Introduction

Abstract

The ultimate goal of physics is to understand the contents of our Universe and the rules governing its evolution starting from just a few underlying principles and assumptions. Small but important steps towards this rather ambitious goal are achieved taking a number of diverse physical phenomena and describing them within the context of a single model or theory. In this introductory chapter we list some of the successes of physics in this direction. We then argue that mathematics plays an integral role in our quest to understand Nature. Conversely, physics discovers structures in Nature that provide motivation and inspiration for mathematics. Major breakthroughs in physics have led to paradigm shifts that turned physicists’ understanding of the Universe on its head, leading invariably to deep and long term consequences for science and society as a whole. When this happens, the old paradigms cannot simply be discarded. New theories must contain within them the seeds of previous theories that were successful within their realm of validity. These seeds must be recoverable within the new paradigm by carefully taking appropriate limits of parameters or scales.