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Impure Systems and Ecological Models (I): Axiomatization

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Abstracts

Building models as a practical aspect of ecological theory has as a principal purpose the determination of relations in formal (mathematical) language. In this paper, the authors provide a formalization of ecological models based on impure systems theory. Impure systems contain objects and subjects: subjects are human beings. We can distinguish a person as an observer (subjectively outside the system) that by definition is the subject himself and part of the system. In this case he acquires the category of object. Objects (relative beings) are significances, which are the consequence of perceptual beliefs on the part of the subject about material or energetic objects (absolute beings) with certain characteristics. The impure system approach is as follows: objects are perceptual significances (relative beings) of material or energetic objects (absolute beings). The set of these objects will form an impure set of the first order. The existing relations between these relative objects will be of two classes: transactions of matter and/or energy and inferential relations. Transactions can have alethic modality: necessity, possibility, impossibility and contingency. In this work we define measures which let us choose the more suitable variables to relate both the model with the ecosystem and with different models. In this way we define different comparison indexes.

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Notes

  1. Semiotics is the theory and study of signs and symbols.

  2. See more details on the paper: “Conclusive reasons that we perceive sets (MacCallum 2000)”.

  3. The Quine–Putnam mathematical indispensability argument asserts that mathematical entities are on a par with other scientific entities from our best scientific theories. This argument is an argument for mathematical realism. Mathematical entities exist because they are indispensable in our best scientific theories.

  4. Let Ω be an open connected space in the complex plane (Markushevich 1978): we will call H Ω the set of all analytical functions over Ω which have a ring structure with the operations of addition and product. For F \(\subset\) H Ω, A(F) will denote the subring generated by F and \(A*(f)\) will be the set of analytical functions which depend algebraically on some subset of F.

    1. 1.

      \(A*(f)\) ⊂ E

    2. 2.

      If f ∈ E then \(f^{{\prime }}\), Ef, Pf Lf ∈ E being \(f^{{\prime }}\) the derivative, Ef the exponential of f, Pf a primitive and Lf the logarithm of f.

    3. 3.

      If f, g ∈ E, then f + g, f/g, (g ≠ 0), fg [that is exp(glog f)], are elements of E.

    4. 4.

      If f, g ∈ E y f(Ω) ⊂ Ω then \(g \circ f\) ∈ E.

    Given F \(\subset\) H Ω any function like the following one: f 1 o f 2 o…o fn with fi \(\in\) F, \(\forall\) i = 1,2,…,n we will call a transformed function (Usó-Domènech et al. 1997) of order n.

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Authors and Affiliations

  1. Department of Applied Mathematics, University of Alicante, Alicante, Spain

    José-Luis Usó-Doménech, Josué-Antonio Nescolarde-Selva & Miguel Lloret-Climent

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  1. José-Luis Usó-Doménech
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  2. Josué-Antonio Nescolarde-Selva
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  3. Miguel Lloret-Climent
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Correspondence to Josué-Antonio Nescolarde-Selva.

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Usó-Doménech, JL., Nescolarde-Selva, JA. & Lloret-Climent, M. Impure Systems and Ecological Models (I): Axiomatization. Found Sci 23, 297–321 (2018). https://doi.org/10.1007/s10699-017-9522-2

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