Europhysics Letters 32 (8),pp.621-626 (1995) 10 December 1995

(*) Present address: AMS, P.O. Box 7621, Cumberland, RI 02864, USA.

(received 28 August 1995; accepted in final form 31 October 1995).

Introduction. - Topos theory (see ref. [1-6]) offers an independent (of the set theory) approach to the foundations of mathematics. Topoi are categories with "set-like", objects, "function-like" arrows and "Boolean-like" logic algebras. Handling "sets", and "functions", in a topos may differ from that in classical mathematics (i.e. the topos **Set** of sets): there are non-classical versions of mathematics, each with its non-Boolean version of logic. One possible view on topoi is this: abstract worlds, universes for mathematical discourse, "inhabitants" (*researchers*) of which may use non-Boolean logics in their reasoning. From this viewpoint the main business of classical physics is to construct models of the objective (*absolute*) universe with a given "bivalent Boolean" model of the researcher, and choose the most adequate one. In a sense, our task is inverse: with a given model of the absolute universe, to
construct models of the researcher, and find out how the researcher’s *proper* universe changes if his logic is changed. Thus, not the universe itself, but rather its "differential" is what interests us here. We build a functional scheme describing the researcher’s interactions (*actions* and *observations*) with the environment.

Action principle (principle I). - The major intuition-based attribute of actions (elementary influences of the researcher upon the world) is that they can be associatively composed (i.e. performed in sequence), the compositions also being actions, and there is an *identity* action (changing nothing). The set of the researcher’s actions (or *effectors*), together with an associative composition, is his *motor space*. Example: in quantum theory the observer’s actions are represented by operators on a linear space and constitute, together with an associative composition, a semi-group with an identity (monoid).

Superposition principle (principle II). - The major intuition-based property of observations (mental and visual pictures of fragments of reality and appearance) is their ability to be superposed, with some real (later we shall generalize the situation for an arbitrary field F ) "weight" factors, assigned by the observer to each item. Intuitively, they measure the "participatory degree" of observations in a particular observational situation. In formal language, there are two algebraic operations on the set of observations: summation and multiplication (by reals). The set of the researcher’s observations (or *reflexors*), together with the two operations, is his *sensory space*. Example: in relativity the tangent space **T _{a}** at an event

Quantum principle (principle III). - "No elementary phenomenon is a phenomenon until it is an observed phenomenon" (any observation of the researcher is one of his actions). This simply means that obtaining constructive information about reality changes its appearance. The quantum principle couples observations with actions into new entities, called by constructivists *states of knowledge*: any rational researcher performs an action in accordance to, and interprets an observation on the basis of, his particular state of knowledge. It is said that rational knowledge consists of two fundamental (*sensory* and *motor*) components.

Paradigm. - We shall call the set of a researcher’s states of knowledge his *paradigm*. Observations,
then, induce superposition of states of knowledge, with weight factors (*extensive development* or *accumulation* of knowledge), and actions induce associative composition (*intensive development* or *elevation* of knowledge). Thus we have three operations on the researcher’s paradigm, which endows it with an algebraic structure. The linear case of this structure is, of course, a real linear associative algebra **A** with an identity. The sensory space **S _{A}** then is the additive linear space of the algebra, and the motor space

Time. - We employ the constructivistic concept of time: a fundamental attribute of thought process, the basis to distinguish one entity from another. No statement on time being a physical property of the universe is made. Constructivists describe time as a partial order on the set of states of knowledge. So do we, slicing the paradigm with a 1-form on its sensory space, which partially orders states of knowledge by the naturally ordered set **R** of reals. Example: The proper time of an inertial observer in special relativity is a 1-form **t**, such that for any event **a** with components **a**^{n} (n = 0,1,2,3)
in his rest frame, **t(a)=a**^{0}

Metric. - A researcher’s natural ability to "feel" angles and estimate distances between observations is represented by a metric on his sensory space **S _{A}** . We do not force metrics into the scheme because a natural metric is defined automatically, once the proper time of the researcher is known, as follows. Each real algebra is completely defined with the structure constant tensor

Absolute universe. - The *actual reality
principle* (principle IV) affirms that the *absolute universe* exists and consists of *interacting systems*. Each system is represented by its *states*. Given all states of a system, it is defined completely. Some different systems may have the same states (*common* states). A system **X** is a *subsystem* of a system **Y** if all states of **X** are common to **X** and **Y** . If two systems are subsystems of each other, it is natural to consider them *equal*. Given two systems **X** and **Y** , we can consider a system **Z** (the *union* of ** and Y) whose states are all states of and all states of Y. For two systems X and Y with common states there is a system Z (the intersection of X and Y ) whose states are their common states. A system X that can have only states that Y cannot, is the complement of Y . The behaviour
of the ontological pair "system, state" resembles that of "set, element" in "naïve" set theory, although conceptually they are very different. Two systems interact if states of one system depend on states of the other one, which is described as a function in set theoretical terms. Thus, with systems as sets (of their states) and interactions as functions, the category Set of sets serves as a first-order model of the absolute-universe axioms. **

Proper universe. - The researcher’s actions change states of a system **X** : a state **x** before an action **a** becomes a state **y** after that, so any action **a** induces a function **X->X**, and we have the influence of the researcher with the motor space **M** on a system **X** as a realization of the monoid **M** in the set **X**, i.e. a function J, assigning to each **a** from **M** a function **f _{a}**:

Summary. - We see that only intuition-based concepts, logically prior to physics, are used here. The technique engaged is extremely simple in the sense that it is just several steps from the set and category axioms. To compare, the notion of smooth affine manifold (a starting point for the working physicist) is far more complicated. Of course, some of the notions seem uncertain - it is often inevitable, and now, to improve the situation, we give a strict form of the above outline.

Definition 1. - Let F be a partially ordered field. 1) An *F-xenomorph* is the category **A**[F] of linear algebras over F. 2) *Paradigms* of an F-xenomorph are **A**[F]-objects, his *actions* (or *effectors*) are **A**[F]-arrows. 3) For a paradigm **A**: a) *states of knowledge* are elements of the algebra **A**; b) a *metric* is the algebra-**A**
structure constant tensor **C** together with a 1-form **t** on the additive linear space **S _{A}** of the algebra

Notes. - 1) We generalized the notion of action. Actions in the old sense (i.e. elements of the monoid **M _{A}** or

Theorem. - **R**-xenomorph has a unique classic paradigm **E**, of dimensionality 4, with a unique metric, of signature 2.

Proof. - If the logic of the topos **{M _{E}}** is Boolean then

t_{0}t_{1}t_{2}t_{3}t_{1}-t_{0}-t_{3}t_{2}t_{2}t_{3}-t_{0}-t_{1}t_{3}-t_{2}t_{1}-t_{0}

**G** must be symmetric. Non-trivial symmetry demands t_{1}= -t_{1}, t_{2}= -t_{2}, t_{3}= -t_{3} which yields t_{1}=t_{2}=t_{3}=0. Thus **E** has a unique (up to scalar factor) metric of signature 2, generated by a unique (up to scalar factor) psychological time (t_{0}, 0, 0, 0), which concludes the proof.

Relativity. - Defining (changes of) *viewpoints* of the xenomorph as (isometric changes of) bases e_{n}, we easily obtain a "sensory-motor" version of special relativity. In fact, though the 1-form **t** plays the role of psychological time (no such thing in special relativity), once **t** generates a metric, the latter in turn generates time in its standard sense. If we ignore the motor structure of the paradigm, four-dimensionality and Lorentz metric become a mystery, which is the case in standard physics. Let us call the paradigm **E** the *Einsteinian*
paradigm. It is easily checked that besides the four-dimensional classic paradigm, there are two (and only two) non-trivial classical paradigms (of dimensionalities 1 and 2), both subalgebras of **E** (see ref. [7]). Informally, physics of the Einsteinian paradigm is a "superposition" of three versions of Boolean physics of different dimensionalities, which may account for the existence of three particle generations.

Virtual reality. - Since the "objective" absolute universe **Set** is a Boolean topos, we can separate a Boolean part - the most "objective", in a sense, in any rational paradigm **A**. Obviously, it is the set **expA** of invertible elements of the algebra **A**: they constitute a group. For any finite-dimensional rational paradigm of an **R**-xenomorph (*decoherent* paradigm), its set of invertible elements is a Lie group, so it has a natural topology. The sensory space **S _{A}** is the tangent space of

Definition 2. - For a decoherent paradigm **A**: 1) If **G** is a metric of **A**, a *physical metric* is a tensor field **g** on the Lie group **expA** of invertible elements of the algebra **A**, such that at any point **u** from **expA**, **g(u)=G(u×a, u×b)**, where **a,b** belong to **A**. 2 a) *Spacetime* is **expA** together with all the physical metrics of **A**;
b) elements **a** from **A**, such that **a** belongs to **(A\)expA** are *(non-)physical events*; c) *topology* of spacetime is that of the Lie group. 3) A *cosmology* is a pair **(expA, g)**, where **g** is a physical metric of **A**.

Notes. - 1) *(Non-)physical* observations and actions are defined automatically. Compositions of physical actions are always physical actions, while compositions of non-physical actions may result in physical actions **(A\expA** is not always closed under the composition). 2) In Lie theory terms, **expA** is an exponent-image of the algebra **A**. Then the psychological time **t** generates its exponent-image T (*cosmological* time). In other words, psychological time is a logarithm-image of cosmological time. 3) The *virtual-reality principle* (principle VI) assigns to every decoherent paradigm its spacetime.

Einsteinian paradigm E. - In this
case
spacetime **expE** has a "cylindric" product topology **(S ^{3})xR** (

**E**-mathematics. - The theorem
offers the mathematics of the topos **{SU(2)xR ^{+}}={expE}** as the real-world math. Boolean though, it has an unpleasant property: the axiom of choice fails in

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