Case study: A Green Mathematics
Our science has long excelled at analysis: breaking things apart to understand what makes them tick. To understand is to “break things down”, “find out what’s inside the box”. Chemistry takes our world and sees the atoms that make up each object; physics examines the sub-atomic. Our mathematics—our formal ways of thinking and understanding—follow suit.
But there is another way to understand: through relationships and interaction, building up to larger wholes. A society is made up of people, certainly, but to explore it solely through the lens of atomized individuals is to miss the magic. Interaction and interconnection is present in our biology, our ecosystems, our communities, our economies, our intelligence. At Topos we see that we live in an increasingly interconnected world, and we are discovering the mathematics to help navigate it.
Our expertise has roots in a mathematical subject first articulated almost a hundred years ago, one that strived to formalize analogies between algebraic equations and geometric shapes. This task of translation led to the discovery of category theory, which represented a new focus on the relationships that interconnect a subject, rather than the analysis of individual objects within it. Since then, category theory has quietly revolutionized much of pure mathematics, before underpinning advances in logic, physics, and computer science.
Over the past 10 years, these ideas, capturing the essence of relationship, connection, translation, have been pushed outward again, towards a mathematics of networks. This intellectual movement, whose leaders include many of the core scientific members of Topos, has produced hundreds of scientific papers, at world-leading academic hubs such as MIT, Oxford, and Cambridge.
For example, one leading line of inquiry provides the math for seeing pictures like circuit diagrams and flowcharts as 2-dimensional languages. Just like words in English can be combined into sentences by placing them in a line according to the rules of English grammar, so can circuit diagrams be understood as sentences, where the words are electrical components, and the grammar is the way the components are interconnected. Instead of just placing words to the left and right of each other, however, our language is now two-dimensional: we can use the whole page to describe the structure! Electrical and computer engineers have long harnessed this intuitive power of two dimensional language to design complex machines that we now use every day. Topos lets us take these insights further.
These higher-dimensional languages underpin the work that Topos does in all of the case studies below, and more broadly have led to advances in chemistry, biology, economics, natural language processing, artificial intelligence, and quantum technologies.