Axioms and aesthetics in constructive mathematics and differential geometry.

摘要

We often find theorems or proofs especially pleasing because of their constructive, structural, categorical, or synthetic approaches. In this dissertation, I articulate these four aesthetic ideas and some mathematical programs motivated by them.; Given the aesthetics, we might choose to follow them loosely, as exhortations. Understanding these exhortations clearly may help us to achieve our aesthetic or other mathematical goals. Alternatively, we might find axioms motivated by the aesthetics, and then might pursue the consequences of those axioms. Working within the context of these novel (sometimes oddly restricted) axioms can be and has been a good way of generating ideas useful in ordinary mathematics.; Errett Bishop's constructive mathematics has especially strong aesthetic motivations. His overarching themes are investigating computations and giving existence proofs which provide algorithms for constructing the relevant objects. The corresponding aesthetic principles include "make every concept positive" and "use the relevant definitions". These principles can be important for pedagogy, for axiomatics, and in some areas of differential geometry.; Saunders Mac Lane is a representative of the contrasting structural approach. He has articulated many structural aesthetics, including "use categorical definitions", and "prove representation theorems". Others have applied structural ideas in axiomatic set theory.; These aesthetic approaches to mathematics in the large have parallels in differential geometry, as in the categorical aesthetics of William Lawvere's smooth infinitesimal analysis and the synthetic aesthetics of Herbert Busemann's theory of G-spaces. Lawvere's aesthetics include "define objects categorically" and "work in categories which allow many categorical constructions". Busemann's aesthetics include "make infinitesimal concepts local" and "reduce differentiability hypotheses". These two approaches to differential geometry provide interesting and fruitful contrasts to the more standard manifold-based approach.; Three philosophical doctrines underlie and add importance to the above. According to formalism, modern mathematics relies on our ability to translate it into formal terms. According to pluralism, there are many axiom systems for different pieces of mathematics. According to aestheticism, mathematical aesthetics are important. Correspondingly, we should seek and develop more mathematics that satisfies these descriptions; and this dissertation aims to contribute to that process.