International Union of History and Philosophy of Science Division of Logic, Methodology and Philosophy of Science Bulletin no. 11

Risto Hilpinen
1987 Studia Logica: An International Journal for Symbolic Logic  
To this day Archimedes is regarded as the embodiment of mathematical creativity, his proofs are considered to be models of mathematical rigour, and his theorems are viewed as symbols of mathematical certainty. The lecture sheds some new light on Archimedes's own attitude concerning these matters, illustrates later developments, inspired by Renaissance thinkers like Kepler and Leibniz, and discusses some problems of modern discrete mathematics. The first section deals with Archimedean axiomatics
more » ... and proof standards, especially with the lemma named after him: Let a and b be two arbitrary real numbers. Then there is always a natural number n, such that na is larger than b. The second section is based on Kepler's 'New stereometry'. Kepler explicitly claimed to have used the laws of Archimedes, to have revealed the sense of his theorems, and the hidden strength of his demonstrations. While Guldin spoke of Kepler's 'new method of proving', A. Anderson denied Kepler's claim to have used the same laws as Archimedes. The third section demonstrates how Leibniz laid the rigorous foundation of the theory of infinitely small and infinite quantities by explicitly referring to Archimedes whose method of proof served as a model of rigour. The fourth and last section discusses new proof-checking problems. Since 1999 Th. Hales claims to have proved the so-called Kepler conjecture using a complex series of computer calculations. His proof has not been fully accepted up to now. Yet, mathematics is still considered as the realm of certainty. Nature has been inextricably connected with human life. There is nature's necessity, such as the cosmic order or bodily functions that men cannot change and have to comply with. There is nature's authority. People use nature to build up social norms and to justify social and political actions. There are also nature's ambiguities. There are no unchanging "essential" characteristics of nature and no fixed boundaries separating man and nature. Our concepts about nature and our relationships to it draw very much on the ideas and norms of society in which we are born, socialized, and educated. And the other way round. Nature may be described and understood in very different ways so that it may serve as justification for very different and sometimes even opposite social and political actions. Nature's more and political significances can be seen in all cultures and in all times in human history. This paper is about how about how ancient Chinese conceptualized nature and how they used their conceptions and understandings of nature to think about standards of the good, the just, and the valuable, and to justify their moral and political actions. It will also discuss how the Chinese emphasis on such moral and political meanings affected the style of the inquiry about nature.
doi:10.1007/bf00396908 fatcat:mgrd45qn4zfd3nystwqf34jv6m