The Geometry of History; 032147658
In a hitherto unpublished lecture, delivered in Atlanta, 1975, W.R. Knorr reflects on historical method, its sensitivity to modern work, both in mathematics and in the philosophy of mathematics. Three examples taken from the work of Tannery, Hasse, Scholz and Becker and concerning the study of pre-euclidean geometry are discussed: the mis-described discovery of irrational ‘numbers’, the alleged foundations crisis in the 5th century B.C. and the problem of constructibility.
By taking into account the work of J. Rataj and M. Zähle [Geom. Dedicata 57, 259-283 (1995; Zbl 0844.53050)], R. Schneider and W. Weil [Math. Nachr. 129, 67-80 (1986; Zbl 0602.52003)], W. Weil [Math. Z. 205, 531-549 (1990; Zbl 0705.52006)], an integral formula is obtained here by using the technique of rectifiable currents.This is an iterated version of the principal kinematic formula for sets of positive reach and generalized curvature measures.
The Legendre transformations on differential spaces (in the sense of Sikorski) is studied, and some properties for spaces with singularities are investigated. A mechanical interpretation of the Legendre transformation is also given.