields originally *not states* and superpositions thereof *but* |*α -representations*.

It is these representations (alone) that carry information about statistics (<sup>f</sup>1, f2, ...) through coefficients <sup>a</sup>*j*. The replicative character of c-multipliers and <sup>Σ</sup>-postulate entail, however, that the two vectors 1 *·* |**Ψ** and c *·* |**Ψ** should correspond to the one and same statistics <sup>f</sup>(D) = (1, 0, . . .) = <sup>f</sup>˜(D) under an observation D with the eigen collection {1|Ψ<sup>⎫</sup> <sup>⎭</sup>,...}.

Let us write the equalities

$$\begin{array}{rclcrcl}\mathbf{f}\_{(\boldsymbol{\upbeta})} & \rightsquigarrow & & \underbrace{\mathbf{1}\cdot\bigtriaparrow\mathbf{Y}}\_{\operatorname{cbvarron}\operatorname{co}\boldsymbol{\upbeta}} &=& \underbrace{\mathbf{a}\_{1}\cdot\bigtriaparrow\mathbf{a}\_{1}\bigtriap}\_{\operatorname{cbvarron}\operatorname{ad}\boldsymbol{\upbeta}}\underbrace{\cdots\bigtriap}\_{\operatorname{cbvarron}\operatorname{ad}\boldsymbol{\upbeta}}\end{array} \quad\rightsquigarrow & \begin{array}{rclcrcl}\mathbf{a}\_{(\boldsymbol{\upalpha})}\wedge\mathbf{b}\_{(\boldsymbol{\upalpha})}\wedge\cdots\wedge\mathbf{b}\_{(\boldsymbol{\upalpha})} & & \left(\mathbf{1}\otimes\mathbf{b}\right) \\ & & \mathbf{c}\cdot\mathbf{b}\cdot\mathbf{c}\left(\mathbf{a}\_{1}\cdot\left|\mathbf{a}\_{1}\right.\right.\right)\left.\left.\mathbf{a}\_{2}\cdot\left|\mathbf{a}\_{2}\right.\right.\right)\left.\left.\cdots\left.\right.\right|} \quad\rightarrow & \mathbf{f}\_{(\boldsymbol{\upalpha})} \\ \end{array} \tag{100}$$

and look at them in the following order: the first line from right to left and the second in the reverse direction. Their right-hand sides are the carriers of some statistics <sup>f</sup>(A ) and <sup>f</sup>˜(A ). The frequencies <sup>f</sup>(A ) = (<sup>f</sup>1, f2, ...) come from the number set (<sup>a</sup>1, a2, ...) under the same environments S, **<sup>M</sup>**,... that give rise to the statistics <sup>f</sup>(D). However, it is also generated by the representative c *·* |**Ψ** , which is associated with the same S, **<sup>M</sup>**,... ; hence, <sup>f</sup>˜(D) = <sup>f</sup>(D). By virtue of the second equal sign in (100), the same S, **<sup>M</sup>**,... are associated with the second A -collection (c \* a1,c \* a2, ...). Therefore, the frequencies <sup>f</sup>˜(A ) that emanate from it have to be identical to those emanating from the first collection (<sup>a</sup>1, a2, ...). That is to say <sup>f</sup>˜(A ) = <sup>f</sup>(A ), and the scale stretches |**Ψ** c *·* |**Ψ** are not recognized by any A -instruments. A more concise reasoning is that the quantum replication c = n + im may be viewed as onedimensional replications *n* (a|*α* ⎫ ⎭), ı (a|*α* ⎫ ⎭) of all the brace <sup>a</sup>*s*|*<sup>α</sup>s* ⎫ ⎭-images and of sums such as *n* (a|*α* ⎫ ⎭) + (ı ◦ *m* )(a|*α* ⎫ ⎭). These replications do not change the superposition statistics as a whole.

The aforesaid gives birth to a universal—stronger than ≈ and irrespective of instruments—observational equivalence relation

$$|\Psi\rangle \nrightarrow \text{const} \cdot |\Psi\rangle$$

on the space H, i.e., the "physical" indistinguishability (Section 2.4).

The basis vectors |*<sup>α</sup>s* and their ( ∼∼∼)-equivalents will be referred to as *eigen vectors/states of instrument* A . Clearly, the concepts of instrument and of (macro)-observation ( O) should now be distinguished. Accordingly, the spectral constructions (51) and (52) should be corrected. Call the data set

$$\left\{ \left| \mathfrak{a}\_1 \right\rangle\_{\left| a\_1 \prime} \left| \mathfrak{a}\_2 \right\rangle\_{\left| a\_2 \prime} \dots \right\} \right. =: \left[ \left. \varkappa \right\prime \right] \tag{101}$$

the **[**A **]**-*representative* of instrument A in H. The add-on (101) does not touch on H-space since the spectral labels 1*αj* are the self-contained objects independent of vectors |*αk* . These labels and their degenerations determine internal properties of the formalized notion of an instrument (101). Conversely, any state vector |**Ψ** or c *·* |**Ψ** may be treated as a **[**C **]**- representative for an imagined/actual instrument C (spectrum is arbitrary) and is a certain (+)ˆ -sum of the eigen elements for any other **[**A **]**-representative.

Remembering (23), we arrive at the quantum "*kinematic framework*" [69], i.e., at the ultimate conclusion that determines the pre-dynamical theory of macroscopic data on micro-events.

#### •*The core first theorem of quantum empiricism:*

(1) The mathematical representatives of physical observations and of preparations are the quantum states |**Ψ** and statistical mixtures of eigen |*α* -states

$$\left\{ |\mathfrak{a}\_1\rangle^{(q\_1)}, |\mathfrak{a}\_2\rangle^{(q\_2)}, \dots \right\}, \qquad \varrho\_1 + \varrho\_2 + \dots = 1 \,\,. \tag{102}$$


The words "complete separable" have been supplemented here for mathematical reasons. This point is partly commented on in [6] and more fully in [130]. Indeed, the algebra constructed above calls for some amendments of a topological nature because the construction contains three infinities: continuum C, continuum H, and dimension D. In this connection, see the book [182]. The term "categorical" may require some explanation, and it is fully given in [130]. Here, one suffices to mention the point that one mathematical axiomatical system can in general have different inequivalent realizations/models [105,106,136]. In turn, the only thing that distinguishes two vector-space models between themselves is their dimension D.

Now, having considered the micro-destruction arrays with empirical rather than a formal take on arithmetic, the ideology of creating the quantitative theory leads to the key feature of quantum states—addition thereof—and the quantities under addition "do amount to" the complex numbers.

**Remark 16.** *As in Section 7.3, we draw attention to a hidden and (logically) unremovable extension of the physical units' concept.*

• *". . .* units*. Despite the rudimentary nature of units, they are probably the most inconsistently understood concept in all of physics . . . where do units come from?".*

> *S. Gryb and F. Mercati ([102], p. 91)*

> > .

*Surprisingly, the naïve and straightforward conjunction of this concept with an abstract number seems to contravene the multiplication arithmetic but not the addition one. The typical example illustrates the point:*

(2kg) × (3kg) = 6kg (3sheep + 5*ψ* = 8Stück)

*(see also [172] (items (4) and (5) on p. 16)). On the other hand,* 2 × (3kg) = 6kg *and* (2kg) + (3kg) = 5kg*, and the* kg *may be replaced here with any other entity: the classical meters, the abstract "Quanten Stücke ψ", and the like. They have no any operational significance, but one cannot get by without them.*

*The numeral characters acquire their usual abstract-numerical meaning—mathematization [58] —only when we throw (Section 7.3) the "units"* ,Stück*,* ◦*C,* sheep*, ψ,* ...- *out of data like* ,5Stück*,* 5 ◦*C, 5*sheep*,* 5*ψ,* ...-*. The symbol "*5*" in* 5 ◦*C is the very same "5" as in* 5 *·* |*α . It is pointless without such a matching/abstracting. In the Newtonian spirit (epigraph to Section 7), the symbol could be defined as follows:*

$$\left[ \begin{array}{c} \text{the abstraction 5} \\ \end{array} \right] := \frac{\left\| \begin{array}{c} \text{ $ \textbf{5}$ 'C} \\ \hline \text{ $ \textbf{1} \text{$  \textbf{Y} $}} \end{array} \right\|}{\left\| \begin{array}{c} \text{$  \textbf{5} $'C} \\ \hline \text{$  \textbf{1} \text{ $ \textbf{Y}$ }} \end{array}} = \frac{\left\| \begin{array}{c} \text{ $ \textbf{5} \text{ }} \text{unit} \\ \hline \text{$  \textbf{1} \text{ }} \text{unit} \end{array} \right\|}{\left\| \begin{array}{c} \text{ $ \textbf{5} \text{ }} \\ \hline \text{$  \textbf{1} \text{ }} \text{unit} \end{array} \right\|} .$$

*It may be even said that creating the number* n *(from its operator n) as an abstracted entity reflects a kind of covariance with respect to attaching the various language tags regardless of whether they are real* Stück*,* sheep*, or the abstract ones such as* |**Ψ** *. At the same time, the inversion of this abstraction—attaching the* ,Stück*,* ◦*C,* sheep*, ψ,* ...- *to the character* 5*—is always* an

interpretation of abstraction*: interpreting "the* Stück*", "*◦*C-interpretation the Celsius", etc. It is not improbable that this is the only point when the completed QT—QM/QFT/quantum-gravity yet to be constructed will resort to word interpretation. See also comments by D. Darling on "sheep, fingers, tokens, numbers, things, to "add" things, abstraction—the process of addition" and the like on p. 178 in the book [2] and in [23] (pp. 263–264).*

Incidentally, within this physical and quantum context:

• The LVS itself should be regarded as no less a primary math-structure than the numbers themselves. Empiricism gives birth to both these structures together. Neither of them is more/less abstract/necessary than the other. Behind them is certainly a commutative group with operator automorphisms over it, and "numbers" is just a shortened term for that operators. Therein lies their nature (Section 7.2).

The habitual physics' construct ,number- × ,physical unit- exemplifies in effect the simplest (one-dimensional) LVS. However, the structure "the LVS", in contrast to the "bare" arithmetic, simply "does not forget and keeps" an operator nature (unary multiplication *·*) of the structure "the number" and its empirical inseparability from the notion of the unit:

$$\underbrace{2\left(3\text{ unit}\right)}\_{\text{vector space}} \iff 2\cdot \left(3\cdot \text{unit}\right) \quad \longleftrightarrow \cdot \cdot \text{ abstracting}\cdot \cdot \cdot \longmapsto \underbrace{\left(2\cdot \otimes\right)\bullet\text{s}\bullet}\_{\text{arithmetic}}.$$

A direct corollary of this point is the fact that principium II can in no way be given up or disregarded. This would be tantamount to impossibility to introduce the further empirical (and classical) notion of a physical unit. The "forgetfulness" of arithmetic about measuring units even leads to a new way of looking at the classical Pythagoras theorem ([130], Section 6).

At the moment, it is worthwhile to summarize where we stand. As we have seen, nothing above and beyond what was used in constructing the mathematics (96)–(99) is required to explain the nature and meaning of the quantum state. Moreover, we have obtained not merely a completion of construction (11):

$$\oplus(\mathfrak{a}\_1, |\mathfrak{a}\_1\rangle; \mathfrak{a}\_2, |\mathfrak{a}\_2\rangle; \dots) = \mathfrak{a}\_1 \cdot |\mathfrak{a}\_1\rangle \oplus \mathfrak{a}\_2 \cdot |\mathfrak{a}\_2\rangle \oplus \dots \oplus \mathfrak{a}\_n$$

In the first place, one establishes *a genesis of the quantal discreteness. Discriminating is an isolated act in the very nature of the perception process*: "one thing is distinct from another", "the controlling the minimal begins with a distincting of something the two", and the like (Section 2.2). Accordingly, "indivisibility, or "individuality", characterizing the elementary processes" ([131], p. 203; N. Bohr) must be formalized into the *"elemental" click*.

• The classical continuality of the perceptual reality—the (<sup>3</sup>+<sup>1</sup>)-space, fields {*u*(*<sup>x</sup>*, *t*), *ψ*(*<sup>x</sup>*, *t*), ...}, and the R-numbers—is a theorization act, whereas the nature of the perception fundamentally "contains an element of discontinuity" ([4], p. 179). The continuality of the classical-physics mathematics we are used to is a "quantum effect".

The theorization also bears on preparation S, **<sup>M</sup>**,... . For example, smoothly reducing the interferometer intensity is not an empiricism but an *imagination* of abstracta the continuity/infinity. Clearly, such an (incorrect) substitution of the perception process should somewhere be replaced with a "correct understanding", such as the introduction of the categories: ,isolated micro-events- + ,(myriads) assemblages thereof-. Granted, the natural language is able to describe the discontinuity only in the classical (the energy) terms—Plank's quantum of action *h*¯, although the *quantum discreteness is not a discretization of* something classical but a discreteness on its own account.

We also clarify the formalization of measurement/preparation and of genesis of the C-numbers. The well-known (∗)-conjugation operation also finds its origin. Moreover, it is supplemented with a transposition 6(a) 7(a) of the real/imaginary part of the C-number, and this transposition should be regarded just as natural operation as the conjugation. The emergen<sup>t</sup> concepts of spectra and of their degenerations and eigen-states

provide a nearly comprehensive mathematical image of physical observables. The state becomes devoid of its mysteriousness [21,31,186] since it is explicitly built in terms of the unique model of the "statistical" |*α* -representatives supplemented with macroscopic mixtures (102).

#### **9. Numbers, Minus, and Equality; Revisited**

. . . quas decet numeris negativis exprimantur, additio et subtractio consueto more peracta nullis premitur difficultatibus—L. Euler (1735)

(. . . if we represent the notions, which are necessary, by negative numbers, then addition and subtraction . . . are executed without any difficulty.)

#### *9.1. Separation of the Number Matters*

The empirical adequacy of QT can be based only on empirical ensembles (Sections 2.5 and 4). The creation of their mathematics tells us, then, that the "quantity of something" (68)–(70) turns into a formal operational algebra through labeling the operator replications (Sections 7.1 and 7.2) and properly yields the numbers. At first, they appear merely as


and then as internal objects of theory:

These steps are necessary and mean that not only are the complex numbers far from self-evident, but even the negative ones are; a key place (50), (55) wherein a group arises. All the other structural points, first and foremost the observational quantities, may be further produced (even as concepts) only by way of certain mathematical mappings:


.

In other words, if a concept is a numerical one already in empiricism—frequencies, spectra, etc.—then its meaningful formalization by means of a mathematical definitio can only resort to mathematics that we have at our disposal: LVS and algebra of numbers.

Thus, numerical quantities in the entire theory are initially divided up by their emergence mechanism (II): the intrinsic abstracta and reifications (103). Without such a division, the *circular logic is inevitable*, and the above-mentioned "unit" treatment of numbers would still be supplemented with the task of their observational interpretation complicated by two-dimensionality. This task would be present in formalism not merely as a problem but as an inherently intractable challenge. Actually, *any entity can be identified with numbers*,

and this is why, the quantum empiricism and principium II—paradigm of the very number in the physical theory—insist on the need to pay the closest possible attention to all these things.

**Remark 17.** *In this regard, the situation is parallel with the familiar history of electrodynamics of moving bodies, as was pointed out just before principium* III*. Lorentz's contraction theory is inconsistent, if the space-time tags to events are not linked up to the empirically precise and operationally defined concepts in different reference frames: clocks, simultaneity, rigid rods, distances, and the like.*

*In the quantum case, the chief subject of empirical definition is a concept of the number and of the "numerical value of . . . ". Otherwise, the meaning given to the conception of a quantitative theory itself has been blurred. The "quantum numbers"* C *are built up from the reals, and the latter have an operator nature (Section 7.2). However, the complexes* C*, being also operators and unlike the reals, never act (operationally) on the reified quantities. They do act on the abstract* |**Ψ** *-elements of the abstract commutative group* H*. Recall that this group and superposition principle were arising before the numbers.*

We have seen now that "it is quite wrong to try founding a theory on observable magnitudes[/categories] alone" [8] (p. 504; Einstein, in a talk (1926) to Heisenberg), and resorting to the physical notions—the camouflaged M-observations—is prohibited (see also Remark 2). The attempts to use statistics at the very beginning of the theory are known [29,34,87,90,93,187,188], and rightly so; they were initiated by H. Margenau (1936) [33] (Ch. 15). However, the scheme just given is rigid. To obviate the premature appearance of the very need for an interpretation, the scheme must not be varied. Being a sequence of steps, it provides in essence an answer to principium II.

#### *9.2. Operations on Numbers*

The last step in this scheme contains, in particular, the map a %→ f, i.e., measurement (48). Its form should be established in its own right—Born's rule [6]. To illustrate, the naïve transformation of negative numbers p into the actually perceived quantities by a "seemingly natural" rule such as |±p| is not correct and does not follow from anywhere. For the built algebra (96)–(98), the operation ±p %→ |±p| is extrinsic and illegal. According to ideology of Section 1.3, not only objects—numbers, vectors, quantities, characteristics, etc.—but also all the math operations should be created because one without the other is meaningless. The numerical object of the theory—the complex pair (<sup>±</sup>p, <sup>±</sup>q)—is as ye<sup>t</sup> single, and it contains a principally "non-materializable" ingredient (Section 7.4) and behaves as a whole. With regards to empiricism, the negative and C-numbers are equally "nonexistent, fictitious entities" since the state's mathematics (96)–(98) has not been supplemented with the doctrine of "empirically perceived" quantities (103). As a matter of fact, the step-by-step transformation of the binary operation ∪ to symbols 0, +, + and, finally, to operations {+<sup>ˆ</sup> , *·*, <sup>⊕</sup>, \*} does not terminate at states. Algebra (96)–(98) will be further required to create the mathematically correct calculation rules of the proper observational quantities.

The foregoing is amplified by the fact that pr. II has been involved in the classical description and in vindication/refutation of, say, the hidden variable theories. Here, numbers are identified with the reified quantities, and subtraction is taken for granted from the outset. However, the negative quantities are also being created here, and they are constructed in the same manner as the "quantum zero" for the H-group in Section 6.3.

Indeed,

•the instrument indications and physical quantities are not numbers, nor the ("pointer") states;

"detector . . . does not measure a field or an *S*-matrix-element" (R. Haag (2010)). They are no more than notches, and "negative notches" are introduced prior to mathematics of symbols according to the following subconsciously intended scheme. The self-apparent physical

conventionality, which has been called "an addition" of two such notches, must produce, in accord with the supra-mathematical requirements of physics, what is named "nought, zero". Two waves at a point, for instance, compensate each other. The result is asserted to be identical with a mathematical zero, and that is the subtraction.

The classical "explanations" are the ones that use compensations/subtractions (see Remark 13), whereas *the minus* we have become accustomed to *is a fairly abstract construction in its own right*. J. Baez and J. Dolan best reflected the situation, observing on page 37 of [85] that "half an apple is easier to understand than a negative apple!"; on the same page, a good discussion of division is given. In this respect, one might state that the very classical physics needs an interpretation in terms of strictly positive "the number of Stücke". The mathematization of empiricism into numbers is not a distinctive feature of a quantum description. However, comprehending "abstracting the minus sign" is not confined to this. A word of explanation is necessary with regard to the situation.

Mathematics formalizes [134] the positive/negative ±p into the pairs' classes (*<sup>m</sup>*, *n*) being equivalent with respect to an "adding" of the class (, ) (the "zero"):

$$\begin{aligned} (+\mathfrak{m}) &:= (\mathfrak{m}, 0) \approx (\mathfrak{m} + \ell, 0 + \ell), & (-\mathfrak{n}) &:= (0, \mathfrak{n}) \approx (0 + \ell, \mathfrak{n} + \ell) \end{aligned} \tag{104}$$

$$\begin{aligned} \pm \mathfrak{p} &\iff & (\mathfrak{m} - \mathfrak{n}) := (\mathfrak{m}, \mathfrak{n}) \approx (\mathfrak{m} + \ell, \mathfrak{n} + \ell) \end{aligned} \tag{104}$$

where *m*, *n*, are to be seen as "something strictly positive". This "adding" is ye<sup>t</sup> another tacitly assumed and much more abstract action: the addition of objects of some other kind—"positive couples" (*<sup>m</sup>*, *<sup>n</sup>*). Technically, at an appropriate place in Section 7, we had to introduce such classes and to assign their own algebraic operations for them. The result might be called the "genuine" arithmetic (of "the positives") and could be enlarged to the "complete arithmetic" with multiplication and division.

As a consequence, the single-token object (+ m) or (−<sup>n</sup>), which we perceive as selfevident (cf. pr. II), is a highly unobvious construction—the generic equivalence-class of two-token (*<sup>m</sup>*, *n*)-abstractions (104). The essence of the symbol of a negative number (−<sup>n</sup>) is revealed only when contrasting the two positive ones. Exactly the same situation has occurred when deriving the superposition principle in (49), (50), and (55).

It is clear that once all the ±p-numbers and the "normal" positive +p's among them have been formalized into the equivalences (104), the fact that they possess any "natural meanings", such as the "operation of the quantity p %→ |p|" invented above, becomes more than unnatural; the abstract class operations appear out of nowhere. Similarly with Q-numbers and their R-extension: classes of equivalent pairs (n/m) := (*<sup>n</sup>*, *m*) ≈ (*n*, *<sup>m</sup>*).

#### *9.3. Naturalness of Abstracta*

We thus infer that the rejection or disregard of the similar "naturally abstract" settheoretic models would be tantamount to the rejection of the minus sign even in elementary physics. This is absurd, but its root is a need for abstracting. On the other hand, the motivated deduction of these models cannot be replaced with (hidden) axiomatic assumptions or with ready-made math-structures. Such an ambivalence is, in our view, one reason why the problem with "decrypting" quantum postulates is so difficult; it touches on the metamathematical and metaphysical aspects of the very thinking [84,105,148,169,179]. The stream of subconsciously abstractive homomorphisms

$$\begin{aligned} \{\mathsf{let},\mathsf{S},\mathsf{t}\} &\longmapsto \{\mathsf{t},\mathsf{X}\} \longmapsto \{\mathsf{t}\} \mathrel{\mathop{\mathsf{in}}\nolimits \longmapsto \{\mathsf{t}\}} \{\mathsf{t}\} \mathrel{\mathop{\mathsf{in}}\nolimits \} \longrightarrow \cdots \longrightarrow \{\{\mathsf{t}\}\mathsf{t}\} \mathrel{\mathop{\mathsf{in}}\nolimits \longmapsto \{\mathsf{t}\}} \longrightarrow \cdots \longrightarrow \\ &\longmapsto \{\mathsf{a}\mathsf{,}|\mathsf{A}\rangle;\mathsf{T},\mathsf{\{b}\}\mathsf{t}\} \mathrel{\mathop{\mathsf{in}}\nolimits \longmapsto \{\mathsf{t}\}} \{\mathsf{t}\} \end{aligned} \longmapsto \begin{aligned} \mathsf{let} \triangleq \mathsf{t}\mathsf{,}\mathsf{,}\mathsf{,}\mathsf{\{b}\} \mathrel{\mathop{\mathsf{in}}\nolimits \} \longrightarrow \mathsf{t}\mathsf{,}\mathsf{,}\mathsf{,}\mathsf{,}\mathsf{,}\mathsf{} \\ \mathsf{let} \triangleq \mathsf{t}\mathsf{,}\mathsf{,}\mathsf{,}\mathsf{,}\mathsf{,}\mathsf{,}\mathsf{} \} \longrightarrow \mathsf{let} \end{aligned} \tag{105}$$

is considerable and is always larger than it seems. In Sections 3–9, we have described not all of them. Each such homomorphism is a mapping into a representation by a model, and for a philosophical discussion of these representations and the origin of these models, see pages 1–230 in [160]. For another comment concerning abstracting/realism, we refer the reader to the first half of a letter from A. Einstein to H. Samuel in [189] (pp. 157–160); see also [180].

Thus, "difficulties" with complex numbers, stricto sensu, should already be attributed to the level of the usual negative ones. Bearing in mind that the minus comes from the equal sign = [86] and the equality comes from the scheme (49) (50), both principia II and III are very important (and functioning) in the classical case. In quantum case, they are just fundamentally unavoidable for the very creation of the theory. The nature of QM theory, of arithmetic, of complex numbers, and of their algebras is one and the same.

Transferring the reasoning above to the natural numbers N, the degrees of classical and quantum abstractions become indistinguishable. Empirical motivation leads, in one way or another, to the standard von Neumann's representation for ordinals

$$0 := \mathcal{Q}, \qquad 1 := \{\mathcal{Q}\}, \qquad \mathcal{Q} := \{\mathcal{Q}, \{\mathcal{Q}\}\}, \qquad \mathcal{3} := \{\mathcal{Q}, \{\mathcal{Q}\}, \{\mathcal{Q}, \{\mathcal{Q}\}\}\}, \qquad \dots \text{ (106)}$$

i.e., to using the ZF-axiom of union: *n* + 1 := {*n*} ∪ *n* [134]. Therefore, the N-numbers are less obvious themselves, followed by the ordering <, topologies, extensions, generalizations, etc. The formal characterization of all the experimental values reduces to the successive creation of the set-theoretic atoms—unions of sets—some direct products thereof and mappings into other constructions of the same kind. Hence, both the physical images "being under a ban above" and auxiliary structures—dimensionalities/orders, etc.—should equally become homomorphisms onto certain formal constructions regardless of the description's classicality/quantumness. The presence of, say, non-binary operations (88) does not stand out because their nature does not differ from the one of habitual subtraction and of division. All of these are involutory structures that have been mathematically inherited from the empirical meta-requirements: repetitions (M-paradigm), experimental context S, **<sup>M</sup>**,... , and covariance III (Section 5.4).

To close the section, we add that the distancing of concepts of state/DataSource and of a physical property is the continuation of a more primary idea—detaching the proper macro-perceptions from what is being *represented* theoretically [87,93,160] and from conceptualizing the notions [84]. As B. Mazur has noted in [86] (p. 2), "This issue has been with us . . . forever: the general question of *abstraction*, as separating what we want from what we are presented with", i.e., the separating the "bare" empiricism from mathematics with Σ-limit and the number.

The atomic constituent of perception—sensory experience—is an elementary quantum event [93,97,123,126], and it begins and terminates in the (≈A )-distinguishability of *α*-clicks (Section 2.1). Any continual is a "speculative theory" (act of abstracting), not the underlying empiricism. Therefore, all the further matters—numbers, arithmetic, cause/effect, (non)inertial reference frames, the notions of an observer, of a classical event in the Minkowski space, the spacetime concept itself and coordinates in the relativity theory (a quantum view of the equivalence principle), device read-out, tensors, composite systems, symbols ⊗, and the like—self-evident as they may seem, are the math add-ons, which could originate only in the "∪-theory" of Section 5. Following von Weizsäcker, it might be coined the name "Ur-theory". There are no contradictions in the observations themselves, regardless of whether we call them macro- or micro-scopic. Contradictions do arise in the "mathematicae being constructed".

#### **10. About Interpretations**

It is . . . not . . . a question of a re-interpretation . . . quantum mechanics would have to be objectively false, in order that another description . . . than the statistical one be possible—J. von Neumann ([25], p. 325)

. . . one begins to suspect that all the deep questions about the meaning of measurement are really empty—S. Weinberg

"At this point in time it appears that a stalemate has been reached with regard to the interpretation of quantum mechanics" (E. Tammaro [190] (p. 1)). A "stalemate in which each side refuses to cede territory but is unable to produce a defining argumen<sup>t</sup> that would change the hearts and minds of the opponents" (M. Schlosshauer [16] (p. 227)).

#### *10.1. Click, Again*

The source of the "foundational skirmishes" [16] (p. 227) and the numerous treatments of QM [7,8], [9] (Ch. 10), [30] (Ch. 10), [33,112]—"the Copenhagen" among them—is the fact that the *α*-event and intuitive sense of the Ψ-primitive (pt. S) are a priori endowed with physical properties, observational/determinative characteristics of the DataSource, and operationality of the canonical QM-concept of the ket-vector |**Ψ** . A representative example in this regard is one of the first sentences from Everett's PhD: "The state function *ψ* is thought of as objectively characterizing the physical system . . . at all times . . . independently of our state of knowledge of it" [107] (p. 73), [124] (p. 3), and also, on p. 48, "The general validity of pure wave mechanics, *without any statistical assertions*, is assumed for *all* physical systems, including observers and measuring apparata". Furthermore, again the Everett's: "The physical 'reality' is assumed to be the wave function of the whole universe itself" [67] (p. 100), [107] (p. 70). However, none of these initially exist. The primitive *α*-events' abstractions Ψ <sup>A</sup> *α* are all there is.

An important point is that the eigen *α*-click (of a photon/electron in the EPR-experiment, say) should not be identified with an |*α* -state. The latter is re-developable with respect to eigen-states associated with other click-sets of any other instruments:

$$|\mathfrak{a}\_1\rangle = \mathfrak{b}\_1 \cdot |\mathfrak{b}\_1\rangle \xleftarrow{} \mathfrak{b}\_2 \cdot |\mathfrak{b}\_2\rangle \xleftarrow{} \cdots = \mathfrak{c}\_1 \cdot |\gamma\_1\rangle \xleftarrow{} \mathfrak{c}\_2 \cdot |\gamma\_2\rangle \xleftarrow{} \cdots = \cdots \ , \ \mu$$

which is why it is logically meaningless to attribute one *α*-click to that which carries the statistics of other clicks *β<sup>k</sup>*, *γ<sup>k</sup>*, . . . and has nothing in common with *α*. All the more so the click may not be related with the physical texts or physically descriptive collocations, such as "the measuring act on Bob's electron reveals the spin-up state". As in the "cat case" (Section 6.3.1), the spin-up here is a click-up rather than a state |↑ . Similarly, a click (allegedly of a photon) with Alice/Bob has nothing to do with distance between photons (the locality "problem"), with speed of light, nor with a kinematic "understanding" of the photon.

The quantum-detection micro-event is not a classical one as we have been understanding it, say, in the special relativity. The "click does not establish the presence of something" [126] (p. 761), it "is an elementary act of "fact creation."" [23] (Wheeler). The facts and phenomena are made up of clicks. That is to say, the distinguished *α*-clicks are not the events spaced at some distance from each other or at different points in time. These are just clicks without accounting to them such descriptive notions, such as distance, coordinates, point of time *t*, or the picturesque words, such as "dead/alive/. . . /cat"; of course, the click itself has no size/duration. Exempli gratia, the particles at accelerators and their physical properties are observed not as "material bodies in the proper sense of the word" [109] (p. 62)—this is impossible—but rather through the abstract detector-snaps. Neither the electron in interferometer nor the Higgs boson at a collider are observed in a detector as objects that are finite in extent; they are not observable entities. "A Higgs" is just a frequency 5*σ*-histogram at LHC.

Likewise, the math-properties of the eigen |*α* and of the abstract |**Ψ** can in no way be "syncretized" with *α*-events when they are still being accumulated. The screen scintillation is not a photon and a photon is not a scintillation. Similarly, "the arrival of an electron" [164] (p. 3) at the screen does not mean "what is here at this point in time with a given coordinate is the materialized particle-electron". |*α* -states and *α*-clicks are accompanied by the phenomenological and dramatic words "up/down/. . . /alive/dead", and this has nothing to do with physics, which is ye<sup>t</sup> to be created. The click should not be an element of the language in which |**Ψ** -terminology, numbers, and physical properties have been employed whatsoever.

#### *10.2. Abstraction the State*

Then, something subsequently referred to as a state (the abstract) and a measurement (the concrete) is created. However, as already stressed in Section 2.4, the process of abstracting is a rather multistage one (Sections 3–9), and a reduction in the long sequence (105) "for physical reasons" does always contain phenomenological axioms a priori. Clearly, in the reverse direction, we confront hard-to-disentangle assumptions and the well-known axiomatic cycle. The physical considerations and phenomenology should not be present in fundamentals of quantum mathematics.

To avoid paradoxes with "quantum cats", "state vector does not describe . . . a single cat" [68] (p. 37)), and "One cannot think about it as in a superposition" [16] (p. 134; D. Greenberger), with "the presence of a particle here and there", or with "quantum bombtesting" (Elitzur–Vaidman) [27,33]:

• It is imperative to keep a severe conceptual differentiation [3] (first column) between the term "the state" and "physically sounding" adjectives/verbs and the spatiotemporal or cause-effect images.

Similarly T. Maudlin: ". . . we need to keep the distinction between mathematical and physical entities sharp. Unfortunately, the usual terminology makes this difficult" [151] (p. 129). Even indirect usage of the terminology borrowed from the classical description can be a source of confusion. For example, a so-called exchange interaction as a "cause of correlation" between identical parts of system.

It seems preferable to radicalize the non-connectivity of these categories, i.e., to proclaim it a postulate. For instance, boldface italics in Remark 12 or the selected thesis on page 41. At least, the differentiation between them should not be neglected in reasoning, inasmuch as it seems unrealistic to change the deeply ingrained [59] (p. 7) and ill-defined terminological locutions, such as a "photon is in a certain *state of polarization* ... *one* photon being in a particular place" [26] (p. 5, 9), "an observable has/acquires a (numerical) value when being measured" [92] (p. 310; criticism), "outcome of a measurement" [9,30,33], "quantum parallelism" [30] (p. 282), or "simultaneous measurability"; see Section 2.1 and pr. I. (As if we have had some micro-physics prior to math; there is nothing a priori.) With this mixing, the circular logic pointed out in Remark 10 will be present at all times. See, for instance, pages 29–30 in the work [74] and notably an emphasized warning by D. Foulis about "*a mistake, and a serious one*!", including criticism addressed to von Neumann on p. 29. This

• trap of the "*braket*ting the ClassPhys'—|physical words or |in /|out —is the very "somewhere . . . hidden a concept" that M. Born spoke of (Section 1.3, p. 4), i.e., the mistaken "*physicality of* |*ψ and of* +" in (1).

Again (see p. 23 and Section 6.4), even the indirect attempts to physically characterize the state function or "reconcile" its non-classicality with any *observational prototypes* are hopeless. "The wave function is in the head and not in nature" ascribed to A. Zeilinger (2014) by A. Khrennikov. The function is the very information DataSource around which all sorts of words on physics—readings, frequencies, objects, phenomena, particles, events, and other entities—are only slated to create.

• "We cannot . . . manage to make do with such old, familiar, and seemingly indispensible terms" (Schrödinger (1933)) as the "" physikalische Realität " .... " Realität der Aussenwelt ", " Real-Zustand eines Systems "" [89] (p. 34) in the way we are doing this in classical physics, even philosophically. To put it both informally and more precisely, the automatic speech—stereotype—"the system in a state" (pt. S) [93,94] (criticism) should be dismissed from QT-fundamentals because the microscopy of quantum *α*-clicks shows that this colloquial habit is an unmeaning collocation.

This term may only be a theoretical conventionality in the follow-up *physical* theory. See also the first sentence in [61] and the selected theses on page 23 and at the beginning of Section 2.4.

The principled abstractness of the |**Ψ** -object [13] (pp. 27–28) is a core attribute of quantum theory as contrasted to the classical one. This abstraction cannot be "struggled"; it is not an idealization of something phenomenological. It is absolute. An interpretative comprehension such as |dead +ˆ |alive , even if it is permissible, may issue only from the |*α* -representations a1 *·* |*<sup>α</sup>*1 +ˆ a2 *·* |*<sup>α</sup>*2 +ˆ ···, i.e., from a treatment of the (+)ˆ -addition (of quantum amplitudes) as an accumulation of *α*-microevents—many "cat boxes".

•In other words, *the* interpretation of the quantum state *is its very definiendum* (96)–(98). Even with the physical terminology created, there may be only one paraphrase for the meaning to the state: an abstract element of the abstract, linear (not Hilbert [130]) vector space over C<sup>∗</sup>. (Point (4) in *Theorem* determines a supplement—the number add-on over the utterly abstract LVS.)

The "not Hilbert" here is because the norm and inner-product are the extra, nonessential math add-ons over H [130], which come from the follow-up introducing the Born statistics [6]. In and of itself, the state needs none and knows nothing of them. These concepts, similarly to the descriptive physical notions and a measurement, will be required further but not now for the calculation of observable quantities: math-calculus of statistics f*k* and of means.

We may not blend the fundamentally abstract part of quantum mathematics—prephysics and the structural properties of H—with those in charge of its observational/physical constituent, i.e., we may not ascribe the ontological status [95,186] to everything. In the strict sense, the ontology of/and physics, the classical one included, cannot arise before the statistical processing of quantum micro-events. (Parenthetically, the sixth Hilbert problem on "Mathematical Treatment of the Axioms of Physics" [191] becomes an ill-posed problem ([130], Section 8).) The processing itself begins with the Born rule [6].

Continuing Scheme (68), a certain parallel takes place between the following couples:

.


Just as we are not raising the question about the abstractness/treatment of the ,<sup>R</sup>numbers- in isolation from the ,physical quantities/. . . - (Remark 16), so also we should not question a treatment or the physical meaning of the ,C-, |**Ψ** -objects-. By analogy, being torn away from the \$-symbol in \$5, the number 5 in and of itself may carry neither the financial nor any other ("bank/(non)commuting/. . . ") treatment, nor does it contain some hidden "microeconomic" content. The number has no a "retrograde memory".

The first "summands" in the aforementioned (+)-conjunctions are the abstracta of principle. They may exist as the "math-things-in-themselves", and we know that they really do just as we are comprehending the existence of the N-arithmetic that has been constructed in Section 7. The second "summands" are the interpretative supplementations in their own rights. If a theory does not spell out a nature of accounting the second to the first (pr. II), then it is impossible to find-out/guess the "true" interpretation or nonexistent physical "protosource" of the abstracta n, a, and |**Ψ** "ab intra" their algebras (57), (71)–(74), (79)–(81), (88), and (96)–(98) or from the Hilbert-space mathematics. See again Remarks 2 and 16, and warnings by Ludwig of "a mistake. . . . false notion that "mathematical objects" must be pictures of physical objects" [94] (p. 228) and of a "reality [of the] word "state," a reality in which one must not believe!" [58] (p. 78). A. Peres also makes special note of the analogous: ". . . physicists have been tempted to elevate the state vector *ψ* to the status of substitute of reality" [113] (p. 645); D. Mermin puts this as "a regretable atavistic tendency to reify the quantum state" [23] (p. 144).

*10.3. Measurement "Problem"*

The most representative example is the (in)famous problem of measurement [25] (Section V.4 and Ch. VI)—"tyranny of thinking of von Neumann measurements" [23] (p. 534) with the collapse postulate. This is the subject of an "endless stream of publications suggesting new theories . . . unending discussions . . . symposia" [91] (p. 519) and of "the mountains of literature" [94] (p. 118) containing opposing opinions [91] (Ch. 11), [9,18,19,30,45,56,159]. It is, indeed, the source of questions around locality in QM. As we have seen, this problem "is simply not a problem at all!" [50] (p. 1013). It is a nonexistent—"the alleged . . . does not exist as a problem of quantum theory" [12] (p. 15)—as well as a pseudo problem and a non-issue [93] (p. 79 (!)), [94] (p. 118 (!)) [110], because

• in measurements, nothing propagates (much less at superluminal speeds) or interacts; nothing collapsed [92] (Section XVII.4.3), [9] (p. 328), [58,68], nullified, or localized; there are no such things as quantum jumps [161]; no "pieces" of the wave function are "cut out" [192] (p. 57, 158).

It is no exaggeration to say that the need to projective postulate—"a fruit of realist thinking" [4] (p. 172)—is much the same as the necessity for the world ether supporting the electromagnetic waves. All the more so because such a view of the theme has been present in the literature for quite a while [4,13,15,32,37,93,94,124,188] even as appeals.

"There is nothing . . . problematic about measurement"

> L. Ballentine (1996)

". . . there is no collapse of wave packets in reality. Do not believe in fairy tales!" G. Ludwig [58] (p. 104)

"A state vector . . . does not evolve continuously between measurements, nor suddenly "collapse" into a new state vector whenever a measurement is performed" A. Peres [113] (p. 644)

"This "reduction" . . . is not a new fundamental process, and, . . . has nothing . . . to do with measurement"

> L. Ballentine [34] (p. 244)

"The mystifying notions arise from attributing physical reality to the "jump" at a given time *t*"

```
G. Ludwig [92] (p. 327)
```
"Really bad books . . . claim that the state of the physical system . . . collapses into the corresponding *un*. This is sheer nonsense. (Finding appropriate references is left as an exercise for the reader.)"

```
A. Peres (2003)
```
Englert [12] (p. 8) does particularly object to the "folklore that "a measurement leaves the system in the relevant eigenstate" . . . It is puzzling that some textbook authors consider it good pedagogy to elevate this folklore to an "axiom" of quantum theory". See also the second epigraph to Section 2.

The point here, put very briefly, is that the measuring the "problem" is one of principle not of practice. Expressed by Bell's words, "the word [measurement] has had such a damaging effect on the discussion, that . . . it should now be banned altogether in quantum mechanics" [28] (p. 216). J. Bub and I. Pitowsky do insist in the book [8] (p. 453) that presumptions "about the ontological significance of the quantum state and about the dynamical account of how measurement outcomes come about, should be rejected as unwarranted dogmas about quantum mechanics".

Another example of circular logic is the critiqued [30,71,193] meaning of the phrase "an ensemble of similarly prepared systems" [8,30,90]. The revision of this (by and large correct) idea, as was set forth above, does actually demonstrate that, like in the ensemble approaches, "quantum mechanics is a *statistical* theory" [4] (p. 2), [58] (p. 123), [129]

(p. 223), [5,34,40,58,64,90,93,187], with a frequency content of randomness and the classical logic [58] but with a different math-calculus of the statistical weights. The "different" is due to the fact that the theory is not tied, as in the classical description, to the notion of an observable quantity, and the f's are calculated from the "other/abstract" numbers [6]. However, for the same reason, emphasizing a close resemblance with the statistical mechanics [11,194], ref. [124] (p. 72) and "explanations" with playing cards/dice, coins/balls/. . . /urns/"socks" [28] (Ch. 16) or with the classical phenomena—unusual as it may sound—are in error. The case in point is not a drastic dismissal of the classical ideas, but rather a "quantum audit" of the classical-physics language [130] (Section 8). The correct "audit" of the classical is a *re*creation of the classical:

,✿✿✿✿✿✿✿✿ classical ✿✿✿✿✿✿✿✿✿✿✿ phenomena- ,classical events/objects- ,micro world- 

$$\begin{array}{l} \left[ \begin{array}{l} \left[ \text{micro-event} \right] \longrightarrow \left[ \begin{array}{l} \text{quantum micro-event} \right] \stackrel{\left(!\right)}{\longrightarrow} \left[ \begin{array}{l} \text{abstract clock} \right] \longrightarrow \end{array} \right] \longrightarrow \\ \left[ \begin{array}{l} \left[ \text{abstraction} \right] \stackrel{\left(!\right)}{\longrightarrow} \left[ \begin{array}{l} \left[ \text{"\right]} \text{-brane (24)}, ... \right] \longrightarrow \end{array} \right] \longrightarrow \end{array} \right] \longrightarrow \end{array} \tag{107}$$
  $\begin{array}{l} \left[ \text{abstavorable concepts} \right] \longrightarrow \left[ \begin{array}{l} \text{abservable numbers} \end{array} \right] \longrightarrow} \cdots \longrightarrow$ 

,statistics, the concept of a mean- ,state, objects, ..., physics- 

,✿✿✿✿✿✿✿✿ classical ✿✿✿✿✿✿✿✿✿✿✿ phenomena-

and, consequently, the creation of the classical concept of a measuring process. Thus, this scheme along with quanta's statistics and LVS-mathematics all add up to a positive answer to Wheeler's question: "Is the entirety of existence, rather than being built on particles or fields of force or multidimensional geometry, built upon billions upon billions of elementary quantum ...,... acts of "observer-participancy," . . . ?" [16] (p. 286), [23].

#### *10.4. Interpretations and Self-Referentiality*

Although we have not ye<sup>t</sup> touched on other significant attributes—the means over statistics, operators, and products of H-spaces will be considered in their own rights—it is clear that the need to quest for a description in terms of hidden variables is also eliminated. Even from a formalistic perspective, the proof of the presence/absence [25,27,195] of these "physical" quantities should be attributed to the semantic conclusions of meta-theory (=physics) [106], i.e., to theorems *about* formal theory rather than to theorems of its *inner* calculus. In our case, and more generally, the formal theory is the syntactical axioms of QM. The corollaries of such axioms are inherently unable to lead to statements about interpretations [106] since theorems *about* object-theory itself is not provable by means *of* its object-language [106,120]. In a word, the nature and interpretation of axioms are not recovered from the very axioms or from the replacement thereof by the other ones.

A similar line of reasoning has accompanied QT for quite a while: "claim that the formalism by itself can generate an interpretation is unfounded and misleading" [68] (p. 38). It is known that even the mathematics itself cannot be grounded in a self-contained way [98] (p. 201), [105,149], [173] (!). All of this stands in stark contrast with the known statement of DeWitt to the effect that "mathematical formalism of the quantum theory is capable of yielding its own interpretation" [80] (pp. 160, 165, 168) or that "conventional statistical interpretation of quantum mechanics thus emerges from the formalism itself" [80] (p. 185). In particular, if we take account of the fact that it is not the theory itself but only its formal interpretation that determines the very semantic terms truth/falsity/provability of sentences (K. Gödel). In turn, "interpretation . . . allows a certain freedom of choice" [78] (p. 310). See also [96] and specifically Ch. III in [105]. In other words, the subconscious striving for "to interpret" and transporting the macro into the micro is the very thing that prevents us from truly gaining an understanding of quantum mathematics.

In any case, the fact that we were initially constructed the set-theoretic model (cf. [96]) rather than an interpretation simply eliminates the problem or, at most, transfers it into the domain of questions about micro-transitions <sup>A</sup> and T-family as entities being employed (see Remark 2). This is the domain of questions that invoke the set theory and touch on

the ontological status of sets at all [149] (Sections V.8 and 9 (!)). Be this as it may, logic— formalized or not—does not allow us to make statements about statements, much less a statement that refers to itself. The self-referentiality ("von Neumann catastrophe") is almost the chief trouble [4,18] encountered in quantum foundations.

All of this, of course, does not depend on whether interpretation is built in a strictly formalized form [120] or in a physically natural one. In effect, the issue of interpretations— in the rigorous definition sense [106] (Ch. 2), [120], ref. [136]—is simply nonexistent. Accordingly, the demystification of the known and the quest for ontological interpretations to a-coordinates of the |**Ψ** -vector [33,165,188]—the wave function—is no longer a problem, and with it, disappears the Feynman question of "the *only* consistent interpretation of this quantity" [164] (p. 22). See also M. Leifer's review [186] and extensive list of references therein.

#### **11. Closing Remarks**

. . . quantum mechanics has been a rich source for the invention of fairy tales— G. Ludwig and G. Thurler ([58], p. 122)

I simply do not know how to change quantum mechanics by a small amount without wrecking it altogether . . . any small change . . . would lead to logical absurdities—S. Weinberg (1994)

#### *11.1. Language and "Philosophy of Quanta"*

Remembering and continuing Section 1.3, it is generally tempting to infer that when creating the theory, we may not rest on any meanings that are tacitly associated with the typical terminology—no matter physical or mathematical—and on the tacit assumption that customary concepts are substantially correct [98].

One should also be very cautious about the wording of statements concerning the phenomena outside the everyday experience. One means that even the very natural utterances—"here/there, electron *with* Alice/Bob" (locality), "big(ger)/small(er)", "let there be a two-particle S" (quantitative statements), "subsystem S1 in such-and-such system S, consisting of ..." (statements about structure)—are de facto "(apparently) "plausible" conclusions from the observed phenomena" [92] (p. 334). These have comprised an equivalent of a measurement/preparation ([4] (pp. 195–196) and Section 3.1) and of physical (pre)imagery and thereby imitate the way of thinking and schemes of classical mechanics; see also the second epigraph to Section 6.

•The "particle, here/there, big/small, this/that/another one, before/after", and the like are *already* "illegal" observations, numbers of sorts, and a premature arithmetization, i.e., this is *already* the subconscious quantifying the micro-events or the arrays thereof by a theory and classical (18) and (19) at that.

Reality's attributes are only slated to create. Say, when we decrease the particles in experiments and reach the atomic level, we still stay in the atomistic paradigm of the particle and of numbers: the objects having mass, their coordinates, degrees of freedom, etc. This is a mistaken intuition. Very informally, we should "religionize themselves" to the quantum micro-events, while the return to the words "particle/. . . /macro" must be performed by a new reasoning mechanism. It comprises, apart from the quantum-LVS apparatus (Section 8.1), the re-creation of the very classical concept of the particle, as schematized in (107).

At the other extreme is an attempt to "hurry up" and bring the reasoning to a Hilbertspace theory or to the quantum mixtures (102). As in Section 6.5, all this may well be incorrect [151]. A source of antinomies is implicit, implying, i.e., in the eclectic—this we stress once again—confounding the observations, clicks, numbers, physics, time, math, and imagination, followed by the *uncontrollable* lexical-"branching", such as *replacing the symbol* +ˆ *with a meaning* taken from reality. For instance, the emerging the word "simultaneously" in the sequence ,the (+)ˆ -superposition of multiple states- ,simultaneously - ,quantum parallelism- [152] (p. 26). W. James has underscored that the "*viciously*

*privative employment of abstract characters and class names* is, . . . , one of the grea<sup>t</sup> original sins of the rationalistic mind" [23] (p. 547). This results in the sense messes, well-known no-go (meta)theorems [28,44], the locality "problem" in QM, and paradoxes such as the EPR [92] (Section XVII.4.4) or the jocular Bell question: "Was the world wave function waiting to jump for thousands of millions of years . . . for some more highly qualified measurer—with a Ph.D.?" [28] (p. 117), [9] (p. 18), [15,37]. As to the no-go theorems, Ballentine remarks that "the growing number [thereof], combined with some peculiar terminology, has led to confusion . . . A woefully common feature, . . . each protagonist had some interpretation of the quantum state in mind, but never stated clearly what it was" [133] (pp. 2 and 6). Ludwig, echoing Ballentine, asks: "However, what do we mean by the notion of a state?" [87] (p. 5).

Clearly, the quantum-clicks do not depend on whether personified homo sapiens interpret the arrays thereof or a biological observer such as a "Heisenberg–Zeilinger dog" [196] (pp. 171–174) [12,65] does simply perceive. The observer—without their "subjective features" [98] (pp. 55, 137) or "the anthropomorphic notions "specifying" and "knowing" "[113] (p. 645)—is just a formally logical element O in theory. Without numbers, solely a quantitative theory is not possible (Section 6.5) because the entire terminology becomes indefinite.

Thus, once a mathematics and *unambiguous* language—spectra, means, and macroscopic dynamical models—have been created, not only is there no longer a need to call on the "otherworldly", eccentric, or anthropic explanations, but the very presence of a certain share of mysticism, of subjectivity, and of (circum-)philosophy [192]—"a philosophical *Überbau*" [12] (p. 12)—in quantum foundations becomes extremely questionable. Ludwig is much more thoroughgoing in his assessment of the language games, which he refers to as the "philosophical gymnastics" [93] (p. 79).

Eventually, we no longer have any freedom to invent exegeses of the quantumpostulate as "a Bible" or "a sacred text" [23] (p. 1038). Moreover, the liberty to ask questions is no longer there since the created object-language of states, of spectra, and of frequencies narrows down the entire admissible lexicon. It is able to generate questions that are not only ill-posed but must, as in Section 6.5, be qualified as "meaningless" [14] (p. 422). For example, those that are based on (human-beings') *intuition* taking the term observation or questions about "the underlying nature of reality". As we mentioned earlier, the notion of "a physical level of rigor" (in reasoning) and the physical justification will not help us with regard to the grounds of QT. Another example is the attempts at (or "to refrain from") "tying description to a clear hypothesis about the real nature of the world" (Schrödinger (1933)) and, in general, the question of "how it should function" at the micro-level. See also [58] (p. 100) on "reality".

In the classical framework, the language sentences are always interrelated since *all* of them, one way or another, handle the observational notions. In the famous Como address, N. Bohr had remarked that "every word in the language refers to our ordinary perception". These notions, in medias res, form our natural speech when describing experiments but are inadequate in the quantum [98] (!). That is, these concepts do not make clear the fact that behind the QT are some structureless abstracta, rather than an "improved" physicomathematical axiomatics or sophisticated math vehicles, e.g., non-commutative calculus; we believe that these are {Ψ <sup>A</sup> *α*, ∪} and procedures (105)—rather than an 'improved' physicomathematical axiomatics or sophisticated math vehicles; e.g., non-commutative calculus.

Language intuition usually makes it easy for us to do away with paradoxes the semantic closedness causes. However, the quantum situation is just one of a misuse of the vocabulary, i.e., when contradictions are inevitable, and this unlimited source of confusion demands *control over the language itself*. One does create the other ("relative") languages within itself [132]: at first, the language of quantum mathematics and thereafter the language of math-physical description and of classical physics, followed by the language of the semantic interpretations. This is just what we call the metamathematics and mathlogic [105], discriminating between metamathematics and philosophy [106]. If this is not

the case—the "quantum conclusions" from thinking (even if partly/implicitly) in terms of physical influences between the classical objects (Deutsch's "bad philosophy")—then we obtain an everlasting source of paradoxes since human intuition has roots in the classical world and is a rather problematic and personal category. A. Stairs calls upon "Do not trust intuition" [73] (p. 256) because it is not meant for QM.

Inasmuch as the conceptual autonomy in quantum fundamentals is minimal (Section 2), the quantum scheme of things must commence with an extremely "ascetic" language (Remark 10), and it should be independent of our intuitive knowledge, which "tend to declare war on our deductions" (van Fraassen). To avoid collisions between theory and meta-language, the subconscious striving of the natural language to include one in the other has to be limited. Einstein adds also the situations when "er führt dazu, überhaupt alle sprachlich ausdrückbaren Sätze als sinnleer zu erklären" [89] (p. 33). A. Leggett's comments on "pseudoquestions" and "gibberish" at the end of Section 1.2 may then be strengthened so that the meaninglessness by itself should become a constitutive element of language, including the language of "philosophy of quanta".

• The rudimentary quantum (meta)mathematics creates the notion of a *prohibited* statement/phrase/question, one that is devoid of meaning. These are sentences that involve the classical analogies in the circumvention of 1) the |*α* -representatives to the non-interpretable abstraction |**Ψ** and of 2) the numerical quantities' nature (Section 9.1).

It is appropriate at this point to quote the 't Hooft remark: "I go along with everything [Copenhagen] says, except for one thing, and the one thing is you're not allowed to ask any questions" and the Einstein reasoning on page 669 in the collected articles [131]: "One may not merely ask . . . not even ask what this . . . *means*". See also Heisenberg's discussion of the problem ,language concepts- on pages 48–54 in [109], their work [197], the pages 234–235 in [131] with Bohr's appeals regarding the "necessity of a radical revision of basic principles for physical explanation . . . revision of the foundation for the unambiguous use of elementary concepts", and their comments on words "phenomena", "observations", "attributes", and "measurements" on p. 237.

The literature on this subject, even taking only the qualified sources into account, is vast [1–3,8,9,16,24,27,29–31,33,41,44,57,64,71,77,119,151,154,159,165] and abounds with terminology—"words, *ostensibly* English" (A. Leggett [9] (p. 300; emphasis ours))—that defies translation into the language of events or of concretization: observer's consciousness, parallel/branching universes/worlds, free will, catalogue of knowledge, world branch, and also such collocations as rational agents, information (*"Whose"* and "about *what*?" [28], by "Bell's sardonic comments" [30] (p. 262)) has been recorded/transmitted/(not)reached an observer (Wigner's friend), teleporting a state, many-minds/worlds/words, quantum psychology, psycho-physical parallelism (in this connection, see [148] (p. 86 (!))), and many other "bad words" by Bell. He italicizes them on p. 215 of [28].

Of course, "without philosophy, science would lose its critical spirit and would eventually become a technical device" [33] (p. 800), but, on the other hand, "the concept of the free will cannot be defined by indications on devices" [94] (p. 151), and "one must not confuse physics with philosophy" [12] (p. 12). Furthermore, yet, we should like to remember a Heisenberg attitude [197] on "a misconception . . . [and 'possibility'] to avoid philosophical arguments . . . and the way of thinking of . . . physicists who insisted on not dealing with philosophy". Namely, "[w]e can not avoid using a language bound up with the traditional philosophy". One cannot but mention the Rovelli article [198] that is entirely devoted to this topic. Therefore, "[i]t must be our task to adapt our thinking and speaking—indeed our scientific philosophy—to the new situation" with regard to the *abstract* meaning of the linear quantum addition +ˆ and quantum math altogether; all of the quotations are from pages 32 and 37–38 of the work [197].

As concerns the attitudes towards QM—at the suggestion of M. Tegmark in the 1990s, polls and statistical analysis of their correlations were even carried out [7]. There are also known attempts to involve here the biology of consciousness/brain [71], [119] (Ch. 9), [125],

[199] (Section 6). Regarding them, however, there have been not merely skeptical but quite the opposite opinions [94] (Section XII.5 (!)), [200] (Sections 17.5–6). Of special note are Ballentine's remark "to stop talking about "consciousness" or "free will"" on the last page of the preprint [133] and Popper's criticism of "the alleged . . . *intrusion of the observer, or the subject,* [or of consciousness] *into quantum theory* . . . based on bad philosophy and on a few very simple mistakes" [108] (pp. 11, 17, 42; everything as in the original) with an appeal "to exorcize the ghost called "consciousness" or "the observer" from quantum mechanics" [108] (p. 7). "[Q]uantum mechanics is a physical theory, not psychology" [4] (p. 83).

#### *11.2. Math-"Assembler" of Quantum Theory*

As a result, we gain "a contribution to philosophy, but not to physics" [82] (p. 86). At the same time, the proposed math "∪-assembler"

$$\begin{array}{cccc} \mathsf{u}\_{\mathsf{i}} \not\models \mathsf{u}\_{\mathsf{i}}, & \mathsf{\varXi} \ \mathsf{\dashv} \end{array}, \qquad \begin{array}{cccc} \mathsf{\varXi} \ \mathsf{\dashv} \end{array}, \qquad \begin{array}{cccc} \{\varXi\}\text{-brane (32)}, & \qquad (\mathsf{\varXi})\text{-algebra (37)-} \end{array} \text{(40)}$$

is quite sufficient for creating the object-language. Giving a natural form to it would be acceptable; however, it is clear that the set-theoretic ∪-base of the language cannot be avoided [96,149]. Nevertheless, the syntactically more formal description of the sequence ,transitions brace numbers- is surely of interest until the way of looking at quanta's mathematics is harmonized with the math-logic. This would turn, however, all the above material into a pure-logic text, which we eschew in the present work. It is probably for this reason that the very important and extremely thorough works (Pre-theories, 76 axioms [93] (p. 241), ordered sets, morphisms, absence of the word superposition in monographs [87,93], the (valid) criticism of "theories of . . . so-called states" [58] (p. 78), etc.) by Günther Ludwig [87,92–94] and by their school are often left out of the literature on quantum foundations. Among other things, in spite of explicitly pointing out a "solution in principle of the measuring problem" in [93] (p. V) and "Derivation of Hilbert Space Structure" of [93], this author has not been mentioned in the detailed reviews [112,118,186] or even in the books [2,5,8,18,31,119].

#### *11.3. Well, Where is Probability?*

An answer to this question in *quantum* elements is brief enough—nowhere. "There is no probability meter" [8] (p. 185; S. Saunders), and the relationship of this concept with empiricism is unique [34] (p. 46)—the statistical proportions f*k*. Cf. the famous de Finetti's (1970) claim that "probability does not exist" [2] and A. Khrennikov's remarks to the effect that "*the only bridge between "reality" and our subjective description is given by relative frequencies*" [23] (p. 139) and that "Experimenters are only interested in . . . frequencies" [150] (p. 36). Moreover, more carefully stated by von Mises' words,

"If we base the concept of probability, *not on* the notion of relative frequency, . . . at the end of the calculations, the meaning of the word 'probability' *is silently changed* from that adopted at the start to a definition based on the concept of frequency" ([129], p. 134; all the emphasis ours).

Indeed, suppose that the word "frequencies" has been banned [19] (p. 44) in substantiating the QT-elements and so have the usage of the words "over/repetition/. . . /statistics". Then, the questions do immediately arise: why the Kolmogorovian axiomatic, and why does it have this very quantification? In other words, why zero/one/. . . /positive? Why not the (−<sup>1</sup> ... 1)-interval? Whence the single-case probability postulates? . . . subjectivity? Well, what is the quantification thereof, and what does subjectivity do in the natural-scientific theory?

". . . it is very doubtful that quantum probabilities can be introduced as a measure of our personal belief. Well, it may be belief, but belief based on frequency information"

> A. Khrennikov; Växjö Conference (2001)

One way or the other, quantum foundations *would demand an interpretation* of Kolmogorov's axioms (besides, these are not categorical in contrast to LVS), and the latter, in turn, demand interpreting the concept of the number—an axiomatic add-on *over* the ZF theory [134].

Bearing in mind the primary nature of numbers and nontriviality of their emergence in physical theory (Section 7.2), it is not just impossible to avoid the statistical weights f*k* [121] (p. 25). Logic also forbids them from being subsidiary with reference to probability in any definition: "probability is the picture for reproducible frequencies; and it is the [only] *prescription* for a *correct* experiment" [94] (p. 144). Pauli, among the few, had been "convinced that

•the concept of 'probability' should not occur in the fundamental laws of a satisfying physical theory".

#### (an excerpt from their 1925 letter to Bohr)

Ensemble empiricism, for its part, is self-sufficient, and the only conventionality within it is an infinite number of repetitions. In this connection, we cannot agree with a statement of theorem III in van Kampen's work [56] (p. 99) and with further comment as to "a single system" and "calculation of spectra". At the same time, for formalizing the infinite, there is an appropriate axiom *in* the ZF-theory [134,135].

To say all this still informally, any non-statistical/non-ensemble framework for what we have been calling QM-probability does explicitly or implicitly—if the expression may be tolerated—"parasitize" on statistics by addressing the words "repetitions, multiplied, ..." and, at the same time, does "attract the empirically vague justifications" in terms of anthropomorphic surrogates: potentiality, tendency, propensity, the amount of ignorance, subjective uncertainty [8] or likelihood, degree of belief, and the like [165]. However, even from a philosophical point of view "probability is a deeply troublesome notion" [16] (p. 78; L. Hardy), which is supported by the vast literature on this subject [17,24,33], [66] (!), [81] (pp. 41–43), [170] (Chs. 3–4), [196], [201] (!). According to Deutsch, D. Papineau calls "this state of affairs . . . a scandal" [8] (p. 550).

An Einsteinian "scientific instinct" [30] (p. 174) against the probability is very well known [78], and Pauli, again, had been recollecting their (Einstein's) frequent remarks in this regard: "One cannot make a theory out of a lot of "maybe's" [= probably] . . . deep down it is wrong, even if it is empirically and logically right". More to the point, the question of what exactly is meant by a probability *event*, i.e., "Probability of *what* exactly?" [28] (p. 228), is also a matter of principle. The answer to it, as seen above, is this: "not of the classical events", i.e., "[n]ot of the . . . *being*" [28] (p. 228) such as "QM-cats", "particle is here/there", "roll of the dice", and the like. An excellent text about probability and the aspects of the probability-physical constructs is the work [202]. Its "verdict" concerning the treatment of this concept [202] (Sections 4.5 and 8) is clearly Misesian [129], i.e. the "*ensemble* and *frequency*" [66] (p. xiii).

Thus, to sum up, the philosophy/axioms of probability or its "quantum deformations" should not be present in *quantum* foundations. There cannot be hidden details underlying the quantum probability because the "details" imply some terminology with a classical content. Quantum probability is the statistical regularity. It comes from Kollektivs [129] of abstracta (32) and may only be a shortened term for the relative "frequencies in long runs" (von Neumann) or "the Einstein hypothesis" by M. Jammer [91] (p. 441). The realistic/physical/. . . /pictorial adjectives and descriptive supplementations to the term "long runs" are prohibited. This is why the conventional tractability of the quantumpostulates' mathematics, i.e., the calculation of *probabilities for the classical* events to occur in the reality—"alive/cat/. . . /imploded/bomb"—"is not adequate" neither as a doctrinal point of departure nor as a post-math interpretation. It presents us with a circulus vitiosus of re-exegeses. Fuchs, referring to de Finetti's words in an interview with Quanta Magazine (4 June 2015), prognosticates that this conception "will go the way of phlogiston". The "not adequate" is a R. Haag quotation, and he expresses this "conviction", applying it even to "the conceptual structure of standard Quantum Theory" [101] (p. 743).

The ultimate conclusion completes Remarks 2 and 7. If we accept the set-theoretic eye on things, Section 5.1, by all appearances, provides a positive answer to the question about the rigidity of QT [103]—"change any one aspect, and the whole structure collapses" [57] (p. 1); see also the second epigraph to this section. At least, it is hard to imagine *what any other* axiom-free way of turning empiricism into quantum mathematics would look like, as soon as we abandon the primitive minimality of the scheme

> ,distinguishable micro *α*-events- + ,ensembles of abstracta Ψ <sup>A</sup> *α*- .

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Acknowledgments:** The author wishes to express their gratitude to the QFT-department staff of TSU for stimulating conversations. A special word of thanks is due to Ivan Gorbunov and to Professor V. Bagrov, who initiated considering the matters on quantum logic [72,74,111]. The work was supported by the Tomsk State University Development Programme (Priority–2030).

**Conflicts of Interest:** The authors declare no conflict of interest.
