In early times, the primary driving forces for the development and application of new mathematics were cartography and navigation, astronomy and architecture. From around the 16th century, the main drivers were physics and physics-based engineering. In the twenty-first century, biology and the human sciences will become the primary driving forces for the development and application of new mathematics. So far, we have seen some applications of mathematics in these fields, some quite substantial. But that has involved old mathematics, developed for other purposes. What we have not yet seen to any great extent are new mathematics and new branches of mathematics developed specifically in response to the needs of those disciplines. In my view, that is where we will see much of the mathematical action in the coming decades. I suspect that some of that new mathematics will look quite different from most of today’s mathematics. But I really don’t have much idea what it will look like.
Keith Devlin is author of books including he Math Instinct: Why You’re a Mathematical Genius (Along with Lobsters, Birds, Cats and Dogs) (buy this book from Amazon (UK) or Amazon (USA)), and Sets, Functions, and Logic: An Introduction to Abstract Mathematics (buy this book from Amazon (UK) or Amazon (USA)). See his website