Space and knowledge. Elements of an epistemic theory of the use of diagrams (Q902466)

There has been a tendency among certain philosophers mathematics to reduce the role of diagrams. Once you have a result, it was thought, the figures are no longer needed and maybe, if you are smart enough you don't need figures at all. Russell wrote for example about geometry: ``in the best books there are no figures at all''. In this book, the author argues in great detail that diagrams can not only beautifully represent knowledge, but, more importantly, generate new knowledge. He shows convincingly that very often a theory develops in a subtle interaction between diagrams and text. In the first chapter, the author argues that the fact that a diagram exists in space has four characteristics or dimensions. A diagram can be manipulated outside of the body, it often possesses a structural analogy with what it represents, we have direct visual access to the diagram, and one can often immediately literally see new knowledge in the diagram. Obviously, a text in writing can also be manipulated outside the body, but there is less direct access to its meaning, no structural analogy and no immediate access to new knowledge. Take the example ``All \(A\) are \(B\)'' and represent it in an Euler-diagram: a circle \(A\) inside a circle \(B\). We can immediately conclude that `All non-\(B\) are non-\(A\)'. The second chapter of the book is on the role of diagrams in Euclidean geometry. One of the author's conclusions is that what one is allowed to see in a Euclidean diagram is not arbitrary but determined by rules. They should be used in the right way and as such they have an important epistemological function. The third chapter concentrates on Oresme (1320--1382), who famously represented qualities by means of line segments. This was spectacular and made a quantitative discussion of instantaneous velocities possible. After Oresme, not only distances, areas, volumes, masses and time could be quantified, diagrammatically represented, and analyzed but in principle everything.