This Will Make You Better at Mathematics FOREVER

Summary

This video delves into the transformative power of learning mathematical proof writing. The speaker explains how mastering proofs can fundamentally change the way one thinks about mathematics, making it more logical and structured. The importance of proof writing is emphasized as a key skill for advancing in mathematics, with recommendations for three essential books: 'How to Read and Do Proofs' by Daniel Solow, 'Book of Proof' by Richard Hammack, and 'How to Prove It: A Structured Approach' by Daniel Velleman. These books, ranging from free resources to more advanced texts, help learners grasp the intricacies of proofs and enhance their mathematical reasoning.

Takeaways

• 😀 Proof writing will completely change the way you think about mathematics by helping you understand the logical structure behind mathematical concepts.
• 😀 Mathematics, especially in proof writing, offers an unshakable certainty unlike the ambiguity of the real world; once you prove something, it is true.
• 😀 Learning to write proofs is a challenging but rewarding skill. While it’s tough, with the right resources and practice, anyone can master it.
• 😀 Proof writing is not only useful for math majors; it is a critical skill that can be learned by anyone interested in deepening their understanding of mathematics.
• 😀 Every definition in mathematics is essentially an 'if and only if' statement, which is foundational in building logical arguments for proofs.
• 😀 Proof writing is a skill that can be learned independently through books, making it accessible even without formal classes.
• 😀 'How to Read and Do Proofs' by Daniel Solow is a recommended beginner-friendly book that introduces the foundational concepts of proof writing.
• 😀 'Book of Proof' by Richard Hammack is a highly recommended free resource that focuses on the practical aspects of writing proofs, especially useful for beginners.
• 😀 'How to Prove It: A Structured Approach' by Daniel Velleman is considered one of the best books for learning proof writing due to its clear explanations and multiple approaches to teaching complex ideas.
• 😀 Vacuous truths—like the statement 'all unicorns are purple'—are a fun example of how logical structures in proofs can lead to surprising but valid conclusions.
• 😀 The best way to learn proof writing is by actually writing proofs yourself. Reading others’ proofs is helpful, but actively solving them helps solidify your understanding.

Q & A

What is the main argument presented in the video about mathematics?

The main argument is that learning how to write proofs in mathematics can radically change the way you think about the subject. Once you learn proof writing, you can approach math with a deeper understanding and clarity, as it involves creating logical arguments that justify mathematical truths.

How does the speaker describe their experience with calculus and proof writing?

The speaker shares that their love for mathematics, especially calculus, was solidified when they studied infinite series. However, it wasn’t until they delved into proof writing that they truly appreciated the logical rigor and beauty of math, which allowed them to tackle complex concepts and become 'unstoppable' in the subject.

Why is proof writing considered a transformative skill in mathematics?

Proof writing is transformative because it teaches you to think logically and justify every step mathematically. By writing proofs, you gain a deep understanding of the subject and can explore any mathematical concept with confidence, making mathematics a more structured and understandable discipline.

What are the main books recommended by the speaker for learning proof writing?

The speaker recommends three books: 'How to Read and Do Proofs' by Daniel Solow, 'Book of Proof' by Richard Hammack, and 'How to Prove It: A Structured Approach' by Daniel Velleman. Each offers different strengths in teaching proof writing, with the latter being the speaker’s personal favorite.

What is the significance of 'How to Read and Do Proofs' by Daniel Solow?

'How to Read and Do Proofs' by Daniel Solow is praised for its focus on understanding definitions and the logical structure of mathematical proofs. The book introduces fundamental concepts of proof writing, making it a great starting point for beginners.

Why is 'Book of Proof' by Richard Hammack considered a valuable resource?

'Book of Proof' by Richard Hammack is considered valuable because it is freely available online, offering accessible explanations of various proof techniques, such as direct proofs, proof by contradiction, and contrapositive proofs. Its affordability and clarity make it an ideal resource for anyone learning proof writing.

What makes 'How to Prove It: A Structured Approach' by Daniel Velleman stand out from the other books?

'How to Prove It: A Structured Approach' is particularly noted for its well-structured explanations and its ability to explain mathematical concepts from multiple perspectives. The book covers advanced topics like vacuous truths in a clear and engaging way, making complex ideas more understandable.

What is the role of vacuous truth in mathematical proofs, and how does the speaker explain it?

Vacuous truth is a concept where a statement about a set is considered true because the set is empty. The speaker uses examples like 'all unicorns are purple' to illustrate vacuous truth, explaining that since there are no unicorns (an empty set), the statement is vacuously true.

How does proof writing relate to other fields, like programming or logic?

Proof writing shares similarities with programming and logic because, like writing code, creating a proof requires logical reasoning and step-by-step justification. The speaker compares reading other people’s proofs to reading someone else’s code—it's harder to understand than writing your own.

What personal insight does the speaker offer about the process of learning mathematics?

The speaker emphasizes that the best way to learn mathematics is to actively engage with it by writing your own proofs. This process helps solidify your understanding and allows you to truly 'grind through' mathematical concepts, ultimately leading to a more profound and lasting mastery of the subject.