The Importance of Empirical Evidence for Mathematics

What is it about?

After a general introduction, the essay focuses specifically and selectively on the philosophy of mathematics of Italian Renaissance scholars, and the limitations that their views imposed on the application of statical principles to dynamics, which, in turn, stimulated efforts of an empirical nature that had a major impact on the further development of mathematics. In the second part the essay turns to the circumstances, theoretical and philosophical, that led to the introduction of empirical considerations and even early efforts at experimentation that challenged earlier assumptions about mathematics and its relation to nature.

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Why is it important?

The principal aims of this paper are to illustrate the consequences of empirical considerations and some experiments for mathematical theory, the understanding of continua and infinitesimals, and the relation between mathematics and the physical universe. In particular, the principal claims of this paper are the following: 1. The historical record complicates the distinction between applied and pure mathematics. 2. The tension between deductive rigor and empirical, material, and physical conditions stimulated mathematical developments with regard to infinitesimal analysis and limit theory. 3. There were major contributions to ‘pure’ mathematics with respect to the relation between discrete numbers and geometrical magnitudes, ratios and proportions, and the completeness of proofs.

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