only doubt with your gut. All Cartesian doubt is visceral.

It’s surprising that Wittgenstein’s philosophy isn’t more well known outside of specialized circles, as it holds a very practical life lesson: we should accept going step by step, clarifying our language as we move forward, and being regularly surprised. It’s an effective antidote for paranoia.

How do you teach math to someone who believes that their intuition and perception of reality are given and impossible to reprogram? It’s exactly like teaching swimming to someone who is convinced their body is as dense as rock and will sink. A prelude to any successful teaching is getting rid of such beliefs.

Mathematics gives us examples of truths in which we can have absolute confidence. It’s not just about superficial truths, like 2 + 2 = 4, but also profound truths, truths that are extremely interesting and not at all immediately apparent. We’ll give several striking examples in the next chapter. It’s only through a relentless confrontation with dou

At a profound level, math is the only successful attempt by humanity to speak with precision about things that we can’t point to with our fingers. This is one of the central themes of this book and we’ll come back to it a number of times. In a math book, the most important passages aren’t the theorems or the proofs: they are the definitions. Mathem

undeniable that mathematical creation feels magical and supernatural. But behind all that there’s necessarily a human reality that is neither supernatural nor magical.

an adult, I’ve developed my own way of making use of the special state of mind just before falling asleep. Rather than focusing on subjects that preoccupy me, I’ve learned to simply let myself be filled with them. The nuance is subtle but fundamental. Focusing is thinking intensely, in search of solutions. It never works and it keeps you from sleep

In an interview with the New York Times, Thurston summed up things as follows: “People don’t understand how I can visualize in four or five dimensions. Five-dimensional shapes are hard to visualize—but it doesn’t mean you can’t think about them. Thinking is really the same as seeing.”

The big ideas are always intuitive and always simple. They’re even ridiculously simple. We only ever really understand things that are obvious. When it’s not obvious, it’s because we haven’t really understood. This is a universal law of human cognition. It states that our science was invented by humans and that humans are, at the deepest level, all