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The foundations of statistics have evolved over many centuries, perhaps millennia, with major paradigm shifts of the form described in Kuhn (1962) . We briefly consider these important transitions and how they have led to major shifts in the foundations of statistical inference. Clearly there is no conventional mathematical or axiomatic basis. But there is a progressive clarification in the processes of statistical inference so that current theory can now coherently and definitively handle adoi:10.2991/jsta.2018.17.2.1 fatcat:jqtza5swxjgfnafzviltbkbm54