If I could give my high school self only one math book, it would have to be Seven Sketches in Compositionality by Fong and Spivak. Did every exercise over winter break in college and realized along the way that I had been hustling through math courses and olympiad problems without appreciating any beauty in the structure of mathematics. It completely changed my life and, at least in my eyes, dissolved the assumption that “applied” math must be less rigorous or “pure” math must be less practical. Not only did it immediately recast my basic intuition about what math “is” (and what numbers “are” or what processes “do”) but with a bit more effort toward studying category theory, I came to see my previous encounters with more advanced topics like forcing in set theory or the Legendre-Fenchel transform used in physics/economics in a completely new light. What is truly wild to me is that Seven Sketches has no real prerequisites, and I could have just as easily read it when I was 14. This book should be the basis of a mandatory course for a math-loving high schooler. Instead of rushing to learn linear algebra and real analysis in high school, I wish I had gained the wonderful perspective of Fong and Spivak—I would have fallen truly in love with math much sooner, found a deeper perspective in my courses much faster, and enjoyed all of it so much more along the way.
Hope someone sees this and shares the book with a high schooler—it’s also available for free online!
Hope someone sees this and shares the book with a high schooler—it’s also available for free online!