Three absent facts of mathematical structuralism
Philosophy "Mathematical objects, structures and proofs" (thematic issue)
DOI:
https://doi.org/10.17072/2078-7898/2022-3-389-398Keywords:
mathematical structuralism, mathematical object, structure, identity, ontological relativityAbstract
The mathematical structuralism of M. Resnik is a possible solution to the problems posed by P. Benacerraf for any adequate philosophy of mathematics. In the paper, the epistemological part of the concept is briefly outlined, and it is shown how, according to M. Resnik, we can obtain the knowledge of mathematical objects by disregarding the perceptual data. The ontological part, according to which consistency is taken as a criterion of existence, is also considered. To understand positions, M. Resnik uses the metaphor of a geometrical point. Thus, positions cannot be compared with one another in case they belong to different structures just as points cannot be individuated in case they do not belong to the same plane. Mathematical structures can be in relations of congruence, occurrence, and definitional equivalence, but there is no identity relation for them since the positions of the structures do not necessarily coincide. In addition, the paper compares M. Resnik’s conception of structural relativity with W.V.O. Quine’s ontological relativity. From the conception of structural relativity naturally follows what can be called the doctrine of three types of absent facts. Each type of absent facts is explained separately, then it is demonstrated that some interpretations of P. Benacerraf’s identification problem are incorrect.References
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