“Le complément supérieur”: On the Poetics of Mathematics

This chapter considers mathematics, with a special emphasis on geometry, as combining mathematical and artistic thinking. By artistic thinking I do not refer to aesthetic aspects of mathematics or mathematical aspects of art or aesthetics, which subjects are only marginally addressed in the article. Instead, I focus on creative aspects of mathematics as the invention of new concepts, theories, or fields. As my subtitle indicates, this argument, beginning with the term “poetics,” follows that of Aristotle’s Poetics [Peri poietike~s], a treatise on ancient Greek poetry, the title of which is derived from the ancient Greek word poeien meaning “making,” putting something together. Aristotle’s Poetics is about how literature is made or composed. Aristotle did not apply the term poetics to mathematics and did not consider mathematics in this way, focusing instead, in his other works, on logical aspects of mathematics. By contrast, I argue, under the heading of the composition principle, that, as a creative endeavor, mathematics is primarily defined by its compositional nature, rather than by its logical or calculational aspects, essential as the latter are. Of course, while compositional, mathematics is not the same as literature and art. In particular, the poetics of mathematics is the poetics of concepts, which play a more limited role in literature and art, and ally mathematics or mathematical sciences, such as physics, more with philosophy. Two other principles define mathematics, as well as science, in the present view: the continuity principle (found in literature and art as well) and the unambiguity principle (not necessary in literature and art). The chapter introduces yet another principle in considering the nature of reality, the “reality without realism” (RWR) principle, which in the case of mathematics, where the primary reality considered in mental, becomes the “ideality without idealism” (IWI) principle.