I explore my methodological assumptions by showing why I put so much emphasis on analysis and its interplay with reference in my investigation of ampliative reasoning in mathematics and science. Reviewing recent important work by philosophers of mathematics Karine Chemla, Carlo Cellucci and Danielle Macbeth, I defend my approach and choice of case studies. I also turn to the work by two influential mathematics educators, Kieran and David Egan, in order to revisit my arguments about the circle, and to prepare the ground for my case studies in Chap. 4 and Chap. 5, where number theory is developed in relation to geometry, algebra, complex analysis, and topology.