With their intensive use of data to classify and price risk, insurers have often been confronted with data-related issues of fairness and discrimination. This paper provides a comparative review of discrimination issues raised by traditional statistics versus machine learning in the context of insurance. We first examine historical contestations of insurance classification, showing that it was organized along three types of bias: pure stereotypes, non-causal correlations, or causal effects that a society chooses to protect against, are thus the main sources (...) of dispute. The lens of this typology then allows us to look anew at the potential biases in insurance pricing implied by big data and machine learning, showing that despite utopic claims, social stereotypes continue to plague data, thus threaten to unconsciously reproduce these discriminations in insurance. To counter these effects, algorithmic fairness attempts to define mathematical indicators of non-bias. We argue that this may prove insufficient, since as it assumes the existence of specific protected groups, which could only be made visible through public debate and contestation. These are less likely if the right to explanation is realized through personalized algorithms, which could reinforce the individualized perception of the social that blocks rather than encourages collective mobilization. (shrink)