Justifying assumptions in mathematical descriptions: a modeling practice methodized by biological theory
Résumé
A mathematical description relies on assumptions about some aspects of things which are not under focus in the description. In models, they are manifested by symmetries, initial and boundary conditions, parameters, etc. Those things are externally determining in the sense that they contribute to the determination of the mathematically formalized unknowns (state), but cannot be affected by them. Now, most assumptions rely on pieces of knowledge regarding the relative invariance of those determining aspects of things. Yet, those invariances are, in fact, bounded in time. Thus, assumptions and therefore mathematical descriptions relying on them cease to be valid at some point.
Based on this epistemological difficulty, we introduce a modeling framework to justify the undetermined knowns (assumptions) of a mathematical description by the determined unknowns of other mathematical descriptions. Notably, we introduce objects defined by mathematical descriptions, oriented relations of determination between them, and discuss some elementary diagrams with examples. We distinguish synchronic from diachronic relations of determinations, which allows us to consider the beginning, change, and end of lasting relations of determinations.
Apart from shedding new lights on the object of physics, this work formalizes a modeling practice that is theoretically relevant in other sciences resorting in part to the same epistemology. To emphasize this aspect, we show how it integrates a theory in biology via closure of constraints. By positing as fundamentally invariant the closure of a self-determining set of objects describing a biological system, the possible variations of the latter are unbounded as long as it remains organizationally closed. This work therefore paves the way for modeling the open-ended evolution of an organized entity using organized mathematical descriptions.
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