6 ideas
Classical probabilists treated their calculus centrally as a formal model of how an enlightened, reasonable person reasons under uncertainty, alongside physical and frequency interpretations that also appear in their work. Where the mathematics diverged from the intuitions of that reasoner, the theory was often adjusted to fit good sense rather than the reverse.
For Huygens and the first generation of probabilists, expectation (the fair price of an uncertain prospect, modelled on aleatory contracts, gambles, and legal shares) was the primary concept, with probability derived from it. This tied the early mathematics to legal notions of equity and fair exchange, though later probabilists reversed the order and defined expectation as probability multiplied by the value of the outcome.
A coin game with infinite mathematical expectation is one no reasonable person would pay more than a small sum to play, so the calculus and good sense openly disagreed. Rather than overrule intuition, Daniel Bernoulli and others revised the mathematics, introducing 'moral expectation' with diminishing utility of wealth, showing the theory was being bent to fit the reasonable man.
Eighteenth-century probabilists could slide between probability as degree of belief and probability as observed frequency because associationist psychology held that the mind's strength of belief naturally tracks the frequency of experienced conjunctions. This psychological bridge made the classical interpretation's ambiguity between epistemic and statistical probability seem harmless rather than confused.
Classical probabilists extended the calculus to the credibility of witnesses and the optimal size and voting rules of juries and tribunals, treating legal certainty as a quantifiable function of independent reliable judgements. These moral-science applications, pursued by Condorcet and Laplace and later Poisson, were central to the program, not peripheral curiosities.
After the French Revolution and in the early nineteenth century, the belief in a single homogeneous 'reasonable man' whose judgements could serve as the standard dissolved, and new social statistics revealed stable mass regularities independent of individual reasoning. Stripped of its psychological and social foundation, probability split into frequentist and subjectivist interpretations, and applications to moral questions like jury verdicts were denounced as the scandal of mathematics.