![]() Boltzmann’s computational style in mathematics and Meinong’s criticism of the confusion between representation and judgment give prominence to an indirect mode of presentation, adopted in a state of suspended belief which is characteristic of assumptions and which enables one to grasp objects that cannot be reached through direct representation or even analogies. An orientation other than the one which contributed to shape twentieth-century philosophy of science is explored through the analysis of the role given to assumptions in Boltzmann’s research strategy, where assumptions are contrasted to hypotheses, axioms, and principles, and in Meinong’s criticism of the privileged status attributed to representations in mental activities. Both are shown to rest on two core issues: the attitude of the subject and the mode of presentation chosen to display a domain of phenomena. Two complementary debates of the turn of the nineteenth and twentieth century are examined here: the debate on the legitimacy of hypotheses in the natural sciences and the debate on intentionality and ‘representations without object’ in philosophy. They reached opposite conclusions: Poincaré argued that physicists must work with a continuous representation of nature, and thus with differential equations, while Boltzmann argued that physicists must ultimately take nature to be discrete. For this reason, their ideas about continuity and discreteness in nature were entangled with epistemology and philosophy of mathematics. However, for Boltzmann and Poincaré, the applicability of mathematics in physics depended on whether there is a basis in physics, intuition or experience for the fundamental axioms of mathematics – and this meant that to determine the status of differential equations in physics, they had to consider whether there was a justification for these mathematical continuity conditions in physics. Through this development, differential calculus was made independent of empirical and intuitive notions of continuity, and based instead on strictly mathematical conditions of continuity. The development of rigorous foundations of differential calculus in the course of the nineteenth century led to concerns among physicists about its applicability in physics.
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