A Combination of Nonstandard Analysis and Geometry Theorem Proving, with Application to Newton's Principia

The theorem prover Isabelle is used to formalise and reproduce some of the styles of reasoning used by Newton in his Principia. The Principia's reasoning is resolutely geometric in nature but contains "infinitesimal" elements and the presence of motion that take it beyond the traditional boundaries of Euclidean Geometry. These present difficulties that prevent Newton's proofs from being mechanised using only the existing geometry theorem proving (GTP) techniques. Usin

'A new sense of place?' Mobile 'wearable' information and communications technology devices and the geographies of urban childhood

In this paper we describe a new research initiative, 'A New Sense of Place?', which involves the collaboration of private- and public-sector partners. Its purpose is to explore and develop the interface between children and new mobile 'wearable' computing and communication devices. The research team is particularly interested in how these new technologies might be applied to help children (re-)engage with urban spaces. In the paper we give a description of wearable computing devices; briefly set out some contexts of children's geographies into which they are emerging; and describe the rationale and objectives of the project. We then give an account of a two-day workshop in which 10 children were introduced to and enabled to experience, work with and respond to these new technologies. The research shows that children are capable of handling and exploiting these technologies and are able to conceptualise their incorporation into their everyday lives. Also, it reveals that the creation of 'virtual' digital landscapes, which these technologies allow, has the potential to represent adult-ordered spaces in more 'child-friendly' forms. Lastly, the programme opens up new questions of power, surveillance and childhood-technology relations

Proving Newton’s Propositio Kepleriana using geometry and nonstandard analysis in Isabelle

The approach previously used to mechanise lemmas and Kepler's Law of Equal Areas from Newton's Principia [13] is here used to mechanically reproduce the famous Propositio Kepleriana or Kepler Problem. This is one of the key results of the Principia in which Newton demonstrates that the centripetal force acting on a body moving in an ellipse obeys an inverse square law. As with the previous work, the mechanisation is carried out through a combination of techniques from geometry theorem proving (GTP) and Nonstandard Analysis (NSA) using the theorem prover Isabelle. This work demonstrates the challenge of reproducing mechanically Newton's reasoning and how thecombination of methods works together to reveal what we believetobe awin Newton's reasoning