If the model itself is to be assigned any kind of ‘existence’, then this existence is located within the Platonic world of mathematical forms. Of course, one might take a contrary viewpoint: namely that the model is itself to have existence only within our various minds, rather than to take Plato’s world to be in any sense absolute and ‘real’. Yet, there is something important to be gained in regarding mathematical structures as having a reality of their own. For our individual minds are notoriously imprecise, unreliable, and inconsistent in their judgements. The precision, reliability, and consistency that are required by our scientific theories demand something beyond any one of our individual (untrustworthy) minds. In mathematics, we find a far greater robustness than can be located in any particular mind. Does this not point to something outside ourselves, with a reality that lies beyond what each individual can achieve?
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From the perspective of Systemic Functional Linguistic Theory, mathematical models "exist" as semiotic systems, potential and instance, that are projections of mental and verbal processes. From this perspective, 'real' refers to the ideational meanings of semiotic systems, and so 'real' does not transcend semiotic systems. Reality is the meaning we construe of experience.
Mathematical models do have 'a reality of their own' in the sense that mathematics is a distinct semiotic system.
The opposition of 'imprecise judgements of individual minds' and the 'precision found in mathematics' is, in this context, the opposition of mentally projecting meanings of non-mathematical systems and of mentally projecting meanings of mathematical systems.
The sense in which mathematics is 'something that is outside ourselves' lies in it being an evolved social semiotic system.
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