This chapter is devoted to introducing the theories of interval algebra to people who are interested in applying the interval methods to uncertainty analysis in science and engineering. In view of this purpose, we shall introduce the key concepts of the algebraic theories of intervals that form the foundations of the interval techniques as they are now practised, provide a historical and epistemological background of interval mathematics and uncertainty in science and technology, and finally describe some typical applications that clarify the need for interval computations to cope with uncertainty in a wide variety of scientific disciplines.
Top1. Introduction: Can We Know About The World?
Are we imprisoned in Descartes’ dualism? That is, are we stuck between two worlds: the world of physical reality and the world of mental reasoning? The answer may be: “Yes, we are”. Moreover, does the way, in which we learn about the physical world, meet our persistent seeking for deterministic scientific knowledge? With the fact that the machinery we employ to acquire our knowledge about the world is not absolute, the answer to this question may be negative. So, how can we get certain knowledge about the world? Heidegger1 adds another version of this dilemma by claiming that the question should not be only about how we acquire knowledge about the world, but also we should ask: “How does the world reveal itself to us through our encounters with it?” (Heidegger, 1962).
In many cases, we can definitely achieve qualitative knowledge with absolute certainty, but quantitative knowledge, about the present and future of the physical world, is completely different: quantitative knowledge is acquired through intuition, experiments, measurements, and proposed hypotheses2. That is, quantitative inquiries can be regarded as the main source of uncertainty, and therefore our whole knowledge cannot be completely certain. Pascal3, within this dilemma, says, in (Pascal, 1995): “We sail within a vast sphere, ever drifting in uncertainty, driven from end to end” . At this point, a crucial question should be posed: Is uncertainty anti-knowledge or pro-knowledge? Hoppe4, in his article “On Certainty and Uncertainty” (Hoppe, 1997), gives us a way out from this dilemma by a brilliant mental experiment. He begins his article by saying:
It is possible to imagine a world characterized by complete certainty. All future events and changes would be known in advance and could be predicted precisely. There would be no errors and no surprises. We would know all of our future actions and their future outcomes. In such a world, nothing could be learned, and accordingly, nothing would be worth knowing. Indeed, the possession of consciousness and knowledge would be useless. For why would anyone want to know anything if all future actions and events were completely predetermined and it would not make any difference for the future course of events whether or not one possessed this or any knowledge? Our actions would be like those of an automaton, and an automaton has no need of any knowledge. Thus, rather than representing a state of perfect knowledge, complete certainty actually eliminates the value of all knowledge.
Thus uncertainty is pro-knowledge, not anti-knowledge. Obviously, our scientific knowledge is not perfect and we commit errors. But, indeed, we can grasp, measure, and correct our errors, and develop ways to deal with uncertainty. All of these add to our knowledge and make it valuable. Knowledge is not the absolute certainty. Knowledge, however, is the tools we develop to purposely get better and better outcomes through our learning about the world. In other words, uncertainty should always exist; for us to develop the weapons against it. This understanding of “what knowledge is” coincides with our understanding of “what science is”. Popper5, in (Popper, 2002), uses the notion of falsification as a criterion of demarcation to distinguish between those theories that are scientific and those that are unscientific, that is, “a theory is scientific if, and only if, it is falsifiable”.