# Antoni Malet

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## Towards a history of mathematical consistency: tacit knowledge and conceptual change in early modern mathematics

In the crucial early modern period, fundamental notions like number, geometrical magnitude, and ratio, essentially defined à la Euclid in 1500, had been abandoned by 1700. By then infinitesimals, a notion inconsistent with traditional mathematics, were widely used in mathematics and rational mechanics. No satisfactory account exists for such a revolution, although traditional histories of mathematics stressed the "lack of rigour" of early modern mathematicians. In fact, the new notions of number and magnitude and the legitimation of infinitesimal thinking provide striking evidence that in critical episodes the evolution of mathematics is not guided by deductive rules nor by mathematical consistency. Motivations and rationale must be social in nature.

This project studies the emergence of implicit novel mathematical notions in mathematical practice, as opposed to mathematical texts. In particular, we focus on practical geometry and the practitioners of the *art of measuring* as sources for tacitly introducing some form of arithmetical continuum. We want to reevaluate the criticism of "lack of rigour" in early modern calculus by situating the foundational discussions within the wider social context in which the sceptical crisis of the early enlightenment discredited traditional axiomatico-deductive structures of knowledge. In particular we think we will be able to show how fruitful turns out to be a Wittgensteinian account of 17th century philosophy of mathematics.

## biographie

Antoni Malet is Professor of History of Science at the Universitat Pompeu Fabra (Barcelona). He majored in Mathematics (Universitat de Barcelona) and got a doctorate in History at Princeton University. He has been research fellow or visiting professor at the universities of Princeton, California (San Diego), Toronto, and París VII, and Marie Curie senior Fellow at the Max Planck Institut for the History of Science, Berlin (2013-2015). Editorial Board member of Annals of Science and Historia Mathematica, where he served as Book Review Editor (2006-2011), he is Fellow of the Académie Internationale d'Histoire des Sciences, and currently the President of the European Society for the History of Science (2016-2018). He is working on the sociological understanding of conceptual change in early modern mathematics.