Abstract
The main topic of this essay is symbolic mathematics or the method of symbolic construction, which I trace to the end of the sixteenth century when Franciscus Vieta invented the algebraic symbolism and started to use the word ‘symbolic’ in the relevant, non-ontological sense. This approach has played an important role for many of the great inventions in modern mathematics such as the introduction of the decimal place-value system of numeration, Descartes’ analytic geometry, and Leibniz’s infinitesimal calculus. It was also central for the rigorization movement in mathematics in the late nineteenth century, as well as for the mathematics of modern physics in the 20th century.However, the nature of symbolic mathematics has been concealed and confused due to the strong influence of the heritage from the Euclidean and Aristotelian traditions. This essay sheds some light on what has been concealed by approaching some of the crucial issues from a historical perspective. Furthermore, I argue that the conception of modern mathematics as symbolic mathematics was essential to Wittgenstein’s approach to the foundations and nature of mathematics. This connection between Wittgenstein’s thought and symbolic mathematics provides the resources for countering the still prevalent view that he defended an uttrely idiosyncratic conception, disconnected from the progress of serious science. Instead, his project can be seen as clarifying ideas that have been crucial to the development of mathematics since early modernity.
Highlights
Sören Stenlund CC-BY what has been concealed by approaching some of the crucial issues from a historical perspective
The disagreement about the origin of proof by mathematical induction is only a particular case of a more general historiographical debate, or quarrel, about the history of ancient Greek mathematics that started with Sabatai Unguru’s article “On the Need to Rewrite the History of Greek Mathematics”, published in 1975.4 Unguru claims that lack of historical sense is a common feature of most mathematicians’ readings of ancient mathematical texts in that they tend to read the ancient texts only from the point of view of modern mathematics
The Greek mathematician who seems to have come closest to the conceptual transformation in which algebraic symbolism originated, was Diophantus of Alexandria
Summary
The rise of symbolic mathematics in the seventeenth century was not just a more or less continuous course of development, or an extension, of ancient Greek and medieval mathematical thought. The normal interest in the history of mathematics (among mathematicians who write history of mathematics) is an interest in our mathematical heritage This interest tends to be conditioned by the contemporary situation and is not always an interest in what happened in mathematics of the past regardless of the contemporary situation. History in the latter sense deserves to be called history.. History and heritage are often confused and one consequence of this kind of confusion is that the transformation of mathematics at the beginning of modern times is concealed. By taking a closer look at some examples from the history of mathematics it will be argued that what happened in mathematics in the past supports the view that the essence of modern mathematics is symbolic mathematics
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