Tata Institute of Fundamental Research
Non-euclidean Algebra
Seminar
Speaker:
Vaughan Pratt
Stanford University
Computer Science Department
2215 Old Page Mill Road
Palo Alto, CA 94304
Date:
Monday, 14 Feb 2011 (all day)
Venue:
AG-66 (Lecture Theatre)
(Scan to add to calendar)
Abstract:
Euclidean geometry today is most naturally understood as affine geometry equipped with a suitable bilinearity as a basis for notions of length, angle, area, rectangles, circles, etc. In this talk we express the affine content of Euclid's Postulates 2, 5, and 1 algebraically in that order. In our account Euclid's Fifth or Parallel Postulate is expressed as an equation parametrized with the number 6. Parameters in the range 3 to 5 yield discrete notions of elliptical geometry while 7 and beyond do the same for hyperbolic geometry. Had the tools of the past half century of universal algebra been available to Euclid, the first of his thirteen books could have been both simpler and more general, and the conceptual obstacles to hyperbolic geometry overcome more easily.
| Speaker: | Vaughan Pratt Stanford University Computer Science Department 2215 Old Page Mill Road Palo Alto, CA 94304 |
| Date: | Monday, 14 Feb 2011 (all day) |
| Venue: | AG-66 (Lecture Theatre) |
(Scan to add to calendar)
Abstract:
Euclidean geometry today is most naturally understood as affine geometry equipped with a suitable bilinearity as a basis for notions of length, angle, area, rectangles, circles, etc. In this talk we express the affine content of Euclid's Postulates 2, 5, and 1 algebraically in that order. In our account Euclid's Fifth or Parallel Postulate is expressed as an equation parametrized with the number 6. Parameters in the range 3 to 5 yield discrete notions of elliptical geometry while 7 and beyond do the same for hyperbolic geometry. Had the tools of the past half century of universal algebra been available to Euclid, the first of his thirteen books could have been both simpler and more general, and the conceptual obstacles to hyperbolic geometry overcome more easily.