A methodological note on proving agreement between the Elementary Process Theory and modern interaction theories Marcoen J.T.F. Cabbolet∗ Center for Logic and Philosophy of Science, Vrije Universiteit Brussel Abstract - The Elementary Process Theory (EPT) is a collection of seven elementary process-physical principles that describe the individual processes by which interactions have to take place for repulsive gravity to exist. One of the two main problems of the EPT is that there is no proof that the four fundamental interactions (gravitational, electromagnetic, strong, and weak) as we know them can take place in the elementary processes described by the EPT. This paper sets forth the method by which it can be proven that the EPT agrees with the knowledge that derives from the successful predictions of a modern interaction theory T . This determines a fundamentally new research program in theoretical physics. Keywords: scientific method, categorical model theory, relativity, interaction 1 Introduction The Elementary Process Theory (EPT) is a first-order theory together with a (speculative) physical interpretation given in the form of interpretation rules, which yields the view that the axioms of the EPT are non-classical, non-quantum, elementary process-physical principles that give an abstract yet exact description of what has to happen in the individual processes at supersmall scale by which interactions take place for massive antiparticles to be repulsed by the gravitational field of bodies of ordinary matter [1, 2, 3]. It follows that these elementary processes are in essence all the same, regardless of the type of interaction that takes place. The question is then: is this relevant for physics? There are then two main issues with the EPT, both mentioned in [1], which are causes for a genuine concern that the answer to that question is 'no': (i) the EPT has in essence been developed from a Gedankenexperiment with an outcome (matter-antimatter repulsive gravity) that cannot possibly be true from the perspective of modern physics; (ii) thus far there is no proof that the four fundamental interactions-at least as far as we know them-can take place in the individual processes described by the EPT. Concerning the first issue, the crux is that the theoretical arguments against a matter-antimatter repulsive gravity-see [4, 5] for an overview-all lean on the assumption that theories of modern physics are valid beyond their established area of application. But as Feynman already remarked, "experiment is the sole judge of scientific truth" [6]. The issue whether or not repulsive gravity exists will thus have to be decided by experimental research; the current state of affairs is then that there are at least four sizeable experimental projects going on to establish the coupling of massive antimatter particles with the gravitational field of the earth: three projects at CERN using anti-hydrogen, AEgIS [7], GBAR [8], and ALPHA [9], and one at the PSI using muonium (an exotic atom made up of an antimuon and an electron) [10]. ∗e-mail: Marcoen.Cabbolet@vub.ac.be 1 Concerning the second issue, the crux is that the EPT is an abstract physical theory formalized within the framework of set matrix theory (SMT)-that is, the EPT consists of (i) the language L(EPT ) of the EPT, which is a sublanguage of L(SMT ) determined by • a nonempty set UEPT , whose elements are individual constants of the EPT; • a nonempty set REPT , whose elements are the unary relation ME ⊂ UEPT and the ternary relation R ⊂ (UEPT )3 of the EPT; (ii) a collection of formal axioms of the EPT: • for every individual constant Φ ∈ UEPT , an axiom ∃x(x = Φ); • for every n-ary relation ρ ∈ REPT , an axiom ∃x(x = ρ ∧ ρ ⊂ (UEPT )n); (iii) a collection of well-formed formulas in L(EPT ), to be called the process-physical axioms of the EPT; (iv) a collection of statements in ordinary language, called the interpretation rules of the EPT, which give a physical meaning to the individual constants and relations of the EPT. Furthermore, let ΣEPT be the total collection of formal and process-physical axioms of the EPT; a theorem of the EPT is then any formula Ψ that can be inferred from ΣEPT within the framework of SMT as in ΣEPT `SMT Ψ (1) All theorems that can inferred from the physical axioms of the EPT by eliminating all quantifiers, are then expressed in terms of abstract formal objects Φ ∈ UEPT . That said, as Halvorson also noted in [11], without interpretation rules the process-physical axioms would have no physical meaning whatsoever. That means that in the context of the EPT, we have to distinguish between the material object, i.e. the (postulated) thing in the physical universe that is referred to, and the formal object, i.e. the thing in the mathematical universe that refers to the material object. Furthermore, an essential feature of the EPT is that the individual constants that refer to ultimate constituents of the physical world are abstract sets, i.e. sets whose elements are not specified: these stand in contrast to concrete sets, i.e. sets whose elements are specified (such as the empty set ∅ or the set ω = {∅, {∅}, {{∅}}, . . .}). Now the ontological status of abstract constants can probably be debated forever, but here the following position is taken. A formal axiom ∃x(x = φ) of the EPT guarantees that there is an object in the mathematical universe whose name is φ, but without elaborating on which object that precisely is. The important point here is that we do not have assumed new objects: therefore the language of the EPT is merely a sublanguage of the language of mathematics-in casu the language L(SMT ) of SMT, a generalization of ZF [13]-in which we have assumed new symbols for existing objects. At the degree of abstractness of the EPT, a formal object thus designates a material object without representing its state, that is, without containing information of (expectation values of) quantitative properties of the material object-this allows to express elementary process-physical principles without reference to a coordinate system of an observer. As a consequence, the successful predictions of modern interaction theories cannot ever be reproduced quantitatively by deriving theorems directly from the EPT: this impossibility gives rise to the issue (ii) mentioned in the beginning of this section. The purpose of this paper is to develop a method by which this issue can be solved, that is, by which it can be proved that the fundamental interactions as we know them can take place in the processes described by the EPT. The next section describes this method, the final section describes the research program thereby determined. 2 2 Method of proof To get verifiable predictions on the basis of the EPT, a standard tool is to develop a concrete set-theoretic model. The notion of a model of a first-order theory is well-defined, see e.g. [14], so we can give a definition of a concrete set-theoretic model of the EPT: Definition 2.1. A concrete set-theoretic model M of the EPT, or shorter: a set-theoretic model of the EPT, is a mathematically concrete structure 〈|M |, I(ME), I(R)〉 for the EPT specified in a language L(M), with interpretation function I : L(EPT )→ L(M), such that (i) every abstract formal object Φ ∈ UEPT that designates a material object is interpreted as a concrete formal object I(Φ) in the universe |M | of individuals of M representing the state of that material object in the reference frame of an observer; (ii) the unary existence relation ME of the EPT is interpreted as a relation I(ME) ⊂ |M | for which Φ ∈ME ⇔ I(Φ) ∈ I(ME) (2) (iii) the ternary relation R of the EPT is interpreted as a relation I(R) ⊂ |M | × |M | × |M | for which 〈Φ1,Φ2,Φ3〉 ∈ R⇔ 〈I(Φ1), I(Φ2), I(Φ3)〉 ∈ I(R) (3) (iv) for any axiom Ψ of the EPT, its interpretation I(Ψ) is true in M : M |= I(Ψ) (4)  There are, however, concrete set-theoretic models of the EPT that are not interesting for physics. The following is an example thereof. Example 2.2. Let, for integers n ∈ ZN = {0, . . . , N − 1} and k ∈ Sω(n) = {1, . . . , ω(n)}, the interpretation function I for the EPT be partially given by: • I(EPΦnk) = {1 + nπ + kπ2} • I(NWΦnk) = {2 + nπ + kπ2} • I(NPΦnk) = {3 + nπ + kπ2} • I(LWΦnk) = {4 + nπ + kπ2} • I(SΦnk) = {5 + nπ + kπ2} • I(Φ1 + Φ2) = I(Φ1) ∪ I(Φ2) In other words, individual constants of the EPT referring to ultimate constituents of the physical world are interpreted as singletons containing a unique number, and individual constants referring to superpositions of ultimate constituents are interpreted as sets of these numbers; with some creativity the structure can then be completed to yield a concrete set-theoretic model in which the axioms of the EPT are true. There is then a concrete set-theoretic model M for each choice of integers N,ω(0), ω(1), . . . , ω(N − 1), so that the universe |M | of individuals of M contains finite sets of these numbers a + nπ + kπ2 for which 1 ≤ a ≤ 5 , n ∈ ZN , k ∈ Sω(n). However, physically this makes no sense since these numbers do not really represent states of particles-these numerical representations don't yield verifiable predictions.  3 This example establishes that specifying just any concrete set-theoretic model of the EPT is insufficient as a method to prove that the fundamental interactions as we know them can take place in the processes described by the EPT. But not only that: specifying a single set-theoretic model M of the EPT-even one that does yield verifiable predictions-is still insufficient because it cannot ever reproduce relativity. To see that, suppose that in a model M of the EPT the initial state of an individual process is modeled as a point-particle with position X0 in the coordinate system of an observer O and with momentum ~p0, and suppose that M predicts that the final state produced by that process is a point-particle with position X1 in the coordinate system of O and with momentum ~p1: this is a verifiable prediction. However, for another observer O′ the initial state of that same system will have to be modeled as a point-particle with some position X ′0 in the coordinate system of O′ and with momentum ~p′0, and the predicted final state of the process will be a point-particle with a position X ′1 in the coordinate system of O′ and with momentum ~p′1. The one model M , however, does not contain the initial state of the process in the coordinate system of O′: it only contains the initial state of the process in the coordinate system of O-for the observer O′ another model m′ of the EPT is required. Moreover, the model M is incapable of predicting what the values of the aforementioned position X ′1 and momentum ~p ′ 1 will be: a single set-theoretic model of the EPT is thus insufficient because it can never predict relativity of spatiotemporal characteristics of motion (time dilation, length contraction). This provides the motivation for introducing the notion of a categorical model of the EPT : a model of the EPT is then identified with a category, which does contain a set-theoretic model of the EPT for every observer. The notion of a categorical model of a first-order theory has already been discussed in the literature, see e.g. [11] and the references therein; applied to the EPT, this gives the following definition: Definition 2.3. A categorical model of the EPT is a (small) category C such that (i) the collection of objects of C is a family {Mi}i∈F1 of set-theoretic models of the EPT, so that any Mp in {Mi}i∈F1 is a structure 〈|Mp|, Ip(ME), Ip(R)〉 for the EPT, specified in a sublanguage L(Mp) of a common background language L(C ), which satisfies Def. 2.1; (ii) the collection of arrows of C is a family {Tj}j∈F2 of structure isomorphisms, so that for any arrow Tk in {Tj}j∈F2 there is a domain Mp ∈ {Mi}i∈F1 and a codomain Mq ∈ {Mi}i∈F1 such that • Tk bijectively maps the universe |Mp| to the universe |Mq|; • Tk(α) ∈ Iq(ME)⇔ α ∈ Ip(ME); • (Tk(α1), Tk(α2), Tk(α3)) ∈ Iq(R)⇔ (α1, α2, α3) ∈ Ip(R).  But what has been said earlier about set-theoretic models of the EPT remains the case for categorical models of the EPT: there are categorical models of the EPT that are not interesting from the point of view of physics-it's a tedious exercise to do so, but it is not difficult to specify such an uninteresting categorical model along the lines of Ex. 2.2. To single out categorical models that are interesting, the notion of empirical reduction, introduced by Rosaler in [15], can be applied to compare a categorical model C of the EPT to an existing interaction theory T : Definition 2.4. Let C be a categorical model of the EPT; then C reduces empirically to an existing interaction theory T if and only if for every experiment that has confirmed a prediction of T , the experimentally successful predictions of T can be reproduced by C .  Note that T does not have to be an axiomatized theory: Def. 2.4 holds for a scientific theory in the sense of a generally accepted body of explanatory principles that has been tested by the 4 scientific method-in that sense, e.g. quantum electrodynamics (QED) is a scientific theory although it is not axiomatized. That brings us to the following formulation of the method by which issue (ii) as stated in the Introduction can be solved: to prove that a fundamental interaction as we know it from the scientific theory T can take place in the processes as described by the EPT, the method is to specify a categorical model C of the EPT such that C reduces empirically to T . The idea is thus that the knowledge that derives from the successful predictions of T has to be incorporated in a (categorical) model C of the EPT: a negative result is then that no such categorical model exists. Note that only the empirically successful predictions of T have to be reproduced by C : it is, thus, not the case that a categorical model C of the EPT has to be developed such that C reduces formally to T , that is, such that T emerges from the mathematical formulation of C by applying some limiting procedure. 3 A new research program in theoretical physics The method set forth in the preceding section determines in a natural way a fundamentally new research program in theoretical physics-a 'research program' as meant by Lakatos [16]. In this section we specify its hard core, positive and negative heuristics, and aims. Hard core The hard core of the research program consists of the EPT and its mathematical foundations. In this research program, the EPT is then considered to be fundamental-that is, the physical axioms of the EPT are the fundamental laws in this research program. This is supplemented by the examples of how the EPT applies, at an abstract level, to real world problems: the application to the Planck era of the universe and the application to the mindbody problem. This hard core already corresponds to what Kuhn called a paradigm (disciplinary matrix) [17]. Heuristics First of all, an 'initial' categorical model C0 of the EPT proving agreement with SR has to be fully specified; such a categorical model is already in the works [18]. The natural positive heuristic is then to develop successors C1, C2, ... of C0 that are theoretically and empirically progressive. Lakatos has defined notions of theoretical progression and empirical progression for theories [16], but these notions can be defined similarly for categorical models of the EPT: Definition 3.1. A categorical model Cn+1 of the EPT is theoretically progressive compared to a categorical model Cn of the EPT when not only all observations, which could be expressed as predictions in the language of Cn, can also be expressed as predictions in the language of Cn+1 but also some observations, which could not be expressed as predictions in the language of Cn, can be expressed as predictions in the language of Cn+1. Likewise, a categorical model Cn+1 of the EPT is empirically progressive compared to a categorical model Cn of the EPT when in the framework of Cn+1 predictions can be formulated that are impossible in the framework of Cn and some of these predictions have been verified.  The natural negative heuristic is to refrain from developments that are inconsistent with the physical axioms of the EPT in the hard core. For example, the EPT is inconsistent with standard QM: it is therefore not interesting to (attempt to) develop a categorical model of the EPT that unifies the EPT and standard QM. Note that this is something else than developing a categorical model of the EPT that reduces empirically to standard QM! Likewise, the EPT is inconsistent with the classical concept of continuous motion: it is therefore not interesting to (attempt to) develop a categorical model of the EPT that unifies the EPT and a theory that applies this concept of continuous motion (e.g. GR)-again, this is something else than developing a categorical model of the EPT that reduces empirically to GR. 5 Aims The short-term aim is to develop a categorical model of the EPT that reduces empirically to GR: that would be both theoretically and empirically progressive compared to C0. This comes down to developing a (relativistic) model of an elementary process in which a gravitational interaction takes place, such that it not only predicts a matter-antimatter gravitational repulsion, but also quantitatively reproduces the empirically successful predictions of GR. If that aim can be achieved, and that's a big 'if', the medium-term aim becomes to develop a categorical model of the EPT that reduces empirically to GR and to QED: although a unification of QED and GR-in the sense of a single theoretical framework in which QED and GR are both universally valid-is impossible, the EPT could then be called a unifying scheme with the unifying principles (the physical axioms of the EPT) at a more abstract level. If that medium-term aim can be achieved, and that's an even bigger 'if', the long-term aim becomes to develop a categorical model of the EPT that in addition reduces empirically to quantum chromodynamics (QCD) and electroweak theory (EW). The EPT could then be called a Grand Unifying Scheme. This notion can thus be defined as follows: Definition 3.2. The EPT is a Grand Unifying Scheme if and only if it has a categorical model C that reduces empirically to GR, QED, QCD, and EW-that is, if and only if it has a categorical model C such that all observations on physical systems governed by the fundamental interactions can be formulated as predictions in the language of that category C .  This notion of a Grand Unifying Scheme is thus related to Van Fraassen's idea of empirical adequacy, introduced in [19]: the EPT is a Grand Unifying Scheme if and only if it has a categorical model that is empirically adequate when applied in the area of physical systems of elementary particles whose behavior is governed by the fundamental interactions. This notion of a Grand Unifying Scheme should, thus, absolutely not be confused with the idea of a Grand Unified Theory: a Grand Unified Theory is a merger of the three gauge interactions of the Standard Model (electromagnetic, weak, strong) in a single interaction model. So, a Grand Unified Theory is thus confined to the framework of the Standard Model, while the above definition of a Grand Unifying Scheme does not assume that objects of the category (which are models of the EPT) have to be formalized in the framework of quantum field theory. Concluding, a method has been presented by which it can be proven that the physically abstract EPT agrees with the knowledge of the physical world that derives from the successful predictions of modern interaction theories. This method gives in a natural way rise to a fundamentally new research program in theoretical physics aimed at establishing whether the EPT is a Grand Unifying Scheme-which is, ultimately, its intended relevance for physics. References [1] M.J.T.F. Cabbolet, Ann. Phys. (Berlin) 522, 699-738 (2010) [2] M.J.T.F. Cabbolet, Ann. Phys. (Berlin) 523, 990-994 (2011) [3] M.J.T.F. Cabbolet, Ann. Phys. (Berlin) 528, 626-627 (2016) [4] M.M. Nieto, T. Goldman, Phys. Rep. 205, 221-281 (1992) [5] M.M. Nieto, T. Goldman, Phys. Rep. 216, 343 (Erratum) (1992) [6] R.P. Feynman, The Feynman Lectures on Physics, The Millenium Edition, vol 1 (New York: Basic Books), p 1-1 (2011) [7] G. Testera et al., Hyperfine Interact. 233(1), 13-20 (2015) [8] P. Indelicato et al., Hyperfine Interact. 228(1), 141-150 (2014) 6 [9] N. Madsen, Hyperfine Interact. 228(1), 37-45 (2014) [10] K. Kirch, K.S. Khaw, Int. J. Mod. Phys. Conf. Ser. 30, 1460258 (2014) [11] H. Halvorson, Scientific Theories, in: P. Humphries (ed.), The Oxford Handbook of Philosophy of Science (Oxford University Press, Oxford), pp. 585-608 (2016) [12] M. Tegmark, Found. Phys. 38(2), 101-150 (2008) [13] M.J.T.F. Cabbolet, H.C.M. de Swart, Log. Anal. 225, 59-82 (2014) [14] J.R. Shoenfield, Mathematical Logic, Natick: AK Peters Ltd, (2001) [15] J. Rosaler, Topoi 34, 325-338 (2015) [16] I. Lakatos, Falsification and the methodology of scientific research programmes, in: I. Lakatos, A. Musgrave (eds.), Criticism and the Growth of Knowledge (Cambridge University Press, Cambridge), pp. 91-195 (1970) [17] T.S. Kuhn, The Structure of Scientific Revolutions, Chicago: University of Chicago Press (1962) [18] M.J.T.F. Cabbolet, A categorical model of the Elementary Process Theory incorporating Special Relativity, Preprint: philpapers.org/rec/CABROS (forthcoming in 2018) [19] B. Van Fraassen, The Scientific Image, Gloucestershire: Clarendon Press, p. 44 (1980)