Coordinate Free Numerics

ii Foreword These notes are being written during seminars in Department of Mathematics, University of Bergen, during the spring 2008. I appreciate any comments from the readers. I will pose ques-tions/comments to the readers of draft versions of the manuscript in margin notes like this one. iii iv FOREWORD Preface A fundamental dichotomy in mathematics and computer science is the distinction between abstract and concrete definitions and formulations of ideas and objects. Whereas a concrete definition of an object is based on a particular representation of the object, an abstract definition concentrates on the properties existing independently of particular representations. For example, a concrete definition of natural numbers could be based on a decimal representation , or on a binary representation. We realize that numbers have fundamental properties that are independent of the particular chosen representation, but it is not evident what these properties are. Peanos abstract definition based on the idea that " every natural number has a unique successor ... " says something essential about numbers that is not evident from a concrete definition. In pure mathematics abstract formalisms have been the main working tools since the early 20'th century. An abstract definition of a mathematical object defines the essential properties of the object independently of particular representations , and concetrates on how the object interacts with other objects. The mathematical language of category theory purifies this view; " the only interesting property of a mathematical object is its interactions with other objects ". In the area of differential equations, coordinate free abstractions concentrate on geometrical properties which can be defined and studied independently of particular coordinate systems. This is also the basis for modern theoretical physics. Einstein emphasized that " physics is the same in all coordinate systems " , and theories that cannot be naturally formulated in a coordinate free language are considered to be of minor physical importance. Indeed, many results in modern mathematics and theoretical physics cannot be formulated and studied without abstract formulations. The distinction between 'specification' and 'implementation' is also of fundamental importance in computer science. Kristen Nygaard and Ole Johan Dahl, who invented the first object oriented programming language SIMULA 67, say: " Programming is what + how ". Abstractions should determine the large scale structure of programs (what should the different program classes do?), while more arbitrary implementation details (how should it be done?) are hidden inside the private parts …