The infinite in early modern philosophy

Quinn, Nicola Bridget (2019) The infinite in early modern philosophy. Masters thesis, Birkbeck, University of London.

Abstract

The calculus was developed in the seventeenth and eighteenth centuries in order to solve certain mathematical problems. In ‘The Analyst’ Berkeley gave his objections to the calculus. In particular, he objected to its dependence upon infinitesimals. He argued that these infinitely small distances were contradictory. Further he objected that the calculus, despite leading to true conclusions was not genuine science. So, although the calculus worked, it raised philosophical problems. I will look at Berkeley’s objections, both logical and metaphysical and explore how philosophically problematic they are. I will then consider what consequences arise from these objections. Specifically, I will look at the consequences for mathematical realism, and aim to answer the following questions: If Berkeley’s philosophical arguments are valid how can we explain that the calculus enables us to derive useful results? If we rely on non-entities such as infinitesimals in our mathematical proofs, does this mean that platonism cannot be the true metaphysical framework? If we reject platonism, does the dependence upon nonentities in the proofs of the calculus mean that all forms of mathematical realism are off the table? Or, is this methodology consistent with non-platonic forms of mathematical realism such as truth-value realism. Or, must we look to an alternative picture, such as fictionalism?

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