## Metadata - Author: [[Paul Lockhart]] - Full Title: Measurement - Category: #books ## Highlights - What is a math problem? To a mathematician, a problem is a probe—a test of mathematical reality to see how it behaves. It is our way of “poking it with a stick” and seeing what happens. We have a piece of mathematical reality, which may be a configuration of shapes, a number pattern, or what have you, and we want to understand what makes it tick: What does it do and why does it do it? So we poke it—only not with our hands and not with a stick. We have to poke it with our minds. ([Location 60](https://readwise.io/to_kindle?action=open&asin=B00AFS6LSC&location=60)) - Now here’s where the art comes in. In order to explain we have to create something. Namely, we need to somehow construct an argument—a piece of reasoning that will satisfy our curiosity as to why this behavior is happening. ([Location 87](https://readwise.io/to_kindle?action=open&asin=B00AFS6LSC&location=87)) - This is abstract art, pure and simple. And art is always a struggle. There is no systematic way to create beautiful and meaningful paintings or sculptures, and there is also no method for producing beautiful and meaningful mathematical arguments. Sorry. Math is the hardest thing there is, and that’s one of the reasons I love it. ([Location 99](https://readwise.io/to_kindle?action=open&asin=B00AFS6LSC&location=99)) - the best problems are your own. You are the intrepid mental explorer; it’s your mind and your adventure. Mathematical reality is yours—it’s in your head for you to explore any time you feel like it. What are your questions? Where do you want to go? I’ve enjoyed coming up with some problems for you to think about, but these are merely seeds I’ve planted to help you start growing your own jungle. Don’t be afraid that you can’t answer your own questions—that’s the natural state of the mathematician. Also, try to always have five or six problems you are working on. It is very frustrating to keep banging your head against the same wall over and over. (It’s much better to have five or six walls to bang your head against!) Seriously, taking a break from a problem always seems to help. ([Location 109](https://readwise.io/to_kindle?action=open&asin=B00AFS6LSC&location=109)) - collaborate. If you have a friend who also wants to do math, you can work together and share the joys and frustrations. It’s a lot like playing music together. Sometimes I will spend six or eight hours working on a problem with a friend, and even if we accomplish next to nothing, we still had fun feeling dumb together. ([Location 115](https://readwise.io/to_kindle?action=open&asin=B00AFS6LSC&location=115)) - What makes a mathematician is not technical skill or encyclopedic knowledge but insatiable curiosity and a desire for simple beauty. ([Location 119](https://readwise.io/to_kindle?action=open&asin=B00AFS6LSC&location=119)) - Do not ignore symmetry! In many ways, it is our most powerful mathematical tool. ([Location 132](https://readwise.io/to_kindle?action=open&asin=B00AFS6LSC&location=132)) - A proof is simply a story. The characters are the elements of the problem, and the plot is up to you. The goal, as in any literary fiction, is to write a story that is compelling as a narrative. In the case of mathematics, this means that the plot not only has to make logical sense but also be simple and elegant. No one likes a meandering, complicated quagmire of a proof. We want to follow along rationally to be sure, but we also want to be charmed and swept off our feet aesthetically. ([Location 142](https://readwise.io/to_kindle?action=open&asin=B00AFS6LSC&location=142)) - improve your proofs. Just because you have an explanation doesn’t mean it’s the best explanation. Can you eliminate any unnecessary clutter or complexity? Can you find an entirely different approach that gives you deeper insight? Prove, prove, and prove again. Painters, sculptors, and poets do the same thing. ([Location 146](https://readwise.io/to_kindle?action=open&asin=B00AFS6LSC&location=146)) - critique your work. Subject your arguments to scathing criticism by yourself and by others. That’s what all artists do, especially mathematicians. As I’ve said, for a piece of mathematics to fully qualify as such, it has to stand up to two very different kinds of criticism: it must be logically sound and convincing as a rational argument, and it must also be elegant, revelatory, and emotionally satisfying. I’m sorry that these criteria are so painfully steep, but that is the nature of the art. ([Location 206](https://readwise.io/to_kindle?action=open&asin=B00AFS6LSC&location=206)) - What is measuring? What exactly are we doing when we measure something? I think it is this: we are making a comparison. We are comparing the thing we are measuring to the thing we are measuring it with. In other words, measuring is relative. Any measurement that we make, whether real or imaginary, will necessarily depend on our choice of measuring unit. ([Location 322](https://readwise.io/to_kindle?action=open&asin=B00AFS6LSC&location=322)) - So one thing to always be aware of is whether you’ve pinned down your objects enough to get any information out of them. ([Location 363](https://readwise.io/to_kindle?action=open&asin=B00AFS6LSC&location=363)) - The solution to a math problem is not a number; it’s an argument, a proof. We’re trying to create these little poems of pure reason. Of course, like any other form of poetry, we want our work to be beautiful as well as meaningful. Mathematics is the art of explanation, and consequently, it is difficult, frustrating, and deeply satisfying. ([Location 522](https://readwise.io/to_kindle?action=open&asin=B00AFS6LSC&location=522))