Continuity in nature and in mathematics: Boltzmann and Poincaré

Synthese 192 (10):3275-3295 (2015)
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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. Through this development, differential calculus was made independent of empirical and intuitive notions of continuity, and based instead on strictly mathematical conditions of continuity. 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. For this reason, their ideas about continuity and discreteness in nature were entangled with epistemology and philosophy of mathematics. 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.



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Marij Van Strien
Radboud University

References found in this work

Reconsidering Logical Positivism.Michael Friedman - 1999 - New York: Cambridge University Press.
Wandering Significance: An Essay on Conceptual Behavior.Mark Wilson - 2006 - Oxford, GB: Oxford: Clarendon Press.
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Science and method.Henri Poincaré - 1914 - Mineola, N.Y.: Dover Publications.
Mathematical Thought from Ancient to Modern Times.M. Kline - 1978 - British Journal for the Philosophy of Science 29 (1):68-87.

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