Of all the demands that mathematics imposes on its practitioners, one of the most fundamental is that proofs ought to be correct. It has been common since the turn of the twentieth century to take correctness to be underwritten by the existence of formal derivations in a suitable axiomatic foundation, but then it is hard to see how this normative standard can be met, given the differences between informal proofs and formal derivations, and given the inherent fragility and complexity of the latter. This essay describes some of the ways that mathematical practice makes it possible to reliably and robustly meet the formal standard, preserving the standard normative account while doing justice to epistemically important features of informal mathematical justification.
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Hamami (in press) does a particularly good job of clarifying the distinction between an informal foundation for mathematical reasoning and a formal one.
In his book, Re-engineering Philosophy for Limited Beings (Wimsatt 2007), William Wimsatt raises a similar concern about the robustness of long chains of reasoning in the context of scientific practice. See Chapter Four, titled “Robustness, Reliability, and Overdetermination.”
Pelc (2009) raises a more fundamental concern, namely, that complex theorems of mathematics may not have formal counterparts that can ever be verified mechanically by any processing device, given physical constraints such as the number of particles in the universe. But contemporary work in formal verification and interactive theorem proving establishes that they do, rather conclusively.
For an example of a fruitful interplay along these lines, see Hamami et al. (2019).
For example, even Philip Johnson-Laird, a foremost proponent of mental models, seems to acknowledge the use of inferential chains in solving logic problems (Johnson-Laird 2010, p. 5).
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I am grateful to Silvia De Toffoli, Yacin Hamami, and three anonymous reviewers for helpful comments and suggestions. This work has been partially supported by AFOSR Grant FA9550-18-1-0120 and the Sloan Foundation.
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Avigad, J. Reliability of mathematical inference. Synthese (2020) doi:10.1007/s11229-019-02524-y
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