Semantic Models as Mediators
Presumably, one could say that semantic models represent fragments of the real world as mediators between them and theories.[103] This account of models became quite popular in the recent philosophy of science (see Morrison and Morgan 1999; Suarez 1999; Morrison 2015, Chap.
4). Mediating model have three principal properties:(a) They are not directly accessible from theories;
(b) They are not forced by observational data;
(c) The replace investigated phenomena.
From a very abstract point of view, mediating models in the contemporary understanding can be presented as algebraic structure, but such an approach would be at odds with their functions. In fact, the proponents of the idea of models as mediators, do not apply this notion to semantic models. They speak about models as results of mathematical modeling. The same lack of references to algebraic structures or semantic models occurs in typical surveys, like Zeigler (1976), Meyer (1984), Weisberg (2013), Pohljolainen (2016). Mathematical modeling is a branch of applied mathematics (see Higham 2015, Part V).
Although theoreticians and practitioners of mathematical modeling do not use semantic terminology, this attitude does not result from a anti-philosophical prejudice. In fact, the representational strength of semantic models is very weak. Even if one were incline to say that the structure represents the fact that Warsaw is a Polish city, this would be a trivial observation. Similarly, the structure, although is not trivial, provides no interesting information, because is entirely ex post.[104] On the other hand, the actual mathematical modeling provides powerful tools for description of empirical phenomena (see Pincock 2012), for instance, in order to represent them as discrete or continuous, measurable by ordering, additive or quotient scales, subjected to various statistical distributions, etc.
The Higgs boson would not be discovered without a proper simulation, the history of quantum and atomic physics is inherently associated with various models (see Cook 2006), etc. In general, a radically abstract character of contemporary physics makes it practically incomprehensible without modeling and simulating (see Falkenburg 1997; Lewis Peter 2016 for more philosophical reports about this situation).Mathematical modeling is essentially based on inputs coming from genuine empirical investigations. Even if (mathematical) models are regarded as more or less fictional, their users can precisely explain what is fictional and what is non-fictional in a given construction. Observe that comprehending the word ‘fiction’ and its various derivatives, assumes that we are able to explain ‘non-fictional’ as its antonym. There is no other way to learn how to use ‘fictional’ and ‘non-fictional’ as appealing to the world and its cognition. Clearly, a skeptic or a radical anti-realist can, using general philosophical arguments, always deny that cognitive acts are directed to real objects, but if one says something like that, he or she announces an understanding of the word ‘real’ and ‘unreal’. I do not claim that this observation refutes skepticism, but only that this view is much weaker than it is customary asserted. To conclude this paragraph, mathematical modeling could be difficult to understand it without asserting that it produces models responding to empirical inputs, which generate criteria for qualifying something as real or non-fictional and something else, if any, as just unreal or fictional. Mathematical models are mediators between theories and the world just in this sense.[105]
The scheme of mathematical modeling distinguishes the following main stages (see Pohjolainen and Heilio 2016, p. 2): real world problem, mathematical model, a solution for the model, interpretation of the solution, return to real world problem. Empirical ingredients of this schematization are evident.
Thus, mathematical modeling has its links with the world by definition. If mathematical models mediate between theories and empirically accessible facts, they must be also somehow related to the former, although this aspect does not occur in the quoted schematization. Yet ways of the interplay between theories and mathematical models are very different and depend on various circumstances. However, it seems that theories as generators of models are considered as correct or even true on a certain class of phenomena. On the other hand, qualifications of particular statement made on the occasion of modeling seems to be a secondary matter. Even if one were insist that, for instance, the sentence ‘changes of velocity as modeled by calculus are continuous' is true, this qualification remains on the level of (TE) and can be skipped without any loss of content. As a matter of fact, mathematical modeling only rarely becomes a subject of philosophical reflection perhaps except a general question of applied mathematics and its relation to pure mathematics.7