If there is an area of discourse in which disagreement is virtually absent, it is mathematics. After all, mathematicians justify their claims with deductive proofs: arguments that entail their conclusions. But is mathematics really exceptional in this respect? Looking at the history and practice of mathematics, we soon realize that it is not. First, deductive arguments must start somewhere. How should we choose the starting points (i.e., the axioms)? Second, mathematicians, like the rest of us, are fallible. Their ability to (...) recognize whether a putative proof is correct is not infallible. In most cases, disagreement over the correctness of a putative proof is, however, evanescent. Once an error is spotted and communicated, the disagreement disappears. But this is not always the case. Sometimes it is recalcitrant; that is, it persists over time. In order to zoom in on this type of disagreement and explain its very possibility, we focus on a single case study: a decades-long (1921-1949) controversy between Federigo Enriques and Francesco Severi, two prominent exponents of the Italian school of algebraic geometry. We suggest that the instability of the mathematical community to which they belonged can be explained by the gap between an abstract criterion of rigor and local criteria of acceptability. It is this instability that made the existence of recalcitrant disagreement over putative proofs possible. We do not condemn speculative mathematics but rather its pretense of being rigorous mathematics. In this respect, we show that the overly self-confident Severi and the more intuitive, visionary Enriques had a completely different attitude. (shrink)

In this paper, I explore how the ideal rationality-based account of propositional justification impacts our understanding of the relation between propositional and doxastic justification. The ideal rationality-based account sits uncomfortably with the widely accepted claim that propositional justification is necessary for doxastic justification. In particular, the combination of the necessity claim and the ideal rationality-based account of propositional justification entails that some plausible doxastic attitudes are doxastically unjustified and thereby severs epistemic justification from connections with epistemic responsibility and the competent (...) formation of beliefs. Thus, those who endorse both the necessity claim and the ideal rationality-based account of propositional justification will be forced to give up either their account of propositional justification or the necessity claim. I argue that there is good reason to give up the ideal rationality-based account of propositional justification instead of the necessity claim. This result indicates that an adequate account of propositional justification should be sensitive to the agent’s cognitive capacities. If propositional justification should be understood in this way, then the widely accepted explanation of doxastic justification in terms of propositional justification is not promising either. This is because an agent’s cognitive capacities and limitations play a prominent role in understanding propositional justification. Instead, I argue that, at the explanatory level, propositional justification relies on doxastic justification. (shrink)