In Quantum theory, groups and representations, Peter Woit writes:
A fundamental principle of modern mathematics is that the way to understand a space $M$, given as some set of points, is to look at $F(M)$, the set of functions on this space.
I was wondering what some examples of this "fundamental principle" is across different fields in mathematics.
Woit, Peter, Quantum theory, groups and representations. An introduction, Cham: Springer (ISBN 978-3-319-64610-7/hbk; 978-3-319-64612-1/ebook). xxii, 668 p. (2017). ZBL1454.81004.