The intuitions of higher dimensional algebra for the study of structured space

Higher dimensional algebra frees mathematics from the restriction to a purely linear notation, in order to improve the modelling of geometry and so obtain more understanding and more modes of computation. It gives new tools for noncommutative, higher dimensional, local to global problems, through the notion of «algebraic inverse to subdivision». We explain the way these ideas arose for the writters, in extending first the classical notion of abstract group to abstract groupoid, in which composition is only partially defined, as in composing journeys, and which brings a spatial component to the usual group theory: An example from knot theory is used to explain how such algebra can be used to describe some structure of a space. The extension to dimension 2 uses compositions of squares in two directions, and the richness of the resulting algebra is shown by some 2-dimensional calculations. The difficulty of the jump from dimension 1 to dimension 2 is also illustrated by the comparison of the commutative square with the commutative cube-discussion of the latter requires new ideas. The importance of category theory is explained, and a range of current and potential applications of higher dimensional algebra indicated.     

The Great Physician: mainline Protestant ministers’ expectations of doctors, 1840–1900

I argue that mainline Protestant clergymen adapted to the scientific and cultural developments of the post-Darwin era by encouraging a separation of bodily healing and spiritual healing that was unprecedented in Christianity. I use ministers’ depictions of Jesus in their addresses to physicians to illustrate this change. In 1840 ministers insisted that doctors’ value lay in their willingness to mimic Jesus’s example by healing both bodies and souls. As doctors gained authority in the late 1800s, the clergyman’s new ideal doctor stayed abreast of scientific developments and stuck to healing bodies. Ministers increasingly portrayed Jesus as an exclusively spiritual healer.