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Talking About Models: The Inherent Constraints of Mathematics

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Abstract

In this article my primary intention is to engage in a discussion on the inherent constraints of models, taken as models of theories, that reaches beyond the epistemological level. Naturally the paper takes into account the ongoing debate between proponents of the syntactic and the semantic view of theories and that between proponents of the various versions of scientific realism, reaching down to the most fundamental, subjective level of discourse. In this approach, while allowing for a limited discussion of physical and positive science models, I am primarily focused on the structure and ontology of mathematical models, in particular Cohen’s forcing models and to a lesser extent Gödel’s constructible universe, to the extent that these were designed to answer questions bearing on the scope, the capacity and ultimately the ontology of models themselves (e.g., the question of continuum), therefore influencing in one or the other way the status of models in general. This status, it is argued, is largely defined by the way models subsume a set-theoretical structure whose constraints, reducible to an extra-linguistic level of discourse, may implicitly condition the epistemic status of models as representations of axiomatic theories. In the last section I deal extensively with the inner constraints of theories (or of corresponding models for that matter) as subjectively originated in a less technical philosophically oriented discussion with certain prompts from phenomenology.

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Notes

  1. Of course this has largely to do with the debated position that the real numbers system is beyond the expressional means of at least the standard set theory.

  2. Concerning the debated possibility of identification of models with theories which is a position discarded by the author, I point to van Fraassen’s view that a scientific theory though not identifiable with, is still identifiable through a class of models properly conceived (Van Fraassen 1985a, p. 25).

  3. Some references to F. W. Lawvere’s work are: Lawvere (1975, 2002, 2017)

  4. By an eidetic law in the world of phenomena one can roughly communicate to a non-phenomenologist what is implied as a regularity by essential necessity and not by mere facticity. The interested reader may consult E. Husserl’s Ideas I, (Husserl 1976, Engl. transl., pp. 12–15).

  5. This is the case of the color model (i.e., Quantum Chromodynamics, QCD) by which were not only compatibly elaborated the symmetry characteristics of quarks and their composites but was also found to be observably equivalent to the paraparticle counterpart (da Costa and French 2003, p. 117).

  6. In a model-theoretic approach the notion of truth is rigorously defined by means of a formal language \({\mathcal {L}}\) and an interpretation of \(\mathcal {L}\) in a mathematical structure \(\mathcal {A}\). A ‘simple pragmatic’ structure is called a partial structure of the form \({\mathcal {A}}=<A, R_{i}, Q>_{i\in I}\) where A is a nonempty set, \(R_{i}, i\in I\), is a partial relation defined on A for every \(i\in I\), where I is an appropriate index set, and Q is a set of sentences of the language \(\mathcal {L}\) having the same similarity type as that of \(\mathcal {A}\). For some i, \(R_{i}\) may be empty and Q may also be empty. The \(R_{i}\) are called partial because any relation \(R_{i}, i\in I\) of arity \(n_{i}\) is not necessarily defined for all \(n_{i}\)-tuples of elements of A. The set A is the domain of the individuals of ‘observation’ modeled in the structure \(\mathcal {A}\) and the set Q of sentences may include observation statements referring to the individuals of the domain. See: da Costa and French (2003, p. 18).

  7. Krause and Coelho (2005) claim however, that the mathematical structure of quantum mechanics should have a non-trivial rigid expansion (i.e. not one obtained by trivially adjoining the ordinal structure) whose physical intuition is that quantum objects are somehow ‘intrinsically’ distinguishable.

  8. For definitions of empirical adequacy and empirical equivalence of theories one may look into van Fraassen’s The Scientific Image; van Fraassen (1980, pp. 45, 64).

  9. In Turney’s words, ‘there is a symmetry between syntax and semantics, which makes it unlikely that one is better than the other for an account of science’. By this token van Fraassen’s view that theory and observation cannot be distinguished by syntactic methods, in which case he makes up for this distinction by semantic methods, is undermined by Turney’s alleged construction of a syntactic method which is equivalent to van Fraassen’s semantic one (Turney 1990, p. 449).

  10. The transitivity property could be roughly described as the preservation under the inclusion relation \(\in\) of the individuality of the zero-level elements of a theory, i.e., for any set A if \(x\in A\) and \(x\in y\) then \(y\in A\).

  11. A formal system any two models of which are isomorphic is called categorical. It follows from the SLT that no formal system that has a non-denumerable model is categorical because this model is reducible to a denumerable one.

  12. For such a case, see: Isham (1994, pp. 29–32).

  13. This turns out to be the case in the proof of existence of nonstandard models for Peano arithmetic, (PA), where by the application of the Completeness Theorem and the upward SL Theorem, PA has been proved to have, except for the standard countable model, infinitely many non-isomorphic ones, i.e., it has models for every cardinal \(\kappa > \aleph _{0}\); see van Dalen (2004, p. 113).

  14. If the exposition proves too mathematically demanding or cumbersome for the generally interested reader he may choose to follow the key steps of the argumentation skipping some technical details.

  15. Da Costa & French state that this relation of representation remains problematic in their version of a semantic approach where broadly ‘theoretical’ structures are related via partial isomorphisms to equally broadly ‘phenomenological’ ones in which case the latter ‘represent’ the phenomena (da Costa and French 2003, fn. 46, p. 211).

  16. The term immanent, widely used in phenomenological and in a broader sense philosophical texts, can be roughly explained as referring to what is or has become correlative (or ‘co-substantial’) to the being of one’s consciousness in contrast to what is ‘external’ or transcendental to it. For instance, a tree is transcendental as such to the consciousness of an ‘observer’ while its appearance in the modes of its appearing within his consciousness is immanent to it.

  17. It is notable, however, that later Schoenfield showed that forcing on partial orders can catch the ‘essence’ of the Boolean approach in a straightforward way, while Boolean models proved also to be cumbersome and unintuitive in the search for new consistency results (Kanamori 2012, p. 56).

  18. It happens that statements with universal and existential quantifier are not exempt themselves from opposing arguments regarding their status within a formal theory, especially with regard to the supposedly ontological claims attributed to the bounded variables. See: Quine (1947, pp. 75–77) and Feferman et al. (1995, p. 341, fns. 19 & 20).

  19. Gödel’s constructible universe L can be roughly described as a subuniverse of the classical universe of sets V constructed in stages along a transfinite generation of ordinals where each stage contains sets definable by well-formed formulas of set theory in the previous stage. The universe L is a prototype of the broader concept of inner models developed in the last decades with the purpose of establishing ever larger infinities.

  20. A broader countability condition, the CCC (Countable Chain Condition), is also necessarily assumed in eliminating an uncountable number of incompatible forcing conditions in the completion of Cohen’s disproof of CH. See: Kunen (1982, pp. 205, 206).

  21. To explain the notion of figural moment Hauser refers to Husserl’s views in as early as the Philosophy of Arithmetic where he proposed that in the intuition of a sensory set must lie immediately comprehensible signs by which the set character can be recognized. Husserl called these ‘immediately comprehensible signs’, e.g. of sethood, figural moments (Hauser 2005, p. 246).

  22. The phenomenological term noema can be roughly considered as a generalization of meaning-giving to all acts, including also all linguistic activities, and is inherently associated with the meaning of a noematic object as intended. This latter object is constituted by certain modes as a well-defined object immanent to the temporal flux of consciousness and it is possibly abstracted, in the sense of a formal-ontological object, as a syntactical object of a formal theory. In Hauser and Öner (2018), Hauser offers a nice parallelism in that the the relation between noema and intentional object is to be construed in terms of satisfaction in the model-theoretic sense, while noema and noesis (i.e., the meaning-giving moment of an act) may stand in a relation of correspondence (Hauser and Öner 2018, p. 32).

  23. The reader may consult in Livadas (2015, 2016) an approach based on a reduction of the infinity of continuum and of any infinity beyond it to the mode of constitution of immanent continuous unity.

  24. A relevant case may be thought of the topological modelization of temporal processes in quantum evolution presented as quantum histories formalism. Again as in set-theoretical structures the issue is the incompatibility, translated in topological terms, between the distinct and the continuous, more precisely, between single-time and continuous-time projection operators. See: Anastopoulos (2001) and Isham (1994).

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Livadas, S. Talking About Models: The Inherent Constraints of Mathematics. Axiomathes 30, 13–36 (2020). https://doi.org/10.1007/s10516-019-09431-4

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