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)

We argue that the later Wittgenstein’s philosophy of language and mathematics, substantially focused on rule-following, is relevant to understand and improve on the Artificial Intelligence (AI) alignment problem: his discussions on the categories that influence alignment between humans can inform about the categories that should be controlled to improve on the alignment problem when creating large data sets to be used by supervised and unsupervised learning algorithms, as well as when introducing hard coded guardrails for AI models. We cast these (...) considerations in a model of human–human and human–machine alignment and sketch basic alignment strategies based on these categories and further reflections on rule-following like the notion of meaning as use. To sustain the validity of these considerations, we also show that successful techniques employed by AI safety researchers to better align new AI systems with our human goals are congruent with the stipulations that we derive from the later Wittgenstein’s philosophy. However, their application may benefit from the added specificities and stipulations of our framework: it extends on the current efforts and provides further, specific AI alignment techniques. Thus, we argue that the categories of the model and the core alignment strategies presented in this work can inform further AI alignment techniques. (shrink)

The conditions involved in the applicability of mathematics in science are the subject of ongoing debates. One of the best‐received approaches is the inferential account, which involves structural mappings and pragmatic considerations in a three‐step model. According to the inferential account, these pragmatic considerations happen in the immersion and interpretation stages, but not during derivation (symbol‐pushing in a mathematical formalism). In this work, I draw inspiration from the later Wittgenstein and make the case that the applicability of mathematics also rests (...) on pragmatic considerations at the mathematical derivation level. I make the further case that pragmatic considerations at the mathematical derivation level may substitute pragmatic considerations in the other two stages, thus showing how they are holistically interrelated. I illustrate these two points with the solution of a simple geometrical problem. (shrink)