Mathematical Methods in Interdisciplinary Sciences

10Universal Intervals: Towards a Dependency‐Aware Interval Algebra

Hend Dawoodand¹ and Yasser Dawood²

¹Department of Mathematics, Faculty of Science, Cairo University, Giza, 12613, Egypt

²Department of Astronomy, Faculty of Science, Cairo University, Giza, 12613, Egypt

The most reliable way of carrying out a proof, obviously, is to follow pure logic, a way that, disregarding the particular characteristics of objects, depends solely on those laws upon which all knowledge rests.

–Gottlob Frege (1848–1925)

10.1 Introduction

Scientific knowledge is not perfect exactitude: it is learning with uncertainty, not eliminating it. We commit errors. But, indeed, we can grasp, measure, and correct our errors and develop ways to deal with uncertainty, which add to our knowledge and make it valuable. Knowledge, then, is not absolute certainty. Knowledge, per contra, is “the tools we develop to purposely get better and better outcomes through our learning about the world” [1]. Many approaches were developed with a view to cope with uncertainty and get reliable knowledge about the world. As examples of these approaches, we can mention probabilitization, fuzzification, and intervalization. For the great degree of reliability it provides, and because of its great importance in many practical applications, intervalization is usually a part of all other methods that deal with uncertainty (see [1] and [2]). Whenever uncertainty exists, there is always a need for interval computations and intervals ...