Mathematical Thinking & Logic Textbooks Recommendation

Question

I'm an incoming college freshman on a gap year due to family/personal reasons. During this time, I would love to continue strengthening my mathematical background and keeping my brain busy :D. Currently, I've taken courses in introductory number theory, intro to statistics, calculus 1, 2, 3, differential equations but not yet done with linear algebra. The primary goal would be focused on improving logic, reasoning, mathematical thinking, and to learn more about computation and algorithm. I will go into machine learning research/start-ups/engineering after graduation if this provides additional contexts.

Here're a brief book sequence that I've put together after doing some research online. Does this sequence and what it is comprise of make sense in maximize my understanding and bridging the knowledge gaps? If you have any suggestions and additional recommendation, feel free to drop them down below as well! Thank you so much <3

1. How to Prove It: A Structured Approach" by Daniel J. Velleman
2. The Art of Problem Solving" series by Richard Rusczyk with "Introduction to Algebra" and "Introduction to Counting & Probability
3. Discrete Mathematics and Its Applications" by Kenneth H. Rosen
4. Linear Algebra Done Right" by Sheldon Axler
5. Introduction to the Theory of Computation" by Michael Sipser
6. Concrete Mathematics: A Foundation for Computer Science" by Ronald L. Graham, Donald E. Knuth, and Oren Patashnik
7. Introduction to Counting & Probability (continued from "The Art of Problem Solving" series)
8. Computational Complexity: A Modern Approach" by Sanjeev Arora and Boaz Barak
9. Gödel, Escher, Bach: An Eternal Golden Braid" by Douglas R. Hofstadter

Answer 1

I would suggest to expand the booklist with some online colleges, such as:

• https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/, see also https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/resources/lecture-1-the-geometry-of-linear-equations/ and youtube.
• https://www.youtube.com/playlist?list=PLelIK3uylPMGzHBuR3hLMHrYfMqWWsmx5 Abstract algebra

Answer 2

I believe there are alot of book that are really good for making foundation in mathematics.
I am currently also working on mathematical thinking and mathematical logic. And according to my own experience I will suggest you to work through the book How to think like a Mathematician by Kevin Houston. This book will motivate you and reveal many secrets about thinking like a mathematician. I personally love this book and I will suggest to study this book before jumping in formal proof writing. After this book I will suggest to work on How to Prove it: a structural approach by Velleman. This is really amazing book for learning logics and proof writing. I believe these two books will help you alot. And you will able to study through books like\

1. Linear algebra done right by Alex Sheldon.
2. Linear algebra by Friedberg.
3. Linear algebra by Hoffman.
4. Contemporary abstract algebra by Gallian.
5. A first course in abstract algebra by John fraleigh. Etc.