Velleman’s How to Prove It vs. Hamkins’ Proof and the Art of Mathematics:
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Velleman appears as a logician interested in propositional and first order predicate logic for its own sake, although he uses these tools to to cover basic facts that concern sets, relations, and functions. Much ink is devoted to learning the language and consequences of first order logic.
Hamkins’ book is, I think, the “missing link” I have been searching for between the lower and advanced mathematics. 🧮
Hamkins seems very much to be preparing his students for mathematical analysis, and, as such, spends more time covering methods of proving mathematical facts. Indeed, the final chapter of his textbook is titled “real analysis” and we might assume that a student will be prepared for it by the time they have covered the rest of the book.