Why did later philosophers reject John Stuart Mill’s explanation of mathematical certainty?

John Stuart Mill argued that mathematical truths were empirical generalizations derived from long-standing, unbroken experience, suggesting that their apparent necessity was merely a habit of thought. Later philosophers, including logical positivists and those following the tradition of Gottlob Frege and Bertrand Russell, rejected this view for several key reasons:1234

• Lack of Necessity and Universality: Critics argued that Mill’s empiricist account failed to explain why mathematical truths are necessary and universal. They maintained that no amount of sense experience could guarantee that a proposition must be true or that it will always hold in the future.2
• Analytic Nature: Logical positivists contended that mathematical statements are analytic propositions—true solely by virtue of the meanings of the terms involved and the rules of logic. Because they are a priori and independent of experience, they cannot be refuted by observation.32
• Logical Foundations: Philosophers like Frege sought to demonstrate that mathematics could be reduced to fundamental logical laws, establishing it as a formal system rather than an empirical science.34

Consequently, the view that mathematics is merely an empirical statement of high generality was largely abandoned, even by many empiricists, as it failed to account for the certainty inherent in mathematical reasoning.2