I have been reading’s Field’s “Mathematical Objectivity and Mathematical Objects” again as part of the process of doing a review (Bueno, 2018; Field, 2003). I still favour an informational Aristotelian immanent or in re realism about numbers according to which different concepts of number (Zermelo, Von Neumann, structuralist, Platonist, fictionalist...) just involve alternative encodings of partial information in/from natural structures (and patterns therein).
One can do this with subjectivist-cognitivism supporting constructivism/intuitionism about numbers, or else have an objective conception - or both. In other words – one can include the brain-mind as a necessary encoding mechanism that is intrinsic in a constructivist sense, or omit it and model the situation in objective terms (including the brain-mind as encoder or not.)
This fits, or is easily reconcilable, with an informational metaphysics something like Floridi's informational structural realism (Floridi, 2008; Long, 2018). One can also characterise the incomplete mathematical objects that Field discusses (cit. Resnik, Shapiro) in terms of partial encoding of information from natural patterns (patterns occurring in structures in nature.) (Resnik, 1997)
From an informational perspective Field's ambitious nominalist approach then starts to look like a process of identifying patterns in natural structures and encoding partial information from them to establish properties for doing math (without reference to Platonic entities) which is similar in many ways to Aristotelian realism (Franklin, 2014). This is curious since Aristotelian in re realism is usually thought to be close to a kind of very weak Platonism.
References
Bueno, O. (2018). Putnam’s indispensability argument revisited, reassessed, revived. Theoria (Spain). https://doi.org/10.1387/theoria.18473
Field, H. (2003). Mathematical Objectivity and Mathematical Objects. Truth and the Absence of Fact, 315–331. https://doi.org/10.1093/0199242895.003.0011
Floridi, L. (2008). A defence of informational structural realism. Synthese. https://doi.org/10.1007/s11229-007-9163-z
Franklin, J. (2014). An aristotelian realist philosophy of mathematics: mathematics as the science of quantity and structure. Basingstoke, U.K: Palgrave Macmillan.
Long, B. (2018). ISR is Still a Digital Ontology. Erkenntnis. https://doi.org/10.1007/s10670-018-00415
Resnik, M. D. (1997). Mathematics as a science of patterns. Oxford;New York; Clarendon Press.
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