Now and then one of my friends asks me for recommendations for books to use to re-engage with mathematics. I am always happy to help and I try to come up with recommendations that will be "goldilocks hard" but also interesting and fun for the person who is asking. I also selfishly think a little about what it will be like for me to talk about the book's content with my friend.

I look for books that motivate the material well, but also actually prove things and I always make a point of telling my friends that they have to read and understand the proofs, ideally well enough that they can reproduce them. Today I asked myself why I always make such a big deal of this. Here's what I came up with:

I look for books that motivate the material well, but also actually prove things and I always make a point of telling my friends that they have to read and understand the proofs, ideally well enough that they can reproduce them. Today I asked myself why I always make such a big deal of this. Here's what I came up with:

- Mathematics is a progression of ideas, expressed in arguments that prove results. The results are often useful in solving problems, but the arguments are really where the mathematics is. If you understand the arguments, you will be able to do problems - often in many different ways.
- Proofs show exactly what is needed to get results. They show you, therefore, exactly where the limits are in applying the results. When you understand a proof, you know exactly what you have to work with.
- Proofs are rigorous and often hard. If your goal is to be able to apply mathematical concepts and hone your mathematical intuition, working through hard proofs is much better training than just practicing algorithms.
- What you can prove in mathematics is what you know. Results that you can apply, but don't understand the proofs for, are what Plato would call "true opinions," not knowledge. Not grounded in proof, your understanding of algorithms and their valid application might "fly away" at any time (See Plato's
*Meno*97-98, for a delightful account of this). - Go back to 1. The beautiful thing about studying mathematics "for fun" is that you get to actually do it. Under time / grade / getting applications done pressure, the mathematics ends up taking a back seat to the applications. It's like fast-forwarding the show and watching the commercials. Coming back now, you can actually watch the show.

So read the proofs!

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