Two scientists more than anyone else have contributed in defining our understanding of gravity: Newton in 1679 and Einstein in 1915. The mathematical frameworks the two have developed and proposed, however, are very different. Newton’s gravity is the one we learn at school and is normally taught at university. It provides a very natural interpretation of what we experience - the apple falls from the tree because the Earth attracts it! Einstein’s gravity is studied only in the most advanced courses at the university and provides a very counterintuitive explanation, requiring the concepts of spacetime and curvature. This chapter will provide a first description of the Einstein equations and, although it will not enter into the mathematical aspects of the equations, it will explain the basic concepts behind them. Acquiring a first qualitative understanding of Einstein equations will be useful to comprehend better the concept of spacetime curvature discussed in Chapter 4.

This chapter looks at the mathematization of the study of nature by focusing on how practical mathematicians from the sixteenth century onward understood mathematics as primarily devoted to solving problems through mathematical construction. This constructive understanding of the nature of mathematics is then related to the double movement of physicalizing mathematics (giving physical interpretations to mathematical constructions) and mathematizing physics (understanding physics as basically involving the solution of problems). The work of seventeenth-century thinkers like Galileo, Descartes, and Mersenne is used to further illustrate these ideas, which led to the establishment of mathematical physics as characterized by its problem-solving nature.

Sidereus Nuncius was a seminal text of the Scientific Revolution. It reported celestial observations whose implications upended the geocentric cosmology few had ever doubted. But Galileo’s treatise also combined topics and methodologies that traditionally had been assigned separately to mixed mathematics and natural philosophy. Whereas the bounds between these disciplines had been weakened by earlier controversies, particularly about the regressus method and about the certainty of mathematics (i.e., the Quaestio de Certitudine), Sidereus Nuncius broke them down altogether, to the delight and dismay of readers. Galileo’s application of mathematical methods—empiricism and quantification—to natural philosophy framed the ensuing discussions, such that even those who disagreed with his conclusions responded on those grounds. Thus, the book was pivotal in the process of disciplinary admixture that reorganized the study of nature into modern science.

The discovery and implementation of telescopy and microscopy extended the power of the senses, with often surprising results. The discoveries made with these instruments challenged existing theories of nature, while simultaneously demanding new theories to account for their operation, especially of vision. Moreover, early modern epistemology had to accommodate experiential evidence made by these “artificial” means. The chapter also address advances in the literature since the twenty-year-old classic, The Invisible World by Catherine Wilson.