Dimitris Chiotis, PhD Student, School of Mathematics, Statistics and Physics, Newcastle University

PhD project title: Superoptimal analytic approximation and exterior products

“My research area is Pure Mathematics and there is no better example than this for one to see how an almost 2500-year old piece of work affects the way of thinking in this vast field of mathematics. It is known that the elements begin with the axioms which are doubtless facts and cannot be proven with logic arguments, e.g. given two points, we can draw a straight line that connects them (we can do it but we cannot prove it). Then there are the definitions and, based on those two, follow the propositions, theorems and corollaries. The proves are categorized to logic and structural. Logic ones use a series of arguments that are based on the axioms, on the definitions and on previous theorems and/or propositions. Structural ones involve the observation as part of the proof, one needs to draw geometrical objects under certain rules and either compare them or examine each one of them.

What makes the “Elements” such a special piece of work is the axiomatic generalization of specific knowledge to abstract knowledge. For example, in Mesopotamia almost 1500 years before Euclid, they knew precisely how to apply the Pythagorean theorem numerically for a specific triangle, but they never came to the point of generalizing this idea. Pythagoras did it first and we can find a proof in Euclid’s elements. That is exactly the purpose of pure mathematics, to generalize particular spects to more general ones through a series of logic arguments that are coherent and are justified by the axioms, the definitions and the previous propositions and/or theorems. The way we write and think about pure mathematics nowadays is exactly the same with Euclid’s. We begin with specific axioms, we introduce the definitions and we start building our ideas which are the propositions and theorems. The form of a pure mathematics publication is identical to an arbitrary page from the “Elements”: Definition-Proposition-proof-Theorem-proof-Corollary.”

There were two small displays in the Liaison Spaces on Levels 3 and 4 of the Philip Robinson Library which included resources that depict how geometry is still relevant in many disciplines today.