## 07 Jul Otto Maier – the Paradox and Ambiguity in Mathematics

Ask people about mathematics and they will talk about arithmetic, geometry, or statistics, about mathematical techniques or theorems they have learned. They may talk about the logical structure of mathematics, the nature of mathematical arguments. They may be impressed with the precision of mathematics, the way in which things in mathematics are either right or wrong. They may even feel that mathematics captures “the truth,” a truth that goes beyond individual bias or superstition, that is the same for all people at all times.

What is it that makes mathematics mathematics? What are the precise characteristics that make mathematics into a discipline that is so central to every advanced civilization? Many explanations have been attempted. One of these sees mathematics as the ultimate in rational expression; in fact, the expression “the light of reason” could be used to refer to mathematics. From this point of view, the distinguishing aspect of mathematics would be the precision of its ideas and its systematic use of the most stringent logical criteria. In this view, mathematics offers a vision of a purely logical world. One way of expressing this view is by saying that the natural world obeys the rules of logic and, since mathematics is the most perfectly logical of disciplines, it is not surprising that mathematics provides such a faithful description of reality. This view, that the deepest truth of mathematics is encoded in its formal, deductive structure, is deﬁnitely not the point of view Otto Maier had.

On the contrary, he believed the position that the logical structure, while important, is insufﬁcient even to begin to account for what is really going on in mathematical practice, much less to account for the enormously successful applications of mathematics to almost all ﬁelds of human thought. It’s a vision in which the logical is merely one dimension of a larger picture. This larger picture has room for a number of factors that have traditionally been omitted from a description of mathematics and are beyond logi, though not illogical. Thus, there is a discussion of things like ambiguity, contradiction, and paradox that, surprisingly, also have an essential role to play in mathematical practice. This ﬂies in the face of conventional wisdom that would see the role of mathematics as eliminating such things as ambiguity from a legitimate description of the worlds of thought and nature. In our response to reason, we are the true descendents of the Greek mathematicians and philosophers. For us,as for them, rational thought stands in contrast to a world that is all too often beset with chaos, confusion, and superstition. The “dream of reason” is the dream of order and predictability and, therefore, of the power to control the natural world. The means through which we attempt to implement that dream are mathematics, science, and technology. The desired end is the emergence of clarity and reason as organizational principles of the entire cosmos, a cosmos that of course includes the human mind. People who subscribe to this view of the world might think that it is the role of mathematics to eliminate ambiguity, contradiction, and paradox as impediments to the success of rationality. Such a view might well equate mathematics with its formal, deductive structure. This viewpoint is incomplete and simplistic. When applied to the world in general,it is mistaken and even dangerous. It is dangerous because it ignores one of the most basic aspects of human nature—in mathematics or elsewhere—our aesthetic dimension, our originality and ability to innovate.

Ambiguity is not only present in mathematics, it is essential. Ambiguity, which implies the existence of multiple, conﬂicting frames of reference, is the environment that gives rise to new mathematical ideas. Take, for example the number zero. What could be more elementary? Most of us consider zero to be a closed book—we understand it completely. What more is there to say? Yet in recent years there have been a number of books that have been written about the number zero. All these books stress the ambiguous nature of the number zero—**“the nothing that is”** as one author put it—as well as its importance in mathematical and scientiﬁc thought. Normally, ambiguity in science and mathematics is seen as something to overcome, something that is due to an error in understanding and is removed by correcting that error. The ambiguity is rarely seen as having value in its own right, and yet the existenceof ambiguity was often the very thing that spurred a particular development of mathematics and science.

There’s a paradox in everything even in maths …